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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Chemical Engineering, 2010., Cataloged from PDF version of thesis., Includes bibliographical references (p. 269-275)., A framework and code have been developed to simulate fiber and film processes; the code can handle three-dimensional, isothermal, incompressible, creeping flow of a Giesekus fluid with free surfaces at infinite capillary number. The code is an extension of the parallel methods developed by Caola et al. from two- to three-dimensional flows, a restructured and improved version of the three-dimensional code of Phillips based on the subproblem formulation. The free surface motion used is an extension of the axially-constrained finite difference method of Phillips from two to three dimensions; internal domain deformation is upgraded from an elliptic mapping technique, only suitable for two-dimensional geometries, to the treatment of the domain as a fictitious elastic solid that conforms to the free surface, similar to the methods of Cairncross et al. The verification of the code and its utility are tested against the ability to capture experimental behavior in a set of benchmark problems which benefit from its ability to compute fully three-dimensional flow fields. Fluids with a nonzero second normal stress difference in simple shear flow, such as Giesekus fluids, exhibit secondary flows in non-circular, non-annular pipes. The code predicts the fully-developed three-dimensional velocity field in flow of a square duct in agreement with Yue et al. Elastically-driven flow transitions from a two-dimensional, steady flow to a three-dimensional, time periodic flow also motivate the need for three-dimensional solvers, which should be able to capture the onset directly. Flow around a closely-spaced linear periodic array of cylinders in a rectangular channel, a flow studied experimentally by Liu, exhibits this kind of flow transition. Numerical simulations have been conducted in this geometry with the Oldroyd-B model, [beta] = 0.67, wall separation distances between 4-32 radii, and Deborah number 0.1 < De < 2.0. The range of bounding wall edge effects on the flow field is shown t, by David Joseph Adrian., Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Chemical Engineering, 2010., Cataloged from PDF version of thesis., Includes bibliographical references (p. 269-275)., A framework and code have been developed to simulate fiber and film processes; the code can handle three-dimensional, isothermal, incompressible, creeping flow of a Giesekus fluid with free surfaces at infinite capillary number. The code is an extension of the parallel methods developed by Caola et al. from two- to three-dimensional flows, a restructured and improved version of the three-dimensional code of Phillips based on the subproblem formulation. The free surface motion used is an extension of the axially-constrained finite difference method of Phillips from two to three dimensions; internal domain deformation is upgraded from an elliptic mapping technique, only suitable for two-dimensional geometries, to the treatment of the domain as a fictitious elastic solid that conforms to the free surface, similar to the methods of Cairncross et al. The verification of the code and its utility are tested against the ability to capture experimental behavior in a set of benchmark problems which benefit from its ability to compute fully three-dimensional flow fields. Fluids with a nonzero second normal stress difference in simple shear flow, such as Giesekus fluids, exhibit secondary flows in non-circular, non-annular pipes. The code predicts the fully-developed three-dimensional velocity field in flow of a square duct in agreement with Yue et al. Elastically-driven flow transitions from a two-dimensional, steady flow to a three-dimensional, time periodic flow also motivate the need for three-dimensional solvers, which should be able to capture the onset directly. Flow around a closely-spaced linear periodic array of cylinders in a rectangular channel, a flow studied experimentally by Liu, exhibits this kind of flow transition. Numerical simulations have been conducted in this geometry with the Oldroyd-B model, [beta] = 0.67, wall separation distances between 4-32 radii, and Deborah number 0.1 < De < 2.0. The range of bounding wall edge effects on the flow field is shown t, by David Joseph Adrian., Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Chemical Engineering, 2010., Cataloged from PDF version of thesis., Includes bibliographical references (p. 269-275)., A framework and code have been developed to simulate fiber and film processes; the code can handle three-dimensional, isothermal, incompressible, creeping flow of a Giesekus fluid with free surfaces at infinite capillary number. The code is an extension of the parallel methods developed by Caola et al. from two- to three-dimensional flows, a restructured and improved version of the three-dimensional code of Phillips based on the subproblem formulation. The free surface motion used is an extension of the axially-constrained finite difference method of Phillips from two to three dimensions; internal domain deformation is upgraded from an elliptic mapping technique, only suitable for two-dimensional geometries, to the treatment of the domain as a fictitious elastic solid that conforms to the free surface, similar to the methods of Cairncross et al. The verification of the code and its utility are tested against the ability to capture experimental behavior in a set of benchmark problems which benefit from its ability to compute fully three-dimensional flow fields. Fluids with a nonzero second normal stress difference in simple shear flow, such as Giesekus fluids, exhibit secondary flows in non-circular, non-annular pipes. The code predicts the fully-developed three-dimensional velocity field in flow of a square duct in agreement with Yue et al. Elastically-driven flow transitions from a two-dimensional, steady flow to a three-dimensional, time periodic flow also motivate the need for three-dimensional solvers, which should be able to capture the onset directly. Flow around a closely-spaced linear periodic array of cylinders in a rectangular channel, a flow studied experimentally by Liu, exhibits this kind of flow transition. Numerical simulations have been conducted in this geometry with the Oldroyd-B model, [beta] = 0.67, wall separation distances between 4-32 radii, and Deborah number 0.1 < De < 2.0. The range of bounding wall edge effects on the flow field is shown t, by David Joseph Adrian., Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Chemical Engineering, 2008., Includes bibliographical references (p. 191-200)., Dissipative particle dynamics (DPD) is a mesoscale simulation technique which uses soft potentials between large particles to reproduce liquid behavior. In form, DPD is similar to molecular dynamics, as all matter is represented by point particles which interact with each other via, pairwise forces. The method was first introduced in the early 1990's, and has since undergone a number of refinements which have put it on a firm thermodynamic footing. DPD is notable for the flexibility it presents the modeler for building complex fluid systems. DPD has been used to study simple molecular liquids, polymer and colloid solutions, and phase behavior of block copolymer melts. Recently, a number of workers have used DPD to study the flow of polymer solutions in various geometries such as microchannels, pores, and sudden contractions. While these types of flows are well-suited to DPD's relative strengths, an important step has been skipped. Before the results of these complex flows can be accepted, it is necessary to demonstrate that the rheological predictions made by DPD are generally reliable. The principle aim of this thesis is to demonstrate that the rheology of polymer solutions can be simulated successfully with DPD. The rheology of a solution of DPD dumbbells using a FENE spring force law is studied in the first part of this thesis via simulation of steady shear flow and steady planar elongational flow. The rheological results are compared to dilute Brownian dynamics simulations of the same FENE dumbbell model. The level of coarse-graining of the DPD fluid is varied by changing the length of the DPD dumbbell relative to the particle size, while maintaining a constant extensibility parameter. Broadly speaking, the viscosity, first normal stress coefficient, and dumbbell extension in shear flow calculated with DPD are in agreement with the BD results. The two methods are not perfectly alike however, and two systematic differences between the DPD and BD results are obser, (cont.) The effect is more powerful in DPD dumbbells which are more coarse-grained. At high shear rates in the power-law regime, DPD systematically overpredicts the rate of shear-thinning, with the greatest deviation occurring in the most coarse-grained dumbbells. This is hypothesized to be a result of hydrodynamic interaction which comes naturally out of DPD's explicit treatment of the solvent. The HI effect is analyzed using the Giesekus anisotropic drag tensor. Shortly after its introduction, the complaint was made that DPD's dynamic results are suspect because it has a very low, gas-like Schmidt number, meaning that momentum and mass are transported through the DPD medium at similar rates. This is in contrast with physical liquids, which have large Schmidt numbers. The use of the Lowe-Anderson formulation of DPD allows the Schmidt number of a solution to be varied for the same polymer model. Shear flow simulations of identical dumbbells under different Schmidt number conditions give results in excellent agreement with each other, indicating that the Schmidt number is not an important factor in determining polymer rheology with DPD. Steady planar elongational flow is simulated for the first time in DPD using the Kraynik and Reinelt boundary conditions, which are periodic in both space and time, allowing for simulations of planar elongational flow for an unlimited period of time. The planar elongational flow results of FENE dumbbells are also in agreement with BD, butshow the same systematic deviations observed in shear flow. The second portion of this thesis examines a more complex polymer solution using DPD, with simulations of semidilute solutions of longer N = 20 bead-spring chain polymers undergoing shear and planar elongational flow. In addition to concentration effects, the importance of the solvent quality is also examined with simulations of polymer solutions in both good and theta solvents. In order to capture concentration dependency, a spring -spring r, (cont.) In planar elongational flow, each solution goes through a coil-stretch transition at the theoretically predicted strain rate De = 0.5. In shear flow, the rheological results are in qualitative agreement with theory, showing a plateau at low De, and a transition into a shear-thinning regime beginning at De = 1. While the planar elongational flow results show clear dependence on the solution relaxation time, the shear results show a mixed dependence on the overall solution relaxation time, which reflects the concentration dependence, and the relaxation rate of an isolated chain, suggesting that only some aspects of the shear rheology are affected by the concentration. The conclusion of this thesis is that DPD is able to faithfully reproduce reliable rheological behavior with bead-spring polymer models. We find however, that the computational costs associated with the explicit simulation of the solvent put DPD at a disadvantage for systematic rheology studies when compared to Brownian dynamics. The high costs of the spring-spring repulsion force implementation are particularly limiting. In complex systems where DPD's natural flexibility in molecular architecture and chemistry make it the best choice, rheological results can now be accepted with more confidence., by Theis Forman Clarke., Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Chemical Engineering, 2008., Includes bibliographical references (p. 191-200)., Dissipative particle dynamics (DPD) is a mesoscale simulation technique which uses soft potentials between large particles to reproduce liquid behavior. In form, DPD is similar to molecular dynamics, as all matter is represented by point particles which interact with each other via, pairwise forces. The method was first introduced in the early 1990's, and has since undergone a number of refinements which have put it on a firm thermodynamic footing. DPD is notable for the flexibility it presents the modeler for building complex fluid systems. DPD has been used to study simple molecular liquids, polymer and colloid solutions, and phase behavior of block copolymer melts. Recently, a number of workers have used DPD to study the flow of polymer solutions in various geometries such as microchannels, pores, and sudden contractions. While these types of flows are well-suited to DPD's relative strengths, an important step has been skipped. Before the results of these complex flows can be accepted, it is necessary to demonstrate that the rheological predictions made by DPD are generally reliable. The principle aim of this thesis is to demonstrate that the rheology of polymer solutions can be simulated successfully with DPD. The rheology of a solution of DPD dumbbells using a FENE spring force law is studied in the first part of this thesis via simulation of steady shear flow and steady planar elongational flow. The rheological results are compared to dilute Brownian dynamics simulations of the same FENE dumbbell model. The level of coarse-graining of the DPD fluid is varied by changing the length of the DPD dumbbell relative to the particle size, while maintaining a constant extensibility parameter. Broadly speaking, the viscosity, first normal stress coefficient, and dumbbell extension in shear flow calculated with DPD are in agreement with the BD results. The two methods are not perfectly alike however, and two systematic differences between the DPD and BD results are obser, (cont.) The effect is more powerful in DPD dumbbells which are more coarse-grained. At high shear rates in the power-law regime, DPD systematically overpredicts the rate of shear-thinning, with the greatest deviation occurring in the most coarse-grained dumbbells. This is hypothesized to be a result of hydrodynamic interaction which comes naturally out of DPD's explicit treatment of the solvent. The HI effect is analyzed using the Giesekus anisotropic drag tensor. Shortly after its introduction, the complaint was made that DPD's dynamic results are suspect because it has a very low, gas-like Schmidt number, meaning that momentum and mass are transported through the DPD medium at similar rates. This is in contrast with physical liquids, which have large Schmidt numbers. The use of the Lowe-Anderson formulation of DPD allows the Schmidt number of a solution to be varied for the same polymer model. Shear flow simulations of identical dumbbells under different Schmidt number conditions give results in excellent agreement with each other, indicating that the Schmidt number is not an important factor in determining polymer rheology with DPD. Steady planar elongational flow is simulated for the first time in DPD using the Kraynik and Reinelt boundary conditions, which are periodic in both space and time, allowing for simulations of planar elongational flow for an unlimited period of time. The planar elongational flow results of FENE dumbbells are also in agreement with BD, butshow the same systematic deviations observed in shear flow. The second portion of this thesis examines a more complex polymer solution using DPD, with simulations of semidilute solutions of longer N = 20 bead-spring chain polymers undergoing shear and planar elongational flow. In addition to concentration effects, the importance of the solvent quality is also examined with simulations of polymer solutions in both good and theta solvents. In order to capture concentration dependency, a spring -spring r, (cont.) In planar elongational flow, each solution goes through a coil-stretch transition at the theoretically predicted strain rate De = 0.5. In shear flow, the rheological results are in qualitative agreement with theory, showing a plateau at low De, and a transition into a shear-thinning regime beginning at De = 1. While the planar elongational flow results show clear dependence on the solution relaxation time, the shear results show a mixed dependence on the overall solution relaxation time, which reflects the concentration dependence, and the relaxation rate of an isolated chain, suggesting that only some aspects of the shear rheology are affected by the concentration. The conclusion of this thesis is that DPD is able to faithfully reproduce reliable rheological behavior with bead-spring polymer models. We find however, that the computational costs associated with the explicit simulation of the solvent put DPD at a disadvantage for systematic rheology studies when compared to Brownian dynamics. The high costs of the spring-spring repulsion force implementation are particularly limiting. In complex systems where DPD's natural flexibility in molecular architecture and chemistry make it the best choice, rheological results can now be accepted with more confidence., by Theis Forman Clarke., Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Chemical Engineering, 2007., Includes bibliographical references (p. 214-228)., In this thesis, the unusual physics of rodlike liquid crystals are explored through simulations of rigid-rod kinetic theory discretized by the finite element method. As their name suggests. these substances retain crystalline. anisotropic microstructure but deform as a liquid: the microstructure results from the anisotropic shape and interactions of the constituent molecules. The molecules are modeled as Brownian. interacting, rigid rods in a kinetic theory formulation. A solution of such rods undergoes a phase transition from a disordered (isotropic) state to an ordered (nematic) state as the density of rods is increased: such a solution also exhibits interesting, complex. non-Newtonian rheological behavior. Liquid-crystalline substances are used in a number of high-performance industrial applications because the aligned, liquid-crystalline structure yields superior mechanical properties in the final product. The properties can be compromised by any interfaces or defects present in the system. The processing of these systems is poorly understood, and further progress requires accurate modeling of the coupled evolution of the microstructure and non-Newtonian flow field. The main goal of this thesis is the simulation of liquid-crystalline phenomena featuring sharp nonhomogeneities in structure on the length scale of a single rod. Such critical phenomena include the interfaces and defects that are prevalent in isotropic-nematic coexistence, phase transitions. aligning boundaries such as walls. and nonhomogeneous flows. In order to capture this level of detail, simulations evolve the rod distribution function, which describes the distribution of rod positions and orientations, coupled with the velocity field. The evolution equation for the distribution function is known as the Doi diffusion equation., Prior studies of rigid rod behavior have typically avoided this level of detail: instead. a series of approximations and assumptions have been used to simplify the diffusion equation and avoid evolving the distribution function since it is a quantity that varies in both spatial dimensions and orientation dimensions. These prior studies evolve the order tensor (the second moment of the distribution function) and invoke closure approximations to obtain a closed form for an evolution equation for this tensor. However, these techniques suffer from inaccuracy, often of unknown magnitude, because of the closure approximations, and they lose the ability to describe phenomena where gradients in structure on the length scale of a single rod are important or where variations in density are important. This thesis avoids these approximations and their attending limitations. A parallel, finite-element method is used to discretize the rod distribution function and the rod interaction potential. However, these simulations differ from traditional finite-element-based simulations of complex fluids because of the coupling between the physical space for rod location and orientation space for rod alignment and because of the difficulty of computing extended rod-rod interactions in a tractable manner. Instead of approximating the rod-rod interaction potential via Taylor expansions as previous studies have done, novel numerical methods are developed to meet these challenges and allow rapid, parallel computation of the full, nonhomogeneous Doi diffusion equation. These methods are applied to the equilibrium phase behavior of solutions of rigid rods in both periodic systems and system bounded by hard walls. Newton's method is used to solve the nonlinear set of equations for extrema of the system free energy, and the properties of the Jacobian matrix describe a local free energy surface., Simulation results show that periodic, isotropic-nematic coexistence states are stable in a given concentration range; the bulk phase properties and interfacial properties are independent of the average concentration or size of the system if the solution is stable. Results for confined systems show a number of surprising results when hard wall boundary conditions are used; in particular, the phase behavior of an extremely large system bounded by walls does not converge to the phase behavior of a homogeneous system without bounds, because the walls fundamentally alter the free energy surface of the system. Spinodal decomposition, the phase transition from an unstable isotropic state to a stable nematic state, is also studied. Prior studies of this process have been limited to linear stability analyses and have yielded sharp conflicts about the dominant mechanism of the process. The linear stability analysis is re-examined and prior conflicts are resolved by numerically showing that the process mechanism is a strong function of the system concentration and diffusivities for rotational and translational rod motion. A semi-implicit, finite-element-based time stepper is then used for full dynamic simulations of the spinodal decomposition process. These simulations mark the first results for the nonhomogeneous Doi diffusion equation in the spinodal decomposition process, and the results show how restricted rod motion can cause the process to become kinetically trapped in nonhomogeneous intermediate states. The method is also applied to the related problem of coarsening between large nematic domains of aligned rods, and results show the effects of domain misalignment on the final structure. This dynamic method is then extended to simulations of rigid rods in wall-driven, rectilinear shear flow and pressure-driven flow between parallel plates. These simulations mark the first results for the full nonhomogeneous Doi diffusion equation in nonhomogeneous shear flow., Silmulation results show that the accurate treatment of wall-rod interactions is critically important: aphysical anchoring conditions at walls can suppress out-of-plane instabilities in planar shear flow. Also. simulations of pressure-driven flow show that misaligned domains marked by "'twist" interfaces can spontaneously form as a result in gradients in the shear rate: this concept lends important insight into the formation of domains in macroscopic rheological experiments. Results also show that miisaligned. tumnbling domains are not stable. Phase diagrarns for wall-driven. rectilinear shear flow and pressure-driven. rectilinear shear flow are given for varying shear rate and system size., by Micah James Green., Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Chemical Engineering, 2008., This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections., Includes bibliographical references (p. 249-261)., State-of-the-art methods for simulating viscoelastic flows couple the conservation equations for mass and momentum with a model from kinetic theory that describes the microstructural state of the polymer. Introduction of appropriate numerical discretization and boundary conditions for these equations leads to a hybrid simulation for studying the dynamic behavior of polymeric liquids in complex geometries. This approach represents a rare example of a successful multiscale solution of a physical problem, as it allows investigation of arbitrary models of kinetic theory. The simulations, however, are not amenable to standard numerical techniques for system-level stability, bifurcation, and control analysis as this requires closed form equations. These simulation either use stochastic descriptions for the polymer microstructure that cannot be reduced to closed form, or involve equations for the evolution of a distribution of polymer conformations, which can only be written in closed form by invoking mathematical closure approximations that can have a significant qualitative impact on the predictive ability of these simulations. The focus of this thesis was to develop a novel numerical method that can enable hybrid simulations to perform system-level analysis of polymeric flows. This numerical approach has been applied directly to kinetic theory models and hybrid simulations to obtain stationary states and associated bifurcations and stability information. The method is general in its applicability in that it treats kinetic theory models and hybrid simulations as black boxes that are then used to obtain system-level information without any modification. The methods developed here are illustrated in a variety of problems., (cont) Steady state results have been obtained for the non-interacting rigid dumbbell model in steady shear, and for the free-draining bead-spring chain model in both steady shear and uniaxial elongation that are in excellent agreement with previous studies and steady state computed from direct integration. The method is also applied to a hybrid simulation for the pressure-driven flow of non-interacting rigid dumbbells in a planar channel with a linear array of equally spaced cylinders. The computed steady state is in agreement with direct integration and qualitatively matches previous computations with closed models. Bifurcation analysis has been performed for the Doi model at equilibrium with the Onsager excluded volume potential. This analysis agrees with previous studies and accurately predicts the isotropic-nematic transition and turning point for the unstable to stable transition on the prolate solution branch. Bifurcation analysis has also been performed for the Doi model in the weak shear flow limit for the Maier-Saupe excluded volume potential. It is found that stable stationary solutions are lost at a limit point beyond which time-periodic tumbling orbits are the only stable solution. This transition occurs via an infinite period global bifurcation, while the limit point approaches a threshold value as the shear rate approaches zero. This result matches a recently published scaling analysis and demonstrates the ability of the method to provide general bifurcation analysis of kinetic theory models. Stability analysis of the fiber-spinning process for polymeric fluids has also been performed by using a hybrid simulation that couples the one-dimensional conservation equations for mass and momentum with a stochastic description for the configuration fields of the Hookean dumbbell model. The steady-state velocity profiles are in good agreement with previous studies with the Oldroyd-B model., (cont) The analysis predicts onset of the draw resonance instability via a Hopf bifurcation and subsequent stabilization via second Hopf bifurcation in draw ratio parameter space. This result is in good agreement with experimentally observed behavior during polymer fiber-spinning., by Zubair Anwar., Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Chemical Engineering, 2008., This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections., Includes bibliographical references (p. 249-261)., State-of-the-art methods for simulating viscoelastic flows couple the conservation equations for mass and momentum with a model from kinetic theory that describes the microstructural state of the polymer. Introduction of appropriate numerical discretization and boundary conditions for these equations leads to a hybrid simulation for studying the dynamic behavior of polymeric liquids in complex geometries. This approach represents a rare example of a successful multiscale solution of a physical problem, as it allows investigation of arbitrary models of kinetic theory. The simulations, however, are not amenable to standard numerical techniques for system-level stability, bifurcation, and control analysis as this requires closed form equations. These simulation either use stochastic descriptions for the polymer microstructure that cannot be reduced to closed form, or involve equations for the evolution of a distribution of polymer conformations, which can only be written in closed form by invoking mathematical closure approximations that can have a significant qualitative impact on the predictive ability of these simulations. The focus of this thesis was to develop a novel numerical method that can enable hybrid simulations to perform system-level analysis of polymeric flows. This numerical approach has been applied directly to kinetic theory models and hybrid simulations to obtain stationary states and associated bifurcations and stability information. The method is general in its applicability in that it treats kinetic theory models and hybrid simulations as black boxes that are then used to obtain system-level information without any modification. The methods developed here are illustrated in a variety of problems., (cont) Steady state results have been obtained for the non-interacting rigid dumbbell model in steady shear, and for the free-draining bead-spring chain model in both steady shear and uniaxial elongation that are in excellent agreement with previous studies and steady state computed from direct integration. The method is also applied to a hybrid simulation for the pressure-driven flow of non-interacting rigid dumbbells in a planar channel with a linear array of equally spaced cylinders. The computed steady state is in agreement with direct integration and qualitatively matches previous computations with closed models. Bifurcation analysis has been performed for the Doi model at equilibrium with the Onsager excluded volume potential. This analysis agrees with previous studies and accurately predicts the isotropic-nematic transition and turning point for the unstable to stable transition on the prolate solution branch. Bifurcation analysis has also been performed for the Doi model in the weak shear flow limit for the Maier-Saupe excluded volume potential. It is found that stable stationary solutions are lost at a limit point beyond which time-periodic tumbling orbits are the only stable solution. This transition occurs via an infinite period global bifurcation, while the limit point approaches a threshold value as the shear rate approaches zero. This result matches a recently published scaling analysis and demonstrates the ability of the method to provide general bifurcation analysis of kinetic theory models. Stability analysis of the fiber-spinning process for polymeric fluids has also been performed by using a hybrid simulation that couples the one-dimensional conservation equations for mass and momentum with a stochastic description for the configuration fields of the Hookean dumbbell model. The steady-state velocity profiles are in good agreement with previous studies with the Oldroyd-B model., (cont) The analysis predicts onset of the draw resonance instability via a Hopf bifurcation and subsequent stabilization via second Hopf bifurcation in draw ratio parameter space. This result is in good agreement with experimentally observed behavior during polymer fiber-spinning., by Zubair Anwar., Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Chemical Engineering, 2007., Includes bibliographical references (p. 214-228)., In this thesis, the unusual physics of rodlike liquid crystals are explored through simulations of rigid-rod kinetic theory discretized by the finite element method. As their name suggests. these substances retain crystalline. anisotropic microstructure but deform as a liquid: the microstructure results from the anisotropic shape and interactions of the constituent molecules. The molecules are modeled as Brownian. interacting, rigid rods in a kinetic theory formulation. A solution of such rods undergoes a phase transition from a disordered (isotropic) state to an ordered (nematic) state as the density of rods is increased: such a solution also exhibits interesting, complex. non-Newtonian rheological behavior. Liquid-crystalline substances are used in a number of high-performance industrial applications because the aligned, liquid-crystalline structure yields superior mechanical properties in the final product. The properties can be compromised by any interfaces or defects present in the system. The processing of these systems is poorly understood, and further progress requires accurate modeling of the coupled evolution of the microstructure and non-Newtonian flow field. The main goal of this thesis is the simulation of liquid-crystalline phenomena featuring sharp nonhomogeneities in structure on the length scale of a single rod. Such critical phenomena include the interfaces and defects that are prevalent in isotropic-nematic coexistence, phase transitions. aligning boundaries such as walls. and nonhomogeneous flows. In order to capture this level of detail, simulations evolve the rod distribution function, which describes the distribution of rod positions and orientations, coupled with the velocity field. The evolution equation for the distribution function is known as the Doi diffusion equation., Prior studies of rigid rod behavior have typically avoided this level of detail: instead. a series of approximations and assumptions have been used to simplify the diffusion equation and avoid evolving the distribution function since it is a quantity that varies in both spatial dimensions and orientation dimensions. These prior studies evolve the order tensor (the second moment of the distribution function) and invoke closure approximations to obtain a closed form for an evolution equation for this tensor. However, these techniques suffer from inaccuracy, often of unknown magnitude, because of the closure approximations, and they lose the ability to describe phenomena where gradients in structure on the length scale of a single rod are important or where variations in density are important. This thesis avoids these approximations and their attending limitations. A parallel, finite-element method is used to discretize the rod distribution function and the rod interaction potential. However, these simulations differ from traditional finite-element-based simulations of complex fluids because of the coupling between the physical space for rod location and orientation space for rod alignment and because of the difficulty of computing extended rod-rod interactions in a tractable manner. Instead of approximating the rod-rod interaction potential via Taylor expansions as previous studies have done, novel numerical methods are developed to meet these challenges and allow rapid, parallel computation of the full, nonhomogeneous Doi diffusion equation. These methods are applied to the equilibrium phase behavior of solutions of rigid rods in both periodic systems and system bounded by hard walls. Newton's method is used to solve the nonlinear set of equations for extrema of the system free energy, and the properties of the Jacobian matrix describe a local free energy surface., Simulation results show that periodic, isotropic-nematic coexistence states are stable in a given concentration range; the bulk phase properties and interfacial properties are independent of the average concentration or size of the system if the solution is stable. Results for confined systems show a number of surprising results when hard wall boundary conditions are used; in particular, the phase behavior of an extremely large system bounded by walls does not converge to the phase behavior of a homogeneous system without bounds, because the walls fundamentally alter the free energy surface of the system. Spinodal decomposition, the phase transition from an unstable isotropic state to a stable nematic state, is also studied. Prior studies of this process have been limited to linear stability analyses and have yielded sharp conflicts about the dominant mechanism of the process. The linear stability analysis is re-examined and prior conflicts are resolved by numerically showing that the process mechanism is a strong function of the system concentration and diffusivities for rotational and translational rod motion. A semi-implicit, finite-element-based time stepper is then used for full dynamic simulations of the spinodal decomposition process. These simulations mark the first results for the nonhomogeneous Doi diffusion equation in the spinodal decomposition process, and the results show how restricted rod motion can cause the process to become kinetically trapped in nonhomogeneous intermediate states. The method is also applied to the related problem of coarsening between large nematic domains of aligned rods, and results show the effects of domain misalignment on the final structure. This dynamic method is then extended to simulations of rigid rods in wall-driven, rectilinear shear flow and pressure-driven flow between parallel plates. These simulations mark the first results for the full nonhomogeneous Doi diffusion equation in nonhomogeneous shear flow., Silmulation results show that the accurate treatment of wall-rod interactions is critically important: aphysical anchoring conditions at walls can suppress out-of-plane instabilities in planar shear flow. Also. simulations of pressure-driven flow show that misaligned domains marked by "'twist" interfaces can spontaneously form as a result in gradients in the shear rate: this concept lends important insight into the formation of domains in macroscopic rheological experiments. Results also show that miisaligned. tumnbling domains are not stable. Phase diagrarns for wall-driven. rectilinear shear flow and pressure-driven. rectilinear shear flow are given for varying shear rate and system size., by Micah James Green., Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Chemical Engineering, 2006., Includes bibliographical references (v. 2, leaves 259-268)., The field of computational fluid mechanics of viscoelastic flows has been well explored in the three decades since its inception. Still, even with the vast amount of work detailed in the literature, much remains to be done towards the improvement of models of viscoelastic fluids and the improvement of the numerical methods used to solve the set of governing equations. The work contained in this document is concentrated in the latter of these areas. The main goal of this body of work is to develop a robust, efficient simulation package to model three-dimensional viscoelastic flows. In order to accomplish this goal, improvements to the numerical methods and equation formulation were necessary to help reduce the overall size of the equation set used to describe viscoelastic flows in three-dimensional geometries. In order to test their viability for use in reducing the overall size of the problem, concepts involving changing the formulation of the equations and the numerical methods used to find the solution to the equations were first implemented and analyzed in a previously developed two-dimensional finite element simulation package. Implementation and analysis is discussed of a formulation change involving decoupling the calculation of the velocity gradient interpolant equation and the momentum and mass continuity equations in the DEVSS-G formulation., (cont.) Two different decoupled methods for computing the velocity gradient, one using a global least squares approximation and the other a local patch algorithm, are explored. While both methods reduce to the true velocity gradient with mesh refinement, the patch algorithm is shown to require significantly more mesh refinement than the global least squares approximation to order to attain equivalent refinement of the solution. Comparison of the two methods taking into account the additional refinement requirements of the local patch algorithm makes clear the superiority of the decoupled global least squares approximation for calculation of the velocity gradient interpolant. The versatility and robustness of the decoupled form of the DEVSS-G equations are demonstrated through the addition and modification of the evolution equations describing the stress of the polymer as well as new physical quantities of the flow. A time-dependent, free-surface finite element method is developed in which an evolution equation derived from the kinematic boundary condition is used to describe the height of the free surface as a function of time., (cont.) This new evolution equation is incorporated into the decoupled formulation by simply adding an additional step to the time integration to evaluate the change in the height of the surface during the current timestep and then updating the element locations in the deformable region of the mesh. Application of the new equation in this manner requires no knowledge of the direct dependence of the system on changes in the new quantity, allowing for quick and easy implementation. Incorporation of more advanced constitutive equations is used as further example of the utility of the decoupled form of the DEVSS-G equations. For most continuum based constitutive equations, the dependence of the equations on the flow variables can be expressed explicitly, allowing for the coupled set of equations to be solved with Newton's method. However, the dependence of the stress on the flow cannot be explicitly written for more advanced constitutive equations such as those derived from kinetic theory or those employing Brownian dynamics, greatly hindering the performance of Newton's method in locating the solution to the system. As an illustrative example, incorporation into the decoupled equation formulation of the closed form of the Adaptive-Length-Scale model (ALS-C) is presented., (cont.) Simulations are presented capturing for the first time the pressure drop enhancement with increasing viscoelasticity of the model of the flow of a Boger fluid in the 4:1:4 axisymmetric contraction-expansion geometry observed experimentally (Rothstein et al., 2001). Simulations of the flow of a 4-mode FENE-P model fluid within the geometry are also presented. Though its dependence on the flow field can be expressed analytically, the cost of computation using multimode models is typically prohibitive when using fully coupled equation sets as the overall problem size grows considerably with the addition of each new mode. Incorporation of the 4-mode model within the decoupled equation formulation adds relatively little computational cost to the overall calculation. Employing the formulation and numerical methods developed herein, a new three-dimensional finite element package is described for simulating confined viscoelastic flows. To make the package more robust, a number of different boundary conditions are included for modeling different geometries used in polymer processing. To help reduce the burden associated with mesh refinement in three-dimensional meshes, a commercial meshing package utilizing o-grid refinement for localization of refinement is employed., (cont.) Furthermore, to allow for computation of the large equation sets typically associated with three-dimensional geometries, a parallel implementation of the three-dimensional simulation package is developed based on the two-dimensional parallel method developed by Caola et al. ((Caola et al., 2001), (Caola et al., 2002)). Simulation results demonstrating the accuracy and performance of the method are presented. As a test of the robustness of the three-dimensional method, simulations of the flow of Newtonian and Oldroyd-B fluids through a periodic, linear array of cylinders are presented. Comparisons with previous calculations for the Oldroyd-B flow in an infinitely wide domain with no variations in the direction of the width show the same trend in the drag force on the cylinder with increasing viscoelasticity as well as in the size and shape of the vortices formed in the gap between the cylinders. The study of this flow includes effects of modeling the cross section of the flow as an infinite domain with no variation in the direction of the width, an infinite domain of periodic computational width, an infinite domain of periodic computational width and a symmetric flow above and below the cylinders, and a bounded domain with solid walls located 4 cylinder radii apart., by Scott David Phillips., Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Chemical Engineering, 1994., Includes bibliographical references (v. 2, leaves 365-374 )., by Suresh Ramalingam., Ph.D.

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Chemical Engineering, 2005., Includes bibliographical references (leaves 105-111)., One of the biggest challenges in non-Newtonian fluid mechanics is calculating the polymer contribution to the stress tensor, which is needed to calculate velocity and pressure fields as well as other quantities of interest. In the case of a Newtonian fluid, the stress tensor is linearly proportional to the velocity gradient and is given by the Newton's law of viscosity, but no such unique constitutive equation exists for non-Newtonian fluids. In order to predict accurately a polymer's rheological properties, it is important to have a good understanding of the molecular configurations in various flow situations. To obtain this information about molecular configurations and orientations, a micromechanical representation of a polymer molecule must be proposed. A micromechanical model may be fine scale, such as the Kramers chain model, which accurately predicts a real polymer's heological properties, but at the same time possesses too many degrees of freedom to be used in complex flow simulations, or it may be a coarse-grained model, such as the Hookean or the FENE dumbbell models, which can be used in complex flow analysis, but have too few degrees of freedom to adequately describe the rheology. The Adaptive Length Scale (ALS) model proposed by Ghosh et al. is only marginally more complicated than the FENE dumbbell model, yet it is able to capture the rapid stress growth in the start-up of uniaxial elongational flow, which is not predicted correctly by the simple dumbbell models. The ALS model is optimized in order to have its simulation time as close as possible to that of the FENE dumbbell., (cont.) Subsequently, the ALS model is simulated in the start-up of the uniaxial elongational and shear flows as well as in steady extensional and shear flows, and the results are compared to those obtained with other competing rheological models such as the Kramers chain, FENE chain, and FENE dumbbell. While a 5-spring FENE chain predicts results that are in very good agreement with the Kramers chain, the required simulation time clearly makes it impossible to use this model in complex flow simulations. The ALS model agrees better with the Kramers chain than does the FENE dumbbell in the start-up of shear and elongational flows. However, the ALS model takes too long to achieve steady state, which is something that needs to be explored further before the model is used in complex flow calculations. Understanding of this phenomena may explain why the stress-birefringence hysteresis loop predicted by the ALS model is unexpectedly small. In general, if polymer stress is to be calculated using Brownian dynamics simulations, a large number of stochastic trajectories must be simulated in order to predict accurately the macroscopic quantities of interest, which makes the problem computationally expensive. However, recent technological advances as well as a new simulation algorithm called Brownian configuration fields make such problems much more tractable. The operation count in order to assess the feasibility of using the ALS model in complex flow situations yields very promising results if parallel computing is used to calculate polymer contribution to stress. In an attempt to capture polydispersity of real polymer solutions, the use of multi-mode models is explored., (cont.) The model is fit to the linear viscoelastic spectrum to obtain relaxation times and individual modes' contributions to polymer viscosity. Then, data-fitting to the dimensionless extensional viscosity in the startup of the uniaxial elongational flow is performed for the ALS and the FENE dumbbell models to obtain the molecule's contour length, bmax. It is found that the results from the single-mode and the four-mode ALS models agree much better with the experimental data than do the corresponding single-mode and four-mode FENE dumbbell models. However, all four models resulted in a poor fit to the steady shear data, which may be explained by the fact that the zero-shear-rate viscosity obtained via a fit to the dynamic data by Rothstein and McKinley and used in present simulations, tends to be somewhat lower than the steady-state shear viscosity at very low shear rates, which may have caused a mismatch between the value of ... used in the simulation and the true ... of the polymer solution. As a motivation for using the ALS model in complex flow calculations, the results by Phillips, who simulated the closed-form version of the model in the benchmark 4:1:4 contraction- expansion problem are presented and compared to the experimental results by Rothstein and McKinley [49]. While the experimental observations show that there exists a large extra pres- sure drop, which increases monotonically with increasing De above the value observed for a Newtonian fluid subjected to the same flow conditions, the simulation results with a closed-form version of the FENE dumbbell model, called FENE-CR, exhibit the opposite trend., (cont.) The ALS-C model, on the other hand, is able to predict the trend correctly. The use of the ALS-C model in another benchmark problem, namely the flow around an array of cylinders confined between two parallel plates, also shows very promising results, which are in much better agreement with experimental data by Liu as compared to the Oldroyd-B model. The simulation results for the ALS-C and the Oldroyd-B models are due to Joo, et al. [28] and Smith et al. [50], respectively. Overall, it is concluded that the ALS model is superior to the commonly used FENE dumb- bell model, although more work is needed to understand why it takes significantly longer than the FENE dumbbell to achieve steady state in uniaxial elongational flows, and why the stress birefringence hysteresis loop predicted by the ALS model is much smaller than that of the other rheological models., by Irina Burmenko., S.M.

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Chemical Engineering, 2005., Includes bibliographical references (leaves 105-111)., One of the biggest challenges in non-Newtonian fluid mechanics is calculating the polymer contribution to the stress tensor, which is needed to calculate velocity and pressure fields as well as other quantities of interest. In the case of a Newtonian fluid, the stress tensor is linearly proportional to the velocity gradient and is given by the Newton's law of viscosity, but no such unique constitutive equation exists for non-Newtonian fluids. In order to predict accurately a polymer's rheological properties, it is important to have a good understanding of the molecular configurations in various flow situations. To obtain this information about molecular configurations and orientations, a micromechanical representation of a polymer molecule must be proposed. A micromechanical model may be fine scale, such as the Kramers chain model, which accurately predicts a real polymer's heological properties, but at the same time possesses too many degrees of freedom to be used in complex flow simulations, or it may be a coarse-grained model, such as the Hookean or the FENE dumbbell models, which can be used in complex flow analysis, but have too few degrees of freedom to adequately describe the rheology. The Adaptive Length Scale (ALS) model proposed by Ghosh et al. is only marginally more complicated than the FENE dumbbell model, yet it is able to capture the rapid stress growth in the start-up of uniaxial elongational flow, which is not predicted correctly by the simple dumbbell models. The ALS model is optimized in order to have its simulation time as close as possible to that of the FENE dumbbell., (cont.) Subsequently, the ALS model is simulated in the start-up of the uniaxial elongational and shear flows as well as in steady extensional and shear flows, and the results are compared to those obtained with other competing rheological models such as the Kramers chain, FENE chain, and FENE dumbbell. While a 5-spring FENE chain predicts results that are in very good agreement with the Kramers chain, the required simulation time clearly makes it impossible to use this model in complex flow simulations. The ALS model agrees better with the Kramers chain than does the FENE dumbbell in the start-up of shear and elongational flows. However, the ALS model takes too long to achieve steady state, which is something that needs to be explored further before the model is used in complex flow calculations. Understanding of this phenomena may explain why the stress-birefringence hysteresis loop predicted by the ALS model is unexpectedly small. In general, if polymer stress is to be calculated using Brownian dynamics simulations, a large number of stochastic trajectories must be simulated in order to predict accurately the macroscopic quantities of interest, which makes the problem computationally expensive. However, recent technological advances as well as a new simulation algorithm called Brownian configuration fields make such problems much more tractable. The operation count in order to assess the feasibility of using the ALS model in complex flow situations yields very promising results if parallel computing is used to calculate polymer contribution to stress. In an attempt to capture polydispersity of real polymer solutions, the use of multi-mode models is explored., (cont.) The model is fit to the linear viscoelastic spectrum to obtain relaxation times and individual modes' contributions to polymer viscosity. Then, data-fitting to the dimensionless extensional viscosity in the startup of the uniaxial elongational flow is performed for the ALS and the FENE dumbbell models to obtain the molecule's contour length, bmax. It is found that the results from the single-mode and the four-mode ALS models agree much better with the experimental data than do the corresponding single-mode and four-mode FENE dumbbell models. However, all four models resulted in a poor fit to the steady shear data, which may be explained by the fact that the zero-shear-rate viscosity obtained via a fit to the dynamic data by Rothstein and McKinley and used in present simulations, tends to be somewhat lower than the steady-state shear viscosity at very low shear rates, which may have caused a mismatch between the value of ... used in the simulation and the true ... of the polymer solution. As a motivation for using the ALS model in complex flow calculations, the results by Phillips, who simulated the closed-form version of the model in the benchmark 4:1:4 contraction- expansion problem are presented and compared to the experimental results by Rothstein and McKinley [49]. While the experimental observations show that there exists a large extra pres- sure drop, which increases monotonically with increasing De above the value observed for a Newtonian fluid subjected to the same flow conditions, the simulation results with a closed-form version of the FENE dumbbell model, called FENE-CR, exhibit the opposite trend., (cont.) The ALS-C model, on the other hand, is able to predict the trend correctly. The use of the ALS-C model in another benchmark problem, namely the flow around an array of cylinders confined between two parallel plates, also shows very promising results, which are in much better agreement with experimental data by Liu as compared to the Oldroyd-B model. The simulation results for the ALS-C and the Oldroyd-B models are due to Joo, et al. [28] and Smith et al. [50], respectively. Overall, it is concluded that the ALS model is superior to the commonly used FENE dumb- bell model, although more work is needed to understand why it takes significantly longer than the FENE dumbbell to achieve steady state in uniaxial elongational flows, and why the stress birefringence hysteresis loop predicted by the ALS model is much smaller than that of the other rheological models., by Irina Burmenko., S.M.

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Chemical Engineering, 2005., Includes bibliographical references (leaves 105-111)., One of the biggest challenges in non-Newtonian fluid mechanics is calculating the polymer contribution to the stress tensor, which is needed to calculate velocity and pressure fields as well as other quantities of interest. In the case of a Newtonian fluid, the stress tensor is linearly proportional to the velocity gradient and is given by the Newton's law of viscosity, but no such unique constitutive equation exists for non-Newtonian fluids. In order to predict accurately a polymer's rheological properties, it is important to have a good understanding of the molecular configurations in various flow situations. To obtain this information about molecular configurations and orientations, a micromechanical representation of a polymer molecule must be proposed. A micromechanical model may be fine scale, such as the Kramers chain model, which accurately predicts a real polymer's heological properties, but at the same time possesses too many degrees of freedom to be used in complex flow simulations, or it may be a coarse-grained model, such as the Hookean or the FENE dumbbell models, which can be used in complex flow analysis, but have too few degrees of freedom to adequately describe the rheology. The Adaptive Length Scale (ALS) model proposed by Ghosh et al. is only marginally more complicated than the FENE dumbbell model, yet it is able to capture the rapid stress growth in the start-up of uniaxial elongational flow, which is not predicted correctly by the simple dumbbell models. The ALS model is optimized in order to have its simulation time as close as possible to that of the FENE dumbbell., (cont.) Subsequently, the ALS model is simulated in the start-up of the uniaxial elongational and shear flows as well as in steady extensional and shear flows, and the results are compared to those obtained with other competing rheological models such as the Kramers chain, FENE chain, and FENE dumbbell. While a 5-spring FENE chain predicts results that are in very good agreement with the Kramers chain, the required simulation time clearly makes it impossible to use this model in complex flow simulations. The ALS model agrees better with the Kramers chain than does the FENE dumbbell in the start-up of shear and elongational flows. However, the ALS model takes too long to achieve steady state, which is something that needs to be explored further before the model is used in complex flow calculations. Understanding of this phenomena may explain why the stress-birefringence hysteresis loop predicted by the ALS model is unexpectedly small. In general, if polymer stress is to be calculated using Brownian dynamics simulations, a large number of stochastic trajectories must be simulated in order to predict accurately the macroscopic quantities of interest, which makes the problem computationally expensive. However, recent technological advances as well as a new simulation algorithm called Brownian configuration fields make such problems much more tractable. The operation count in order to assess the feasibility of using the ALS model in complex flow situations yields very promising results if parallel computing is used to calculate polymer contribution to stress. In an attempt to capture polydispersity of real polymer solutions, the use of multi-mode models is explored., (cont.) The model is fit to the linear viscoelastic spectrum to obtain relaxation times and individual modes' contributions to polymer viscosity. Then, data-fitting to the dimensionless extensional viscosity in the startup of the uniaxial elongational flow is performed for the ALS and the FENE dumbbell models to obtain the molecule's contour length, bmax. It is found that the results from the single-mode and the four-mode ALS models agree much better with the experimental data than do the corresponding single-mode and four-mode FENE dumbbell models. However, all four models resulted in a poor fit to the steady shear data, which may be explained by the fact that the zero-shear-rate viscosity obtained via a fit to the dynamic data by Rothstein and McKinley and used in present simulations, tends to be somewhat lower than the steady-state shear viscosity at very low shear rates, which may have caused a mismatch between the value of ... used in the simulation and the true ... of the polymer solution. As a motivation for using the ALS model in complex flow calculations, the results by Phillips, who simulated the closed-form version of the model in the benchmark 4:1:4 contraction- expansion problem are presented and compared to the experimental results by Rothstein and McKinley [49]. While the experimental observations show that there exists a large extra pres- sure drop, which increases monotonically with increasing De above the value observed for a Newtonian fluid subjected to the same flow conditions, the simulation results with a closed-form version of the FENE dumbbell model, called FENE-CR, exhibit the opposite trend., (cont.) The ALS-C model, on the other hand, is able to predict the trend correctly. The use of the ALS-C model in another benchmark problem, namely the flow around an array of cylinders confined between two parallel plates, also shows very promising results, which are in much better agreement with experimental data by Liu as compared to the Oldroyd-B model. The simulation results for the ALS-C and the Oldroyd-B models are due to Joo, et al. [28] and Smith et al. [50], respectively. Overall, it is concluded that the ALS model is superior to the commonly used FENE dumb- bell model, although more work is needed to understand why it takes significantly longer than the FENE dumbbell to achieve steady state in uniaxial elongational flows, and why the stress birefringence hysteresis loop predicted by the ALS model is much smaller than that of the other rheological models., by Irina Burmenko., S.M.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Chemical Engineering, 1986., MICROFICHE COPY AVAILABLE IN ARCHIVES AND SCIENCE., Bibliography: leaves 254-262., by Antony N. Beris., Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Chemical Engineering, 1988., Includes bibliographical references (p. 240-244)., by David Shu Chung Lee., Ph.D.

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Chemical Engineering, 2000., Includes bibliographical references (p. 306-325)., Efficient parallel computation of complex viscoelastic flows is essential to bring modem computer power to bear on fluid calculations where complicated constitutive descriptions are required. We have developed an efficient, parallel time integration scheme for calculating viscoelastic flows. The method is a synthesis of an operator splitting and time integration method that decouples the calculation of the polymer portion of the stress by solution of a hyperbolic constitutive equation from the temporal updating of the velocity and pressure fields by solution of a generalized Stokes problem. Finite element discretizations using the DEVSS-G 1 method are used; here a direct interpolation of the components of the velocity gradient is introduced. The key to the parallelization is the incorporation of iterative matrix solution for each portion of the split calculation. The linear system that results from the generalized Stokes problem is asymmetric, indefinite, and block singular. At each timestep, this system is solved using an algorithm which combines a parallel preconditioner with a Krylov subspace method. The parallel preconditioner, called the Block Complement and Additive Levels Method (BCALM) preconditioner, is based on treating pressure unknowns separately from the variables velocities and gradients. A pressure preconditioner is constructed from a factor of the Schur complement of the pressures using a Jacobi iteration. The viscous operator is treated using the additive Schwarz method. The number of Krylov iterations required per implicit step with increasing numbers of processors is stabilized using a two-level additive Schwarz method. The coarse grid is generated using addition as a restriction operator and insertion for extension. The resulting iterative method is demonstrated to have high parallel efficiency, subject to effective domain decomposition. Robustness and good performance result from using geometric partitioning of the unknowns by the nested bisecti, by Anthony Edward Caola., Ph.D.

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Chemical Engineering, 1998., Includes bibliographical references (v. 2, leaves 358-374)., The focus of this thesis was twofold. First, efficient numerical methods were developed for the modeling of the molecular orientation distribution function, f, of rigid rod polymers in flow. These methods were used to investigate the liquid crystal polymer (LCP) system. Second, the distribution function calculations were incorporated into a discontinuous Galerkin finite element framework for the modeling of complex LCP flows. Due to the localized nature of f under flow, the Daubechies D6 wavelets were used as basis functions for the approximation of f, and resulted in an efficient numerical technique to model LCPs. Bifurcation analysis of the Doi LCP model in shear flow showed that steady state solutions are lost due to the formation of a limit point in the plot of structure parameter S vs. concentration parameter N, beyond which time-periodic solutions are the only stable solutions. At low De, the time-periodic solutions onset as tumbling states at a global bifurcation. At high De, they onset as wagging states at a Hopf bifurcation, with a subsequent second transition to tumbling as N is increased. The period of oscillation of all tumbling states follows the experimentally observed scaling with strain. However, the wagging states do not follow this scaling, indicating that tumbling and wagging states belong to different solution. families. Computation of f in pressure-driven channel flow of LCPs using the wavelet­finite element method showed that the range of tumbling periods, combined with the existence of an infinite period tumbling streamline at the centerline, provides a mechanism for texture generation and refinement. The analysis also demonstrated the possibility of predicting biphasic solutions to LCP problems. Simulations of LCP flow in a tapering contraction geometry showed that the Doi model predicts inter­mediate concentrations to be most effective in producing flow-alignment of molecular distributions. At higher concentrations, tumbling is found to be w, by Radha Nayak., Ph.D.

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Chemical Engineering, 1979., Vita., Includes bibliographical references., by Kenji Yasuda.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Chemical Engineering, 1997., Includes bibliographical references (leaves 278-299)., by Alice Weimin Liu., Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Chemical Engineering, 1996., Includes bibliographical references (leaves 296-312)., by Michael Joseph Szady., Ph.D.

Thesis (Sc.D.)--Massachusetts Institute of Technology, Dept. of Chemical Engineering, 1981., MICROFICHE COPY AVAILABLE IN ARCHIVES AND SCIENCE., Vita., Bibliography: leaves 288-296., by Steven Minh Dinh., Sc.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Chemical Engineering, 1996., Includes bibliographical references (p. 241-249)., by Jeffrey Alan Byars., Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Chemical Engineering, 1985., MICROFICHE COPY AVAILABLE IN ARCHIVES AND SCIENCE., Bibliography: leaves 254-256., by Saad Akhtar Khan., Ph.D.

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Chemical Engineering, 1998., Includes bibliographical references (v. 2, leaves 358-374)., The focus of this thesis was twofold. First, efficient numerical methods were developed for the modeling of the molecular orientation distribution function, f, of rigid rod polymers in flow. These methods were used to investigate the liquid crystal polymer (LCP) system. Second, the distribution function calculations were incorporated into a discontinuous Galerkin finite element framework for the modeling of complex LCP flows. Due to the localized nature of f under flow, the Daubechies D6 wavelets were used as basis functions for the approximation of f, and resulted in an efficient numerical technique to model LCPs. Bifurcation analysis of the Doi LCP model in shear flow showed that steady state solutions are lost due to the formation of a limit point in the plot of structure parameter S vs. concentration parameter N, beyond which time-periodic solutions are the only stable solutions. At low De, the time-periodic solutions onset as tumbling states at a global bifurcation. At high De, they onset as wagging states at a Hopf bifurcation, with a subsequent second transition to tumbling as N is increased. The period of oscillation of all tumbling states follows the experimentally observed scaling with strain. However, the wagging states do not follow this scaling, indicating that tumbling and wagging states belong to different solution. families. Computation of f in pressure-driven channel flow of LCPs using the wavelet­finite element method showed that the range of tumbling periods, combined with the existence of an infinite period tumbling streamline at the centerline, provides a mechanism for texture generation and refinement. The analysis also demonstrated the possibility of predicting biphasic solutions to LCP problems. Simulations of LCP flow in a tapering contraction geometry showed that the Doi model predicts inter­mediate concentrations to be most effective in producing flow-alignment of molecular distributions. At higher concentrations, tumbling is found to be w, by Radha Nayak., Ph.D.

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Chemical Engineering, 2000., Includes bibliographical references (p. 306-325)., Efficient parallel computation of complex viscoelastic flows is essential to bring modem computer power to bear on fluid calculations where complicated constitutive descriptions are required. We have developed an efficient, parallel time integration scheme for calculating viscoelastic flows. The method is a synthesis of an operator splitting and time integration method that decouples the calculation of the polymer portion of the stress by solution of a hyperbolic constitutive equation from the temporal updating of the velocity and pressure fields by solution of a generalized Stokes problem. Finite element discretizations using the DEVSS-G 1 method are used; here a direct interpolation of the components of the velocity gradient is introduced. The key to the parallelization is the incorporation of iterative matrix solution for each portion of the split calculation. The linear system that results from the generalized Stokes problem is asymmetric, indefinite, and block singular. At each timestep, this system is solved using an algorithm which combines a parallel preconditioner with a Krylov subspace method. The parallel preconditioner, called the Block Complement and Additive Levels Method (BCALM) preconditioner, is based on treating pressure unknowns separately from the variables velocities and gradients. A pressure preconditioner is constructed from a factor of the Schur complement of the pressures using a Jacobi iteration. The viscous operator is treated using the additive Schwarz method. The number of Krylov iterations required per implicit step with increasing numbers of processors is stabilized using a two-level additive Schwarz method. The coarse grid is generated using addition as a restriction operator and insertion for extension. The resulting iterative method is demonstrated to have high parallel efficiency, subject to effective domain decomposition. Robustness and good performance result from using geometric partitioning of the unknowns by the nested bisecti, by Anthony Edward Caola., Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Chemical Engineering, 1997., Includes bibliographical references (leaves 278-299)., by Alice Weimin Liu., Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Chemical Engineering, February 2003., Includes bibliographical references (v. 2, leaves 351-363)., Knowing and understanding the dynamics and molecular configurations of polymer molecules is important for efficient process design and novel product development. Much research has been focused on combining molecular simulations and traditional fluid mechanics computations to simulate the behavior of polymeric liquids in a fabrication process. These simulation approaches require solution of the coupled set of the equations of change, the governing equations from kinetic theory, and the flux expressions that map molecular configurations to macroscopic quantities. Most complex flow simulations so far make use of a mixed finite element method to calculate the velocity field, with stress tensor evaluated by using a stochastic simulation method. This so-called CONNFFESSIT approach suffers from both a large memory requirement and stochastic noise. This thesis focuses on the development and application of a fully deterministic numerical approach for computing viscoelastic flows with constitutive descriptions based directly on diffusion equations from kinetic theory. The numerical approach is based on an operator splitting time integration method that decouples the calculation of microstructure by solution of a hyperbolic diffusion equation from the velocity and pressure field evolution, which is obtained by solution of a generalized Stokes problem. The generalized Stokes problem is written in the DEVSS-G formulation, where a direct interpolation of the components of the velocity gradient tensor is introduced. The efficiency and robustness of this numerical method is demonstrated through calculating the viscoelastic flows of a modified Doi model for liquid crystalline polymer and a number of reptation models for polymer melts in different flow geometries., (cont.) Simulations of the original Doi model with the Maier-Saupe potential in a pressure-driven channel flow by Nayak showed disclination formation due to the shear-rate-dependent frequencies of the tumbling/wagging states of the Doi model in a shear or mixed shear flow. The lack of an instrinsic length scale in the model leads to an infinitesimal structure refinement that eventually causes numerical instabilities. In this thesis, the effect of concentration variation is incorporated to develop a modified Doi model for introducing an intrinsic length scale through translational diffusion. This changes the mathematical characteristics of the spatial variation of the underlying diffusion equation from that of a hyperbolic equation to that of an elliptic equation. The resulting elliptic diffusion equation is then solved by using a local discontinuous Galerkin method, where an auxillary variable is introduced to rewrite the elliptic diffusion equation into a pair of formal, hyperbolic equations, which in turn is solved by the standard discontinuous Galerkin method. Unlike the original Doi model, a steady state is reached for a variety of De. Although there is structure variation across the channel width, the director profiles point uniformly along the flow direction. The lack of disclination formation may be rectified by introducing Frank elasticity into the modified Doi model ..., by Jason Ka-Chun Suen., Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Chemical Engineering, 1996., Includes bibliographical references (leaves 296-312)., by Michael Joseph Szady., Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Chemical Engineering, 1987., Bibliography: v. 2, leaves 299-303., by Miguel Angel Bibbó., Ph.D.

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Chemical Engineering, 1979., Vita., Includes bibliographical references., by Kenji Yasuda.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Chemical Engineering, 1988., Includes bibliographical references (p. 240-244)., by David Shu Chung Lee., Ph.D.

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Chemical Engineering, 2000., Includes bibliographical references (leaves 275-280)., The need to predict the interactions between the microstructure of polymeric fluids and the macroscopic flow field in polymer processing operations has lead to the development of many numerical methods for the simulation of viscoelastic flows. The two main obstacles to the use of these tools in the quantitative analysis of polymer processing operations are the need for accurate constitutive equations and the computational efficiency of the numerical methods. The goal of the first part of this thesis is to develop constitutive equations appropriate for modeling nonisothermal viscoelastic flows and to examine the influence of temperature variations on the stress in the fluid. The pseudotime method developed by Crochet and Naghdi (1978) and by Sarti (1977) is used to obtain thermodynamically consistent constitutive equations for the internal energy, the stress, and the heat flux vector. Fourier"s Jaw is assumed for the thermal conductivity, and the internal energy is chosen to depend only on the temperature of the fluid. The non isothermal analog to the Giesekus constitutive equation for the polymer contribution to the stress is then de1ived with the pseudotime approach. The constitutive equations are combined with conservation equations for mass. momentum, and energy to model the nonisothermal viscoelastic extrudate swell problem. For contraction-shaped dies, the final thickness of the extrudate is shown to depend on both the thermal and the strain histories of the fluid emerging, at the die exit. In this thesis second part of this thesis, efficient time integration algorithms are developed within the framework of examining the linear stability of isothermal viscoelastic two dimensional steady flows in complex geometries to three-dimensional perturbations. In this approach, a steady base state is found by discretizing in two dimensions with the discrete elastic-viscous split stress gradient (DEVSS-G) finite element formulation of the momentum-continuity pair combined, by Mark Davis Smith., Ph.D.

Thesis (Sc.D.)--Massachusetts Institute of Technology, Dept. of Chemical Engineering, 1981., MICROFICHE COPY AVAILABLE IN ARCHIVES AND SCIENCE., Vita., Bibliography: leaves 288-296., by Steven Minh Dinh., Sc.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Chemical Engineering, 1996., Includes bibliographical references (p. 241-249)., by Jeffrey Alan Byars., Ph.D.

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Chemical Engineering, 2000., Includes bibliographical references (leaves 275-280)., The need to predict the interactions between the microstructure of polymeric fluids and the macroscopic flow field in polymer processing operations has lead to the development of many numerical methods for the simulation of viscoelastic flows. The two main obstacles to the use of these tools in the quantitative analysis of polymer processing operations are the need for accurate constitutive equations and the computational efficiency of the numerical methods. The goal of the first part of this thesis is to develop constitutive equations appropriate for modeling nonisothermal viscoelastic flows and to examine the influence of temperature variations on the stress in the fluid. The pseudotime method developed by Crochet and Naghdi (1978) and by Sarti (1977) is used to obtain thermodynamically consistent constitutive equations for the internal energy, the stress, and the heat flux vector. Fourier"s Jaw is assumed for the thermal conductivity, and the internal energy is chosen to depend only on the temperature of the fluid. The non isothermal analog to the Giesekus constitutive equation for the polymer contribution to the stress is then de1ived with the pseudotime approach. The constitutive equations are combined with conservation equations for mass. momentum, and energy to model the nonisothermal viscoelastic extrudate swell problem. For contraction-shaped dies, the final thickness of the extrudate is shown to depend on both the thermal and the strain histories of the fluid emerging, at the die exit. In this thesis second part of this thesis, efficient time integration algorithms are developed within the framework of examining the linear stability of isothermal viscoelastic two dimensional steady flows in complex geometries to three-dimensional perturbations. In this approach, a steady base state is found by discretizing in two dimensions with the discrete elastic-viscous split stress gradient (DEVSS-G) finite element formulation of the momentum-continuity pair combined, by Mark Davis Smith., Ph.D.

Thesis (Sc.D.)--Massachusetts Institute of Technology, Dept. of Chemical Engineering, 1979., MICROFICHE COPY AVAILABLE IN ARCHIVES AND SCIENCE., Vita., Bibliography: leaves 381-382., Sc.D.

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Chemical Engineering, 1979., Vita., Includes bibliographical references., by Kenji Yasuda.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Chemical Engineering, 1996., Includes bibliographical references (p. 241-249)., by Jeffrey Alan Byars., Ph.D.

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Chemical Engineering, 2000., Includes bibliographical references (p. 306-325)., Efficient parallel computation of complex viscoelastic flows is essential to bring modem computer power to bear on fluid calculations where complicated constitutive descriptions are required. We have developed an efficient, parallel time integration scheme for calculating viscoelastic flows. The method is a synthesis of an operator splitting and time integration method that decouples the calculation of the polymer portion of the stress by solution of a hyperbolic constitutive equation from the temporal updating of the velocity and pressure fields by solution of a generalized Stokes problem. Finite element discretizations using the DEVSS-G 1 method are used; here a direct interpolation of the components of the velocity gradient is introduced. The key to the parallelization is the incorporation of iterative matrix solution for each portion of the split calculation. The linear system that results from the generalized Stokes problem is asymmetric, indefinite, and block singular. At each timestep, this system is solved using an algorithm which combines a parallel preconditioner with a Krylov subspace method. The parallel preconditioner, called the Block Complement and Additive Levels Method (BCALM) preconditioner, is based on treating pressure unknowns separately from the variables velocities and gradients. A pressure preconditioner is constructed from a factor of the Schur complement of the pressures using a Jacobi iteration. The viscous operator is treated using the additive Schwarz method. The number of Krylov iterations required per implicit step with increasing numbers of processors is stabilized using a two-level additive Schwarz method. The coarse grid is generated using addition as a restriction operator and insertion for extension. The resulting iterative method is demonstrated to have high parallel efficiency, subject to effective domain decomposition. Robustness and good performance result from using geometric partitioning of the unknowns by the nested bisecti, by Anthony Edward Caola., Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Chemical Engineering, 1988., Includes bibliographical references (p. 240-244)., by David Shu Chung Lee., Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Chemical Engineering, February 2003., Includes bibliographical references (v. 2, leaves 351-363)., Knowing and understanding the dynamics and molecular configurations of polymer molecules is important for efficient process design and novel product development. Much research has been focused on combining molecular simulations and traditional fluid mechanics computations to simulate the behavior of polymeric liquids in a fabrication process. These simulation approaches require solution of the coupled set of the equations of change, the governing equations from kinetic theory, and the flux expressions that map molecular configurations to macroscopic quantities. Most complex flow simulations so far make use of a mixed finite element method to calculate the velocity field, with stress tensor evaluated by using a stochastic simulation method. This so-called CONNFFESSIT approach suffers from both a large memory requirement and stochastic noise. This thesis focuses on the development and application of a fully deterministic numerical approach for computing viscoelastic flows with constitutive descriptions based directly on diffusion equations from kinetic theory. The numerical approach is based on an operator splitting time integration method that decouples the calculation of microstructure by solution of a hyperbolic diffusion equation from the velocity and pressure field evolution, which is obtained by solution of a generalized Stokes problem. The generalized Stokes problem is written in the DEVSS-G formulation, where a direct interpolation of the components of the velocity gradient tensor is introduced. The efficiency and robustness of this numerical method is demonstrated through calculating the viscoelastic flows of a modified Doi model for liquid crystalline polymer and a number of reptation models for polymer melts in different flow geometries., (cont.) Simulations of the original Doi model with the Maier-Saupe potential in a pressure-driven channel flow by Nayak showed disclination formation due to the shear-rate-dependent frequencies of the tumbling/wagging states of the Doi model in a shear or mixed shear flow. The lack of an instrinsic length scale in the model leads to an infinitesimal structure refinement that eventually causes numerical instabilities. In this thesis, the effect of concentration variation is incorporated to develop a modified Doi model for introducing an intrinsic length scale through translational diffusion. This changes the mathematical characteristics of the spatial variation of the underlying diffusion equation from that of a hyperbolic equation to that of an elliptic equation. The resulting elliptic diffusion equation is then solved by using a local discontinuous Galerkin method, where an auxillary variable is introduced to rewrite the elliptic diffusion equation into a pair of formal, hyperbolic equations, which in turn is solved by the standard discontinuous Galerkin method. Unlike the original Doi model, a steady state is reached for a variety of De. Although there is structure variation across the channel width, the director profiles point uniformly along the flow direction. The lack of disclination formation may be rectified by introducing Frank elasticity into the modified Doi model ..., by Jason Ka-Chun Suen., Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Chemical Engineering, 1993., Includes bibliographical references (leaves 285-299)., by Aparna Vasant Bhave., Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Chemical Engineering, 1985., MICROFICHE COPY AVAILABLE IN ARCHIVES AND SCIENCE., Bibliography: leaves 254-256., by Saad Akhtar Khan., Ph.D.

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Chemical Engineering, 1998., Includes bibliographical references (v. 2, leaves 358-374)., The focus of this thesis was twofold. First, efficient numerical methods were developed for the modeling of the molecular orientation distribution function, f, of rigid rod polymers in flow. These methods were used to investigate the liquid crystal polymer (LCP) system. Second, the distribution function calculations were incorporated into a discontinuous Galerkin finite element framework for the modeling of complex LCP flows. Due to the localized nature of f under flow, the Daubechies D6 wavelets were used as basis functions for the approximation of f, and resulted in an efficient numerical technique to model LCPs. Bifurcation analysis of the Doi LCP model in shear flow showed that steady state solutions are lost due to the formation of a limit point in the plot of structure parameter S vs. concentration parameter N, beyond which time-periodic solutions are the only stable solutions. At low De, the time-periodic solutions onset as tumbling states at a global bifurcation. At high De, they onset as wagging states at a Hopf bifurcation, with a subsequent second transition to tumbling as N is increased. The period of oscillation of all tumbling states follows the experimentally observed scaling with strain. However, the wagging states do not follow this scaling, indicating that tumbling and wagging states belong to different solution. families. Computation of f in pressure-driven channel flow of LCPs using the wavelet­finite element method showed that the range of tumbling periods, combined with the existence of an infinite period tumbling streamline at the centerline, provides a mechanism for texture generation and refinement. The analysis also demonstrated the possibility of predicting biphasic solutions to LCP problems. Simulations of LCP flow in a tapering contraction geometry showed that the Doi model predicts inter­mediate concentrations to be most effective in producing flow-alignment of molecular distributions. At higher concentrations, tumbling is found to be w, by Radha Nayak., Ph.D.

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Chemical Engineering, 2000., Includes bibliographical references (p. 306-325)., Efficient parallel computation of complex viscoelastic flows is essential to bring modem computer power to bear on fluid calculations where complicated constitutive descriptions are required. We have developed an efficient, parallel time integration scheme for calculating viscoelastic flows. The method is a synthesis of an operator splitting and time integration method that decouples the calculation of the polymer portion of the stress by solution of a hyperbolic constitutive equation from the temporal updating of the velocity and pressure fields by solution of a generalized Stokes problem. Finite element discretizations using the DEVSS-G 1 method are used; here a direct interpolation of the components of the velocity gradient is introduced. The key to the parallelization is the incorporation of iterative matrix solution for each portion of the split calculation. The linear system that results from the generalized Stokes problem is asymmetric, indefinite, and block singular. At each timestep, this system is solved using an algorithm which combines a parallel preconditioner with a Krylov subspace method. The parallel preconditioner, called the Block Complement and Additive Levels Method (BCALM) preconditioner, is based on treating pressure unknowns separately from the variables velocities and gradients. A pressure preconditioner is constructed from a factor of the Schur complement of the pressures using a Jacobi iteration. The viscous operator is treated using the additive Schwarz method. The number of Krylov iterations required per implicit step with increasing numbers of processors is stabilized using a two-level additive Schwarz method. The coarse grid is generated using addition as a restriction operator and insertion for extension. The resulting iterative method is demonstrated to have high parallel efficiency, subject to effective domain decomposition. Robustness and good performance result from using geometric partitioning of the unknowns by the nested bisecti, by Anthony Edward Caola., Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Chemical Engineering, 1997., Includes bibliographical references (leaves 278-299)., by Alice Weimin Liu., Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Chemical Engineering, 1986., MICROFICHE COPY AVAILABLE IN ARCHIVES AND SCIENCE., Bibliography: leaves 254-262., by Antony N. Beris., Ph.D.

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Chemical Engineering, 1979., Vita., Includes bibliographical references., by Kenji Yasuda.