Heat and Mass Transfer of Hydrodynamic Boundary Layer Flow along a Flat Plate with the Influence of Variable Temperature and Viscous Dissipation (original) (raw)
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PLoS ONE, 2013
The steady boundary layer flow of a viscous and incompressible fluid over a moving vertical flat plate in an external moving fluid with viscous dissipation is theoretically investigated. Using appropriate similarity variables, the governing system of partial differential equations is transformed into a system of ordinary (similarity) differential equations, which is then solved numerically using a Maple software. Results for the skin friction or shear stress coefficient, local Nusselt number, velocity and temperature profiles are presented for different values of the governing parameters. It is found that the set of the similarity equations has unique solutions, dual solutions or no solutions, depending on the values of the mixed convection parameter, the velocity ratio parameter and the Eckert number. The Eckert number significantly affects the surface shear stress as well as the heat transfer rate at the surface.
Laminar boundary layer in low Prandtl number flows with variable thermal properties on a flat plate
Journal of Engineering Mathematics, 1974
The laminar boundary layer flow of a low Prandtl number fluid with arbitrary thermal properties past a flat plate is studied by the method of matched asymptotic expansions. By assuming power law relations for the viscosity, density and Prandtl number, the first order results for the skin friction, the recovery factor and the heat transfer rate at the wall are obtained. It turns out that the outer flow in the thermal boundary layer is governed by a simple nonlinear differential equation of second order, which is correct to all orders in Prandtl number. Exact and approximate solutions to this outer equation are obtained. Further, it is shown that the first order terms for the recovery factor are independent of the thermal properties, while the heat transfer terms have a complicated dependence. The skin friction result shows the dependence on thermal properties, Mach number and heat transfer rate.
Communications in Nonlinear Science and Numerical Simulation, 2010
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.
The steady, laminar boundary layer flow with a convective boundary condition, to a continuously moving flat plate is studied taking into account the variation of viscosity with temperature in the presence of a magnetic field, heat generation and thermal radiation. The fluid viscosity is assumed to vary as a linear function of temperature. The resulting, governing equations are non-dimensionalized and transformed using a similarity transformation and then solved numerically by sixth order Runge-Kutta method alongside with shooting method. Comparison with previously published work is performed and there was a perfect agreement at large value of the Biot number. A parametric study of all the embedded flow parameters involved is conducted, and a representative set of numerical results for the velocity and temperature profiles as well as the skin-friction parameter and the Nusselt number is illustrated graphically to show typical trend of the solutions. It is worth pointing out that, when the variation of viscosity with temperature is strong in the presence of the effect of a magnetic field, radiation, heat generation, the results of the present work are completely different from those that studied the same problem in the absence of magnetic field, thermal radiation and the heat generation. It is interesting to note that higher the values of Prandtl number lesser the effects of Biot number and the magnetic field intensity.
Fluid Dynamics, 2017
A rational asymptotic theory is proposed, which describes the turbulent dynamic and thermal boundary layer on a flat plate under zero pressure gradient. The fact that the flow depends on a finite number of governing parameters makes it possible to formulate algebraic closure conditions relating the turbulent shear stress and heat flux with the gradients of the averaged velocity and temperature. As a result of constructing an exact asymptotic solution of the boundary layer equations, the known laws of the wall for velocity and temperature, the velocity and temperature defect laws, and the expressions for the skin friction coefficient, Stanton number, and Reynolds analogy factor are obtained. The latter makes it possible to give two new formulations of the temperature defect law, one of which is identical to the velocity defect law and contains neither the Stanton number nor the turbulent Prandtl number, and the second formulation does not contain the skin friction coefficient. The heat transfer law is first obtained in the form of a universal functional relationship between three parameters: the Stanton number, the Reynolds number, and the molecular Prandtl number. The conclusions of the theory agree well with the known experimental data.
The steady magneto hydrodynamic (MHD) boundary layer flow and combined heat and mass transfer of a non-Newtonian fluid over an inclined stretching sheet have been investigated in the present analysis. The effects of the flow parameters on the velocity, temperature, species concentration, local skin friction, local Nusselt number, and Sherwood number are computed, discussed and have been graphically represented in figures and tables for various values of different parameters. The numerical results are carried out for several values of the combined effects of magnetic parameter M, stretching parameter λ, Prandtl number Pr, Eckert number Ec, Schmidt number Sc, Soret number S0, slip parameter A and Casson parameter n on velocity, temperature and concentration profiles and also the skin-friction coefficient f "(0) local Nusselt number -θ'(0) and local Sherwood number -ψ'(0) are discussed and presented in tabular form. The results pertaining to the present study indicate that the velocity profiles decrease as the increase of magnetic field parameter, but reverse trend arises for the effect of Casson parameter and stretching ratio parameter for both Newtonian and non-Newtonian fluids. The temperature profiles increase forthe effect of magnetic parameter, Prandtl number and Eckert number in case of Newtonian and non-Newtonian fluids. The concentration profile increases for the effect of Soret number while concentration profile decreases for the increasing values of Schmidt number, magnetic parameter, Prandtl number and Eckert number for both Newtonian and non-Newtonian fluids. By considering the cooling plate the numerical results for the skin-friction coefficient f "(0) , local Nusselt number -θ'(0) and local Sherwood number -ψ'(0) are presented in Tables 1-3.
Unsteady thermal boundary layer flow of a non-newtonian fluid over a flat plate
International Journal of Engineering Science, 1981
The transient thermal response of a power law type non-Newtonian, laminar boundary layer flow past a flat plate is investigated. Consideration is given to the case of a step change in surface temperature. The transient heat flux and details of the temperature field are obtained and have been illustrated graphically. The range of Prandtl numbers investigated was 5-1000 while the viscosity index was allowed to vary 0.1-5.0. Unsteady thermal boundary layer Row of a non-Newtonian fluid over a flat plate n= 1.0 1397 10 -
MHD Boundary Layer Flow and Heat Transfer over a Moving Non-Isothermal Flat Plate
Computational Mathematics and Modeling, 2014
The steady two-dimensional laminar incompressible electrically conducting viscous fluid on a moving flat plate in the presence of a transverse magnetic field is studied. Taking suitable similarity variables, we transform the governing boundary layer equations into ordinary differential equations and solve them numerically by standard techniques. The effects of moving and magnetic parameters, Prandtl number, and Eckert number on the velocity and temperature as well as on the skin-friction coefficient and Nusselt number are studied.
World Journal of Mechanics, 2012
An unsteady boundary layer flow of viscous incompressible fluid over a stretching plate has been considered to solve heat flow problem with variable thermal conductivity. First, using similarity transformation, the velocity components have been obtained, and then the heat flow problem has been attempted in the following two ways: 1) prescribed stretching surface temperature (PST), and 2) prescribed stretching surface heat flux (PHF) Flow and temperature fields have been analyzed through graphs. The expressions for skin friction and coefficient of convective heat transfer Nusselt number in PST and PHF cases have been derived.