Viscous dissipation effects on thermal transport characteristics of combined pressure and electroosmotically driven flow in microchannels

This study investigates the influence of viscous dissipation on thermal transport characteristics of the fully developed combined pressure and electroosmotically driven flow in parallel plate microchannels subject to uniform wall heat flux. Closed form expressions are obtained for the transverse distributions of electrical potential, velocity and temperature and also for Nusselt number. From the results it is realized that the Brinkman number has a significant effect on Nusselt number. Generally speaking, to increase Brinkman number is to decrease Nusselt number. Although the magnitude of Joule heating can affect Brinkman number dependency of Nusselt number, however the general trend remains unchanged. Depending on the value of flow parameters, a singularity may occur in Nusselt number values even in the absence of viscous heating, especially at great values of dimensionless Joule heating term. For a given value of Brinkman number, as dimensionless Debye–Huckel parameter increases, the effect of viscous heating increases. In this condition, as dimensionless Debye–Huckel parameter goes to infinity, the Nusselt number approaches zero, regardless of the magnitude of Joule heating. Furthermore, it is realized that the effect of Brinkman number on Nusselt number for pressure opposed flow is more notable than purely electroosmotic flow, while the opposite is true for pressure assisted flow.     

Forced convection heat and mass transfer flow of a nanofluid through a porous channel with a first order chemical reaction on the wall

This study is devoted to investigate the fully developed forced convection heat and mass transfer in a horizontal porous channel filled with a nanofluid. It is assumed that the walls of the channel are subject to a constant heat flux. It is also assumed that the first order catalytic reaction takes place on the walls and that the viscous dissipation term in the energy equation is taken into account. Brinkman model is used for the flow in the porous media and “clear fluid compatible” viscous dissipation model is considered. Thermal effect is taken also into account in the concentration equation. Closed form analytical solutions are presented for the governing dimensionless momentum, energy and concentration equations. The effects of nanoparticle volume fraction, Darcy, Brinkman, Damkohler and Soret numbers are investigated on the Nusselt number, velocity, temperature and concentration distributions.     

Thermally developing Brinkman–Brinkman forced convection in rectangular ducts with isothermal walls

The Extended Weighted Residuals Method (EWRM) is applied to investigate the effects of viscous dissipation on the thermal development of forced convection in a porous-saturated duct of rectangular cross-section with isothermal boundary condition. The Brinkman flow model is employed for determination of the velocity field. The temperature in the flow field was computed by utilizing the Green’s function solution based on the EWRM. Following the computation of the temperature field, expressions are presented for the local Nusselt number and the bulk temperature as a function of the dimensionless longitudinal coordinate. In addition to the aspect ratio, the other parameters included in this computation are the Darcy number, viscosity ratio, and the Brinkman number.     

Forced convection heat transfer of Williamson fluid flow in porous media over horizontal plate with constant heat flux

Forced convection of Williamson fluid flow in porous media under constant surface heat flux conditions is investigated numerically. A model of Darcy–Forchheimer–Brinkman is used and the corresponding governing equations are expressed in dimensionless forms and solved numerically using bvp4c with MATLAB package. Boundary layer velocity, shear stress, and temperature profiles, in addition to the local Nusselt number parameter over a horizontal plate, are found. The effects of the Forchheimer parameter, Nusselt number, Darcy parameter, porous inertia, and Williamson parameter on the velocity profiles, temperature profiles, coefficient of friction, and coefficient of heat transfer are investigated. The results showed that as the Darcy parameter increases, boundary layer velocity and shear stress increase, while the temperature and Nusselt number decrease. In addition, as Williamson's parameter increases, velocity within the boundary layer, shear stress, and Nusselt number decrease while the temperature profile increases. Also, with larger values of the Forchheimer parameter, the velocity of the boundary layer, shear stress, temperature, and Nusselt number increase. Furthermore, the Nusselt number and the coefficient of friction are obtained on the surface of the horizontal plate.     

Non-Darcy natural convection of a non-Newtonian fluid in a porous cavity

A numerical study of non-Darcy natural convection in a porous enclosure saturated with a power-law fluid is presented. Hydrodynamic and heat transfer results are reported for the configuration in which the enclosure is heated from a side-wall while the horizontal walls are insulated. The flow in the porous medium is modeled using the modified Brinkman–Forchheimer-extended Darcy model for power-law fluids, which accounts for both inertia and boundary effects. The results indicate that when the power law index is decreased, the circulation within the enclosure increases leading to a higher Nusselt number and these effects are enhanced as the Darcy number is increased. Consequently as the power law index decreases, the onset of the transitions from Darcy regime to Darcy–Forchheimer–Brinkman regime to asymptotic convection (boundary layer) regime shift to higher corresponding values of the Darcy number. An increase in Rayleigh number produces similar effects as a decrease in power law index.     

A numerical study of single-phase convective heat transfer in microtubes for slip flow

The steady-state convective heat transfer for laminar, two-dimensional, incompressible rarefied gas flow in the thermal entrance region of a tube under constant wall temperature, constant wall heat flux, and linear variation of wall temperature boundary conditions are investigated by the finite-volume finite difference scheme with slip flow and temperature jump conditions. Viscous heating is also included, and the solutions are compared with theoretical results where viscous heating has been neglected. For these three boundary conditions for a given Brinkman number, viscous effects are presented in the thermal entrance region along the channel. The effects of Knudsen and Brinkman numbers on Nusselt number are presented in graphical and tabular forms in the thermal entrance region and under fully developed conditions.     

Laminar film condensation on an elliptical tube embedded in porous media

Laminar film condensation of saturated vapor flowing over an isothermal elliptical tube embedded in a porous medium is analyzed for conditions of free and forced convection. The flow field in the porous medium is described by the Darcy–Brinkman–Forchheimer model. The effect of vapor shear on condensation is determined by simultaneous solution of the two phase vapor boundary layer and condensate film momentum equations. The numerical results, which are presented in the form of local film thickness and local Nusselt number, show a dependence of these physical parameters on practical dimensionless parameters such as Reynolds number, Darcy number, Bond number and eccentricity.     

Heat and fluid flow characteristics of gases in micropipes

In this study, laminar forced convective heat transfer of a Newtonian fluid in a micropipe is analyzed by taking the viscous dissipation effect, the velocity slip and the temperature jump at the wall into account. Hydrodynamically and thermally fully developed flow case is examined. Two different thermal boundary conditions are considered: the constant heat flux (CHF) and the constant wall temperature (CWT). Either wall heating (the fluid is heated) case or wall cooling (the fluid is cooled) case is examined. The Nusselt numbers are analytically determined as a function of the Brinkman number and the Knudsen number. Different definitions of the Brinkman number based on the definition of the dimensionless temperature are discussed. It is disclosed that for the cases studied here, singularities for the Brinkman number-dependence of the Nusselt number are observed and they are discussed in view of the energy balance.     

Thermally developing forced convection in a porous medium: parallel plate channel with walls at uniform temperature, with axial conduction and viscous dissipation effects

A modified Graetz methodology is applied to investigate the thermal development of forced convection in a parallel plate channel filled by a saturated porous medium, with walls held at uniform temperature, and with the effects of axial conduction and viscous dissipation included. The Brinkman model is employed. The analysis leads to expressions for the local Nusselt number, as a function of the dimensionless longitudinal coordinate and other parameters (Darcy number, Péclet number, Brinkman number).     

Effects of viscous dissipation on forced convective heat transfer in a channel embedded in a power-law fluid saturated porous medium

The convective heat transfer analysis in a channel embedded in a power-law fluid saturated porous medium subject to uniform heat flux is presented and compared with a Newtonian fluid concerning the effects of viscous dissipation. Governing momentum and energy equations for non-Newtonian fluids which accounts for the viscous dissipation effects are solved numerically. The temperature profiles of the non-Newtonian fluids are found to relate closely to the velocity profiles. When viscous dissipation is taken account of, Nusselt numbers for non-Newtonian fluid are found to deviate more from Newtonian fluid with increasing Brinkman number for a certain range of the Darcy number.     

Magnetohydrodynamic forced convection in bidisperse porous medium: Analysis of flow and temperature distributions in a parallel-plate channel

This study aims to explore magnetohydrodynamic forced convection in a parallel-plate channel filled with a bidisperse porous medium, while emphasizing the significance of viscous dissipation. The study utilizes the two-velocity two-temperature model to analyze the flow and temperature distributions in both the fluid phase and solid phase. Convective boundary conditions at the channel walls are considered, and momentum slip is incorporated into the analysis. By nondimensionalizing the governing equations and employing the Homotopy Analysis Method, the velocity and temperature profiles for both phases are determined. Notably, the findings of the study highlight a notable discrepancy in the temperature increase between the solid phase and the fluid phase. Furthermore, the study investigates the impact of various parameters, such as the Darcy number, Biot number, slip parameter, Hartmann number, and Brinkman number, on velocity, temperature, Nusselt number, and skin friction.     

Convective heat transfer over a heated square porous cylinder in a channel

A numerical study is made of the unsteady flow and convection heat transfer for a heated square porous cylinder in a channel. The general Darcy–Brinkman–Forchheimer model is adopted for the porous region. The parameters studies including porosity, Darcy number, and Reynolds number on heat transfer performance have been explored in detail. The results indicate that the average local Nusselt number is augmented as the Darcy number increases. The average local Nusselt number increases as Reynolds number increases; in particular, the increase is more obvious at a higher Darcy number. In contrast, the porosity has slight influence on heat transfer.     

Effect of local thermal non-equilibrium on thermally developing forced convection in a porous medium

The classical Graetz methodology is applied to investigate the effect of local thermal non-equilibrium on the thermal development of forced convection in a parallel-plate channel filled by a saturated porous medium, with walls held at constant temperature. The Brinkman model is employed. The analysis leads to an expression for the local Nusselt number, as a function of the dimensionless longitudinal coordinate, the Péclet number, the Darcy number, the solid-fluid heat exchange parameter, the solid/fluid thermal conductivity ratio, and the porosity.     

Viscoelastic boundary layer analysis of constant surface temperature plate embedded in saturated porous media

Mathematical models and numerical solutions of Williamson fluid flow under influences of various boundary conditions provide important support to experimental studies in the solar energy field. Therefore, the present study is concerned with the effects of forced convection of the viscoelastic boundary layer on a horizontal plate embedded in saturated porous media subjected to constant surface temperature. The study explores the profiles of shear stress, velocity, temperature, and heat transfer coefficient. The governing equations in nondimensional forms are obtained by using a model of Darcy–Forchheimer–Brinkman and finally are solved numerically by using bvp4c with MATLAB package. The results of the numerical solution show an insignificant rise in the distribution of the velocity boundary layer and shear stress profile as the Darcy parameter is increased, while a decrease in the temperature and Nusselt numbers are found. On the other hand, as the viscoelastic parameter is increased, the Darcy parameter shows a reverse response. Finally, insignificant increases in profiles of boundary layer velocity, temperature, shear stress, and Nusselt number are observed at high values of the Forchheimer number.     

Analysis of heat transfer and entropy generation for a thermally developing Brinkman–Brinkman forced convection problem in a rectangular duct with isoflux walls

Heat transfer and entropy generation analysis of the thermally developing forced convection in a porous-saturated duct of rectangular cross-section, with walls maintained at a constant and uniform heat flux, is investigated based on the Brinkman flow model. The classical Galerkin method is used to obtain the fully developed velocity distribution. To solve the thermal energy equation, with the effects of viscous dissipation being included, the extended weighted residuals method (EWRM) is applied. The local (three-dimensional) temperature field is solved by utilizing the Green’s function solution based on the EWRM where symbolic algebra is being used for convenience in presentation. Following the computation of the temperature field, expressions are presented for the local Nusselt number and the bulk temperature as a function of the dimensionless longitudinal coordinate, the aspect ratio, the Darcy number, the viscosity ratio, and the Brinkman number. With the velocity and temperature field being determined, the second law (of thermodynamics) aspect of the problem is also investigated. Approximate closed form solutions are also presented for two limiting cases of MDa values. It is observed that decreasing the aspect ratio and MDa values increases the entropy generation rate.     

Effects of viscous dissipation on the heat transfer in a forced pipe flow. Part 2: Thermally developing flow

In this part of the study, consideration is given to thermally developing laminar forced convection in a pipe including viscous dissipation. The axial heat conduction in the fluid is neglected. Two different thermal boundary conditions are considered: the constant heat flux (CHF) and the constant wall temperature (CWT). Both the wall heating (the fluid is heated) case and the wall cooling (the fluid is cooled) case are considered. The distributions for the developing temperature and local Nusselt number in the entrance region are obtained. Results show that the temperature profiles and local Nusselt number are influenced by the Brinkman number (Br) and the thermal boundary condition used for the wall. Significant viscous dissipation effects have been observed for large Br.     

Analytical Study on Forced Convection of Nanofluids With Viscous Dissipation in Microchannels

This article presents an analytical study on forced convection of laminar fully developed flow of incompressible, constant-property nanofluids in microchannels. Closed-form solutions for the temperature distributions in the radial direction with the incorporation of viscous dissipation are obtained under isoflux boundary condition. The effects of the governing parameters, including modified Brinkman number, thermal conductivity ratio, and nanoparticle volume fraction of the nanofluids, on the temperature distributions are investigated and analyzed for both heating and cooling processes. The heat transfer performance characterized by the Nusselt number is investigated based on the effects induced by these parameters. In the comparison between the models with and without viscous dissipation, it is found that the thermal performance of a microchannel is overrated when viscous dissipation is excluded in the analysis. It is concluded that these governing parameters are intimately interrelated in the flow and thermal analyses of nanofluids in microchannels. The interrelationship of the viscous dissipation effect and the nanoparticle volume fraction is examined in a contour deviation map of Nusselt numbers between the model with and without considering the viscous dissipation.     

Soret Effect on Darcy–Brinkman Convection in a Binary Viscoelastic Fluid‐Saturated Porous Layer

A linear and weakly nonlinear stability analyses is performed to study the onset of Darcy–Brinkman double diffusive convection in a binary viscoelastic fluid‐saturated porous layer in the presence of the Soret effect. The modified Darcy–Brinkman–Oldroyd model including the time derivative term is employed for the momentum equation. The expressions for stationary, oscillatory, and finite amplitude Rayleigh number are obtained as a function of the governing parameters. There is a competition between the processes of the Soret coefficient, viscoelasticity, thermal diffusion, and solute diffusion that causes the convection to set in through an oscillatory mode rather than a stationary mode. The effects of the Soret parameter, Darcy number, relaxation and retardation parameters, and Darcy–Prandtl number on the stationary, oscillatory, and finite amplitude convection is shown graphically. The weakly nonlinear theory is based on truncated representation of the Fourier series method and is used to find the Nusselt and Sherwood numbers. Further, the transient behavior of the Nusselt and Sherwood numbers is investigated by solving the nonlinear system of ordinary differential equations numerically using the Runge–Kutta method. © 2013 Wiley Periodicals, Inc. Heat Trans Asian Res, 43(4): 297–320, 2014; Published online 3 October 2013 in Wiley Online Library ( wileyonlinelibrary.com/journal/htj ). DOI 10.1002/htj.21076     

Effects of radial thermal dispersion on fully-developed forced convection in cylindrical packed tubes

The fluid flow and heat transfer characteristics of a fully-developed forced convective flow in a cylindrical packed tube with symmetric heating are analyzed in this paper. The Darcy—Brinkman—Ergun model is used as the momentum equation, with the radial porosity variation of the packed column approximated by an exponential function. The method of matched asymptotic expansions is applied to construct a composite solution for the axial velocity profile of a hydrodynamically fully-developed flow. The interaction of inertial and wall channeling effects on the pressure drop and the axial velocity profile is illustrated. The effects of radial thermal dispersion and variable stagnant thermal conductivity are taken into consideration in the energy equation for a thermally fully-developed flow in the packed tube, which is heated circumferentially with constant heat flux or constant wall temperature. A wall function is used to model the wall effect on the transverse thermal dispersion process, and the predicted Nusselt numbers agree with existing experimental data. Numerical results of the corresponding heat transfer characteristics in the packed tubes, without introducing the wall function, are also presented for comparison.     

Numerical Investigation of Unsteady Natural Convection in an Inclined Square Enclosure With Heat-Generating Porous Medium

The unsteady laminar natural convection in an inclined square enclosure with heat-generating porous medium whose heat varies by a cosine function is investigated by a thermal equilibrium model and the Brinkman–Darcy–Forchheimer model numerically, with the four cooled walls of closure as isothermal. The numerical code based on the finite-volume method has been validated by reference data before it was adopted. Influence of dimensionless frequency and inclination angle on heat transfer characteristics in a square enclosure, such as flow distribution, isotherm, averaged Nusselt number on each wall, and time-averaged Nusselt number, are discussed, with specified value for Rayleigh number = 10⁸, Darcy number = 10^?4, Prandtl number = 7, porosity = 0.4, and specific heat ratio = 1. It is found that when the internal heat source varies by cosine, the Nusselt numbers of the four walls oscillate with the same frequency as the internal heat source; however, phase difference occurs. Moreover, frequency has little impact on time-averaged Nusselt number of the four walls, which is different from the phenomenon discovered in natural convection with suitable periodic varying wall temperature boundary condition. Moreover, inclination angle plays an important role in the heat transfer characteristics of the walls studied.