Forced convection with viscous dissipation using a two-equation model in a channel filled by a porous medium

The effects of viscous dissipation on the temperature profiles for a fully developed forced convection flow between two parallel plates with a constant heat flux boundary condition are studied. A two-equation model that includes viscous dissipation in the fluid phase is solved analytically and exact solutions for the temperature fields are obtained. Based on the solutions, the effects of several parameters on the transverse temperature profiles and Nusselt number are studied. The solution reducing to two respective limiting situations of slug flow and clear fluid flow agrees with the literature. A comparison with a one-equation model is also presented.     

MHD stagnation point flow of Casson fluid over a convective stretching sheet considering thermal radiation,slip condition,and viscous dissipation

This study presents the problem of MHD stagnation point flow of Casson fluid over a convective stretching sheet considering thermal radiation, slip condition, and viscous dissipation. The partial differential equations with the corresponding boundary conditions that govern the fluid flow are reduced to a system of highly nonlinear ordinary differential equations using scaling group transformations. The fourth-order method along shooting technique is applied to solve this system of boundary value problems numerically. The effects of flow parameters on the velocity, temperature, and concentration profiles are presented via graphs. The impact of the physical parameters on the skin friction coefficient reduced Nusselt numbers and reduced Sherwood numbers are investigated through tables. Comparison of the present findings with the previously published results in the literature shows an excellent agreement. It is also noted that a rise in the Eckert number results in a drop in the temperature of the fluid in the thermal boundary layer region of the fluid flow.     

Combined effect of viscous dissipation and joule heating on the coupling of conduction and free convection along a vertical flat plate

An analysis is presented to investigate the influences of viscous dissipation and Joule heating in the entire thermo-fluid dynamic field resulting from the coupling of bouncy forced flow with conduction along one side of a heated flat plate of thickness ‘b’. The governing equations are transformed into dimensionless non-similar equations by using set of suitable transformations and solved numerically by the finite difference method along with Newton's linearization approximation. Numerical results for the velocity profiles, temperature profiles, skin friction coefficient and the surface temperature distributions are shown both on graphs and in tabular form for different values of the parameters.     

Effect of non-uniform slot injection (suction) on a forced flow over a slender cylinder

An analysis is performed to obtain the non-similar solution of a steady laminar forced convection boundary layer flow over a horizontal slender cylinder including the effect of non-uniform slot injection (suction). The effects of transverse curvature and viscous dissipation are also included in the analysis. The governing boundary layer equations along with the boundary conditions are first cast into a dimensionless form using suitable transformations and the resulting system of nonlinear coupled partial differential equations is then solved by an implicit finite difference scheme in combination with the quasilinearization technique. Numerical results for the effect of non-uniform slot injection (suction) on skin friction coefficient and heat transfer rate are presented. The effects of transverse curvature, viscous dissipation and Prandtl number on velocity and temperature profiles and skin friction and heat transfer coefficients are also reported.     

Numerical study of chemical reaction and heat transfer of MHD slip flow with Joule heating and Soret–Dufour effect over an exponentially stretching sheet

A study of Soret–Dufour effects along with chemical reaction, viscous dissipation combining on MHD Joule heating for viscous incompressible flow is presented. It is assumed that fluid is flowing past an angled stretching sheet saturated in porous means. The slip conditions of velocity, concentration, and temperature are accounted for at the boundary. The mathematical expression of the problem contains highly nonlinear interconnected partial differential equations. To convert governing equations into ordinary differential equations, appropriate similarity transformations were utilized. These differential equations with boundary constraints are resolved by homotopy analysis method. Expression for velocity, concentration, and temperature are derived in the form of series. Effects of numerous physical parameters, for example, Schmidt number, Soret number, buoyancy ratio parameter, slip parameter, and so forth, on various flow characteristics are presented through graphs. Numerous values of velocity, concentration, and temperature gradient are tabulated against different parameters. Results show that the fluid velocity increases by enhancing the Soret number, Dufour number, or permeability parameter. The fluid's concentration rises as the Soret number increases, while it falls as the Dufour number, chemical reaction parameter, or permeability parameter increases.     

The effect of viscous dissipation in thermally fully-developed electro-osmotic heat transfer in microchannels

The influence of viscous dissipation on thermally fully-developed, electro-osmotically generated flow has been analyzed for a parallel plate microchannel and circular microtube under imposed constant wall heat flux and constant wall temperature boundary conditions. Such a flow is established not by an imposed pressure gradient, but by a voltage potential gradient along the length of the tube. The result is a combination of unique electro-osmotic velocity profiles and volumetric heating in the fluid due to the imposed voltage gradient. For large ratio of the microtube radius (or microchannel half-width) to Debye length, the wall-normal fluid velocity gradients can be extremely high, which has the potential for significant viscous heating. The solution for the fully-developed, dimensionless temperature profile and corresponding Nusselt number have been determined for both geometries and for both thermal boundary conditions. It is shown that three dimensionless parameters govern the thermal transport: the relative duct radius (ratio of the duct radius or plate gap half-width to Debye length), the dimensionless volumetric source (ratio of Joule heating to wall heat flux), and a dimensionless parameter that relates the magnitude of the viscous heating to the Joule heating. Surprisingly, it is shown that the influence of viscous dissipation is only important at low values of the relative duct radius. For magnitudes of the dimensionless parameters which characterize most practical electro-osmotic flow applications, the effect of viscous dissipation is negligible.     

Influence of viscous dissipation and radiation on unsteady MHD free-convection flow past an infinite heated vertical plate in a porous medium with time-dependent suction

This paper investigates the influence of viscous dissipation and radiation on the problem of unsteady magnetohydrodynamic free-convection flow past an infinite vertical heated plate in an optically thin environment with time-dependent suction. By taking the radiative heat flux in the differential form, and imposing an oscillatory time-dependent perturbation, the coupled nonlinear problem is solved for the temperature and velocity distributions. The effects of the material parameters on the temperature and velocity profiles are discussed quantitatively. The results show that increased cooling (Gr>0) of the plate and the Eckert number leads to a rise in the velocity profile; while increases in magnetic field, radiation and Darcy parameters are associated with decrease in the velocity. Also, an increase in the Eckert number leads to an increase in the temperature, whereas increases in radiation and magnetic field parameters lead to a decrease in the temperature distribution when the plate is being cooled.     

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.     

Impact of viscous dissipation and Dufour on MHD natural convective flow past an accelerated vertical plate with Hall current

The present research work concentrates on viscous dissipation, Dufour, and heat source on an unsteady magnetohydrodynamics natural convective flow of a viscous, incompressible, and electrically conducting fluid past an exponentially accelerated infinite vertical plate in the existence of a strong magnetic field. The presence of the Hall current induces a secondary flow in the problem. The distinguishing features of viscous dissipation and heat flux produced due to gradient of concentration included in the model along with heat source as they are known to arise in thermal-magnetic polymeric processing. The flow equations are discretized implicitly using the finite difference method and solved using MATLAB fsolve routine. Numerical values of the primary and secondary velocities, temperature, concentration, skin friction, Nusselt number, and Sherwood number are illustrated and presented via graphs and tables for various pertinent parametric values. The Dufour effect was observed to strengthen the velocity and temperature profile in the flow domain. In contrast, due to the impact of viscous dissipation, the local Nusselt number reduces. The study also reveals that the inclusion of the chemical reaction term augments the mass transfer rate and diminishes the heat transfer rate at the plate.     

Slip-flow and heat transfer of gaseous flows in the entrance of a wavy microchannel

The present work investigates the developing fluid flow and heat transfer through a wavy microchannel with numerical methods. Governing equations including continuity, momentum and energy with the velocity slip and temperature jump conditions at the solid walls are discretized using the finite-volume method and solved by SIMPLE algorithm in curvilinear coordinate. The effects of creep flow and viscous dissipation are assumed. The numerical results are obtained for various Knudsen numbers. The results show that Knudsen number has declining effect on both the C_f.Re and Nusselt number on the undeveloped fluid flow. Significant viscous dissipation effects have been observed for large Knudsen number. Also, viscous dissipation causes a singular point in Nusselt profiles.     

On laminar hydromagnetic mixed convection flow in a vertical channel with symmetric and asymmetric wall heating conditions

The problem of hydromagnetic fully developed laminar mixed convection flow in a vertical channel with symmetric and asymmetric wall heating conditions in the presence or absence of heat generation or absorption effects is considered. Through proper choice of dimensionless variables, the governing equations are developed and three types of thermal boundary conditions are prescribed. These thermal boundary conditions are isothermal-isothermal, isoflux-isothermal, and isothermal-isoflux for the left-right walls of the channel. Analytical solutions for the velocity and temperature profiles for various special cases of the problem are reported. In addition, closed-form expressions for the Nusselt numbers and reversal flow conditions at both the left and right channel walls are derived. The general problem which includes the effects of both viscous dissipation and Joule heating is solved numerically by an implicit finite-difference scheme. Favorable comparisons of special cases with previously published work are obtained. A selected set of graphical results illustrating the effects of the various parameters involved in the problem including viscous and magnetic dissipations on the velocity and temperature profiles as well as flow reversal situations and Nusselt numbers is presented and discussed.     

Numerical analysis of MHD nanofluid flow over a wedge,including effects of viscous dissipation and heat generation/absorption,using Buongiorno model

The key purpose of this article is to examine magnetohydrodynamics flow, generative/absorptive heat, and mass transfer of nanofluid flow past a wedge in the presence of viscous dissipation through a porous medium. The investigation is completely theoretical, and the present model expresses the influence of Brownian motion and thermophoresis using the nanofluid Buongiorno model. The fundamental model of partial differential equations is reframed into the structure of ordinary differential equations implementing the nondimensional similarity transformation, which are tackled through the fourth–fifth-order Runge–Kutta–Fehlberg algorithm together with the shooting scheme. The analysis of sundry nondimensional controlling parameters, such as magnetic parameter, Eckert number, heat generation/absorption parameter, porosity parameter, Brownian motion parameter, and thermophoresis parameter on velocity, temperature, and concentration profiles are discussed graphically. The effects of the physical factors on the rate of momentum and heat and mass transfer are also determined with appropriate analysis in terms of skin friction, Nusselt number, and Sherwood number. The outcomes illustrate that the local Nusselt number and local Sherwood number are reduced for higher values of the thermophoresis parameter. Besides, it is found that higher estimations of heat generation/absorption and viscous dissipation parameters increase temperature. Moreover, it is found that the temperature profile increases with the involvement of the Brownian motion parameter, while an opposite trend is observed in the concentration profile. A comparison is also provided for limiting cases to authenticate our obtained results.     

Numerical Study of Heat and Mass Transfer MHD Viscous Flow Over a Moving Wedge in the Presence of Viscous Dissipation and Heat Source/Sink with Convective Boundary Condition

In this article, the effects of viscous dissipation and internal heat generation/absorption on combined heat and mass transfer MHD viscous fluid flow over a moving wedge in the presence of mass suction/injection with the convective boundary condition are carried out numerically for the various values of dimensionless parameters. With the help of similarity transformation, the momentum, energy, and concentration equations are reduced to a set of dimensionless non‐linear ordinary differential equations. The significance of the dimensionless velocity, temperature, mass profiles, and their gradients are presented in graphical form. Three types of flows—particularly the flat plate, vertical wedge, and stagnation point flows—in favorable and unfavorable regimes are analyzed. The obtained results confirm that the flow field is substantially influenced by the magnetic, stretching/shrinking, pressure, Prandtl number, heat generation/dissipation, and mass suction/injection parameters. Current results indicate that stretching a wall boundary causes an increase in velocity, temperature, shear stress, temperature, and mass gradients while shrinking causes a decreasing trend with these profiles. The special modified form of the current problem is found to be in good agreement with the other published data. © 2013 Wiley Periodicals, Inc. Heat Trans Asian Res, 43(1): 17–38, 2014; Published online in Wiley Online Library ( wileyonlinelibrary.com/journal/htj ). DOI 10.1002/htj.21063     

Numerical study on the parabolic flow of MHD fluid past a vertical plate in a porous medium

A numerical investigation on MHD fluid flow in parabolic mode has been performed to point out its significant properties. Thermal radiation, porous medium, heat generation, chemical reaction, and thermal diffusion along with variable temperature and concentration are taken into consideration in the analysis. The novelty of the work is the inclusion of heat generation and thermal diffusion along with exponentially varying temperature and concentration. The constituent governing equations are solved by using finite difference schemes in explicit form. The fluctuations in velocity, concentration, and temperature are observed and discussed with the help of graphs as well as numerical data. Their gradients are also calculated and analyzed the flow properties by using numerical tables. The existence of heat generation, as well as viscous dissipation, creates an increment in the temperature. The gradient of heat transfer rises with the impact of Prandtl number and decay in it is examined under the existence of a source of heat and viscous dissipation.     

Extended Graetz problem including streamwise conduction and viscous dissipation in microchannel

Extended Graetz problem in microchannel is analyzed by using eigenfunction expansion to solve the energy equation. The hydrodynamically developed flow is assumed to enter the microchannel with uniform temperature or uniform heat flux boundary condition. The effects of velocity and temperature jump boundary condition on the microchannel wall, streamwise conduction and viscous dissipation are all included. From the temperature field obtained, the local Nusselt number distributions are shown as the dimensionless parameters (Peclet number, Knudsen number, Brinkman number) vary. The fully developed Nusselt number for each boundary condition is obtained also in terms of these parameters.     

Effects of non‐Darcian and temperature dependent heat source on two‐phase flow in a vertical porous space with thermal radiation

In this article, influences of viscous dissipation and thermal radiation on MHD flow of two immiscible fluids in a vertical channel filled with porous materials have been studied theoretically. The equations governing the problem are transformed to a system of ODE and are solved by homotopy analysis method (HAM). The effects of physical parameters on flow and heat transfer characteristics have been discussed with the help of graphs. It is found that viscous dissipation parameter, heat source parameter, thermal parameter lead to enhance velocity as well as temperature field. Also, increasing Brinkmann number and heat source parameter lead to suppress Coefficient of skin friction at the left wall but the opposite is true at the other wall. However, these parameters give reverse trend on Nusselt number distribution. Further, increasing thermal conductivity ratio and fluids height ratio leads to increase heat transfer coefficient significantly at the left wall. In addition, we have compared present HAM solution with analytical solution of the problem (ie, absence of radiation parameter and Brinkmann number).     

Assessment of heat transfer in large parallel plates with a narrow gap maintained at a constant temperature

Investigations are conducted on electromagnetohydrodynamic (EMHD) flow and heat transfer in a third-grade fluid flowing through large parallel plates, which are maintained at constant temperatures. The impact of convective heat transmission is disregarded since the space between the plates is small. The influence of viscous dissipation is considered. Despite being addressed for Newtonian fluids, the conduction problem with the viscous dissipation effect is not examined in third-grade fluids for EMHD flow and heat transfer behavior. The least-square method is adopted to solve nondimensional, nonlinear momentum and energy conservation equations to get the dimensionless velocity, temperature distribution, and heat flux. Temperature and heat flux are investigated in relation to the third-grade fluid parameter, the Hartmann number, the electric field parameter, and the Brinkman number. The findings show a rise in the Brinkman number dramatically increases heat transfer from both walls, necessitating cooling of both plates. The heat flow from both walls increases as the parameters of third-grade fluid increases.     

Unsteady MHD Mixed Convective Boundary‐Layer Slip Flow and Heat Transfer with Thermal Radiation and Viscous Dissipation

A numerical analysis has been carried out to investigate the problem of MHD boundary‐layer flow and heat transfer of a viscous incompressible fluid over a moving vertical permeable stretching sheet with velocity and temperature slip boundary condition. A problem formulation is developed in the presence of radiation, viscous dissipation, and buoyancy force. A similarity transformation is used to reduce the governing boundary‐layer equations to coupled higher‐order nonlinear ordinary differential equations. These equations are solved numerically using the fourth‐order Runge–Kutta method along with shooting technique. The effects of the governing parameters such as Prandtl number, buoyancy parameter, slip parameter, magnetic parameter, Eckert Number, suction, and radiation parameter on the velocity and temperature profiles are discussed and shown by plotting graphs. It is found that the temperature is a decreasing function of the slip parameter S_T. The results also indicate that the cooling rate of the sheet can be improved by increasing the buoyancy parameter. In addition the numerical results for the local skin friction coefficient and local Nusselt number are computed and presented in tabular form. The numerical results are compared and found to be in good agreement with previously published results on special cases of the problem. © 2013 Wiley Periodicals, Inc. Heat Trans Asian Res, 43(5): 412–426, 2014; Published online 3 October 2013 in Wiley Online Library ( wileyonlinelibrary.com/journal/htj ). DOI 10.1002/htj.21086     

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.     

Effect of viscous heating on heat transfer performance in microchannel slip flow region

Based on the superposition principle, an analytical solution for steady convective heat transfer in a two-dimensional microchannel in the slip flow region is obtained, including the effects of velocity slip and temperature jump at the wall, which are the main characteristics of flow in the slip flow region, and viscous heating effects in the calculations. The cases of constant heat flux boundary conditions and one wall as adiabatic and the other wall at constant heat flux input are studied. The solution method is verified for the cases where micro-scale effects are neglected. The effects of viscous heating on the temperature profiles and on the heat transfer performance are analyzed in detail. It is concluded that the effect of viscous heating, like an internal energy source, heats the fluid along the flow direction and severely distorts the temperature profiles. The effects of key parameters, such as the Brinkman and Knudsen numbers, on the Nusselt number, which expresses the heat transfer performance are investigated.