Simultaneous heat and mass transfer under unsteady mixed convection along a vertical slender cylinder embedded in a porous medium

Unsteady laminar mixed convection flow (combined free and forced convection flow) along a vertical slender cylinder embedded in a porous medium under the combined buoyancy effect of thermal and species diffusion has been studied. The effect of the permeability of the medium as well as the magnetic field has been included in the analysis. The partial differential equations with three independent variables governing the flow have been solved numerically using a implicit finite difference scheme in combination with the quasilinearization technique. Computations have been carried out for accelerating, decelerating and oscillatory free stream velocity distributions. The effects of the permeability of the medium, buoyancy forces, transverse curvature and magnetic field on skin friction, heat transfer and mass transfer have been studied. It is found that the effect of free stream velocity distribution is more pronounced on the skin friction than on the heat and mass transfer. The permeability and magnetic parameters increase the skin friction, but reduce the heat and mass transfer. The skin friction, heat transfer and mass transfer are enhanced due to the buoyancy forces and curvature parameter. The heat transfer is strongly dependent on the viscous dissipation parameter and the Prandtl number, and the mass transfer on the Schmidt number.     

Steady mixed convection flow of Maxwell fluid over an exponentially stretching vertical surface with magnetic field and viscous dissipation

The steady mixed convection flow and heat transfer from an exponentially stretching vertical surface in a quiescent Maxwell fluid in the presence of magnetic field, viscous dissipation and Joule heating have been studied. The stretching velocity, surface temperature and magnetic field are assumed to have specific exponential function forms for the existence of the local similarity solution. The coupled nonlinear ordinary differential equations governing the local similarity flow and heat transfer have been solved numerically by Chebyshev finite difference method. The influence of the buoyancy parameter, viscous dissipation, relaxation parameter of Maxwell fluid, magnetic field and Prandtl number on the flow and heat transfer has been considered in detail. The Nusselt number increases significantly with the Prandtl number, but the skin friction coefficient decreases. The Nusselt number slightly decreases with increasing viscous dissipation parameter, but the skin friction coefficient slightly increases. Maxwell fluid reduces both skin friction coefficient and Nusselt number, whereas buoyancy force enhances them.     

Unsteady free convection flow over an infinite vertical porous plate due to the combined effects of thermal and mass diffusion,magnetic field and Hall currents

The unsteady free convection flow over an infinite vertical porous plate, which moves with time-dependent velocity in an ambient fluid, has been studied. The effects of the magnetic field and Hall current are included in the analysis. The buoyancy forces arise due to both the thermal and mass diffusion. The partial differential equations governing the flow have been solved numerically using both the implicit finite difference scheme and the difference-differential method. For the steady case, analytical solutions have also been obtained. The effect of time variation on the skin friction, heat transfer and mass transfer is very significant. Suction increases the skin friction coefficient in the primary flow, and also the Nusselt and Sherwood numbers, but the skin friction coefficient in the secondary flow is reduced. The effect of injection is opposite to that of suction. The buoyancy force, injection and the Hall parameter induce an overshoot in the velocity profiles in the primary flow which changes the velocity gradient from a negative to a positive value, but the magnetic field and suction reduce this velocity overshoot.     

Simultaneous heat and mass transfer in unsteady free convection from two-dimensional and axisymmetric bodies

The unsteady laminar free convection boundary layer flows around two-dimensional and axisymmetric bodies placed in an ambient fluid of infinite extent have been studied when the flow is driven by thermal buoyancy forces and buoyancy forces from species diffusion. The unsteadiness in the flow field is caused by both temperature and concentration at the wall which vary arbitrarily with time. The coupled nonlinear partial differential equations with three independent variables governing the flow have been solved numerically using an implicit finite-difference scheme in combination with the quasilinearization technique. Computations have been performed for a circular cylinder and a sphere. The skin friction, heat transfer and mass transfer are strongly dependent on the variation of the wall temperature and concentration with time. Also the skin friction and heat transfer increase or decrease as the buoyancy forces from species diffusion assist and oppose, respectively, the thermal buoyancy force, whereas the mass transfer rate is higher for small values of the ratio of the buoyancy parameters than for large values. The local heat and mass transfer rates are maximum at the stagnation point and they decrease progressively with increase of the angular position from the stagnation point.     

Unsteady mixed convection with double diffusion over a horizontal cylinder and a sphere within a porous medium

The combined effects of the permeability of the medium, magnetic field, buoyancy forces and dissipation on the unsteady mixed convection flow over a horizontal cylinder and a sphere embedded in a porous medium have been studied. The nonlinear coupled partial differential equations with three independent variables have been solved numerically using an implicit finite-difference scheme in combination with the quasilinearization technique. The skin friction, heat transfer and mass transfer increase with the permeability of the medium, magnetic field and buoyancy parameter. The heat and mass transfer continuously decrease with the stream-wise distance, whereas the skin friction increases from zero, attains a maximum and then decreases to zero. The skin friction, heat transfer and mass transfer are significantly affected by the free stream velocity distribution. The effect of dissipation parameter is found to be more pronounced on the heat transfer than on the skin friction and mass transfer.Untersucht wurden kombinierte Effekte der Permeabilität des Mediums, des magnetischen Feldes, der Auftriebskräfte und der Dissipation auf die instationäre Mischkonvektions-Strömung über einen horizontalen Zylinder und eine Kugel, die in einem porösen Medium eingebettet sind. Die nichtlinearen gekoppelten partiellen Differentialgleichungen mit drei unabhängigen Variablen wurden numerisch unter Benutzung eines impliziten Finite-Differenzen-Verfahrens in Verbindung mit der Quasi-Linearisierungstechnik gelöst. Die Oberflächenreibung und die Wärme- und Stoffübertragung steigen mit der Permeabilität des Mediums, dem magnetischen Feld und dem Auftriebsparameter an. Die Wärme- und Stoffübertragung fällt stetig in Strömungsrichtung ab, wohingegen die Oberflächenreibung von Null ansteigt, ein Maximum erreicht und wieder auf Null abfällt. Die Oberflächenreibung und die Wärme- und Stoffübertragung werden signifikant von der Verteilung der Freistromgeschwindigkeit beeinflut. Es wurde festgestellt, das der Dissipations-Parameter stärker die Wärmeübertragung als die Oberflächenreibung und die Stoffübertragung beeinflut.     

Unsteady mixed convection flow from a rotating vertical cone with a magnetic field

An analysis is developed to study the unsteady mixed convection flow over a vertical cone rotating in an ambient fluid with a time-dependent angular velocity in the presence of a magnetic field. The coupled nonlinear partial differential equations governing the flow have been solved numerically using an implicit finite-difference scheme. The local skin friction coefficients in the tangential and azimuthal directions and the local Nusselt number increase with the time when the angular velocity of the cone increases, but the reverse trend is observed for decreasing angular velocity. However, these are not mirror reflection of each other. The magnetic field reduces the skin friction coefficient in the tangential direction and also the Nusselt number, but it increases the skin friction coefficient in the azimuthal direction. The skin friction coefficients and the Nusselt number increase with the buoyancy force.     

The influence of thermal radiation on hydromagnetic Darcy-Forchheimer mixed convection flow past a stretching sheet embedded in a porous medium

A boundary layer analysis is performed to study the influence of thermal radiation and buoyancy force on two-dimensional magnetohydrodynamic flow of an incompressible viscous and electrically conducting fluid over a vertical stretching sheet embedded in a porous medium in the presence of inertia effect. The governing system of partial differential equations is first transformed into system of ordinary differential equations using self-similarity transformation. A special form for magnetic field is chosen to obtain the similarity solution. The transformed boundary layer equations are solved numerically for some important values of the physical parameters. The present results are compared with the previously published papers and the results are found to be in excellent agreement. The important features of the flow, heat and mass transfer characteristics for different values of thermal radiation, porous permeability, magnetic field and buoyancy parameters are analyzed and discussed. The effects of various physical parameters on the skin friction coefficient, local Nusselt number and local Sherwood number are also presented. It is found that increase in the value of thermal radiation parameter R ₁ increases the skin friction coefficient and Sherwood number whereas reverse trend is seen for the local Nusselt number.     

Steady laminar mixed convection stagnation-point flow of a nanofluid over a vertical permeable surface in the presence of a magnetic field

A similarity solution for a steady laminar mixed convection boundary layer flow of a nanofluid near the stagnation point on a vertical permeable plate with a magnetic field and a buoyancy force is obtained by solving a system of nonlinear ordinary differential equations. These equations are solved analytically by using a new kind of a powerful analytic technique for nonlinear problems, namely, the homotopy analysis method (HAM). Three different types of nanoparticles, namely, copper (Cu), alumina (Al₂O₃), and titanium oxide (TiO₂), with water as the base fluid are considered. The influence of the volume fraction of nanoparticles, permeability parameter, magnetic parameter, and mixed convection parameter on the surface shear stress and surface heat transfer, as well as on the velocity and temperature profiles, is considered. It is observed that the skin friction coefficient and the local Nusselt number increase with the nanoparticle volume fraction for all types of nanoparticles considered in this study. The greatest values of the skin friction coefficient and the local Nusselt number are obtained for Cu nanoparticles.     

Unsteady mixed convection over two-dimensional and axisymmetric bodies

The unsteady laminar incompressible mixed convection flow over a two-dimensional body (cylinder) and an axisymmetric body (sphere) has been studied when the buboyancy forces arise from both thermal and mass diffusion and the unsteadiness in the flow field is introduced by the time dependent free stream velocity. The nonlinear partial differential equations with three independent variables governing the flow have been solved numerically using an implicit finite-difference scheme in combination with the quasilinearization technique. The results indicate that for the thermally assisting flow the local skin friction, heat transfer and mass diffusion are enhanced when the buoyancy force from mass diffusion assists the thermal buoyancy force. But this trend is opposite for the thermally opposing flow. The point of zero skin friction moves upstream due to unsteadiness. No singularity is observed at the point of zero skin friction for unsteady flow unlike steady flow. The flow reversal is observed after a certain instant of time. The velocity overshoot occurs for assisting flows.     

Non-darcy double-diffusive mixed convection from heated vertical and horizontal plates in saturated porous media

The non-darcy mixed convection flows from heated vertical and horizontal plates in saturated porous media have been considered using boundary layer approximations. The flows are considered to be driven by multiple buoyancy forces. The similarity solutions for both vertical and horizontal plates have been obtained. The governing equations have been solved numerically using a shooting method. The heat transfer, mass transfer and skin friction are reduced due to inertial forces. Also, they increase with the buoyancy parameter for aiding flow and decrease for the opposing flow. For aiding flow, the heat and mass transfer coefficients are found to approach asymptotically the forced or free convection values as the buoyancy parameter approaches zero or infinity.     

Unsteady MHD mixed convection flow over an impulsively stretched permeable vertical surface in a quiescent fluid

The unsteady mixed convection flow of an incompressible laminar electrically conducting fluid over an impulsively stretched permeable vertical surface in an unbounded quiescent fluid in the presence of a transverse magnetic field has been investigated. At the same time, the surface temperature is suddenly increased from the surrounding fluid temperature or a constant heat flux is suddenly imposed on the surface. The problem is formulated in such a way that for small time it is governed by Rayleigh type of equation and for large time by Crane type of equation. The non-linear coupled parabolic partial differential equations governing the unsteady mixed convection flow under boundary layer approximations have been solved analytically by using the homotopy analysis method as well as numerically by an implicit finite difference scheme. The local skin friction coefficient and the local Nusselt number are found to decrease rapidly with time in a small time interval and they tend to steady-state values for t*≥5. They also increase with the buoyancy force and suction, but decrease with injection rate. The local skin friction coefficient increases with the magnetic field, but the local Nusselt number decreases. There is a smooth transition from the unsteady state to the steady state.     

Effect of viscous dissipation on the mixed convection heat transfer from an exponentially stretching surface

The mixed convection flow and heat transfer from an exponentially stretching vertical surface in a quiescent fluid is analyzed using similarity solution technique. Wall temperature and stretching velocity are assumed to have specific exponential function forms. The influence of buoyancy along with viscous dissipation on the convective transport in the boundary layer region is analyzed in both aiding and opposing flow situations. The flow is governed by the mixed convection parameter Gr/Re². The velocity and temperature inside the boundary layer are observed to be influenced by the parameters like Prandtl number Pr, Gebhart number Gb. Significant changes are observed in non-dimensional skin friction and heat transfer coefficients due to viscous dissipation in the medium. The flow and temperature distributions inside the boundary layer are analyzed and the results for non-dimensional skin friction and heat transfer coefficients are discussed through computer generated plots.     

Radiative heat transfer in a hydromagnetic nanofluid past a non-linear stretching surface with convective boundary condition

Heat transfer characteristics of a two-dimensional steady hydromagnetic natural convection flow of nanofluids over a non-linear stretching sheet taking into account the effects of radiation and convective boundary condition has been investigated numerically. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis. The local similarity solutions are obtained by using very robust computer algebra software Maple 13. The results corresponding to the dimensionless temperature profiles and the reduced Nusselt number, Sherwood number and skin friction coefficient are displayed graphically for various pertinent parameters. The results show that temperature within the boundary layer is enhanced with the increase of the Biot number, buoyancy due to nanoparticle concentration, strength of the applied magnetic field, Brownian motion parameter, and thermophoresis parameter. An opposite trend is observed for the increase of the buoyancy due to temperature, stretching index, and the radiation parameter. The results also show that the local rate of heat transfer strongly depends on the nonlinear stretching index, radiation parameter, Biot number, Brownian motion parameter, and thermophoresis parameter.     

Effects of hall current on MHD Couette flow in a channel partially filled with a porous medium in a rotating system

MHD Couette flow in a channel with non-conducting walls, partially filled with a porous medium, is investigated in the presence of an inclined magnetic field in a rotating system. It is observed that the MHD flow behaviour in the channel has been influenced significantly by the Coriolis force, the hydromagnetic force with an inclusion of Hall current and the permeability of the porous medium. Effects of the parameters of these forces on the velocity distributions, induced magnetic field distributions and the skin friction have been depicted graphically and discussed.     

Natural convection flow from a continuously moving vertical surface immersed in a thermally stratified medium

 Natural convection boundary layer flow over a continuously moving isothermal vertical surface immersed in a thermally stratified medium has been investigated here. The non-linear coupled partial differential equations governing the non-similar flow have been solved numerically using an implicit finite difference scheme. For small values of the streamwise distance the partial differential equations are solved by using a perturbation expansion procedure and also using the Shanks transformation. The results indicate that the thermal stratification significantly affects both the surface shear stress and the surface heat transfer. The buoyancy parameter and the Prandtl number increase significantly, both the surface shear stress and heat transfer. Also the buoyancy force gives rise to an overshoot in the velocity profile. Received on 1 February 2000     

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.     

Oberbeck convection flow of a couple stress fluid through a vertical porous stratum

The flow and heat transfer characteristics of Oberbeck convection of a couple stress fluid in a vertical porous stratum is investigated. The perturbation method of solution is obtained in terms of buoyancy parameter N valid for small values of N. This limitation is relaxed through numerical solutions using the finite difference technique with an error of 0.1×10^-7. The effect of increase in the values of temperature difference between the plates, permeability parameter and couple stress parameter on velocity, temperature, mass flow rate, skin friction and rate of heat transfer are reported. A new achievement is explored to analyse the flow for strong, weak and comparable porosity with the couple stress parameter. It is noted that both the porous parameter and the couple stress parameter suppress the flow. Higher-temperature difference is required to achieve the mass flow rate equivalent to that of viscous flow.     

Numerical study of nanofluid mixed convection in a horizontal curved tube using two-phase approach

Laminar mixed convection of nanofluid consisting of water/Al₂O₃ in a horizontal curved tube is investigated numerically. Three dimensional elliptical governing equations have been solved to study the simultaneous effect of the buoyancy and centrifugal forces throughout the curved tube. The effects of nanoparticle concentrations on the secondary flow and also on the contours of temperature are presented and discussed. Axial velocity profiles with respect to the horizontal and vertical diameter are shown. In addition, the effects of nanoparticle volume fractions on the axial evolution of the local peripheral average convective heat transfer coefficient and the local peripheral average skin friction coefficient are studied. It is shown that the average convective heat transfer coefficient augments with the nanoparticle concentrations. However, its effect on the average skin friction coefficient is negligible.     

Combined heat and mass transfer along a vertical moving cylinder with a free stream

The mixed convection flow over a continuous moving vertical slender cylinder under the combined buoyancy effect of thermal and mass diffusion has been studied. Both uniform wall temperature (concentration) and uniform heat (mass) flux cases are included in the analysis. The problem is formulated in such a manner that when the ratio λ(= u _w/(u _w + u _∞), where u _w and u _∞ are the wall and free stream velocities, is zero, the problem reduces to the flow over a stationary cylinder, and when λ = 1 it reduces to the flow over a moving cylinder in an ambient fluid. The partial differential equations governing the flow have been solved numerically using an implicit finite-difference scheme. We have also obtained the solution using a perturbation technique with Shanks transformation. This transformation has been used to increase the range of the validity of the solution. For some particular cases closed form solutions are obtained. The surface skin friction, heat transfer and mass transfer increase with the buoyancy forces. The buoyancy forces cause considerable overshoot in the velocity profiles. The Prandtl number and the Schmidt number strongly affect the surface heat transfer and the mass transfer, respectively. The surface skin friction decreases as the relative velocity between the surface and free stream decreases. Received on 17 May 1999     

Development of mixed convection flow over a vertical plate due to an impulsive motion

The development of the mixed convection flow of an incompressible laminar viscous fluid over a semi-infinite vertical plate has been investigated when the fluid in the external stream is set into motion impulsively, and at the same the surface temperature is suddenly raised from its ambient temperature. The problem is formulated in such a way that at time t = 0, it reduces to Rayleigh type of equation and as time t , it tends to Blasius type of equation. The scale of time has been selected such that the traditional infinite region of integration becomes finite which significantly reduces the computational time. The nonlinear coupled singular parabolic partial differential equations governing the unsteady mixed convection flow have been solved numerically by using an implicit finite-difference scheme. The surface shear stress and the heat transfer increase or decrease with time when the buoyancy parameter is greater or less than a certain valve. There is a smooth transition from the initial steady state to the final steady state. The skin friction and heat transfer for the constant heat flux case are more than those of the constant wall temperature case. Also they increase with the buoyancy force.