Thermal convection in rotating spherical annuli—2. Stratified flows

The steady combined thermal convection of a viscous Boussinesq fluid contained between two concentric spheres is considered. The spheres are maintained at different temperatures and rotate about a common axis with different angular velocities. A uniform radial gravitational field acts on the fluid. Approximate solutions to the governing equations are obtained with a modified Galerkin technique for moderate Reynolds numbers. The resulting flow patterns, temperature distributions, and heat-transfer and torque characteristics are presented for several angular velocity ratios and degrees of stratification. It is shown that increasing the buoyancy forces alters the primary and secondary flow patterns as well as the temperature distributions. The total rate of heat transfer and torque are subsequently enhanced.     

Carrier-extraction from two rigid spheres in tandem

Unsteady mass transfer between two rigid spheres in tandem and a surrounding fluid flow with reversible second-order chemical reaction on the surface of the spheres has been analysed. The spheres have the same diameter and initial concentration. The dispersed phase reactant and product are insoluble in the continuous phase. The continuous phase reactant and product are insoluble in the dispersed phase. Axisymmetric, slow, viscous flow (Stokes flow) around the spheres was considered. The mass balance equations were solved numerically in spherical/bispherical coordinates systems by a finite difference nonlinear multigrid method. For moderate Pe number, Pe = 100, modified Hatta modulus, Ha² = 10, 100, 1000, and different values of the diffusion coefficients, the computations focused on the influence of the spheres spacing on the mass transfer rates.     

Natural convection heat transfer from a horizontal cylinder embedded in a porous medium

This paper presents the results of an experimental investigation of heat transfer by natural convection from a horizontal cylinder embedded in porous media consisting of randomly packed glass spheres saturated by either water or silicone oil. It is shown that the overall range of the Rayleigh number, Ra, can be divided into two subregions, called ‘low’ and ‘high’, in each of which the Nusselt number, Nu, behaves differently. It is demonstrated that the low Ra region corresponds to Darey flow and the high to Forchheimer flow. Correlation equations for Nu for the Darcy regime are presented that account for viscous dissipation, and others for the Forchheimer regime that involve the first and second Forchheimer coefficients. The variation of properties with temperature and the wall effect on porosity (and consequently on heat transfer) are considered. The paper includes information concerning the resistance to flow in porous media that was obtained in conjunction with the heat transfer study.     

Numerical solution of power law fluids flow and heat transfer with a magnetic field in a rectangular duct

The steady laminar flow and heat transfer of an incompressible, electrically conducting, power law non-Newtonian fluids in a rectangular duct are studied in the presence of an external uniform magnetic field. The momentum and energy equations are solved iteratively using a finite difference method. Two cases of the thermal boundary conditions are considered; (1) T thermal boundary condition “constant temperature at the wall” and (2) H2 thermal boundary condition “constant heat flux at the wall”. The viscous and Joule dissipations are taken into consideration in the energy equation. A numerical solution for the governing partial differential equations is developed and the influence of the magnetic field on the velocity distribution, the friction factor and the average Nusselt number are discussed.     

Thermal convection in rotating spherical annuli—1. Forced convection

The steady forced convection of a viscous fluid contained between two concentric spheres which are maintained at different temperatures and rotate about a common axis with different angular velocities is considered. Approximate solutions to the governing equations are obtained in terms of a regular perturbation solution valid for small Reynolds numbers and a modified Galerkin solution for moderate Reynolds numbers. The resulting flow pattern, temperature distribution, and heat-transfer characteristics are presented for the various cases considered. The theoretical heat-transfer results for small and moderate Reynolds number flows within a spherical annulus with a stationary outer sphere are compared with previous experimental results for the large Reynolds number flow situation. The difference between conduction, Stokes flow, and boundary-layer convection is shown.     

Second-law analysis of flow and heat transfer inside a microannulus

The purpose of this work is to investigate the entropy generation in a microannulus flow. Fully developed laminar flow is considered with uniform heat flux at the walls. The viscous dissipation effect, the velocity slip and the temperature jump at the wall are taken into consideration. The velocity and temperature profiles are obtained analytically and used to compute the entropy generation rate. The effects of Kn, Br, Br/Ω and r? on velocity, temperature profiles, entropy generation rate and average entropy generation are discussed. The present analytical results for the case with and without the viscous dissipation effect are compared with those available in the literature and an excellent agreement is observed. Entropy generation is shown to decrease with an increase in Kn while increasing Br, Br/Ω and r? results in increasing entropy generation.     

Thermosolutal convective instability and viscous dissipation effect in a fluid-saturated porous medium

The combined effects of the double-diffusion and of the viscous dissipation on the convective instability in a fluid-saturated porous medium with a basic horizontal throughflow are investigated. A horizontal porous layer with an impermeable adiabatic lower wall and an impermeable isothermal upper wall is considered. The parallel boundary walls are assumed to have uniform, but unequal, concentrations of the solute. A linear stability analysis is carried out both numerically and by a first-order perturbation method. General disturbances having the form of oblique rolls are considered, reducing either to longitudinal rolls or to transverse rolls in the special cases of roll axes parallel or orthogonal to the basic flow direction, respectively. It is shown that the combined effects of viscous dissipation and mass diffusion may lead to the instability of the basic horizontal flow. Either the longitudinal rolls or the transverse rolls may be the preferred modes of instability depending on the value of the viscous dissipation parameter Ξ. The longitudinal rolls are the most unstable when Ξ < 61.86657.     

Transient flow of a conducting fluid with heat transfer due to an infinite rotating disk

The unsteady MHD flow of an incompressible viscous electrically conducting fluid above an infinite rotating disk is studied with heat transfer. The effect of an external uniform magnetic field on the velocity and temperature distributions as well as the heat transfer is considered. Numerical solutions of the nonlinear equations which govern the magnetohydrodynamics and energy transfer are obtained.     

On the transition for a generalized Couette flow of a reactive third-grade fluid with viscous dissipation

We study the thermal transition of a reactive flow of a third-grade fluid with viscous heating and chemical reaction between two horizontal flat plates, where the top is moving with a uniform speed and the bottom plate is fixed in the presence of imposed pressure gradient. This study is a natural continuation of earlier work on rectilinear shear flows. The governing equations are non-dimensionalized and the resulting system of equations are not coupled. An approximate explicit solution is found for the flow velocity using homotopy-perturbation technique and the range of validity is determined. After the velocity is known, the heat transport may be analyzed. It is found that the temperature solution depends on the non-Newtonian material parameter of the fluid, Λ, viscous heating parameter, Γ, and an exponent, m. Attention is focused upon the disappearance of criticality of the solution set {β, δ, θ_max} for various values of Λ, Γ and m, and the numerical computations are presented graphically to show salient features of the solution set.     

Effect of pressure stress work and viscous dissipation in some natural convection flows

A regular two-parameter perturbation analysis is presented here to study the effects of both viscous dissipation and pressure stress on natural convection flows. Four different vertical flows have been analyzed, those adjacent to an isothermal surface and uniform heat flux surface, a plane plume and flow generated from a horizontal line energy source on a vertical adiabatic surface, or wall-plume. For high gravity levels, stress work effects may be important for gases at very low temperatures, and for high Prandtl number liquids. Significant changes in heat transfer and flow quantities are observed even at moderate values of the perturbation parameters. For the entire range of Prandtl number values considered, the viscous dissipation term is seen to inhibit heat transfer from the surface for heated upward flows. The pressure term enhances heat transfer from the surface for lower Prandtl numbers. However, this effect is seen to reverse at Pr = 100, for both the isothermal and uniform flux surfaces. It is observed that viscous dissipation effects on heat transfer are much smaller than those due to the pressure stress, for many practical circumstances.     

On fully-developed mixed convection and flow reversal of a power-law fluid in a vertical channel

The fully-developed mixed convection of power-law fluids is investigated for laminar flow in a parallel-plate vertical channel. The boundary condition of uniform and unequal temperatures prescribed at the channel walls is considered. The velocity field, the viscous stress field and the temperature field are obtained by solving analytically the momentum and energy balance equations as well as the stress-strain constitutive equation. An expression which allows the evaluation of the friction factors is presented. The condition for the occurrence of flow reversal is obtained.     

Flow and heat transfer of an electrically conducting fluid of second grade over a stretching sheet subject to suction and to a transverse magnetic field

An analysis is performed for flow and heat transfer of a steady laminar boundary-layer flow of an electrically conducting fluid of second grade subject to suction and to a transverse uniform magnetic field past a semi-infinite stretching sheet. The governing partial differential equations are converted into ordinary differential equations by a similarity transformation and an analytical solution for this flow is utilized. The effects of viscous dissipation and work due to deformation are considered in the energy equation and the variations of dimensionless surface temperature and dimensionless surface temperature gradient with various parameters are graphed and tabulated. Two cases are studied, namely, (i) the sheet with constant surface temperature (CST case) and (ii) the sheet with prescribed surface temperature (PST case).     

A note on the thermal stability of a reactive non-Newtonian flow in a cylindrical pipe

The effect of variable viscosity and viscous dissipation on the thermal stability of a one-step exothermic, reactive non-Newtonian flow in a cylindrical pipe was considered assuming negligible reactant consumption. The governing equations; momentum and energy equations, which are coupled due to the dissipative term in the energy equation, were solved by closed-form and approximate techniques respectively. We obtained closed-form solutions for the momentum equation via the Homotopy analysis method(HAM), while the resulting energy equation was analysed for thermal stability via the variational technique. Parametric analysis of the solutions were conducted on the dimensionless velocity (v(r)), the critical Frank-Kamenetskii parameter(δ_cr) and the critical temperature(θ_cr) and expressed graphically and in tabular forms.     

Mixed convection with viscous dissipation and pressure work in a lid-driven square enclosure

Buoyant laminar flow in a square lid-driven enclosure is analysed. The vertical sides are kept isothermal at different temperatures, while the horizontal sides are insulated. Assisting mixed convection flow due to uniform motion of the top side is considered. The governing balance equations are solved numerically by employing a Galerkin finite element method. The effects of viscous dissipation and pressure work are taken into account. In order to investigate the influence of these effects, the Nusselt number is evaluated with respect to the heat fluxes at both vertical sides, for different values of the Rayleigh number and of the Péclet number based on the lid velocity. Two sample fluids are considered: a gas and a highly viscous liquid. In the framework of the Oberbeck–Boussinesq approximation, a comparison is made between three different energy balance models: (A) enthalpy formulation (pressure work and viscous dissipation are included); (B) internal-energy formulation (viscous dissipation is included); (C) both pressure work and viscous dissipation are neglected. It is shown that, in the absence of a lid motion, the three models yield substantially the same predictions. On the other hand, when the forced flow induced by the lid motion becomes sufficiently large, the three models yield discrepant results, thus implying that pressure work and viscous dissipation are not negligible. Moreover, it is shown that, in this case, model (A) yields unphysical results, while model (B) leads to reasonable predictions.     

On the effectiveness of viscous dissipation and Joule heating on steady MHD flow and heat transfer of a Bingham fluid over a porous rotating disk in the presence of Hall and ion-slip currents

The combined effects of viscous dissipation and Joule heating on steady magnetohydrodynamics (MHD) flow of an electrically conducting viscous incompressible non-Newtonian Bingham fluid over a porous rotating disk in the presence of Hall and ion-slip currents is studied. An external uniform magnetic field is applied in the z-direction and the fluid is subjected to uniform suction. Numerical solutions are obtained for the governing momentum and energy equations. Results for the details of the velocity as well as temperature are shown graphically and the numerical values of the skin friction and the rate of heat transfer are entered in tables.     

Mixed convection within a porous heat generating horizontal annulus

A numerical investigation of mixed convection in a horizontal annulus filled with a uniform fluid-saturated porous medium in the presence of internal heat generation is carried out. The inner cylinder is heated while the outer cylinder is cooled. The forced flow is induced by the cold outer cylinder rotating at a constant angular velocity. The flow field is modeled using a generalized form of the momentum equation that accounts for the presence of porous medium viscous, Darcian and inertial effects. Discretization of the governing equations is achieved using a finite element scheme based on the Galerkin method of weighted residuals. Comparisons with previous works are performed and the results show excellent agreement. The effects of pertinent parameters such as the internal Rayleigh number, the Darcy number, the annulus gap, and the Richardson number on the flow and heat transfer characteristics are considered in the present study. The obtained results depict that the Richardson number plays a significant role on the heat transfer characterization within the annulus. The present results show that an increase in Reynolds number has a significant effect on the flow patterns within the annulus with respect to two-eddy, one-eddy and no-eddy flows. Categorization of the flow regimes according to the number of eddies is established on the Ra-Re plane for various Rayleigh numbers.     

Mixed convection boundary layer flow along vertical thin needles: Assisting and opposing flows

The problem of steady laminar mixed convection boundary layer flow of an incompressible viscous fluid along vertical thin needles for both assisting and opposing flow cases is considered in this paper. The transformed boundary layer equations are solved numerically using an implicit finite-difference scheme known as the Keller-box method. Numerical computations are carried out for various values of the dimensionless parameters of the problem, namely the mixed convection parameter λ and the parameter a representing the needle size, with Prandtl number, Pr = 0.7. It has been found that the flow and heat transfer characteristics are significantly influenced by these parameters.     

CFD study on flow distribution uniformity in fuel distributors having multiple structural bifurcations of flow channels

This work studied the issues of uniform flow distribution for general application in fuel cells, fuel processing chemical reactors, and other industrial devices. A novel method for uniform flow distribution was proposed, in which multiple levels of flow channel bifurcations were considered to uniformly distribute a flow into 2ⁿ flow channels at the final stage, after n levels of bifurcation. To study the effect of the flow channel bifurcation structure and dimensions on the flow distribution uniformity, numerical analysis was conducted. Parameters such as the flow channel length and width at each level of bifurcation as well as the curvature of the turning area of flow channels were particularly investigated. Important results concerning the geometrical design of flow distributors for better flow distribution and uniformity are presented. The best structure of a flow distributor was selected based on the criterion of flow distribution uniformity and low pressure loss. Since the studied novel flow distributor distributes a flow into a number of parallel channels in a remarkable uniformity, the flow distribution structure is expected to be widely used in fuel cells, fuel cell systems, and variety of industrial reactors and heat exchangers to significantly improve the performance of these devices. The studied flow regime is limited to laminar flow. A CFD tool FLUENT^@ was used for the simulation. The numerical treatment of convection terms in governing equations was based on the QUICK scheme, and the coupled computation solving for pressure and velocity fields was based on the SIMPLE algorithm.     

Numerical simulation of drop deformation and breakup in shear flow

Three‐dimensional numerical simulation of the deformation and breakup of an isolated liquid drop suspended in immiscible viscous fluid under shear flow was performed with diffuse interface method. The governing equations of the model were described by Navier– Stokes– Cahn– Hilliard equations. The surface tension was treated as a modified stress. In this paper, a uniform staggered Cartesian grid was used. The transient Navier– Stokes equations were solved by an approximation projection method based on pressure increment formulation, while the Cahn– Hilliard equations were solved by a nonlinear full approximation multigrid method. The numerical results of the drop deformation and breakup were in good agreement with the experimental measurements. Therefore, the present model could be perfectly applied to study the mechanism of drop deformation and breakup. © 2007 Wiley Periodicals, Inc. Heat Trans Asian Res, 36(5): 286– 294, 2007; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/htj.20160     

Laminar free-convection in water with variable physical properties adjacent to a vertical plate with uniform heat flux

The steady laminar boundary layer flow of water along a vertical stationary uniform flux plate is studied. The working fluid is water whose density-temperature relationship is nonlinear at low temperatures and viscosity and thermal conductivity are functions of temperature. The results are obtained with the numerical solution of the boundary layer equations and cover the temperature range between 40 and 0 °C taking into account the temperature dependence of μ, k and ρ. Both upward and downward flow is considered. The variation of μ, k and ρ with temperature has a strong influence on the results.