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.     

Stagnation point flow and heat transfer over a non-linearly moving flat plate in a parallel free stream with slip

An analysis is presented for the steady boundary layer flow and heat transfer of a viscous and incompressible fluid in the stagnation point towards a non-linearly moving flat plate in a parallel free stream with a partial slip velocity. The governing partial differential equations are converted into nonlinear ordinary differential equations by a similarity transformation, which are then solved numerically using the function bvp4c from Matlab for different values of the governing parameters. Dual (upper and lower branch) solutions are found to exist for certain parameters. Particular attention is given to deriving numerical results for the critical/turning points which determine the range of existence of the dual solutions. A stability analysis has been also performed to show that the upper branch solutions are stable and physically realizable, while the lower branch solutions are not stable and, therefore, not physically possible.     

Laminar flow heat transfer from a flat plate with variable thermal conductivity

Zusammenfassung Es wird der Einfluss einer mit der Temperatur linear veränderlichen Leitfähigkeit auf die Wärmeübertragung an einer ebenen Platte untersucht.     

Friction law for turbulent boundary layers on a flat plate with suction

A universal friction law for turbulent boundary layers on a flat plate with suction is established. Experimental skin-friction distributions obtained for various suction factors and Reynolds numbers are described in similarity variables by a single universal curve. The law is valid for the entire range of suction velocities from zero one till the limiting values corresponding to asymptotic-suction boundary layers. The analysis is not based on any particular turbulence model. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

Similarity solutions for flow and heat transfer over a permeable surface with convective boundary condition

The steady laminar boundary layer flow over a permeable flat plate in a uniform free stream, with the bottom surface of the plate is heated by convection from a hot fluid is considered. Similarity solutions for the flow and thermal fields are possible if the mass transpiration rate at the surface and the convective heat transfer from the hot fluid on the lower surface of the plate vary like x^−1/2, where x is the distance from the leading edge of the solid surface. The governing partial differential equations are first transformed into ordinary differential equations, before being solved numerically. The effects of the governing parameters on the flow and thermal fields are thoroughly examined and discussed.     

Mixed convection boundary layer flow over a permeable vertical surface with prescribed wall heat flux

The effects of suction/injection on the laminar mixed convection boundary-layer flow on a vertical wall with a prescribed heat flux are considered. The conditions which allow the equations to be reduced to similarity form are derived and numerical solutions of the resulting equations are obtained for a range of values of the suction/injection and buoyancy parameters. Two specific cases, corresponding to a stagnation point flow and uniform wall heat flux, are treated in detail. Results are presented in terms of the skin friction and wall temperature with a selection of velocity and temperature profiles also being given. Dual solutions are found to exist for assisting flow, these are an addition to what has been reported previously for opposing flows. Solutions for some limiting values of the parameters are also derived.      

Mixed convection boundary layer flow over a permeable vertical surface with prescribed wall heat flux

The effects of suction/injection on the laminar mixed convection boundary-layer flow on a vertical wall with a prescribed heat flux are considered. The conditions which allow the equations to be reduced to similarity form are derived and numerical solutions of the resulting equations are obtained for a range of values of the suction/injection and buoyancy parameters. Two specific cases, corresponding to a stagnation point flow and uniform wall heat flux, are treated in detail. Results are presented in terms of the skin friction and wall temperature with a selection of velocity and temperature profiles also being given. Dual solutions are found to exist for assisting flow, these are an addition to what has been reported previously for opposing flows. Solutions for some limiting values of the parameters are also derived.     

The forced convection flow of a uniform stream over a flat surface with a convective surface boundary condition

The forced convection heat transfer resulting from the flow of a uniform stream over a flat surface on which there is a convective boundary condition is considered. In previous papers [5], [6], [7], [8] it was assumed that the convective heat transfer parameter h_f associated with the hot surface depended on x, where x measures distance along the surface, so that problem could be reduced to similarity form. Here it is assumed that this heat transfer parameter h_f is a constant, with the result that the temperature profiles and overall heat transfer characteristics evolve as the solution develops from the leading edge. The heat transfer near the leading edge (small x), which we find to be dominated by the surface heat flux, the solution at large distances along the surface (large x), which dominated by the surface temperature, are discussed. A numerical solution to the full problem is then obtained for a range of values of the Prandtl number to join these two solution regimes.     

Explicit solution for a Stefan problem with variable latent heat and constant heat flux boundary conditions

In Voller, Swenson and Paola [V.R. Voller, J.B. Swenson, C. Paola, An analytical solution for a Stefan problem with variable latent heat, Int. J. Heat Mass Transfer 47 (2004) 5387-5390], and Lorenzo-Trueba and Voller [J. Lorenzo-Trueba, V.R. Voller, Analytical and numerical solution of a generalized Stefan problem exhibiting two moving boundaries with application to ocean delta formation, J. Math. Anal. Appl. 366 (2010) 538-549], a model associated with the formation of sedimentary ocean deltas is studied through a one-phase Stefan-like problem with variable latent heat. Motivated by these works, we consider a two-phase Stefan problem with variable latent of fusion and initial temperature, and constant heat flux boundary conditions. We obtain the sufficient condition on the data in order to have an explicit solution of a similarity type of the corresponding free boundary problem for a semi-infinite material. Moreover, the explicit solution given in the first quoted paper can be recovered for a particular case by taking a null heat flux condition at the infinity.     

Group solution for unsteady free-convection flow from a vertical moving plate subjected to constant heat flux

The problem of heat and mass transfer in an unsteady free-convection flow over a continuous moving vertical sheet in an ambient fluid is investigated for constant heat flux using the group theoretical method. The nonlinear coupled partial differential equation governing the flow and the boundary conditions are transformed to a system of ordinary differential equations with appropriate boundary conditions. The obtained ordinary differential equations are solved numerically using the shooting method. The effect of Prandlt number on the velocity and temperature of the boundary-layer is plotted in curves. A comparison with previous work is presented.     

Perturbation analysis of unsteady magnetohydrodynamic convective heat and mass transfer in a boundary layer slip flow past a vertical permeable plate with thermal radiation and chemical reaction

An analytical study for the problem of unsteady mixed convection with thermal radiation and first-order chemical reaction on magnetohydrodynamics boundary layer flow of viscous, electrically conducting fluid past a vertical permeable plate has been presented. Slip boundary condition is applied at the porous interface. The classical model is used for studying the effect of radiation for optically thin media. The non-linear coupled partial differential equations are solved by perturbation technique. The results obtained show that the velocity, temperature and concentration fields are appreciably influenced by the presence of chemical reaction, thermal stratification and magnetic field. It is observed that the effect of thermal radiation and magnetic field decreases the velocity, temperature and concentration profiles in the boundary layer. Also, the effects of the various parameters on the skin-friction coefficient and the rate of heat transfer at the surface are discussed.     

A moving-wall boundary layer flow of a slightly rarefied gas free stream over a moving flat plate

In the current work, the boundary layer flow of a slightly rarefied gas free stream over a moving flat plate is presented and solved numerically. The first-order slip boundary condition is adopted in the derivation. The dimensionless velocity and shear stress profiles are plotted and discussed. A theoretical derivation of the estimated solution domain is developed, which will give a very close estimation to the exact solution domain obtained numerically. The influences of velocity slip at the wall on the velocity and shear stress are also addressed.     

Natural convection boundary layer flow of fluid with temperature-dependent viscosity from a horizontal elliptical cylinder with constant surface heat flux

This study deals with the temperature-dependent viscosity effects on the natural convection boundary layer on a horizontal elliptical cylinder with constant surface heat flux. The mathematical problem is reduced to a pair of coupled partial differential equations for the temperature and the stream function, and the resulting nonlinear equations are solved numerically by cubic spline collocation method. Results for the heat transfer characteristics are presented as functions of eccentric angle for various values of viscosity variation parameters, Prandtl numbers and aspect ratios. Results show that an increase in the viscosity variation parameter tends to accelerate the fluid flow near the surface and increase the maximum velocity, thus decreasing the velocity boundary layer thickness. As the viscosity variation parameter is increased, the surface temperature tends to decrease, thus increasing the local Nusselt number. Moreover, the local Nusselt number of the elliptical cylinder increases as the Prandtl number of the fluid is increased.     

The compressible gas-solid particle flow over a semi-infinite flat plate

Zusammenfassung Es wird gezeigt, dass die Grenzschicht-Approximation auf die Zwei-Phasen-Strömung über eine Halbebene anwendbar ist, wenn die Teilchendichte in der Grenzschicht von der Ordnung ist. Die Bedeutung eines neuen Parameters _m /x wird herausgestellt, wo _m die Länge ist, auf der die Teilchen ihre Geschwindigkeit ume ^–1 verringern, und der Abstand von der Vorderkante. Reihenentwicklungen in diesem Parameter werden benutzt, derart, dass eine Ähnlichkeitsgrösse für die Halbebene gefunden werden kann. Dies reduziert ein System partieller Differentialgleichungen auf ein System gewöhnlicher Differentialgleichungen. Letztere werden numerisch und wenn möglich analytisch gelöst, so dass Schubspannungskoeffizienten und Wärmeübergangskoeffizienten für begrenzte Teile der Wand berechnet werden können. Hierbei wird die Gasdichte nicht als konstant angenommen.