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Mixed convection in a square enclosure arising from thermal buoyancy force and a rotating flat plate is investigated in the paper. The moving boundary problem is solved with the implicit virtual boundary method on a fixed non-staggered Cartesian grid system. Computations are performed for Prandtl number 0.71 at various Rayleigh numbers Ra and rotational Reynolds numbers Re. For a rotor of length d = 0.6 rotating at Re = 430 , the numerical results show that thermal oscillation occurs when the Rayleigh number exceeds 0.55 × 10 6 . There is a critical Rayleigh number ( 0.13 × 10 6 ). Below that the rotor enhances the heat transfer. By contrast, the rotor would suppress the heat transfer beyond the critical Rayleigh number. In the case of Ra = 10 6 , the Nusselt number oscillates 23 cycles when the rotor makes 29 revolutions. For a longer rotor d = 0.9 rotating at Re = 430 , thermal oscillation exists for all Rayleigh numbers, and the critical Rayleigh number increases to 0.38 × 10 6 . In the case of Ra = 10 6 , the Nusselt number oscillates twice when the rotor makes one revolution.

Influence of cooling rate on ac LED (light-emitting diode) with multiple quantum wells is investigated in the present study. A semi-empirical model based on existing experimental measurements is proposed to describe the relationship between the current density and the forwards voltage drop across the p–n junction for LEDs with multiple quantum wells. The numerical model then is employed to compute the electrical and temperature fields on ac LED with multiple quantum wells under various cooling rate. The resulting temperature of the LED oscillates under ac current. The temperature increases due to the heat generated in the active layer of the LED when the electrical potential exceeds the threshold voltage. Otherwise, there is no electrical current and thus the temperature decreases due to the effect of the cooling device. Both light-emitting power and maximum temperature increase as expected when the applied ac electrical potential increases. Fortunately, the temperature of the LED can be efficiently controlled by increasing the cooling rate of the cooling device. Although increasing the cooling rate would decrease the light-emitting power, the influence is not significant.

Heat transfer in a unit three-dimensional skin tissue with an embedded vascular system of actual histology structure is computed in the present work. The tissue temperature and the blood temperatures in artery and vein vessels are solved with a multi-grid system. The mean temperature of the tissue over the cross-section of the unit skin area is evaluated. The resulting one-dimensional function is regarded as the temperature of healthy tissue (or injured skin but the blood perfusion is still normally working) for large area of skin in view of the symmetric and periodic structure of the paired artery–vein vessels in nature. A three-dimensional bioheat equation then is formulated by the superposition of the skin burn wound effect and the healthy skin temperature with and without thermal radiation exposure. When this bioheat equation is employed to simulate ADT process on burn wounds, the decaying factor of the skin surface temperature is found to be a sharply decreasing function of time in the self-cooling stage after a thermal radiation heating. Nevertheless, the boundary of non-healing (needing surgery) and healing regions in a large burn wound can be estimated by tracking the peak of the gradient of decaying factor within 30 s after the thermal radiation is turned off. Experimental studies on the full ADT procedure are needed to justify the assumptions in the present computation.

Stresses distribution between the interfaces of the materials is critical for structures like thin film layers, organic/inorganic light emitting diode layers and various kinds of modern optical-electronics components with stacking materials properties. The classical Aleck's model ( 1949 ) can be considered as a very simple and a practical method to analyze these kinds of structures. In order to deal with it, the integration scheme is used to solve the displacement equations with zero displacements on the clamped edge including the clamped corner. The condition of zero traction is imposed at all of the grid points on the free surface except for the clamped corner of layers. The results show a stress singularity at the clamped corner. This agrees with an existing finite element solution performed on a very fine grid. By contrast, the known Kantorovich solution for Aleck problem from Blech and Kantor (1984) did not predict such stress singularity. As remarked by Blech and Kantor, the Kantorovich solution posed difficulties in convergence in the immediate vicinity of the clamped corner, thus is questionable to apply the results for the layer structures through this model.

The transition stage (self-accelerating necking process) generally takes place before a bubble collapses. However, the mechanism is still not well understood. In this article, seven existing experiments dealing with bubble formation on the orifice of a submerged vertical nozzle are examined by solving the Young–Laplace equation. Multiple solution modes are found. Bubble in solution mode 2 has a neck and thus taller than mode 1 at the same volume. The present numerical result along with an experiment from Longuet-Higgins et al. [J. Fluid Mech. 230, 365–390 (1991)] evidences an isometric transition from solution modes 1 to 2. This might account for the self-accelerating necking process before a bubble collapses. Surprisingly, all of the seven existing experiments agree excellently with the bubble shapes from the Young–Laplace equation without the dynamic effect even when the bubble growth rate in the experiment is 2.25 times as large as the critical value. The gas flow rate (the dynamic effect) seems to play a role only after the transition stage. © 2011 Canadian Society for Chemical Engineering

A liquid–air fountain flow due to the downward motion of a rectangular sleeve over a stationary piston is studied in the paper. Two-dimensional incompressible laminar flows are assumed to prevail in both air and liquid regions. A single set of governing equations over the entire physical domain including the liquid, the air, and the liquid–air interface (free surface) is solved with the extended weighting function scheme and the NAPPLE (nonstaggered APPLE) algorithm on a fixed nonstaggered Cartesian grid system. To ensure the required dynamic contact angle, the liquid meniscus near the sleeve wall is corrected by solving the force balance equation with the geometry method. This is equivalent to introducing a slip condition at the contact line, and thus successfully removes the stress singularity. Steady state solution of the velocity and the pressure as well as the shape of the free surface is obtained. The numerical result evidences the existence of a toroidal-like motion on the free surface postulated by Dussan [E.B. Dussan V., Immiscible liquid displacement in a capillary tube: the moving contact line, AIChE J. 23 (1977) 131–133], although it is quite weak and thin. The resulting free surface profile agrees with the existing experimental observation excellently. Influence of the piston on the flow is discussed.

The NAPPLE algorithm for incompressible viscous flow on Cartesian grid system is extended to nonorthogonal curvilinear grid system in this paper. A pressure-linked equation is obtained by substituting the discretized momentum equations into the discretized continuity equation. Instead of employing a velocity interpolation such as pressure-weighted interpolation method (PWIM), a particular approximation is adopted to circumvent the checkerboard error such that the solution does not depend on the under-relaxation factor. This is a distinctive feature of the present method. Furthermore, the pressure is directly solved from the pressure-linked equation without recourse to a pressure-correction equation. In the use of the NAPPLE algorithm, solving the pressure-linked equation is as simple as solving a heat conduction equation. Through two well-documented examples, performance of the NAPPLE algorithm is validated for both buoyancy-driven and pressure-driven flows.

A new numerical formulation is presented in this paper for incompressible viscous free surface flow without smearing the free surface. Laws of conservation for the entire physical domain, including both liquid and air, are formulated with a single set of governing equations. To properly handle the discontinuities of the physical properties across the free surface, an extended weighting function scheme is developed for the numerical solution on a fixed and nonstaggered Cartesian grid system. Unlike existing numerical methods, the force balance equation is imposed on the free surface through the use of the NAPPLE algorithm without smearing the free surface. During the solution procedure, a harmonic function referred as an “extrapolated velocity” from the liquid is computed based on the velocity solution at the grid points adjacent to the free surface on the liquid side. With a migration velocity interpolated from such a harmonic function, the free surface profile for the next time step is estimated. This gives rise to a smooth free surface profile and thus circumvents high frequency noises on the curvature of the resulting free surface. Furthermore, advancement of the free surface is not restricted to one grid mesh in a single time step. Performance of the present method is examined through three well-documented dam-breaking examples. The results reveal the existence of an induced vortex in a layer of air adjacent to the free surface. Good agreement between the computed water front and the experimental data is observed. Although only two-dimensional cases are demonstrated in this paper, concept of the present numerical method is equally applicable to three-dimensional problems with moving free surfaces. It also applies to convective heat and mass transfer problems such as filling process in gravity and die casting processing.

In this paper, the integration scheme is employed to solve a coupling problem of transient heat conduction and displacement due to thermal deformation. Based on the resulting displacement, gap or contact pressure on the interface of two thermoelastic bodies in imperfect contact is estimated. Such information is then used to determine the interfacial heat transfer between the two thermoelastic bodies. This again affects the temperature distribution and thus the thermal deformation. In the course of heat transfer and thermal deformation, effect of thermal rectification also is taken into account. Numerical solutions of transient and steady-state quantities including gap formation, normal stress, and temperature are demonstrated for a pair of stainless steel and aluminum with conforming shapes. Such an analysis of conduction-deformation interaction based on a finite-difference-like scheme has not been studied in the past.

The effect of anisotropy on transport phenomena in anisotropic porous media is studied in this paper. For convenience, a bank of circular cylinders which can be treated as an anisotropic porous medium is employed such that both Darcy-Forchheimer drag and effective thermal conductivity can be accurately determined. Two problems including a forced flow and a natural convection are illustrated to investigate the effect of anisotropy on fluid flow and heat transfer through a bank of circular cylinders. The solutions reveal that inclination of the cylinder bundle could give rise to an influence of more than 100% on the heat transfer rate for both forced convection and natural convection. Hence, the anisotropy of an anisotropic porous medium should not be ignored.

Based on the microscale temperature, the effective thermal conductivity in the principal direction of an anisotropic porous medium is evaluated. A correlation of the effective thermal conductivity then is employed to predict the volume-averaged temperature of a two-dimensional nonhomogeneous anisotropic porous medium. Through the example, the present model is found to possess an excellent performance even when the principal direction of the anisotropic medium is not parallel to the physical coordinates. The results also reveal that using the classic mixing rule could lead to a significant error when the thermal conductivity ratio of the solid and fluid is large. The present numerical procedure is believed to be able to determine the effective thermal conductivity for most porous media as long as their microstructure is regular and known.

In the present investigation, an incompressible fluid flow across a bank of circular cylinders is studied and modeled as a non-Darcy flow through a porous medium. The continuity equation and the momentum equation in pore scale are solved on a Cartesian grid system. To circumvent the numerical difficulties arising from the flow domain of irregular shape, the weighting function scheme along with the APPLE algorithm and the SIS solver is employed. The Darcy-Forchheimer drag is then determined from the resulting volumetric flow rate under a prescribed pressure drop. The result indicates that the permeability approaches zero at the particular porosity of 0.2146 when the fluid flow across the cylinders becomes impossible. In addition, the pressure drag (Forchheimer drag) is found to contribute a significant resistance at large porosities and/or large granule Reynolds numbers. A correlation of Darcy-Forchheimer drag is proposed for 0.2146 ⩽ e ⩽ 1 and 0 ⩽ Red ⩽ 50.

Transient conjugate heat transfer relating the heat conduction inside a solid body of arbitrary shape and the natural convection around the solid is studied in the present investigation. A single set of governing equations covering both solid and fluid is solved by using the weighting function scheme along with the NAPPLE algorithm and the SIS solver. All of the computations are performed on a Cartesian grid system. Numerical results of velocities and isotherms are presented for Pr = 7 and Gr = 10 7 and 10 8 . An interesting observation on formation, growth, merging and decay of several plumes is discussed. The solution shows that there is a dead fluid inside the cavity under the solid body due to an upward temperature gradient. The maximum temperature thus occurs there in a late stage of the heat transfer process. An increase in the Grashof number is found to enhance the heat transfer. Through this study, the present numerical method is shown to have a good performance for conjugate heat transfer on solid body of arbitrary shape without the need of a body-fitted grid system.

A latent heat method is modified in the present study to properly handle the latent heat evolved in the solidification process of a binary alloy system. The eutectic state ( C E , T E ) is imposed on the eutectic front such that no species equation is needed in the pure solid region. Both momentum and species equations in the irregular domain consisting of the mushy zone and the liquid region are successfully solved by using the weighting function scheme along with the NAPPLE algorithm. To trace the interface of the mushy zone and the pure liquid region, an interpolation technique is proposed. Two examples were conducted to examine the performance of the present method. Satisfactory agreement with the existing experimental results is observed on the total volume of the mushy zone.

Forced convection on a rotating cylinder cooled with an air jet is investigated in the present study. To properly handle the jet flow before and after hitting the cylinder, a computational domain having an irregular shape is employed. Results of streamlines, isotherms and Nusselt numbers are presented for jet Reynolds numbers of Re j = 100 , 500 and 1000 under various rotation Reynolds numbers in the range of 0 ≤ Re ω /Re j ≤ 1. It is interesting to find two solution modes in the present problem. For small rotation speeds, the jet flow is separated into two branches upon hitting the cylinder. Each branch has a separation point. With the aid of the rotating cylinder, one of the separation points moves downstream, while the other receives little influence. Hence, the overall heat transfer is enhanced. For high rotation speeds, however, this heat transfer behavior is reversed due to a layer of dead air round the cylinder. Nevertheless, a more uniform heat transfer is achieved. For both cases of Re j = 500 and 1000, dual solutions according to each of the two solution modes exist in a range of rotation Reynolds numbers. Like the transition from laminar to turbulent flow, the heat transfer characteristics jump from one solution mode to the other depending on the flow instability. However, the transition between the two solution modes is not clear for a jet Reynolds number Re j as small as 100.

The discretized pressure-linked equation can be proven to possess a singular coefficient matrix. This implies that the pressure solution corresponding to a given velocity would not exist unless this velocity precisely satisfies the continuity equation. In the present investigation, an artificial source term is added to the pressure-linked equation. This additional term will generate an extra pressure called the ‘artificial pressure’ for each updated velocity to compensate for their nonzero dilation. The use of the artificial pressure is equivalent to creating an extra mass for the need of the updated velocity in satisfying the continuity equation. This treatment guarantees the existence and uniqueness of the pressure solution such that the pressure can be directly solved without recourse to the conventional pressure correction equation. Based on the concept of artificial pressure, the APPLE (Artificial Pressure for Pressure-Linked Equation) and the NAPPLE (Nonstaggered APPLE) algorithms are developed for incompressible flows. Through two well-known examples, the APPLE algorithm is found to produce essentially the same results with only about 40% CPU time as compared with the SIMPLER algorithm. Due to its simplicity, the NAPPLE algorithm has a potential use for problems with arbitrarily shaped domain as long as the grid size is not large.

The static Young-Laplace equation is solved with the geometry method to yield the bubble shape on a horizontal flat surface under various contact angles. Multi-solution modes are found. Among the many possible equilibrium shapes of the bubble, however, only the fundamental solution mode could occur naturally. The value of VAR (volume to contact area ratio) could be a good measure for stability of equilibrium bubbles. The bubble becomes less stable when VAR increases. The numerical result reveals that in the course of bubble growth (i.e. volume increases) the VAR of the bubble increases linearly until the maximum contact area is reached. After that, VAR has a sharp increase due to a decreasing contact area. Beyond the maximum volume, equilibrium bubble does not seem possible. Based on the finding, it is postulated that bubble detachment occurs somewhere between the maximum contact area and the maximum volume according to perturbations from environment. However, the postulation seems to underestimate the stability of the bubble significantly for contact angles of larger than 160 degrees. A correction is proposed in the paper. Numerical result of bubble detachment criterion is fitted with polynomial functions of the contact angle.Copyright © 2009 by ASME

An enthalpy formulation is proposed in the present investigation for a phase change material (PCM) having a distinct freezing temperature. The latent heat is separated from the sensible heat such that there exists a dependent variable (the sensible heat) that is a continuous function over the entire physical domain. Inside each control volume, the latent heat is rigorously evaluated from the fraction of the liquid phase to achieve an even latent heat evolution. In the dimensionless transformation, the characteristic time is denned in terms of the Stefan number. The coefficients of the unsteady terms thus are always less than unity. This will achieve a good numerical stability for any Stefan number. In addition, this particular transformation makes the present enthalpy formulation applicable to single phase problems if an infinite Stefan number is assigned. To account for a thermal diffusivity jump at the liquid-solid interface, a modified weighting function scheme is developed. Through a few examples, the present enthalpy formulation is seen to produce an accurate and smooth liquid-solid interface for PCM having a distinct freezing point.

A numerical formulation based on the weighting function scheme is developed in the present investigation for convection-conduction phase change problems. The solid phase is regarded as a liquid having an infinite viscosity. Such a treatment allows the stream-vorticity formulation to apply over the entire physical domain including the liquid-solid interface. The weighting function scheme then is used to handle the sharp viscosity jump at the liquid-solid interface. A problem dealing with pure tin is employed to test the performance of the present method. The predicted interface profiles are found to agree with available experimental data qualitatively. The computation effort required by the present numerical technique, however, is less than 2% ofthat needed by a two-region method for this same problem. Unlike the existing single-region methods, the present numerical technique produces smooth streamlines and isotherms even in the vicinity of the solidification front.

Growth and detachment of carbon dioxide bubbles on a horizontal porous surface in water is investigated in this paper. Uniform carbon dioxide injection on the porous surface is considered. The Young–Laplace equation is solved with the geometry method to yield the bubble shape. The dynamic pressure on the bubble surface due to bubble expansion is neglected. Multi-solution modes are found. Based on the characteristics of the solution modes, it is postulated that the bubble grows with a monotonically increasing base area until the maximum value is reached according to the fundamental solution mode. After that, the bubble jumps toward the secondary solution mode at a constant volume, and then detaches from the surface. The increasing rate of the bubble volume due to mass diffusion/convection is evaluated by solving the momentum and concentration equations. Evolution of the bubble shape then is determined corresponding to the variation of the bubble volume. The numerical results indicate that hydrophobic surface produces large bubble and thus gives rise to better efficiency for dissolved gas removal.

It is very difficult to fabricate tunable optical systems having an aperture below 1000 micrometers with the conventional means on macroscopic scale. Krogmann et al. (J. Opt. A: Pure Appl. Opt. vol. 8, 2006, pp. S330-S336) presented a MEMS-based tunable liquid micro-lens system with an aperture of 300 micrometers. The system exhibited a tuning range of back focal length between 2.3 mm and infinity by using the electrowetting effect to change the contact angle of the meniscus shape on silicon with a voltage of 0−45V. However, a serious optical aberration was found in their lens system. In the present study, a numerical simulation is performed for this same physical configuration by solving the Young-Laplace equation on the interface of the lens liquid and the surrounding liquid. The resulting meniscus shape produces a back focal length that agrees with the experimental observation excellently. To eliminate the optical aberration, an electric field is applied on the system. The electric field alters the Young-Laplace equation and thus changes the meniscus shape and the quality of the lens system. The numerical result shows that the optical aberration of the lens system can be essentially eliminated when a proper electric field is applied.

The present investigation studies the cooling of a continuous moving sheet of finite thickness. The effect of the buoyancy force is also taken into account. The temperature distribution along the solid-fluid interface is determined by solving a conjugate heat transfer problem that consists of heat conduction inside the sheet and induced mixed convection adjacent to the sheet surface. For a better numerical stability, the weighting function scheme along with an axial coordinate transformation is employed to solve the transformed boundary layer equations. Three parameters are found to exist in the present investigation. They are the Prandtl number of the fluid Pr, the buoyancy parameter Ω and the heat capacity ratio C. Numerical results including the Biot number, the surface temperature and the overall heat transfer rate of the sheet are presented for 0.7 ⩽ Pr ⩽ 100, 0 ⩽ Ω ⩽ 10 and 0.1 ⩽ C ⩽ 1. The buoyancy force is seen to have a significant effect on the results. The heat capacity ratio, however, is the most important parameter. Based on the present results, it is concluded that using a liquid as the cooling medium could obtain a better cooling performance than using a gas. This is because the liquid has a larger heat capacity than a gas. The Prandtl number has only a minor effect.

A strongly implicit solver is proposed in the present investigation for solving a large set of algebraic equations that arise from a discretization of an elliptic differential equation. Based on the guessed θNW and θSE values (solutions at the northwest and the southeast corners of a computational cell), the nine-diagonal coefficient matrix is factored in terms of a lower and an upper triangular matrix with only seven nonzero diagonals. The solution procedure then is iterated with a successive overrelaxation (SOR) factor until the solution converges within a prescribed tolerance. In the present solver, there is no need to evaluate the residual for the guessed solution. The CPU time thus is reduced a great amount for a single iteration. In addition, the storage is only one-half as large as that required by the SIP solver because the original matrix is no longer needed after it is factored. The convergence rate of the SIP solver is very sensitive to the cancellation parameter a. The present solver needs no ...

Flow structure around a rising clean microbubble in a vertical square tube is investigated on a fixed nonstaggered Cartesian grid system. A single set of three dimensional governing equations over the entire physical domain including the liquid, the air, and the bubble surface (the liquid-air interface) is formulated without smearing the bubble surface. There are three step functions in the problem, namely, the density jump, the viscosity jump, and the pressure jump across the bubble surface. A modified NAPPLE algorithm is proposed to handle the pressure jump on a fixed nonstaggered Cartesian grid system, while the density and viscosity jumps appearing in the momentum equations are solved with the extended weighting function scheme. This strategy prevents the occurrence of “parasitic current” on the bubble surface. Solutions of flow structure (velocity and pressure) are obtained for bubble diameters of less than 500 micrometers rising in quiescent pure water. The numerical result evidences that the distorted microbubble under study is of the oblate spheroid type. The distortion parameter is only 1.0006 when the bubble diameter is 250 micrometers. However, the surface tension force due to bubble distortion generates a large pressure jump across the bubble surface that dominates the flow around the bubble. It plays a role similar to the non-transpiration condition on a solid surface. The resulting terminal velocity of the bubble agrees excellently with the existing analytical solutions and numerical results.Copyright © 2007 by ASME

There are still many unanswered questions related to the problem of a capillary surface rising in a tube. One of the major questions is the evolution of the liquid meniscus shape. In this paper, a simple geometry method is proposed to solve the force balance equation on the liquid meniscus. Based on a proper model for the macroscopic dynamic contact angle, the evolution of the liquid meniscus, including the moving speed and the shape, is obtained. The wall condition of zero dynamic contact angle is allowed. The resulting slipping velocity at the contact line resolves the stress singularity successfully. Performance of the present method is examined through six well-documented capillary-rise examples. Good agreements between the predictions and the measurements are observable if a reliable model for the dynamic contact angle is available. Although only the capillary-rise problem is demonstrated in this paper, the concept of this method is equally applicable to free surface flow in the vicinity of a contact line where the capillary force dominates the flow.

A numerical simulation for a filling process in an open tank is performed in this paper. A single set of governing equations is employed for the entire physical domain covering both water and air regions. The great density jump and the surface tension existing at the free surface are properly handled with the extended weighting function scheme and the NAPPLE algorithm. There is no need to smear the free surface. Through the use of a properly defined boundary condition, the method of “extrapolated velocity” is seen to provide accurate migrating velocity for the free surface, especially when the water front hits a corner or a vertical wall. In the present numerical procedure, the unsteady term of the momentum equation is discretized with an implicit scheme. Large time-steps thus are allowed. The numerical results show that when the water impinges upon a corner, a strong pressure gradient forms in the vicinity of the stagnation point. This forces the water to move upward along the vertical wall. The water eventually falls down and generates a gravity wave. The resulting free surface evolution is seen to agree well with existing experimental data. Due to its accuracy and simplicity, the present numerical method is believed to have applicability for viscous free-surface flows in industrial and environmental problems such as die-casting, cutting with water jet, gravity wave on sea surface, and many others.

It is very difficult to fabricate tunable optical systems having an aperture below 1000 micrometers with the conventional means on macroscopic scale. Krogmann et al. (J. Opt. A 8, S330-S336, 2006) presented a MEMS-based tunable liquid micro-lens system with an aperture of 300 micrometers. The system exhibited a tuning range of back focal length between 2.3mm and infinity by using the electrowetting effect to change the contact angle of the meniscus shape on silicon with a voltage of 0-45 V. However, spherical aberration was found in their lens system. In the present study, a numerical simulation is performed for this same physical configuration by solving the Young-Laplace equation on the interface of the lens liquid and the surrounding liquid. The resulting meniscus shape produces a back focal length that agrees with the experimental observation excellently. To eliminate the spherical aberration, an electric field is applied on the lens. The electric field alters the Young-Laplace equation and thus changes the meniscus shape and the lens quality. The numerical result shows that the spherical aberration of the lens can be essentially eliminated when a proper electric field is applied.

Wave instability of mixed convection flow along an isothermal vertical flat plate is analyzed by the linear theory for the entire mixed convection regime (0 ⩽ χ ⩽ 1, χ = [1 + ( Gr x Re x 2 ) 1 4 ] −1 ) for fluids with Prandtl number of 0.7 and 7. In the analysis, the domain beyond the mainflow boundary layer edge (η ⩾ η∞) is solved analytically to provide boundary conditions at η = η∞ instead of those at η = ∞. This treatment is important in the use of the modified Thomas transformation when λ(λ = Re x 1 2 + Gr x 1 4 ) is small. Dual solutions of critical λ ∗ values are seen to exist in the range of 0 ⩽ χ ⩽ 0.5. The results show that the two limiting neutral stability curves, one for the Blasius flow (χ = 1 or Grx = 0, with λ ∗ = 290.6 ) and the other for the pure free convection flow (χ = 0 or Rex = 0, with λ ∗ = 33.33 ), correspond to two different modes.

This paper presents a finite element solution to the problem of low Peclet number fluid flow in the thermal entrance region of a round pipe. The velocity is assumed to be laminar and fully developed throughout the pipe and the fluid temperature is kept uniform atX = —∞. The pipe wall is adiabatic at X ≤ 0 and cooled convectively at X ≥ 0. The solutions include temperature distributions and Nusselt numbers for the parameters, Bi = 0.04, 0.4, 4, 20 and Pe = 1, 3, 5, 10, 20, 30, which are in excellent agreement with the existing analytic solution except in the region near the singular point Δ A temperature discrepancy in the analytic solution at this point is physically impossible. The finite element method overcomes this mathematical difficulty and shows a greater value in the Nusselt number due to a higher wall temperature at X ≥ 0. On presente, dans le travail, une solution utilisant une methode d'elements finis pour le probleme de l'ecoulement d'un fluide a faibles nombres de Peclet dans la region d'entree thermique d'un tube rond. On suppose que la vitesse est laminaire et etablie d'un bout a l'autre du tube et que la temperature du fluide est maintenue uniforme a X = — ∞. La paroi du tube est adiabatique a X ≤ 0 et elle est refroidie d'une maniere convective a X ≥ 0. Les solutions comprennent des distributions de temperatures et des nombres de Nusselt pour les parametres Bi = 0.04, 0.4, 4, 20 et Pe = 1, 3, 5, 10, 20, 30, elles concordent tres bien avec la solution analytique existante, sauf dans la region a proximite du point singulier X = 0, r = 1; il est physiquement impossible d'obtenir une difference de temperature dans la solution analytique en ce point. La methode des elements finis remedie a cette difficulte mathematique et fournit un nombre Nusselt de aleur plus grande, par suite d'une temperature plus elevee de la paroi a X ≥0.

Mixed convection along vertical cylinders and needles with uniform surface heat flux is investigated for the entire mixed convection regime. A single modified buoyancy parameter {chi} and a single curvature parameter {Lambda} are employed in the analysis such that a smooth transition from pure forced convection ({chi} = 1) to pure free convection ({chi} = 0) can be accomplished. For large values of the curvature parameter and/or Prandtl number, the governing transformed equations become stiff. Thus, a numerically stable finite-difference method is employed in the numerical solution in conjunction with the cubic spline interpolation scheme to overcome the difficulties that arise from the stiffness of the equations. Local Nusselt numbers are presented for 0.1 {le} Pr {le} 100 that cover 0 {le} {chi} {le} 1 ({infinity} {ge} {Omega}{sub x} {ge} 0) and 0 {le} {Lambda} {le} 50. For needles ({Lambda} {ge} 5), the local Nusselt numbers (Nu{sub x}/Re{sub x}{sup 1/2} + Gr{sub x}{sup 1/5}*) are found to be nearly independent of the buoyancy parameter {chi}. Correlation equations for the local Nusselt numbers are also presented.

The combined effects of axial conduction and solidification on heat transfer and pressure drop in pipe flows are investigated by the use of a modified Galerkin finite element method. To allow for the upstream heat conduction, the domain of study is extended from X = -∞ to X = -∞ as has been done in previous analyses. As a preliminary study on the effect of axial conduction, the present investigation assumes a superheat ratio that is sufficiently large (To > Tf or Tw ≈ Tf) such that solidification begins at a location near X = 0. For numerical convenience, the infinite domain -∞ ≤ X ≤ ∞ is transformed onto a finite domain -1 ≤ z ≤ 1. The energy equation for the liquid-phase is then solved by a modified Galerkin finite element method. For better numerical stability, a procedure is proposed for controlling the numerical error that might propagate from the singular point (X, R) = (O, Ro). The profile of the solid-liquid interface, the heat transfer rate and the pressure drop are presented for various values of Peclet number, Pe = 1, 3, 5, 10 and 30, and for the modified superheat ratio, c = 0.1, 0.5, 5.0, and ∞. A l'aide d'une methode d'elements finis de type Galerkin modifie, nous avons etudie les effets combines de la conduction axiale et de la solidification sur le transfert de chaleur et la perte de charge dans des ecoulements dans des tubes. Pour permettre la conduction de la chaleur en amont, le champ d'etude a ete etendu de X = ∞ a X = -∞ comme nous l'avons fait dans les analyses anterieures. Comme etude preliminaire sur l'effet de la conduction axiale, les presentes recherches supposent un rapport de surchauffe qui soit suffisamment grand (To ≤ Tf ou Tw ≈ Tf) pour permettre a la solidification de demarrer a un point pres de X = 0. Pour faciliter le calcul numerique, le domaine infini -∞ ≤ X ≤ ∞ est transforme en un domaine fini -1 ≤ z ≤ 1. L'equation d'energie pour la phase liquide est alors resolue par une methode d'elements finis de type Galerkin modifie. Pour obtenir une meilleure stabilite numerique, nous proposons une methode pour contrǒler l'erreur numerique qui peut se propager du point singulier (X, R) = (0, Ro). Le profil de l'interface solides-liquide, le taux de transfert de chaleur et la perte de charge sont presentes pour diverses valeurs du nombre de Peclet, Pe = 1, 3, 5, 10 et 30 et pour le rapport modifie de surchauffe, c = 0, 1, 0,5, 5,0 et ∞.

A non-parallel flow analysis by an order-of-magnitude approach is performed to investigate the linear wave instability of mixed convection flow along an isothermal inclined flat plate. This analysis removes the previous restriction on the weak dependence of the streamwise variation of disturbance quantities. The resulting non-homogeneous, coupled equations for the momentum and temperature disturbances are solved by a superposition technique along with a modified Thomas transformation method. Critical Reynolds numbers are presented for inclination angles in the range of 0° ⩽ γ ⩽ 90° (with γ being measured from the horizontal) covering the bouyancy parameter range of −0.15 ⩽ Grx/Rex2 ⩽ 1 for Prandtl numbers of 0.7 and 7. It is found that the net effect of buoyancy force on the critical Reynolds number is essentially zero at γ = 1.05° when the plate is almost horizontal. For γ > 1.05°, an increase in the value of Grx/Rex2 stabilizes the flow. This behavior is reversed for γ

An order-of-magnitude approach is employed to analyze the linear nonparallel wave instability of boundary-layer flows. This analysis removes the restriction imposed by previous investigators on the weak dependence of the streamwise variation of disturbance quantities. A superposition technique along with a modified Thomas trans-formation is employed to solve the resulting nonhomogeneous equation for the amplitude function of the disturbance without the use of an adjoint eigenfunction, which is needed in conventional approaches. All of the eigenfuncrions are treated in their general forms. The growing rate of the disturbance intensity is found to depend on both streamwise and transverse coordinates. At a local observation point × (or Re×), the growing rate of the disturbance intensity has a maximum value near y = Y = Ylpar;U∞/ vX)½ = 3 for Blasiusflow. The neutral stability curve can then be defined by letting the maximum growing rate of the disturbance intensity be zero. Excellent agreement is found betwe...

Two finite-difference methods are proposed for solving wave instability problems with and without coupling between momentum and energy equations. Neutral critical stability results are compared with those generated by the Runge-Kutta integration method in conjunction with an orthonormalization procedure. The new finite-difference methods are found to be very accurate, timesaving, and easy to program. They can also be applied to solve systems of high-order ordinary differential equations.

On emploie une methode de differences finies pour resoudre les equations d'Orr Sommerfeld modifiees, avec un couplage entre les fonctions d'amplitude pour la perturbation de vitesse et de temperature dans lequel le non-parallelisme de l'ecoulement principal est pris en compte