Heat transfer in a hydromagnetic flow between two porous disks—One rotating and other at rest,under uniform suction

Heat transfer in the flow of a conducting Fluid between two non-conducting porous disks (—one is rotating and other is stationary) in the presence of a transverse uniform magnetic field and under uniform suction, is studied. Asymptotic solutions are obtained for R«M ². The rate of Heat flux from the disks and the temperature distribution are investigated. It is observed that the temperature distribution and heat flux increase with the increase of magnetic field.     

Thermal dispersion effects on non-Darcy natural convection with lateral mass flux

The method of similarity solution is used to study the influence of lateral mass flux and thermal dispersion on non-Darcy natural convection over a vertical flat plate in a fluid saturated porous medium. Forchheimer extension is considered in the flow equations and the coefficient of thermal diffusivity has been assumed to be the sum of molecular diffusivity and the dispersion thermal diffusivity due to mechanical dispersion. The suction/injection velocity distribution has been assumed to have power function form Ax ^l , where x is the distance from the leading edge and the wall temperature distribution is assumed to be uniform. When l=−1/2, similarity solution is possible, and the results indicate that the boundary layer thickness decreases where as the heat transfer rate increases as the mass flux parameter passes from injection domain to the suction domain. The increase in the thermal dispersion parameter is observed to enhance the heat transfer. The combined effect of thermal dispersion and fluid suction/injection on the heat transfer rate is discussed. Received on 9 September 1996     

On the effects of uniform high suction on the steady hydromagnetic flow due to a rotating disc

Summary The effect of uniform high suction on the steady flow of conducting viscous liquid due to a rotating disc, in the presence of a transverse magnetic field is considered. Series solutions for velocity components are obtained in descending powers of the suction parameter (a). The solutions, thus obtained, are valid for small values of m (= B ₀ ² /). Tables for displacement thickness, momentum thickness, and velocity components are given for different values of a and m.     

On thermal boundary layer of a non-Newtonian fluid on a power-law stretched surface of variable temperature with suction or injection

 Heat transfer characteristics of a non-Newtonian fluid on a power-law stretched surface of variable temperature with suction or injection were investigated. Similarity solutions of the laminar boundary layer equations describing heat transfer and fluid flow in a quiescent fluid were obtained and solved numerically. Velocity and temperature profiles as well as the Nusselt number, Nu, were studied for two thermal boundary conditions; uniform surface temperature and variable surface temperature, for different parameters; Prandtl number Pr, temperature exponent b, velocity exponent m, injection parameter d and power-law index n. It was found that decreasing injection parameter d, and power-law index n and increasing Prandtl number Pr and surface temperature exponent b enhance the heat transfer coefficient. Received on 27 April 2000     

Laminar mixed convection from a continuously moving vertical surface with suction or injection

The boundary layer flow over a uniformly moving vertical surface with suction or injection is studied when the buoyancy forces assist or oppose the flow. Similarity solutions are obtained for the boundary layer equations subject to power law temperature and velocity boundary conditions. The effect is of various governing parameters, such as Prandtl number Pr, temperature exponent n, injection parameter d, and the mixed convection parameter λ=Gr/Re², which determine the velocity and temperature distributions and the heat transfer coefficient, are studied. The heat transfer coefficient increases as λ assisting the flow for all d at Pr=0.72 however, for n=−1 it decreases sharply with λ. On the other hand, increasing λ has no effect on heat transfer coefficient for Pr=10 at n=0, and 1 for almost all values of d studied. However, for n=−1 it has similar effect as for Pr=0.72. It is also found that Nusselt number increases as n increases for fixed λ and d. Received on 26 March 1997     

Influence of lateral mass flux on mixed convection heat and mass transfer over inclined surfaces in porous media

 The effect of lateral mass flux on mixed convection heat and mass transfer in a saturated porous medium adjacent to an inclined permeable surface is analyzed. A similarity solution is obtained when surface temperature and concentration, free stream velocity and injection/suction velocity of fluid are prescribed as power functions of distance from the leading edge. The cases when the flow and buoyancy forces are in the same and opposite directions are discussed both for aiding and opposing buoyancy effects. The governing parameters are the mixed convection parameter Gr, the Lewis number Le, the buoyancy ratio N, the lateral mass flux parameter f _w, representing the effects of injection or withdrawal of fluid at the wall, and λ which specifies three cases of the inclined plate. The interactive effect of these parameters on heat and mass transfer rates are presented. It is observed that the diffusion ratio (Le) has a more pronounced effect on concentration field than on flow and temperature fields. It is found that the rates of heat and mass transfer increase with suction and decrease with injection of the fluid. Received on 31 August 2000 / Published online: 29 November 2001     

Thermal dispersion effects on non-Darcy natural convection over horizontal plate with surface mass flux

Summary The effect of surface mass flux on the non-Darcy natural convection over a horizontal flat plate in a saturated porous medium is studied using similarity solution technique. Forchheimer extension is considered in the flow equations. The suction/injection velocity distribution has been assumed to have power function form Bx ^l , similar to that of the wall temperature distribution Ax ⁿ , where x is the distance from the leading edge. The thermal diffusivity coefficient has been assumed to be the sum of the molecular diffusivity and the dynamic diffusivity due to mechanical dispersion. The dynamic diffusivity is assumed to vary linearly with the velocity component in the x direction, i.e. along the hot wall. For the problem of constant heat flux from the surface (n=1/2), similarity solution is possible when the exponent l takes the value −1/2. Results indicate that the boundary layer thickness decreases whereas the heat transfer rate increases as the mass flux parameter passes from the injection domain to the suction domain. The increase in the thermal dispersion parameter is observed to favor the heat transfer by reducing the boundary layer thickness. The combined effect of thermal dispersion and fluid suction/injection on the heat transfer rate is discussed. Received 7 December 1995; accepted for publication 7 January 1997     

Flow of a viscous incompressible fluid of temperature dependent viscosity past a permeable wedge with uniform surface heat flux

 The effect of uniform suction on the steady two-dimensional laminar forced flow of a viscous incompressible fluid of temperature dependent viscosity past a wedge with uniform surface heat flux is considered. The governing equations for the flow are obtained by using suitable transformations and are solved by using an implicit finite difference method. Perturbation solutions are also obtained near the leading edge and in the downstream regime. The results are obtained in terms of the local skin friction coefficient and the rate of heat transfer for various values of the pertinent parameters, such as the Prandtl number, Pr, the velocity gradient parameter, m, the local suction parameter, ξ, and the viscosity variation parameter, ɛ. Perturbation solutions are compared with the finite difference solutions and are found to be in excellent agreement. The effect of ξ, m and ɛ on the dimensionless velocity profiles and viscosity distribution are also presented graphically for Pr = 0.7 and 7.0, which are the appropriate values for gases and water respectively. Received on 22 July 1999     

Friction and Heat Transfer in a Laminar Separated Flow behind a Rectangular Step with Porous Injection or Suction

Results of a numerical study of a laminar separated flow behind a rectangular step on a porous surface with uniform injection or suction are described. Two cases are considered: an unconfined flow past a step and flow evolution in a confined channel (duct). It is shown that mass transfer on the surface causes strong changes in the flow structure and substantially affects the position of the reattachment point, as well as friction and heat transfer. More intense injection leads first to an increase in the separation-zone length and then to its rapid vanishing due to boundary-layer displacement. Vice versa, suction at high Reynolds numbers Re _s > 100 reduces the separation-zone length. The duct flow has a complicated distribution of friction and heat-transfer coefficients along the porous surface owing to the coupled effect of the transverse flow of the substance and changes in the main flow velocity due to mass transfer. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 1, pp. 18–28, January–February, 2006.     

Drag reduction of a nonnewtonian fluid by fluid injection on a moving wall

Summary A boundary layer problem of a nonnewtonian fluid flow with fluid injection on a semi-infinite flat plate whose surface moves with a constant velocity in the opposite direction to that of the uniform mainstream is analyzed. Concluding similarity equations are solved numerically to show the dependence of the problem to the velocity ratio λ of the plate to uniform flow and to the injection velocity parameter C. The critical values of λ and C for each nonnewtonian power-law index n are obtained, and their significance in drag reduction is discussed. Received 26 August 1997; accepted for publication 21 October 1998     

Non-Darcian Forced Convection Flow of Viscous Dissipating Fluid over a Flat Plate Embedded in a Porous Medium

In this study, laminar boundary layer flow over a flat plate embedded in a fluid-saturated porous medium in the presence of viscous dissipation, inertia effect and suction/injection is analyzed using the Keller box finite difference method. The flat plate is assumed to be held at constant temperature. The non-Darcian effects of convection, boundary and inertia are considered. Results for the local heat transfer parameter and the local skin friction parameter as well as the velocity and temperature profiles are presented for various values of the governing parameters. The non-Darcian effects are shown to decrease the velocity and to increase the temperature. It is also shown that the local heat transfer parameter and the local skin friction parameter increase due to suction of fluid while injection reverses this trend. It is disclosed that the effect of the viscous dissipation for negative values of Ec (T _w < T _∞) is to enhance the heat transfer coefficient while the opposite is true for positive values of Ec (T _w > T _∞). The results are compared with those available in the existing literature and an excellent agreement is obtained.     

Axisymmetric stagnation flow obliquely impinging on a moving circular cylinder with uniform transpiration

Laminar stagnation flow, axisymmetrically yet obliquely impinging on a moving circular cylinder, is formulated as an exact solution of the Navier–Stokes equations. Axial velocity is time‐dependent, whereas the surface transpiration is uniform and steady. The impinging free stream is steady with a strain rate k?. The governing parameters are the stagnation‐flow Reynolds number Re=k?a²/2ν, and the dimensionless transpiration S=U₀/k?a. An exact solution is obtained by reducing the Navier–Stokes equations to a system of differential equations governed by Reynolds number and the dimensionless wall transpiration rate, S. The system of Boundary Value Problems is then solved by the shooting method and by deploying a finite difference scheme as a semi‐similar solution. The results are presented for velocity similarity functions, axial shear stress and stream functions for a variety of cases. Shear stresses in all cases increase with the increase in Reynolds number and suction rate. The effect of different parameters on the deflection of viscous stagnation circle is also determined. Copyright © 2010 John Wiley & Sons, Ltd.     

Numerical investigations of asymmetric flow of a micropolar fluid between two porous disks

Numerical solution is presented for the two- dimensional flow of a micropolar fluid between two porous coaxial disks of different permeability for a range of Reynolds number Re （-300≤ Re 〈 0） and permeability parameter A （1.0≤A ≤2.0）. The main flow is superimposed by the injection at the surfaces of the two disks. Von Karman＇s similarity transformations are used to reduce the governing equations of motion to a set of non-linear coupled ordinary differential equations （ODEs） in dimensionless form. An algorithm based on the finite difference method is employed to solve these ODEs and Richardson＇s extrapolation is used to obtain higher order accuracy. The results indicate that the parameters Re and A have a strong influence on the velocity and microrotation profiles, shear stresses at the disks and the position of the viscous/shear layer. The micropolar material constants cl, c2, c3 have profound effect on microrotation as compared to their effect on streamwise and axial velocity profiles. The results of micropolar fluids are compared with the results for Newtonian fluids.     

Hydromagnetic flow between two porous disks rotating about non-coincident axes

Hydromagnetic flow between two porous disks rotating with same angular velocity Ω about two noncoincident axes has been studied in the presence of a uniform transverse magnetic field. An exact solution of the governing equations has been obtained in a closed form. It is found that the primary velocity f/Ωl increases and the secondary velocity g/Ωl decreases with increase in either Reynolds number Re or the Hartman number M. It is also found that the torque at the disk η= 0 increases with increase in either M^2 or K^2. On the other hand there is no torque at the disk η= 1 for large M^2 and K^2. The heat transfer characteristic has also been studied on taking viscous and Joule dissipation into account. It is seen that the temperature increases with increase in either M^2 or K^2. It is found that the rate of heat transfer at the disk η= 0 increases with increase in either M or K. On the other hand the rate of heat transfer at the disk η= 1 increases with increase in K but decreases with increase in M.     

The effect of suction or injection on the boundary layer flows induced by continuous surfaces stretched with prescribed skin friction

The boundary layer flow and heat transfer on a stretched surface moving with prescribed skin friction is studied for permeable surface. Three major cases are studied for isothermal surface (n=0) stretched corresponding to different dimensional skin friction boundary conditions namely; skin friction at the surface scales as (x ^?1/2) at m=0, constant skin friction at m=1/3 and skin friction scales as (x) at m=1. The constants m and n are the indices of the power law velocity and temperature exponent respectively. 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, suction/injection parameter f _w , m and n are studied. The results show that for isothermal surface increasing m enhances the dimensionless heat transfer coefficient for fixed f _w at the suction case and the reverse is true at the injection case. Furthermore, for fixed m, as f _w increases the dimensionless heat transfer coefficient increases. Large enhancements are observed in the heat transfer coefficient as the temperature boundary condition along the surface changes from uniform to linear where the dimensional skin friction is of order (x) at m=1. This enhancement decreases as the suction increases.     

Flow of power-law fluids over a moving wedge surface with wall mass injection

The boundary layer problem of a power-law fluid flow with fluid injection on a wedge whose surface is moving with a constant velocity in the opposite direction to that of the uniform mainstream is analyzed. The free stream velocity, the injection velocity at the surface, moving velocity of the wedge surface, the wedge angle and the power law index of non-Newtonian fluid are assumed variables. The fourth order Runge–Kutta method modified by Gill is used to solve the non-dimensional boundary layer equations for non-Newtonian flow field. Without fluid injection, for every angle of wedge β, a limiting value for velocity ratio λ _cr (velocity of the wedge surface/velocity of the uniform flow) is found for each power-law index n. The value of λ _cr increases with the increasing wedge angle β. The value of wedge angle also restricts the physical characteristics of the fluid to be used. The effects of the different parameters on velocity profile and on skin friction are studied and the drag reduction is discussed. In case of C = 2.5 and velocity ratio λ = 0.2 for wedge angle β = 0.5 with the fluid with power law-index n = 0.5, 48.8% drag reduction is obtained.     

Local nonsimilarity solution for the impact of the buoyancy force on heat and mass transfer in a flow over a porous wedge with a heat source in the presence of suction/injection

Combined heat and mass transfer in free, forced and mixed convection flows along a porous wedge with internal heat generation in the presence of uniform suction or injection is investigated. The boundary-layer analysis is formulated in terms of the combined thermal and solute buoyancy effect. The flow field characteristics are analyzed using the Runge-Kutta-Gill method, the shooting method, and the local nonsimilarity method. Due to the effect of the buoyancy force, power law of temperature and concentration, and suction/injection on the wall of the wedge, the flow field is locally nonsimilar. Numerical calculations up to third-order level of truncation are carried out for different values of dimensionless parameters as a special case. The effects of the buoyancy force, suction, heat generation, and variable wall temperature and concentration on the dimensionless velocity, temperature, and concentration profiles are studied. The results obtained are found to be in good agreement with previously published works.     

Effect of ion slip on the time-varying Hartmann flow of a non-Newtonian viscoelastic fluid with heat transfer

Ion slip in a time-varying Hartmann flow of a conducting incompressible non-Newtonian viscoelastic fluid between two parallel horizontal insulating porous plates is studied with allowance for heat transfer. A uniform and constant pressure gradient is applied in the axial direction. An external uniform magnetic field and uniform suction and injection through the surface of the plates are applied in the normal direction. The two plates are maintained at different but constant temperatures; the Joule and viscous dissipations are taken into consideration. Numerical solutions for the governing momentum and energy equations are obtained with the use of finite differences, and the effect of various physical parameters on both the velocity and temperature fields is discussed.     

Unsteady Flow of an Oscillating Porous Disk and a Fluid at Infinity

An exact analytic solution of the unsteady Navier–Stokes equations is obtained for the flow caused by the non-coaxial rotations of a porous disk and a fluid at infinity. The porous disk is executing oscillations in its own plane with superimposed injection or suction. An increasing or decreasing velocity amplitude of the oscillating porous disk is also discussed. Further, it is shown that a combination of suction/injection and decreasing/increasing velocity amplitude is possible as well. In addition, the flow due to porous oscillating disk and a fluid at infinity rotating about an axis parallel to the z-axis is attempted as a second problem. Sommario. Si studia il flusso non stazionario prodotto dall'oscillazione di un disco poroso in un fluido e si fornisce una soluzione analitica delle equazioni di Navier–Stokes. Si discute l'effetto di una suzione/iniezione e di una variazione sull'ampiezza della velocità' di oscillazione. Infine si studia il flusso dovuto alle oscillazioni non coassiali di un disco poroso e di un fluido all'infinito.     

Effect of suction upon unsteady laminar free convection flow on a vertical infinite flat plate

Summary Unsteady laminar free convection flow past a vertical infinite flat plate subjected to suction is considered. Exact solutions of momentum and energy equations are obtained in two cases: (1) When the plate temperature is proportional to some power of time and (2) when the heat flux at the plate is proportional to some power of time. It is assumed that the suction velocity varies as (time)^–1/2. Expressions for the temperature and velocity profiles are obtained in closed forms in both the cases. Effect of suction on velocity, temperature, skin friction and rate of head transfer is studied for Prandtl numbers 0.02, 0.1, 0.72, 1 and 10.