Effect of variable viscosity on mixed convection boundary layer flow over a vertical surface embedded in a porous medium

The steady mixed convection boundary layer flow over a vertical impermeable surface embedded in a porous medium when the viscosity of the fluid varies inversely as a linear function of the temperature is studied. Both cases of assisting and opposing flows are considered. The transformed boundary layer equations are solved numerically by a finite difference method. Numerical results for the flow and heat transfer characteristics are obtained for various values of the mixed convection parameter ε and the variable viscosity parameter θ_e. It has been found that in the opposing flow case, dual solutions exist and boundary separation occurs.     

Natural Convection Flow with Surface Radiation Along a Vertical Wavy Surface

In this study, natural convection boundary layer flow of thermally radiating fluid along a heated vertical wavy surface is analyzed. Here, the radiative component of heat flux emulates the surface temperature. Governing equations are reduced to dimensionless form, subject to the appropriate transformation. Resulting dimensionless equations are transformed to a set of parabolic partial differential equations by using primitive variable formulation, which are then integrated numerically via iterative finite difference scheme. Emphasis has been given to low Prandtl number fluid. The numerical results obtained for the physical parameters, such as, surface radiation parameter, R, and radiative length parameter, ξ, are discussed in terms of local skin friction and Nusselt number coefficients. Comprehensive interpretation of velocity distribution is also given in the form of streamlines.     

Conjugate thermal and mass diffusion effect on natural convection flow in presence of strong cross magnetic field

Analysis of magnetohydrodynamic natural convection boundary layer flow of an electrically conducting incompressible fluid along a heated vertical flat plate in the presence of strong cross magnetic field has been discussed. The boundary layer equations governing the flow are transformed to a convenient dimensionless form by using stream function formulation (SFF) and the numerical solution of which is obtained by employing an efficient marching order implicit finite difference scheme over the entire range of local Hartmann parameter, ξ. The behavior of ξ is also studied near the leading edge of the plate by embracing series solution method. However, asymptotic solution for large values of ξ is established analytically, based on the inverse coordinate expansion method. Here, consideration has been given to those fluids which served as liquid metals, by taking Pr << 1. Discussion has been carried out over the results obtained for small, large and all ξ regimes, for different physical parameters in terms of shear stress, τ_w, rate of heat transfer, q_w, and rate of mass transfer, m_w, in the strong cross magnetic field. Influence of local Hartmann parameter ξ and Schmidt number, Sc, on velocity, temperature and concentration distributions are also shown graphically. In addition, comprehensive interpretation of energy and species distributions is also given in terms of heatlines and masslines, respectively.     

Transient mixed convection flow arising due to thermal diffusion over a porous sensor surface inside a squeezing horizontal channel

A transient laminar mixed convection flow of viscous incompressible fluid generated by thermal buoyancy force, over a horizontal porous sensor surface placed inside a squeezing channel is analysed. The non-similar boundary layer equations for the flow and energy are solved numerically for different time regimes. The quantities of physical interest like skin friction coefficient, heat transfer coefficient, velocity and temperature profiles are calculated for different values of physical parameters, Pr, b, S and ξ. The implicit finite difference approximation together with Keller box method is employed for the solution for small and all time regime, where as, a series solution is found for large time regimes. A good agreement of the results computed by different methods has been observed.     

The analysis of conjugate problems of conduction and convection for non-Newtonian fluids past a flat plate

In this paper, the conjugate problems of laminar forced convection in non-Newtonian fluid flow and heat conduction inside a heated flat plate is studied. A conjugate parameter ζ is proposed to reflect the characteristics of the conjugate problems. The value of the conjugate parameter lies among 0 and 1 and the two limiting values correspond to the ordinary convection problem with boundary condition of constant wall heat flux (ζ = 0) and constant wall temperature (ζ = 1), respectively. In addition, the power-law model is used for non-Newtonian fluids with exponent n < 1 for pseudoplastics, n = 1 for Newtonian fluids and n > 1 for dilatant fluids. Furthermore, the coordinates and dependent variables are transformed to yield computationally efficient numerical solutions that are valid over the entire range of conjugate problems and the whole regime of the non-Newtonian fluids. The effects of the conjugate parameter, the power-law viscosity index and the generalized Prandtl number on the temperature profiles, as well as on the local heat transfer rate are clearly illustrated.     

Unsteady flow of viscous incompressible fluid with temperature-dependent viscosity due to a rotating disc in presence of transverse magnetic field and heat transfer

This paper studies the effect of a magnetic field and temperature-dependent viscosity on the unsteady flow and heat transfer for a viscous laminar incompressible and electrically conducting fluid due to an impulsively started rotating infinite disc. The unsteady axisymmetric boundary layer equations are solved using three methods, namely, (i) perturbation solution for small time, (ii) asymptotic analysis for large time and (iii) finite difference method together with Keller box elimination technique for intermediate times. The solutions are obtained in terms of local radial skin friction, local tangential skin friction, and local rate of heat transfer at the surface of the disc, for different values of the pertinent parameters: the Prandtl number Pr, the viscosity variation parameter ε and magnetic field parameter m. The computed dimensionless velocity and temperature profiles for Pr=0.72 are shown graphically for different values of ε and m.     

Effect of temperature dependent viscosity on the onset of Bénard–Marangoni ferroconvection

The onset of coupled Bénard–Marangoni convection in a horizontal layer of ferrofluid with viscosity depending exponentially on temperature is investigated. The lower rigid and the upper free boundaries are assumed to be insulated to temperature perturbations and the free boundary at which the surface tension effects are accounted for is assumed to be non-deformable. The resulting eigenvalue problem is solved numerically using the Galerkin technique and also analytically by a regular perturbation technique with a wave number as a perturbation parameter. The analytical and numerically computed results are found to be in concurrence. The combined effect of magnetic number M₁ and the viscosity parameter B is to reinforce together and to hasten the onset of Bénard–Marangoni ferroconvection compared to their presence in isolation. Nonetheless, the effect of increasing B also shows initially some stabilizing effect on the system depending on the strength of magnetic and buoyancy forces. In addition, the nonlinearity of fluid magnetization M₃ is found to have no influence on the criterion for the onset of Bénard–Marangoni ferroconvection.     

Natural convection flow along the vertical wavy cone in case of uniform surface heat flux where viscosity is an exponential function of temperature

The effect of temperature dependent viscosity μ(T), on steady two dimensional natural convection flow along a vertical wavy cone with uniform surface heat flux has been investigated. Viscosity is taken to be an exponential function of temperature. Using the appropriate variables the basic equations are transformed to non-dimensional boundary layer equations and then solved numerically employing implicit finite difference method. The effects of viscosity variation parameter on the velocity profile, temperature profile, velocity vector field, skin friction, average Nusselt number, streamlines and isotherm have been discussed. The results have been shown graphically by utilizing the visualizing software Techplot.     

Effects of flow inertia on vertical,natural convection in saturated,porous media

An analytical study is made for the effect of flow inertia on vertical, natural convection in saturated, porous media. Within the framework of boundary-layer approximations, Forchheimer's model was transformed into a set of non-similar equations. Effects of flow inertia are measured and examined in terms of the dimensionless inertia parameter ξ = Gr_xFo_x where Gr_x is the local Grashof number of determined by the bulk properties of saturated porous media, and Fo_x is a new dimensionless parameter governed by the microstructure of porous matrix. The non-similar solutions are presented and discussed for two types of flow: (1) the uniform heat flux surface; and (2) plane plume flows. Results show that thermal boundary layer in the non-Darcy regime is thicker than the corresponding pure-Darcy flow. In addition, the local wall heat fluxes for the first case and the maximum temperature gradient for the second case decrease with increasing ξ.     

Unsteady forced convection heat transfer on a flat plate embedded in the fluid-saturated porous medium with inertia effect and thermal dispersion

For an unsteady forced convection on a flat plate embedded in the fluid-saturated porous medium with inertia effect and thermal dispersion, this paper presents a precise and rigorous method to obtain the entire solutions from one-dimensional transient conduction (ξ=0) to steady forced convection in porous medium (ξ=1) under conditions of uniform wall temperature and uniform heat flux, respectively. It is worth noted that the rate of unsteady heat transfer can be accelerated by the thermal dispersion, which may be regarded as the effect of mixing or agitating, to enhance the heat transfer in porous medium. Additionally, it is found that the time response, from the transient heat conduction to a steady forced convection in Darcy's flow, is τ=1, and is independent of wall heating condition and thermal dispersion strength (φ).     

Free Convective Flow of Electrically Conducting and Viscous Immiscible Fluid Flow in a Vertical Channel in the Presence of First‐Order Chemical Reaction

In this paper, the effects of chemical reaction on free convective flow of electrically conducting and viscous incompressible immiscible fluids are analyzed. The coupled nonlinear equations governing the heat and mass transfer are solved analytically and numerically with appropriate boundary conditions for each fluid and the solutions have been matched at the interface. The analytical solutions are solved by using regular perturbation method valid for small values of perturbation parameter and numerically by using finite difference method. The numerical results for various values of thermal Grashof number, mass Grashof number, Hartman number, viscosity ratio, width ratio, conductivity ratio, and chemical reaction parameter have been presented graphically in the presence and in the absence of electric field load parameter. In addition, the closed form expression for volumetric flow rate, Nusselt number, species concentration, and total heat rate added to the flow is also analyzed. The solutions obtained by finite difference method and perturbation method agree very well to the order of 10^?4 for small values of perturbation parameter.     

Laminar mixed convection boundary layers induced by a linearly stretching permeable surface

The boundary layer flow on a linearly moving permeable vertical surface is studied when the buoyancy force assists or opposes the flow. Similarity and local similarity solutions are obtained for the boundary layer equations subject to power law temperature and velocity variation. The effect of various governing parameters, such as Prandtl number Pr, injection parameter d, and the mixed convection parameter λ=Gr_x/Re_x², which determine the velocity and temperature distributions, the heat transfer coefficient, and the shear stress at the surface are studied. The heat transfer coefficient increases as λ assisting the flow for all d for uniformly or linearly heated surface and as Pr increases it becomes almost independent of λ. However, as the temperature inversely proportional to the distance up the surface, the buoyancy has no effects on the heat transfer coefficient. Critical buoyancy parameter values are obtained for vanished shear stress and for predominate natural convection. Critical values are also presented for predominate buoyancy shear stress at the surface for assisting or opposing flow. A closed form analytical solution is also presented as a special case of the energy equation.     

Flows in a fluid layer induced by the combined action of a shear stress and the Soret effect

Convection in a horizontal fluid layer of a binary mixture is studied analytically and numerically. In the formulation of the problem, use is made of the Boussinesq approximation. Neumman boundary conditions are specified for the temperature on all walls of the cavity. In addition of the Soret contribution, a shear stress, τ, is applied on the upper free surface of the layer. The flows are found to be dependent of the Darcy–Rayleigh number, R_T, the Lewis number, Le, the solutal to thermal buoyancy ratio, ϕ, the shear stress, τ and the thermal boundary conditions. Numerical results for finite amplitude convection, obtained by solving numerically the full governing equations, are found to be in good agreement with the analytical solution based on the parallel flow approach. For given sets of the control parameters, the occurrence of multiple steady state solutions is demonstrated. The existence of subcritical bifurcations for both stabilizing and destabilizing mass flux is also demonstrated.     

Forced convection heat transfer in microchannel heat sinks

This paper presents an analysis of forced convection heat transfer in microchannel heat sinks for electronic system cooling. In view of the small dimensions of the microstructures, the microchannel is modeled as a fluid-saturated porous medium. Numerical solutions are obtained based on the Forchheimer–Brinkman-extended Darcy equation for the fluid flow and the two-equation model for heat transfer between the solid and fluid phases. The velocity field in the microchannel is first solved by a finite-difference scheme, and then the energy equations governing the solid and fluid phases are solved simultaneously for the temperature distributions. Also, analytical expressions for the velocity and temperature profiles are presented for a simpler flow model, i.e., the Brinkman-extended Darcy model. This work attempts to perform a systematic study on the effects of major parameters on the flow and heat transfer characteristics of forced convection in the microchannel heat sink. The governing parameters of engineering importance include the channel aspect ratio (α_s), inertial force parameter (Γ), porosity (ε), and the effective thermal conductivity ratio (k_r). The velocity profiles of the fluid in the microchannel, the temperature distributions of the solid and fluid phases, and the overall Nusselt number are illustrated for various values of the problem parameters. It is found that the fluid inertia force alters noticeably the dimensionless velocity distribution and the fluid temperature distribution, while the solid temperature distribution is almost insensitive to the fluid inertia. Moreover, the overall Nusselt number increases with increasing the values of α_s and ε, while it decreases with increasing k_r.     

MHD mixed convection flow along a vertically heated sheet

Main objective of this frame work is to establish the modeling and simulation of mix convection flow along a vertically heated sheet filled with water. Two important mechanisms: magneto-hydrodynamics and porous medium are also considered within the restricted domain of the fluid flow. Temperature is controlled with the wall temperature and then mathematical model is constructed in the form of PDEs. To determine the similarity solution results are obtained via two different techniques. Numerically solutions are obtained with the help of shooting technique and then validate with the help of optimal homotopy analysis method (OHAM). Obtained analytical and numerical results are validated graphically. Effect of emerging parameters are plotted for velocity and temperature profiles. It is found that for mixed convection parameter $(\xi <0)\text{,}$ velocity profile depicts the increasing behavior for various values of power index m. However, for $\xi >0\text{,}$ velocity profile shows the decreasing behavior with respect the parameter m. Temperature distribution in the restricted domain depicts the decreasing behavior for both m and $\xi$.     

Heat transfer in laminar flows of extended modified power law fluids in rectangular ducts

This paper reports the results of a numerical study of the Nusselt number for the H1 and T boundary conditions in laminar, fully-developed flows of pseudoplastic and dilatant fluids in rectangular ducts. Equations for the apparent viscosity that span the entire shear rate range were utilized and the Nusselt numbers were calculated as a function of a shear rate parameter that determines the flow regime where the duct is operating. Numerical results for the Nusselt number in all flow regimes are included along with correlation equations. Finally, errors associated with applying power law solutions are discussed.     

Unsteady MHD free convective flow of a compressible fluid past a moving vertical plate in the presence of radiative heat transfer

The paper studies the free convection flow of a compressible Boussinesq fluid under the simultaneous action of buoyancy and transverse magnetic field while the Rosselant approximation has been invoked to describe the radiative flux in the energy equation. The viscosity of the fluid ν and its thermal conductivity k in this model are assumed to be functions of temperature. Under suitable non-dimensionalization the governing non-linear, coupled, partial differential equations are solved employing a perturbation technique based on the assumption that the fluid flow field is made up of a steady part and a transient. Results obtained which compare favourably well with published data show, that the skin friction for a compressible fluid is lower than that for an incompressible fluid.     

Heat transfer by mixed convection in a vertical flow undergoing transition

Heat transfer, for aiding mixed convection from vertical, uniform flux surfaces and for small forced convection effects, is considered here. Simple relations have been proposed to correlate the new experimental data which were obtained in a flow undergoing transition from a laminar regime toward turbulence. Experiments were performed in air at pressures ranging from 4.4 to about 8 bar. The correlation based on experimental data for laminar flow for Pr = 0.7 has been extended to other Prandtl numbers through numerical integration of the transport equations. It is shown that, for both laminar and turbulent mixed convection, the Nusselt number may be successfully correlated, employing suitable combinations of the corresponding heat transfer correlations for forced and for natural convection. The parameter characterizing the mixed convection effect was found to be different in laminar and turbulent flow. However, in each of these regions, the relevant parameter is proportional to the ratio of the applicable characteristic forced and natural convection velocity scales.     

Free convection heat and mass transfer from an isothermal sphere to a micropolar regime with Soret/Dufour effects

An analysis is presented for the steady free convection heat and mass transfer from a spherical body in a micropolar fluid with Soret/Dufour effects included. The governing boundary layer equations are transformed into a non-dimensional form and the resulting non-linear system of partial differential equations are solved using the efficient Keller-box implicit numerical finite difference method. Excellent correlation of the present numerical solutions is achieved for the case of free convection micropolar flow neglecting Sort/Dufour effects, with the earlier study by Nazar et al. (2002) [12]. With increasing Soret number, the local Nusselt number is boosted considerably, with the converse response with an increase in Dufour number. Increasing micropolar vortex viscosity parameter (K) decelerates the flow near the sphere surface but serves to accelerate the flow further from the sphere wall. With increasing K, reverse spin is induced of the micro-elements near the sphere surface. Temperature is strongly boosted throughout the boundary layer regime with increasing K with a similar response computed for the species concentration Buoyancy-assistance (N > 0) accelerates the flow whereas buoyancy-opposition (N < 0) retards flow. Reverse spin of micro-elements (i.e. negative micro-rotation) is reduced near the sphere surface with buoyancy-opposition but exacerbated with buoyancy-assistance. Buoyancy-assistance serves to depress both temperature and concentration values through the boundary layer regime with the reverse effect sustained with buoyancy-opposition. An increase in Schmidt number acts to strongly inhibit the flow and also to stifle micro-rotation of the micro-elements. Conversely both temperatures and concentration values are accentuated with increasing Schmidt number. The study has important applications in buoyancy-driven, chemical engineering flow regimes.