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This work aims to explore the T-channel momentum and heat transfer characteristics with the combined effect of Bingham plastic fluids (0.01 ≤ Bn ≤ 20) behavior and geometrical variation in terms of branching angle (30 deg ≤ α ≤ 90 deg). The problem has been solved over a wide range of Reynolds number (50 ≤ Re ≤ 300) and Prandtl number (10 ≤ Pr ≤ 50). For the momentum flow, qualitative and quantitative features are analyzed in terms of streamlines, structure of yielded/unyielded regions, shear rate contours, plug width and length variation, and local pressure coefficient. These features have been represented in terms of isotherm patterns, temperature profile, Nusselt number, and its asymptotic value for heat transfer characteristics. The recirculating flows have been presented here in the vicinity of T-junction, which promote mixing and heat transfer. Broadly, the size of this zone bears a positive dependence on Re and α. However, fluid yield stress tends to suppress it. The critical Reynolds and Bingham numbers were found to be strong functions of the pertinent parameters like α. The inclination angle exerts only a weak effect on the yielded/unyielded regions and on the recirculation length of main branch. Results show a strong relationship of the plug width and length with key parameters and branches. The Nusselt number exhibits a positive relationship with α, Bn, and Re but for lower Pr in the T-junction vicinity for both branches. Such length indicates the required optimum channel length for thermal mixing.

Laminar Bingham fluid flow in the entrance region of a circular pipe is investigated using a momentum integral model. The fully developed flow is uniform in the core region, while the velocity changes in the annular part of the cross section of the pipe. The inlet-filled region concept is adopted. In the inlet region, the boundary layer thickness increases until the size of the plug flow area reaches the fully developed flow size. The model converges to the fully developed solution in the filled region. The model provides the velocity, pressure drop, and skin friction coefficient profiles. The pressure drop results are in good agreement with published experimental data. The flow results asymptotically converge to the fully developed values. In addition, the results are consistent with published Newtonian fluid flow experimental data and theoretical results for the boundary layer thickness, pressure drop, and centerline velocity for small values of the Bingham number.

A three-dimensional thermohydrodynamic model is developed to predict non-Newtonian lubricant behavior in slider bearings and channel flow. The generalized Reynolds equation is established using the concept of generalized Newtonian fluids (GNF) and the temperature field is determined with the energy equation. The chosen rheological models are the power-law, Bingham, and Hershel–Bulkley models. The last two models hold uniformly in yielded and unyielded regions using the approach proposed by Papanastasiou. The results present the evolution of the velocity, pressure, and thermal fields. The power loss, load capacity, and friction coefficient are analyzed. Comparisons are made with Newtonian lubricants and other recent non-Newtonian computational analyses.

The thermal characteristics of Bingham plastic fluid flows are analyzed in circular microchannels under uniform wall heat flux condition. The analytic approach presented here reveals that the governing parameters are Bingham number, dimensionless radius of the plug flow region, and Brinkman number. The results demonstrate that there is a strong influence of viscous dissipation on heat transfer and entropy generation for Brinkman numbers greater than a specific value. With increasing the Brinkman number and dimensionless radius of the plug flow region, entropy generation is increased, while the Nusselt number is decreased. The influence of these parameters on the entropy generation from heat transfer is strongly higher than the entropy generation from fluid friction. The average dimensionless total entropy shows that the Bingham plastic fluids generate entropy more than Newtonian fluids; also, an increase in the dimensionless radius of the plug flow region results in increasing the average dimensionless total entropy generation. By letting the dimensionless radius of the plug flow region equal to zero, the generalized expressions and results will be simplified to Newtonian fluids.

We propose simple formulas for calibration of rheological parameters of Bingham fluid using a co-axial cylinder rheometer, in which the inner cylinder rotates at a constant speed, and the outer cylinder is stationary. A critical rotational speed exists due to existence of yield stress. If rotation speed exceeds the critical value, all fluid is fully sheared between the concentric cylinders. Plug flow exists only when rotation speed is less than the critical value. The effects of radius ratio and Bingham number are discussed. The rheometer with radius ratio very close to unity is discussed as a limiting case, and the result confirms previous research. Formulas for calibration are derived using Least Squares and perturbation methods for all values of radius ratio, based on measured rotation speed and torque. Two sets of experimental data are used for verification. The validation shows that the formulas derived here yield reasonable and accurate estimates of rheological parameters.

Pressure-driven flows over a square cavity are studied numerically for Bingham plastics exhibiting a yield stress. The problem is encountered whenever pressure measurements are made by a drilled-hole based pressure transducer. The Bingham constitutive equation is used with an appropriate modification proposed by Papanastasiou, which applies everywhere in the flow field in both yielded and practically unyielded regions. Newtonian results are obtained for a wide range of Reynolds numbers (0

The lubrication approximation theory (LAT) is used to provide numerical results for calendering a sheet from an infinite reservoir. The Herschel–Bulkley model of viscoplasticity is employed, which reduces with appropriate modifications to the Bingham, power-law, and Newtonian models. A dimensionless slip coefficient is introduced to account for the case of slip at the roll surfaces. The results give the final sheet thickness as a function of the dimensionless power-law index (in the case of pseudoplasticity), the Bingham number or the dimensionless yield stress calculated at the nip (in the case of viscoplasticity), and the dimensionless slip coefficient in both cases. Integrated quantities of engineering interest are also calculated. These include the maximum pressure, the roll-separating force, and the power input to the rolls. Decreasing the power-law index or increasing the dimensionless yield stress lead to excess sheet thickness over the thickness at the nip. All engineering quantities calculated in dimensionless form increase substantially with the departure from the Newtonian values. The presence of slip decreases pressure and the engineering quantities and increases the domain in all cases.

In the present investigation, an analytical numerical solution is presented for the mass transfer from a rotating disk to a Bingham fluid for the case of laminar boundary layer flow. The analytical approach includes the coupled effects of steady disk rotation and non-Newtonian fluid properties on the mass transfer rate. A dimensionless expression for the wall mass transfer rate based on the Sherwood number, Sh, is obtained in terms of the system parameters (Reynolds number, Rep, and Schmidt number, Scp) which depend on the dimensionless yield stress or Bingham number, By. The analytical relation indicates that an increase in By (up to the limit By⩽1) leads to a slight increase in the wall mass transfer rate, and thereafter, for By>1, the mass transfer rate is reduced.

We present a numerical study of Dean instability for non-Newtonian fluids in a laminar 180deg curved-channel flow of rectangular cross section. A methodology based on the Papanastasiou model (Papanastasiou, T. C., 1987, J. Rheol., 31(5), pp. 385–404) was developed to take into account the Bingham-type rheological behavior. After validation of the numerical methodology, simulations were carried out (using FLUENT CFD code) for Newtonian and non-Newtonian fluids in curved channels of square or rectangular cross section and for a large aspect and curvature ratios. A criterion based on the axial velocity gradient was defined to detect the instability threshold. This criterion was used to optimize the grid geometry. The effects of curvature and aspect ratio on the Dean instability are studied for all fluids, Newtonian and non-Newtonian. In particular, we show that the critical value of the Dean number decreases with increasing curvature ratio. The variation of the critical Dean number with aspect ratio is less regular. The results are compared to those for Newtonian fluids to emphasize the effect of the power-law index and the Bingham number. The onset of Dean instability is delayed with increasing power-law index. The same delay is observed in Bingham fluids when the Bingham number is increased.

The dripping problem of a viscoplastic (yield-stress) liquid running slowly out of a narrow vertical tube is considered. The volume of the drops which break away is determined. A Lagrangian coordinate system is used to analyze the extension of the thread as it sags under its own weight, neglecting inertia and capillarity. The biviscosity model has been used to characterize viscoplastic fluids; the Newtonian and Bingham models can be recovered as limiting cases. The Bingham limit is of special interest.

Miniaturization of traditional devices is in high demand formicro-electro-mechanical systems~MEMS!. Examples includebut are not limited to optics, communication and information sys-tems, fluidics, biotechnology, medicine, automotive, and aero-space. An area of interest in several engineering fields is the con-trol of fluid flow within microchannels and microtubes. Theapplications include, but are not limited to, fields such as vibrationcontrol of structures and systems using small devices, reactors formodification and separation of biological cells, energy systems asa mobile power supply, heat exchangers for micro and macro de-vices, and propulsion engines @1–3#.In recent years, there has been growing attention given to theliquid flow in microchannels in parallel with the development ofminiaturized devices and systems. The understanding of flowcharacteristics, such as velocity distribution and pressure loss isnecessary in design and process control of microfluidic devices.Peng, Peterson, and Wang @4# experimentally studied the flowcharacteristics of water flowing through rectangular microchan-nels having hydraulic diameters of 0.133 to 0.367 mm and heightto width ratios of 0.333 to 1. Their results indicated that the lami-nar flow transition occurred for the range of the Re number be-tween 200 and 700. They also claimed that friction behavior forboth laminar and turbulent flow depart from classical thermofluidcorrelations, and the friction factor is proportional to Re

The problem of stretching a viscoplastic (yield-stress) thread of a liquid hanging vertically is considered. The length of the thread at later times and the time at which it ruptures is determined. A Lagrangian coordinate system is used to analyze the extension of the thread as it sags under its own weight, with negligible inertial effects. The biviscosity model has been used to characterize viscoplastic fluids; the Newtonian and Bingham models can be recovered as limiting cases. The Bingham limit is of special interest.

A constitutive theory is presented for a transversely isotropic, viscoplastic (Bingham) fluid. The theory accounts for threshold (yield) and viscous flow characteristics through inclusion of a potential function serving the dual role of a threshold function and a viscous flow potential. The arguments and form of the potential function derive from the theory of tensorial invariants. The model reduces to a transversely isotropic model of perfect plasticity in the limit of vanishing viscosity. In the limit of isotropy, it reduces to the Hohenemser-Prager generalization of Bingham’s model. A characterization procedure is prescribed based on correlation with experiments conducted under simple states of stress. Application is made to polymer melts filled with talc particles.

Thermally developing laminar flow of a Bingham plastic in a circular pipe with uniform wall heat flux has been studied analytically. Expressions for the fully developed temperature and Nusselt number are presented in terms of the yield stress, Peclet number, and Brinkman number. The solution to the Graetz problem has been obtained by using the method of separation of variables, where the resulting eigenvalue problem is solved approximately by using the method of weighted residuals. The effects of the yield stress, Peclet, and Brinkman numbers on the Nusselt number are discussed. In particular, it is shown that the heat transfer characteristics in the entrance region are significantly affected by the yield stress with the inclusion of viscous dissipation.

A mathematical model is developed for parametric investigation of the influence of the rheological properties of the lubricant on the thermohydrodynamic film conditions which occur in slider and journal bearings. The model is utilized for predicting the thickness of the shear zone in lubricating films with fixed geometry between the stationary and moving surfaces using a Bingham model with a shear yield stress which best fits the experimental pressure. It is found that the shear zone thickness is significantly smaller than the fluid film thickness in the lubrication zone and is of the same order of magnitude as that obtained by the empirical formula proposed by Wang and Seireg (1995) ASME J. Tribol., 117, p. 444. The results also suggest that the shear yield stress for the lubricating film is proportional to the pressure developed in the film within the range of the investigated cases. [S0742-4787(00)05101-8]

Presented are theoretical and experimental results from a project supporting development of a new method for cement vibration in well’s annulus to prevent gas migration. In this method, cyclic pressure pulses are applied at the wellhead and transmitted down the annulus. These pulses cause reciprocation and shear within cement column—a process delaying the loss of hydrostatic pressure and preventing the inflow of gas into the well’s annulus. Field tests in real wells conducted to observe transmission of a single pressure pulse and to measure compressibility of setting cement showed that application of pressure pulses of 100 psi amplitude and 0.1 Hz frequency may be an effective and inexpensive way of preventing gas flow after cementing. The paper presents development of a cement pulsation design method based upon the analytical model of pressure propagation in Bingham plastic fluid and experimental data on rheology of cement slurries subjected to continuous shear. The primary objective of the method is to minimize the likelihood of gas invasion into the cement-filled annulus.

This paper presents a feedback control performance of a full-car suspension system featuring electrorheological (ER) dampers for a passenger vehicle. A cylindrical ER damper is designed and manufactured by incorporating a Bingham model of an ER fluid which is commercially available. After evaluating field-dependent damping characteristics of the ER damper, a full-car suspension system installed with four independent ER dampers is then constructed and its governing equation of motion, which includes vertical, pitch, and roll motions, is derived. A sliding mode controller, which has inherent robustness against system uncertainties, is then formulated by treating the sprung mass as uncertain parameter. For the demonstration of a practical feasibility, control characteristics for vibration suppression of the proposed ER suspension system are evaluated under various road conditions through the hardware-in-the-loop simulation (HILS). [S0022-0434(00)02801-X]

A detailed analysis of the flow characteristics and lubrication effectiveness of grease in journal bearings is presented. Based on the ideal Bingham plastic model, the appropriate governing equations are developed which effectively handle the formation of a floating core as well as a core which may attach itself to either the shaft or the stationary bushing. The formulation and numerical solutions presented pertain to three-dimensional flows in finite journal bearings. Moreover, an extensive amount of computational results are presented, which take thermal effects into account according to the thermohydrodynamic theory.

The flow of a non-Newtonian viscoplastic Bingham fluid over an axisymmetric sudden expansion is studied by numerically solving the governing fully-elliptic continuity and momentum equations. Solutions are obtained for a wide range of Reynolds and yield numbers in the laminar flow regime with constant fluid properties. The present work demonstrates that the finite-difference technique can successfully be employed to obtain solutions to separating/reattaching internal flows of Bingham plastics. The results demonstrate the strong effects of the yield and Reynolds numbers on both the integral and the local structure of the separating and reattaching flow. Higher yield numbers result in larger overall effective viscosities and thus faster flow recovery downstream of the sudden expansion. The reattachment length decreases with increasing yield numbers, eventually reaching an asymptotic nonzero value which, in turn, is dependent on the Reynolds number. The strength of the recirculating flow also decreases with increasing yield numbers.

Electrorheological (ER) materials develop yield stresses on the order of 5–10 kPa in the presence of strong electric fields. Viscoelastic and yielding material properties can be modulated within milli-seconds. An analysis of flowing ER materials in the limiting case of fully developed steady flow results in simple approximations for use in design. Small-scale experiments show that these design equations can be applied to designing devices in which the flow is unsteady. More exact models of ER device behavior can be determined using curve-fitting techniques in multiple dimensions. A previously known curve-fitting technique is extended to deal with variable electric fields. Experiments are described which illustrate the potential for ER devices in large-scale damping applications and the accuracy of the modeling technique.

Electrorheological (ER) materials are suspensions of specialized, micron-sized particles in nonconducting oils. When electric fields are applied to ER materials, they exhibit dramatic changes (within milli-seconds) in material properties. Pre-yield, yielding, and post-yield mechanisms are all influenced by the electric field. Namely, an applied electric field dramatically increases the stiffness and energy dissipation properties of these materials. A previously known cubic equation which describes the flow of fluids with a yield stress through a rectangular duct can be applied to annular flow, provided that certain conditions on the material properties are satisfied. An analytic solution and a uniform approximation to the solution, for the rectangular duct Poiseuille flow case is presented. A numerical method is required to solve the flow in annular geometries. The approximation for rectangular ducts is extended to deal with the annular duct case.

Several materials that are of interest in engineering present a yield stress and behave as viscoplastic fluids. This paper investigates numerically the motion of a Bingham fluid between two coaxial cylinders due to a periodic pressure gradient and/or to the periodic displacement of the internal cylinder. The constitutive equation presents a discontinuity at the zero shear rate. To overcome the difficulty, the rheologic law has been regularized using a smooth function based on the error function. The velocity fields have been calculated using an implicit finite difference method. The procedure has been validated, comparing the numerical results with the analytical solution of the same problem for a Newtonian fluid. The nonlinear behavior of the fluid is emphasized, comparing the effects due to the simultaneous action of the pressure gradient and the displacement of the internal wall with the sum of the effects due to the single actions. In all cases, the mean discharge in a period increases. The comparison between the effects of the forcing agents shows that if the dimensionless frequency is less than 10 the increases of the discharge obtained by applying the pulsatile pressure gradient or moving the internal wall are similar. At low frequencies the action of the gradient exceeds that of the moving wall, whereas for higher frequencies the effect of the moving wall increases rapidly because a fixed displacement of the internal cylinder leads to very great values for the velocity of the internal wall.

In this technical note we discuss the importance of using a generalized Brinkman number definition for laminar pipe flow of a Bingham fluid, when viscous dissipation effects are relevant. We show that adapting the Brinkman number definition commonly used for Newtonian fluids directly to the more general class of non-Newtonian fluids does not calculate correctly the ratio between heat generated by viscous dissipation and heat transfer at the wall and leads to a distortion of the graphical representation of the Nusselt number, Nu, rendering difficult, if not impossible, the comparisons of the Nu behavior between different Brinkman numbers. The use of the proposed generalized Brinkman number removes these problems and simultaneously it has the merit of being independent of any reference apparent viscosities.

An analysis is conducted of the two-dimensional flow of Bingham solids between two rotating plates. The maximum friction law is adopted at the plate surface. An asymptotic analysis of the solution is performed in the vicinity of the friction surface. Its results are used in a numerical procedure to obtain an accurate approximation of the solution near the friction surface. The through thickness distribution of velocities, the equivalent strain rate, and stresses is illustrated. Qualitative features of the solution are emphasized. The results are compared with the solution for rate-independent materials.

An approximate analytical model for the pulsatile flow of an ideal Bingham plastic fluid in both a rigid and a periodically displaced tube has been developed using regular perturbation methods. Relationships are derived for the velocity field and dimensionless flow rate. The solution compares adequately with available experimentally measured oscillatory non-Newtonian fluid flow data. These solutions provide useful analytical models supporting experimental and computation studies of arterial blood flow.

In consideration of the pulsatile blood flow in a conduit, the consitutive equation for the whole human blood of F. J. Walburn and D. J. Schneck (Biorheology, Vol. 13, 1976, pp. 201-210) is utilized. Governing equations are solved numerically yielding the velocity and the shear stress distributions. These results are discussed and compared with the Newtonian fluid, Casson’s fluid, and Bingham fluid applications.

The problem of high-speed wire drawing is solved based on a global Coulomb friction condition on the die surface. The drawing force is determined as a function of the die angle, friction coefficient, reduction in area, and the Bingham number.

The calculation of single-phase and two-phase flowing pressure gradients in a well annulus is generally based on an extension of empirical correlations developed for Newtonian fluids in circular pipes. Various techniques for extending pipe flow correlations to an annular geometry have been presented in the literature which involve the representation of the annular well geometry with an equivalent circular diameter and the representation of non-Newtonian fluid behavior with an apparent Newtonian viscosity. Unfortunately, little experimental data have been available which would allow a comparison of the relative accuracy of the various proposed techniques. In this study, experimental pressure gradient data have been taken in two 6000-ft wells. Frictional pressure losses for single-phase flow (mud only) in two annuli were compared to values predicted by the Bingham plastic and power law models. These calculations utilized the equivalent diameters defined by the Crittendon criteria, the hydraulic diameter, and the slot approximation. Also, total pressure difference for two-phase flow was measured for one annular geometry. This data was compared to that predicted by the Poettmann and Carpenter, Hagedorn and Brown, Orkiszewski, and Beggs and Brill correlations. Comparison of experimental data with the various prediction techniques was favorable, each having advantage in certain situations. For the data investigated, the Crittendon criteria using a Bingham plastic model gave the best results. The two-phase flow data was best predicted by the Hagedorn and Brown correlation utilizing an equivalent hydraulic diameter.

A theoretical investigation of transient free convection in a Bingham plastic on a vertical flat plate with constant wall temperature is presented. Except for a linear variation of density with temperature in the body force term, all fluid properties are assumed to be constants. The parameters of the problem are the Prandtl number Pr and a dimensionless group involving the Hedstrom and Grashof numbers, He/GrL3/4. Solutions to the governing boundary-layer equations are obtained using an explicit finite-difference procedure. Mean Nusselt numbers NuL are presented for a range of the parameters, along with representative velocity profiles, temperature profiles, and friction coefficients. Flow in the Bingham plastic does not start until the buoyancy forces become sufficiently large to cause a shear stress in the material which exceeds the yield stress. Thus, for short times heat is transferred by one-dimensional transient conduction, which has the well-known solution expressed in terms of the complementary error function. A temporal minimum, which becomes more pronounced with increasing He/GrL3/4, is noted in the mean Nusselt number. Steady-state NuL values are higher for Bingham plastics than for Newtonian fluids, but the maximum increase, which decreases with increasing Pr, is noted to be less than 15 percent. Due to the behavior of the velocity gradient at the wall, which reaches a maximum before steady-state conditions are reached, a temporal maximum is observed in the mean friction coefficient. Bingham-plastic friction coefficients are significantly higher than for Newtonian fluids; however, this increase is due primarily to the yield stress rather than as a consequence of a steeper velocity gradient at the wall.

Calcul du nombre de Nusselt local, du nombre de Nusselt moyen et de la temperature d'un fluide plastique de Bingham en ecoulement d'entree laminaire dans une conduite verticale circulaire a temperature de paroi constante en fonction de longueur d'entree sans dimension

The criterion for the initiation of flow of a Bingham material between two concentric circular cylinders is considered. An external torque and an axial pressure gradient are acting on the plastic. For three values of the ratio of outer to inner radii of the cylinders the solutions are presented graphically.