Momentum and heat transfer of a special case of the unsteady stagnation-point flow

This paper investigates the unsteady stagnation-point flow and heat transfer over a moving plate with mass transfer,which is also an exact solution to the unsteady Navier-Stokes(NS)equations.The boundary layer energy equation is solved with the closed form solutions for prescribed wall temperature and prescribed wall heat flux conditions.The wall temperature and heat flux have power dependence on both time and spatial distance.The solution domain,the velocity distribution,the flow field,and the temperature distribution in the fluids are studied for different controlling parameters.These parameters include the Prandtl number,the mass transfer parameter at the wall,the wall moving parameter,the time power index,and the spatial power index.It is found that two solution branches exist for certain combinations of the controlling parameters for the flow and heat transfer problems.The heat transfer solutions are given by the confluent hypergeometric function of the first kind,which can be simplified into the incomplete gamma functions for special conditions.The wall heat flux and temperature profiles show very complicated variation behaviors.The wall heat flux can have multiple poles under certain given controlling parameters,and the temperature can have significant oscillations with overshoot and negative values in the boundary layers.The relationship between the number of poles in the wall heat flux and the number of zero-crossing points is identified.The difference in the results of the prescribed wall temperature case and the prescribed wall heat flux case is analyzed.Results given in this paper provide a rare closed form analytical solution to the entire unsteady NS equations,which can be used as a benchmark problem for numerical code validation.     

Unsteady magnetohydrodynamic stagnation point flow — closed-form analytical solutions

This paper investigates the unsteady stagnation point flow and heat transfer of magnetohydrodynamic(MHD) fluids over a moving permeable flat surface. The unsteady Navier-Stokes(NS) equations are transformed into a similarity nonlinear ordinary differential equation, and a closed form solution is obtained for the unsteadiness parameter of 2. The boundary layer energy equation is transformed into a similarity equation,and is solved for a constant wall temperature and a time-dependent uniform wall heat flux case. The solution domain, velocity, and temperature profiles are calculated for different combinations of parameters including the Prandtl number, mass transfer parameter, wall moving parameter, and magnetic parameter. Two solution branches are obtained for certain combinations of the controlling parameters, and a stability analysis demonstrates that the lower solution branch is not stable. The present solutions provide an exact solution to the entire unsteady MHD NS equations, which can be used for validating the numerical code of computational fluid dynamics.     

Effect of viscous dissipation and heat source on flow and heat transfer of dusty fluid over unsteady stretching sheet

This paper investigates the problem of hydrodynamic boundary layer flow and heat transfer of a dusty fluid over an unsteady stretching surface.The study considers the effects of frictional heating(viscous dissipation) and internal heat generation or absorption.The basic equations governing the flow and heat transfer are reduced to a set of non-linear ordinary differential equations by applying suitable similarity transformations.The transformed equations are numerically solved by the Runge-Kutta-Fehlberg-45 order method.An analysis is carried out for two different cases of heating processes,namely,variable wall temperature(VWT) and variable heat flux(VHF).The effects of various physical parameters such as the magnetic parameter,the fluid-particle interaction parameter,the unsteady parameter,the Prandtl number,the Eckert number,the number density of dust particles,and the heat source/sink parameter on velocity and temperature profiles are shown in several plots.The effects of the wall temperature gradient function and the wall temperature function are tabulated and discussed.     

Melting heat transfer in steady laminar flow over a moving surface

The steady laminar boundary layer flow and heat transfer from a warm, laminar liquid flow to a melting surface moving parallel to a constant free stream is studied in this paper. The continuity, momentum and energy equations, which are coupled nonlinear partial differential equations are reduced to a set of two nonlinear ordinary differential equations, before being solved numerically using the Runge–Kutta–Fehlberg method. Results for the skin friction coefficient, local Nusselt number, velocity profiles as well as temperature profiles are presented for different values of the governing parameters. Effects of the melting parameter, moving parameter and Prandtl number on the flow and heat transfer characteristics are thoroughly examined. It is found that the problem admits dual solutions.     

Flow and heat transfer over a generalized stretching/shrinking wall problem—Exact solutions of the Navier-Stokes equations

In this paper, we investigate the steady momentum and heat transfer of a viscous fluid flow over a stretching/shrinking sheet. Exact solutions are presented for the Navier-Stokes equations. The new solutions provide a more general formulation including the linearly stretching and shrinking wall problems as well as the asymptotic suction velocity profiles over a moving plate. Interesting non-linear phenomena are observed in the current results including both exponentially decaying solution and algebraically decaying solution, multiple solutions with infinite number of solutions for the flow field, and velocity overshoot. The energy equation ignoring viscous dissipation is solved exactly and the effects of the mass transfer parameter, the Prandtl number, and the wall stretching/shrinking strength on the temperature profiles and wall heat flux are also presented and discussed. The exact solution of this general flow configuration is a rare case for the Navier-Stokes equation.     

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     

Melting heat transfer effects on stagnation point flow of micropolar fluid saturated in porous medium with internal heat generation （absorption）

The effect of melting heat transfer on the two dimensional boundary layer flow of a micropolar fluid near a stagnation point embedded in a porous medium in the presence of internal heat generation/absorption is investigated. The governing non-linear partial differential equations describing the problem are reduced to a system of non-linear ordinary differential equations using similarity transformations solved numerically using the Chebyshev spectral method. Numerical results for velocity, angular velocity and temperature profiles are shown graphically and discussed for different values of the inverse Darcy number, the heat generation/absorption parameter, and the melting parameter. The effects of the pertinent parameters on the local skin-friction coefficient, the wall couple stress, and the local Nusselt number are tabulated and discussed. The results show that the inverse Darcy number has the effect of enhancing both velocity and temperature and suppressing angular velocity. It is also found that the local skin-friction coefficient decreases, while the local Nusselt number increases as the melting parameter increases.     

Numerical solutions of free convection boundary layer flow on a solid sphere with Newtonian heating in a micropolar fluid

In this paper, the problem of free convection boundary layer flow on a solid sphere in a micropolar fluid with Newtonian heating, in which the heat transfer from the surface is proportional to the local surface temperature, is considered. The transformed boundary layer equations in the form of partial differential equations are solved numerically using an implicit finite-difference scheme. Numerical solutions are obtained for the local wall temperature, the local skin friction coefficient, as well as the velocity, angular velocity and temperature profiles. The features of the flow and heat transfer characteristics for different values of the material or micropolar parameter K, the Prandtl number Pr and the conjugate parameter γ are analyzed and discussed.     

Variable porosity and thermal dispersion effects on vortex instability of a horizontal natural convection flow in a saturated porous medium

A numerical analysis is made to analyze the variable porosity and thermal dispersion effects on the vortex mode of instability of a horizontal natural convection boundary layer flow in a saturated porous medium. The porosity of the medium is assumed to vary exponentially with distance from the wall. In the base flow, the governing equations are solved by using a suitable variable transformation and employing an implicit finite difference Keller Box method. The stability analysis is based on the linear stability theory and the resulting eigenvalue problem is solved by the local similarity approximations. The results indicate that both effects increase the heat transfer rate. In addition, the thermal dispersion effect stabilizes the flow to the vortex mode of disturbance, while the variable porosity effect destabilizes it.

     

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     

Lie group analysis for thermophoretic and radiative augmentation of heat and mass transfer in a Brinkman-Darcy flow over a flat surface with heat generation

A boundary layer analysis is presented to investigate numerically the effects of radiation,thermophoresis and the dimensionless heat generation or absorption on hydromagnetic flow with heat and mass transfer over a flat surface in a porous medium.The boundary layer equations are transformed to non-linear ordinary differential equations using scaling group of transformations and they are solved numerically by using the fourth order Runge-Kutta method with shooting technique for some values of physical parameters.Comparisons with previously published work are performed and the results are found to be in very good agreement.Many results are obtained and a representative set is displayed graphically to illustrate the influence of the various parameters on the dimensionless velocity,temperature and concentration profiles as well as the local skin-friction coefficient,wall heat transfer,particle deposition rate and wall thermophoretic deposition velocity.The results show that the magnetic field induces acceleration of the flow,rather than deceleration(as in classical magnetohydrodynamics(MHD) boundary layer flow) but to reduce temperature and increase concentration of particles in boundary layer.Also,there is a strong dependency of the concentration in the boundary layer on both the Schmidt number and mass transfer parameter.     

Effects of radiation and heat source on MHD flow of a viscoelastic liquid and heat transfer over a stretching sheet

We study the MHD flow and also heat transfer in a viscoelastic liquid over a stretching sheet in the presence of radiation. The stretching of the sheet is assumed to be proportional to the distance from the slit. Two different temperature conditions are studied, namely (i) the sheet with prescribed surface temperature (PST) and (ii) the sheet with prescribed wall heat flux (PHF). The basic boundary layer equations for momentum and heat transfer, which are non-linear partial differential equations, are converted into non-linear ordinary differential equations by means of similarity transformation. The resulting non-linear momentum differential equation is solved exactly. The energy equation in the presence of viscous dissipation (or frictional heating), internal heat generation or absorption, and radiation is a differential equation with variable coefficients, which is transformed to a confluent hypergeometric differential equation using a new variable and using the Rosseland approximation for the radiation. The governing differential equations are solved analytically and the effects of various parameters on velocity profiles, skin friction coefficient, temperature profile and wall heat transfer are presented graphically. The results have possible technological applications in liquid-based systems involving stretchable materials.     

Dual solutions of a mixed convection flow near the stagnation point region over an exponentially stretching/shrinking sheet in nanofluids

The aim of this paper is to study the development of mixed convection flow near the stagnation point region over an exponentially stretching/shrinking sheet in nanofluids. The external flow, stretching velocity and wall temperature are assumed to vary as prescribed exponential functions. Using the local similarity method, it has been shown that dual solutions of velocity and temperature exist for certain values of suction/injection, mixed convection, nanoparticle volume fraction and stretching/shrinking parameters. The transformed non-linear ordinary differential equations along with the boundary conditions form a two point boundary value problem and are solved using Shooting method, by converting into an initial value problem. In this method, the system of equations is converted into a set of first order system which is solved by fourth-order Runge–Kutta method. Three different types of nanoparticles, namely copper (Cu), aluminum oxide (Al₂O₃) and titanium oxide (TiO₂) are considered by using water-based fluid with Prandtl number Pr = 6.2. It is also found that the skin friction coefficient and the heat transfer rate at the surface are highest for Copper–water nanofluids as compared to Al₂O₃. The effect of the solid volume fraction parameter φ of the nanofluids on the heat transfer characteristics is also investigated. The results indicate that dual solutions exist only for shrinking sheet. The effects of various parameters on the velocity and temperature profiles are also presented here.     

Variable fluid properties and thermal radiation effects on flow and heat transfer in micropolar fluid film past moving permeable infinite flat plate with slip velocity

This work deals with the influence of thermal radiation on the problem of the mixed convection thin film flow and heat transfer of a micropolar fluid past a moving infinite vertical porous flat plate with a slip velocity.The fluid viscosity and the thermal conductivity are assumed to be the functions of temperature.The equations governing the flow are solved numerically by the Chebyshev spectral method for some representative value of various parameters.In comparison with the previously published work,the excellent agreement is shown.The effects of various parameters on the velocity,the microrotation velocity,and the temperature profiles,as well as the skin-friction coefficient and the Nusselt number,are plotted and discussed.     

Unsteady flow with heat and mass transfer due to impingement of slot jet on an inclined plate

The unsteady laminar incompressible boundary layer flow due to a two-dimensional slot jet on a flat plate at an angle of attack has been studied. The unsteadiness in the flow field is due to the free stream velocity distribution or wall temperature (concentration) which varies with time. The governing partial differential equations in primitive variables have been solved numerically using an implicit finite-difference scheme in combination with the quasilinearization technique. The effect of the variation of the free stream velocity distribution with time is found to be more pronounced on the skin friction than on the heat or mass transfer. The Prandtl number and the variation of the wall temperature with time strongly affect the heat transfer. Similarly, the Schmidt number and the variation of the concentration at the wall with time strongly affect the mass transfer. Beyond a certain critical value of the viscous dissipation parameter, the plate gets heated instead of being cooled.     

Unsteady natural convective flow past a moving vertical cylinder with heat and mass transfer

Transient natural convection boundary layer flow of an incompressible viscous fluid past an impulsively moving semi- infinite vertical cylinder is considered. The temperature and concentration of the cylinder surface are taken to be uniform. The unsteady, nonlinear and coupled governing equations of the flow are solved using an implicit finite difference scheme. The finite difference scheme is unconditionally stable and accurate. Numerical results are presented with various sets of parameters for both air and water. Transient effects of velocity, temperature and concentration profiles are analyzed. Local and average skin friction, rates of heat and mass transfer are shown graphically. Received on 1 November 1999     

Simultaneous heat and mass transfer by natural convection from a plate embedded in a porous medium with thermal dispersion effects

The problem of steady, laminar, simultaneous heat and mass transfer by natural convection flow over a vertical permeable plate embedded in a uniform porous medium in the presence of inertia and thermal dispersion effects is investigated for the case of linear variations of both the wall temperature and concentration with the distance along the plate. Appropriate transformations are employed to transform the governing differential equations to a non-similar form. The transformed equations are solved numerically by an efficient implicit, iterative, finite-difference scheme. The obtained results are checked against previously published work on special cases of the problem and are found to be in good agreement. A parametric study illustrating the influence of the porous medium effects, heat generation or absorption, wall suction or injection, concentration to thermal buoyancy ratio, thermal dispersion parameter, and the Schmidt number on the fluid velocity, temperature and concentration as well as the skin-friction coefficient and the Nusselt and Sherwood numbers is conducted. The results of this parametric study are shown graphically and the physical aspects of the problem are highlighted and discussed.     

Second-order effects in unsteady laminar compressible three-dimensional stagnation-point boundary layers

The flow, heat and mass transfer at the stagnation point of a three-dimensional body in unsteady laminar compressible fluid with variable properties have been studied using a second-order boundary-layer theory when the basic potential flow admits selfsimilarity. Both nodal and saddle point regions have been considered. The equations governing the flow have been solved numerically using an implicit finite-difference scheme. It is observed that the enhancement or reduction in the skin friction and heat transfer due to the second-order boundary layers depends upon the values of the parameter characterizing the unsteadiness in the free-stream velocity, the nature of the stagnation point, the variation of the density-viscosity product across the boundary layer, mass transfer and the wall temperature. The suction increases the skin friction and heat transfer whereas injection does the opposite.     

Mixed convection boundary layer flow along vertical moving thin needles with variable heat flux

The problem of steady laminar mixed convection boundary layer flow of an incompressible viscous fluid along vertical moving thin needles with variable heat flux for both assisting and opposing flow cases is theoretically considered in this paper. The governing boundary layer equations are first transformed into non-dimensional forms. The curvature effects are incorporated into the analysis whereas the pressure variation in the axial direction has been neglected. These equations are then transformed into similarity equations using the similarity variables, which are solved numerically using an implicit finite-difference scheme known as the Keller-box method. The solutions are obtained for a blunt-nosed needle (m = 0). Numerical calculations are carried out for various values of the dimensionless parameters of the problem, which include the mixed convection parameter λ, the Prandtl number Pr and the parameter a representing the needle size. It is shown from the numerical results that the skin friction coefficient, the surface (wall) temperature and the velocity and temperature profiles are significantly influenced by these parameters. The results are presented in graphical form and are discussed in detail.