Flow and Heat Transfer of Viscous Fluids Saturated in Porous Media over a Permeable Non-isothermal Stretching Sheet

The present paper deals with the flow and heat transfer of a viscous fluid saturated in a porous medium past a permeable and non-isothermal stretching sheet with internal heat generation or absorption and radiation. Closed-form solutions to steady, two dimensional momentum equations with neglecting quadratic inertia terms and heat transfer equation are found using a similarity transformation. Asymptotic expressions of the temperature functions are also presented valid for both very large and very small modified Prandtl numbers. Attention is focused on the effects of porous parameter K, suction parameter R, radiation parameter Nr, viscosity ratio Λ, internal heat parameter α and Prandtl number P to the characteristics of flow and heat transfer.     

Effect of thermo‐solutal stratification on recirculation flow patterns in a backward‐facing step channel flow

This paper presents results on the combined effect of thermo‐solutal buoyancy forces on the recirculatory flow behavior in a horizontal channel with backward‐facing step and the ensuing impact on heat and mass transfer phenomena. The governing equations for double diffusive mixed convection are represented in velocity–vorticity form of momentum equations, velocity Poisson equations, energy and concentration equations. Galerkin's finite‐element method has been employed to solve the governing equations. Recirculatory flow fields with heat and mass transfer are simulated for opposing and aiding thermo‐solutal buoyancy forces by assuming suitable boundary conditions for energy and concentration equations. The effect of Richardson number (0.1?Ri?10) and buoyancy ratio (?10?N?10) on the recirculation bubble and Nusselt and Sherwood numbers are studied in detail. For Richardson number greater than unity, distinct variations in the gradients of Nusselt number and Sherwood number with buoyancy ratio are observed for flow regimes with opposing and aiding buoyancy forces. Copyright © 2009 John Wiley & Sons, Ltd.     

Magnetohydrodynamic flow over a vertical stretching surface with suction and blowing

An analysis is presented to investigate the flow and heat transfer characteristics of a vertical stretching surface with suction and blowing, and variable magnetic effects. The magnetic field of variable intensity is applied perpendicular to the surface. The range of the magnetic parameter M investigated is 0.1 to 1.0. The flow is considered steady, incompressible, and three-dimensional. The governing momentum and energy equations are solved numerically. Numerical results are presented for velocity distribution, temperature distribution, surface shear stress, and wall heat transfer rate. Discussion is provided for the effect of the magnetic field strength on the velocity and temperature fields. Received on 26 November 1997     

Effects of Knudsen number and geometry on gaseous flow and heat transfer in a constricted microchannel

A flow and heat transfer numerical simulation is performed for a 2D laminar incompressible gas flow through a constricted microchannel in the slip regime with constant wall temperature. The effects of rarefaction, creeping flow, first order slip boundary conditions and hydrodynamically/thermally developing flow are assumed. The effects of Knudsen number and geometry on thermal and hydrodynamic characteristics of flow in a constricted microchannel are explored. SIMPLE algorithm in curvilinear coordinate is used to solve the governing equations including continuity, energy and momentum with the temperature jump and velocity slip conditions at the solid walls in discretized form. The resulting velocity and temperature profiles are then utilized to obtain the microchannel C _f Re and Nusselt number as a function of Knudsen number and geometry. The results show that Knudsen number has declining effect on the C _f Re and Nusselt number in the constricted microchannel. In addition, the temperature jump on wall and slip velocity increase with increasing Knudsen number. Moreover, by decreasing the throttle area, the fluid flow characteristics experience more intense variations in the constricted region. To verify the code a comparison is carried out with available results and good agreement is achieved.     

Unsteady mixed convection on the stagnation-point flow adjacent to a vertical plate with a magnetic field

An analysis is performed to study the unsteady combined forced and free convection flow (mixed convection flow) of a viscous incompressible electrically conducting fluid in the vicinity of an axisymmetric stagnation point adjacent to a heated vertical surface. The unsteadiness in the flow and temperature fields is due to the free stream velocity, which varies arbitrarily with time. Both constant wall temperature and constant heat flux conditions are considered in this analysis. By using suitable transformations, the Navier–Stokes and energy equations with four independent variables (x, y, z, t) are reduced to a system of partial differential equations with two independent variables (, ). These transformations also uncouple the momentum and energy equations resulting in a primary axisymmetric flow, in an energy equation dependent on the primary flow and in a buoyancy-induced secondary flow dependent on both primary flow and energy. The resulting system of partial differential equations has been solved numerically by using both implicit finite-difference scheme and differential-difference method. An interesting result is that for a decelerating free stream velocity, flow reversal occurs in the primary flow after certain instant of time and the magnetic field delays or prevents the flow reversal. The surface heat transfer and the surface shear stress in the primary flow increase with the magnetic field, but the surface shear stress in the buoyancy-induced secondary flow decreases. Further the heat transfer increases with the Prandtl number, but the surface shear stress in the secondary flow decreases.     

Effects of chemical reaction on mixed convection flow of a polar fluid through a porous medium in the presence of internal heat generation

This paper reports a detailed numerical investigation on mixed convection flow of a polar fluid through a porous medium due to the combined effects of thermal and mass diffusion. The energy equation accounts for heat generation or absorption, while the nth order homogeneous chemical reaction between the fluid and the diffusing species is included in the mass diffusion equation. The governing equations of the linear momentum, angular momentum, energy and concentration are obtained in a non-similar form by introducing a suitable group of transformations. The final set of non-similar coupled non-linear partial differential equations is solved using an implicit finite-difference scheme in combination with quasi-linearization technique. The effects of various parameters on the velocity, angular velocity, temperature and concentration fields are investigated. Numerical results for the skin friction coefficient, wall stress of angular velocity, Nusselt number and Sherwood number are also presented.     

Flow and heat transfer over an unsteady stretching surface with Hall effect

The present investigation is concerned with the effect of Hall currents on the flow and heat transfer of an electrically conducting fluid over an unsteady stretching surface in presence of a strong magnetic field. The induced magnetic field is neglected while the electron-atom collision frequency is assumed to be relatively high, so that the Hall effect is assumed to exist. The incorrect similarity transformation of Elbashbeshy and Bazid (Heat Mass Transfer 41:1–4, 2004). is corrected and a physically realistic distribution of the velocity and temperature is obtained. Using a similarity transformation the governing time dependent boundary layer equations for momentum and thermal energy are reduced to a set of coupled ordinary differential equations which are then solved numerically by the shooting method. Effects of the magnetic field, M , Hall parameter, m, and the unsteadiness parameter, S, on the velocity and temperature profiles as well as the local skin friction coefficients and the heat transfer rate are shown graphically.     

Direct and adjoint global stability analysis of turbulent transonic flows over a NACA0012 profile

In this work, various turbulent solutions of the two‐dimensional (2D) and three‐dimensional compressible Reynolds averaged Navier–Stokes equations are analyzed using global stability theory. This analysis is motivated by the onset of flow unsteadiness (Hopf bifurcation) for transonic buffet conditions where moderately high Reynolds numbers and compressible effects must be considered. The buffet phenomenon involves a complex interaction between the separated flow and a shock wave. The efficient numerical methodology presented in this paper predicts the critical parameters, namely, the angle of attack and Mach and Reynolds numbers beyond which the onset of flow unsteadiness appears. The geometry, a NACA0012 profile, and flow parameters selected reproduce situations of practical interest for aeronautical applications. The numerical computation is performed in three steps. First, a steady baseflow solution is obtained; second, the Jacobian matrix for the RANS equations based on a finite volume discretization is computed; and finally, the generalized eigenvalue problem is derived when the baseflow is linearly perturbed. The methodology is validated predicting the 2D Hopf bifurcation for a circular cylinder under laminar flow condition. This benchmark shows good agreement with the previous published computations and experimental data. In the transonic buffet case, the baseflow is computed using the Spalart–Allmaras turbulence model and represents a mean flow where the high frequency content and length scales of the order of the shear‐layer thickness have been averaged. The lower frequency content is assumed to be decoupled from the high frequencies, thus allowing a stability analysis to be performed on the low frequency range. In addition, results of the corresponding adjoint problem and the sensitivity map are provided for the first time for the buffet problem. Finally, an extruded three‐dimensional geometry of the NACA0012 airfoil, where all velocity components are considered, was also analyzed as a Triglobal stability case, and the outcoming results were compared to the previous 2D limited model, confirming that the buffet onset is well detected. Copyright © 2014 John Wiley & Sons, Ltd.     

Two-phase flow in heterogeneous porous media: The method of large-scale averaging

The analysis of two-phase flow in porous media begins with the Stokes equations and an appropriate set of boundary conditions. Local volume averaging can then be used to produce the well known extension of Darcy's law for two-phase flow. In addition, a method of closure exists that can be used to predict the individual permeability tensors for each phase. For a heterogeneous porous medium, the local volume average closure problem becomes exceedingly complex and an alternate theoretical resolution of the problem is necessary. This is provided by the method of large-scale averaging which is used to average the Darcy-scale equations over a region that is large compared to the length scale of the heterogeneities. In this paper we present the derivation of the large-scale averaged continuity and momentum equations, and we develop a method of closure that can be used to predict the large-scale permeability tensors and the large-scale capillary pressure. The closure problem is limited by the principle of local mechanical equilibrium. This means that the local fluid distribution is determined by capillary pressure-saturation relations and is not constrained by the solution of an evolutionary transport equation. Special attention is given to the fact that both fluids can be trapped in regions where the saturation is equal to the irreducible saturation, in addition to being trapped in regions where the saturation is greater than the irreducible saturation. Theoretical results are given for stratified porous media and a two-dimensional model for a heterogeneous porous medium.     

Squeezed flow and heat transfer over a porous surface for viscous fluid

Flow and heat transfer over a permeable sensor surface placed in a squeezing channel is analyzed. A constant transpiration through the sensor surface is assumed. Locally non-similar momentum and energy equations are solved by three different methods, against the transpiration parameter τ, for different values of the squeezing parameter b, and Prandtl number Pr. From the investigation, it is found that when the channel being squeezed, the skin-friction reduces but the heat transfer coefficient increases. Increase in the value of the squeezing parameter onsets reverse flow at the sensor surface when fluid is being injected and the affect is enhanced with the increase of injection through the surface. It is further observed that increase of suction of fluid through the sensor thins the thermal and the momentum boundary layer regions, whereas injection of fluid leads to thickening of both the thermal and the momentum boundary layer regions. Heat transfer from the surface of the sensor increases with the increase of the value of Pr for the entire range of surface mass-flux parameter τ. M. A. Hossain is on leave of absence from University of Dhaka.     

Laminar Mixed Convection Adjacent to Vertical Continuously Stretching Sheets With Variable Viscosity and Variable Thermal Diffusivity

The paper presents a study of the laminar mixed convection adjacent to vertical continuously stretching sheets, taking into account the effects of variable viscosity and variable thermal diffusivity. The similarity solutions are reported for isothermal sheet moving with a velocity of the form u_w=Bx^0.5 and a continuous linearly stretching sheet with a linear surface temperature distribution. The equations of conservation of mass, momentum and energy, which govern the flow and heat transfer, are solved numerically by using the shooting method. The numerical results obtained for the flow and heat transfer characteristics reveal many interesting behaviors. The numerical results show that, variable viscosity, variable thermal diffusivity, the velocity exponent parameter, the temperature exponent parameter and the buoyancy force parameter have significant influences on the velocity and temperature profiles, shear stress and Nusselt number in two cases air and water.     

Two-dimensional solidification of viscous flow over a semi infinite plate with constant heat removal

Two-dimensional solidification of forced viscous flow over a semi infinite flat plate is considered. The plate is initially maintained at a constant temperature above the freezing point of the liquid. At time t>0, heat is removed from the plate so that the liquid freezes. The continuity equation and the two-dimensional, nonsteady, momentum and energy equations in the liquid and solid phases are solved numerically. The growth of the solid layer as a function of time and position along the plate is determined.     

Two-equation (k, ϵ) turbulence computations by the use of a finite element model

A finite element technique is presented and applied to some one- and two-dimensional turbulent flow problems. The basic equations are the Reynolds averaged momentum equations in conjunction with a two-equation (k, ?) turbulence model. The equations are written in time-dependent form and stationary problems are solved by a time iteration procedure. The advection parts of the equations are treated by the use of a method of characteristics, while the continuity requirement is satisfied by a penalty function approach. The general numerical formulation is based on Galerkin's method. Computational results are presented for one-dimensional steady-state and oscillatory channel flow problems and for steady-state flow over a two-dimensional backward-facing step.     

Momentum and heat transfer over a continuous moving surface in a non-Darcian fluid

The momentum and heat transfer characteristics associated with the boundary layer on a continuous moving flat surface in a non-Darcian fluid have been investigated exploiting a local similarity solution procedure. The full boundary layer equations, which describe the effects of convective inertia, solid boundary, and porous inertia in addition to the Darcy flow resistance, were solved using novel transformed variables, deduced from a scale analysis on the momentum and energy conservation equations. Details are provided for the effects of convective inertia and porous inertia on the velocity and temperature profiles. The resulting friction and heat transfer characteristics are found to be substantially different from those of forces convection over a stationary flat plate. Furthermore, useful asymptotic expressions for the local Nusselt number are presented in consideration of possible physical limiting conditions.     

A Modified Nested-Gridding for Upscaling–Downscaling in Reservoir Simulation

This article reports a numerical study of double-diffusive convection in a fluid-saturated vertical porous annulus subjected to discrete heat and mass fluxes from a portion of the inner wall. The outer wall is maintained at uniform temperature and concentration, while the top and bottom walls are adiabatic and impermeable to mass transfer. The physical model for the momentum equation is formulated using the Darcy law, and the resulting governing equations are solved using an implicit finite difference technique. The influence of physical and geometrical parameters on the streamlines, isotherms, isoconcentrations, average Nusselt and Sherwood numbers has been numerically investigated in detail. The location of heat and solute source has a profound influence on the flow pattern, heat and mass transfer rates in the porous annulus. For the segment located at the bottom portion of inner wall, the flow rate is found to be higher, whereas the heat and mass transfer rates are higher when the source is placed near the middle of the inner wall. Further, the average Sherwood number increases with Lewis number, while for the average Nusselt number the effect is opposite. The average Nusselt number increases with radius ratio (λ); however, the average Sherwood number increases with radius ratio only up to λ = 5, and for λ > 5 , the average Sherwood number does not increase significantly.     

MHD effect of mixed convection boundary-layer flow of Powell-Eyring fluid past nonlinear stretching surface

Sufficient conditions are found for the existence of similar solutions of the mixed convection flow of a Powell-Eyring fluid over a nonlinear stretching permeable sur- face in the presence of magnetic field. To achieve this, one parameter linear group trans- formation is applied. The governing momentum and energy equations are transformed to nonlinear ordinary differential equations by use of a similarity transformation. These equations are solved by the homotopy analysis method （HAM） to obtain the approximate solutions. The effects of magnetic field, suction, and buoyancy on the Powell-Eyring fluid flow with heat transfer inside the boundary layer are analyzed. The effects of the non- Newtonian fluid （Powell-Eyring model） parameters ε and δon the skin friction and local heat transfer coefficients for the cases of aiding and opposite flows are investigated and discussed. It is observed that the momentum boundary layer thickness increases and the thermal boundary layer thickness decreases with the increase in ε whereas the momentum boundary layer thickness decreases and thermal boundary layer thickness increases with the increase in δ for both the aiding and opposing mixed convection flows.     

A variational formulation of the coupled thermo-mechanical boundary-value problem for general dissipative solids

A variational formulation of the coupled thermo-mechanical boundary-value problem for general dissipative solids is presented. The coupled thermo-mechanical boundary-value problem under consideration consists of the equilibrium problem for a deformable, inelastic and dissipative solid with the heat conduction problem appended in addition. The variational formulation allows for general dissipative solids, including finite elastic and plastic deformations, non-Newtonian viscosity, rate sensitivity, arbitrary flow and hardening rules, as well as heat conduction. We show that a joint potential function exists such that both the conservation of energy and the balance of linear momentum equations follow as Euler-Lagrange equations. The identification of the joint potential requires a careful distinction between equilibrium and external temperatures, which are equal at equilibrium. The variational framework predicts the fraction of dissipated energy that is converted to heat. A comparison of this prediction and experimental data suggests that α-titanium and Al2024-T conform to the variational framework.     

Non-Newtonian magnetohydrodynamic flow over a stretching surface with heat and mass transfer

An analysis is carried out to study the momentum, mass and heat transfer characteristics on the flow of visco-elastic fluid (Walter's liquid-B model) past a stretching sheet in the presence of a transverse magnetic field.In heat transfer, two cases are considered:

1.

The sheet with prescribed surface temperature (PST case); and

2.

The sheet with prescribed wall heat flux (PHF case).

The solution of equations of momentum, mass and heat transfer are obtained analytically. Emphasis has been laid to study the effects of various parameters like magnetic parameter Mn, visco-elastic parameter k₁, Schmidt number Sc, and Prandtl number Pr on flow, heat and mass transfer characteristics.     

Frequency lock-in phenomenon for oscillating airfoils in buffeting flows

Navier-Stokes based computer simulations are conducted to determine the aerodynamic flow field response that is observed for a NACA0012 airfoil that undergoes prescribed harmonic oscillation in transonic buffeting flows, and also in pre-buffet flow conditions. Shock buffet is the term for the self-sustained shock oscillations that are observed for certain combinations of Mach number and steady mean flow angle of attack even in the absence of structural motion. The shock buffet frequencies are typically on the order of the elastic structural frequencies, and therefore may be a contributor to transonic aeroelastic response phenomena, including limit-cycle oscillations. Numerical simulations indicate that the pre-shock-buffet flow natural frequency increases with mean angle of attack, while the flow damping decreases and approaches zero at the onset of buffet. Airfoil harmonic heave motions are prescribed to study the interaction between the flow fields induced by the shock buffet and airfoil motion, respectively. At pre-shock-buffet conditions the flow response is predominantly at the airfoil motion frequency, with some smaller response at multiplies of this frequency. At shock buffet conditions, a key effect of prescribed airfoil motions on the buffeting flow is to create the possibility of a lock-in phenomenon, in which the shock buffet frequency is synchronized to the prescribed airfoil motion frequency for certain combinations of airfoil motion frequencies and amplitudes. Aerodynamic gain-phase models for the lock-in region, as well as for the pre-shock-buffet conditions are suggested, and also a possible relationship between the lock-in mechanism and limit-cycle oscillation is discussed.     

Free Convection Boundary Layer Flow from a Heated Upward Facing Horizontal Flat Plate Embedded in a Porous Medium Filled by a Nanofluid with Convective Boundary Condition

The steady laminar incompressible free convective flow of a nanofluid over a permeable upward facing horizontal plate located in porous medium taking into account the thermal convective boundary condition is studied numerically. The nanofluid model used involves the effect of Brownian motion and the thermophoresis. Using similarity transformations the continuity, the momentum, the energy, and the nanoparticle volume fraction equations are transformed into a set of coupled similarity equations, before being solved numerically, by an implicit finite difference numerical method. Our analysis reveals that for a true similarity solution, the convective heat transfer coefficient related with the hot fluid and the mass transfer velocity must be proportional to x ^−2/3, where x is the horizontal distance along the plate from the origin. Effects of the various parameters on the dimensionless longitudinal velocity, the temperature, the nanoparticle volume fraction, as well as on the rate of heat transfer and the rate of nanoparticle volume fraction have been presented graphically and discussed. It is found that Lewis number, the Brownian motion, and the convective heat transfer parameters increase the heat transfer rate whilst the thermophoresis decreases the heat transfer rate. It is also found that Lewis number and the convective heat transfer parameter enhance the nanoparticle volume fraction rate whilst the thermophoresis parameter decreases nanoparticle volume fraction rate. A very good agreement is found between numerical results of the present article for special case and published results. This close agreement supports the validity of our analysis and the accuracy of the numerical computations.