A similarity solution for laminar thermal boundary layer over a flat plate with a convective surface boundary condition

This paper considers the classical problem of hydrodynamic and thermal boundary layers over a flat plate in a uniform stream of fluid. It is well known that similarity solutions of the energy equation are possible for the boundary conditions of constant surface temperature and constant heat flux. However, no such solution has been attempted for the convective surface boundary condition. The paper demonstrates that a similarity solution is possible if the convective heat transfer associated with the hot fluid on the lower surface of the plate is proportional to x^?1/2. Numerical solutions of the resulting similarity energy equation are provided for representative Prandtl numbers of 0.1, 0.72, and 10 and a range of values of the parameter characterizing the hot fluid convection process. For the case of constant heat transfer coefficient, the same data provide local similarity solutions.     

Hydrodynamic and thermal slip flow boundary layers over a flat plate with constant heat flux boundary condition

In this paper the boundary layer flow over a flat plat with slip flow and constant heat flux surface condition is studied. Because the plate surface temperature varies along the x direction, the momentum and energy equations are coupled due to the presence of the temperature gradient along the plate surface. This coupling, which is due to the presence of the thermal jump term in Maxwell slip condition, renders the momentum and energy equations non-similar. As a preliminary study, this paper ignores this coupling due to thermal jump condition so that the self-similar nature of the equations is preserved. Even this fundamental problem for the case of a constant heat flux boundary condition has remained unexplored in the literature. It was therefore chosen for study in this paper. For the hydrodynamic boundary layer, velocity and shear stress distributions are presented for a range of values of the parameter characterizing the slip flow. This slip parameter is a function of the local Reynolds number, the local Knudsen number, and the tangential momentum accommodation coefficient representing the fraction of the molecules reflected diffusively at the surface. As the slip parameter increases, the slip velocity increases and the wall shear stress decreases. These results confirm the conclusions reached in other recent studies. The energy equation is solved to determine the temperature distribution in the thermal boundary layer for a range of values for both the slip parameter as well as the fluid Prandtl number. The increase in Prandtl number and/or the slip parameter reduces the dimensionless surface temperature. The actual surface temperature at any location of x is a function of the local Knudsen number, the local Reynolds number, the momentum accommodation coefficient, Prandtl number, other flow properties, and the applied heat flux.     

A Spectral method determination of the first critical Rayleigh number in a cylindrical container

The onset of laminar axisymmetric Rayleigh–Bénard convection is investigated analytically for fluid in a cylindrical container. All surfaces are considered to be solid and no-slip for the flow, whereas for the thermal boundary conditions both a perfectly conducting and an insulated side wall are considered. The governing Boussinesq equations are perturbed and an approximate solenoidal flow field and a temperature field are determined, using the assumption of separation of variables. Subsequently, a Chebysev–Galerkin spectral method is employed to reduce the equations to a system of first-order nonlinear ordinary differential equations. The approximate representation of the flow and temperature fields make it possible to perform the calculations analytically. The first critical Rayleigh number (Ra_cr) is then calculated using local stability analysis. The resulting value of Ra_cr compares favorably with previous numerical and experimental studies. The analytical solution presented here allows for deeper insights into the physics of this extensively studied problem to be identified.     

On multiscale homogenization problems in boundary layer theory

This paper is concerned with the homogenization of the equations describing a magnetohydrodynamic boundary layer flow past a flat plate, the flow being subjected to velocities caused by injection and suction. The fluid is assumed incompressible, viscous and electrically conducting with a magnetic field applied transversally to the direction of the flow. The velocities of injection and suction and the applied magnetic field are represented by rapidly oscillating functions according to several scales. We derive the homogenized equations, prove convergence results and establish error estimates in a weighted Sobolev norm and in C ⁰-norm. We also examine the asymptotic behavior of the solutions of the equations governing a boundary layer flow past a rough plate with a locally periodic oscillating structure.     

Group method analysis of magneto-elastico-viscous flow along a semi-infinite flat plate with heat transfer

The group theoretic method is applied for solving problem of the flow of an elastico-viscous liquid past an infinite flat plate in the presence of a magnetic field normal to the plate. The application of one-parameter transformation group reduces the number of independent variables, by one, and consequently the system of governing partial differential equations with boundary conditions reduces to a system of ordinary differential equations with appropriate corresponding conditions. Numerical solution of the velocity field and heat transfer have been obtained. The effect of the magnetic parameter M on velocity field, shear stress, temperature fields and heat transfer has been discussed.     

Group method analysis of combined heat and mass transfer by MHD non-Darcy non-Newtonian natural convection adjacent to horizontal cylinder in a saturated porous medium

The group theoretic method is applied for solving problem of combined magneto-hydrodynamic heat and mass transfer of non-Darcy natural convection about an impermeable horizontal cylinder in a non-Newtonian power law fluid embedded in porous medium under coupled thermal and mass diffusion, inertia resistance, magnetic field, thermal radiation effects. The application of one-parameter groups reduces the number of independent variables by one and consequently, the system of governing partial differential equations with the boundary conditions reduces to a system of ordinary differential equations with appropriate boundary conditions. The ordinary differential equations are solved numerically for the velocity using shooting method. The effects of magnetic parameter M, Ergun number Er, power law (viscosity) index n, buoyancy ratio N, radiation parameter R_d, Prandtl number Pr and Lewis number Le on the velocity, temperature fields within the boundary layer, heat and mass transfer are presented graphically and discussed.     

Effects of thermal radiation and magnetic field on unsteady mixed convection flow and heat transfer over an exponentially stretching surface with suction in the presence of internal heat generation/absorption

In this paper, the problem of unsteady laminar two-dimensional boundary layer flow and heat transfer of an incompressible viscous fluid in the presence of thermal radiation, internal heat generation or absorption, and magnetic field over an exponentially stretching surface subjected to suction with an exponential temperature distribution is discussed numerically. The governing boundary layer equations are reduced to a system of ordinary differential equations. New numerical method using Mathematica has been used to solve such system after obtaining the missed initial conditions. Comparison of obtained numerical results is made with previously published results in some special cases, and found to be in a good agreement.     

Similarity solutions for mixed convection boundary layer flow over a permeable horizontal flat plate

The effects of suction and injection on steady laminar mixed convection boundary layer flow over a permeable horizontal flat plate in a viscous and incompressible fluid is investigated in this paper. The similarity solutions of the governing boundary layer equations are obtained for some values of the suction and injection parameter f₀, the constant exponent n of the wall temperature as well as the mixed convection parameter λ. The resulting system of nonlinear ordinary differential equations is solved numerically for both assisting and opposing flow regimes using a finite-difference scheme known as the Keller-box method. Numerical results for the reduced skin friction coefficient, the reduced local Nusselt number, and the velocity and temperature profiles are obtained for various values of the parameters considered. Dual solutions are found to exist for the opposing flow.     

Heat transfer in a liquid film over an unsteady stretching surface with viscous dissipation in presence of external magnetic field

This paper presents a mathematical analysis of MHD flow and heat transfer to a laminar liquid film from a horizontal stretching surface. The flow of a thin fluid film and subsequent heat transfer from the stretching surface is investigated with the aid of similarity transformation. The transformation enables to reduce the unsteady boundary layer equations to a system of non-linear ordinary differential equations. Numerical solution of resulting non-linear differential equations is found by using efficient shooting technique. Boundary layer thickness is explored numerically for some typical values of the unsteadiness parameter S and Prandtl number Pr, Eckert number Ec and Magnetic parameter Mn. Present analysis shows that the combined effect of magnetic field and viscous dissipation is to enhance the thermal boundary layer thickness.     

Creeping flow of a second grade fluid in a corner

A variant of Taylor’s (1962) [23] scraper problem, in which, the lower plate rotates is considered. The non-linear partial differential equations governing the flow of a second grade fluid are modeled and solved by using the domain perturbation technique considering the angular velocity of the rotating plate as a small parameter. Also the rheology of the second grade fluid is examined by depicting the profiles of the velocity, stream function, pressure and stress fields.