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.     

Dual solutions of boundary layer flow over an upstream moving plate

The homotopy analysis method is applied to study the boundary layer flow over a flat plate which has a constant velocity opposite in direction to that of the uniform mainstream. The dual solutions in series expressions are obtained with the proposed technique, which agree well with numerical results. Note that, by introducing a new auxiliary function β(z), the bifurcation of the solutions is obtained. This indicates that the homtopy analysis method is a open system, in the framework of this technique, we have great freedom to choose the auxiliary parameters or functions. As a result, complicated nonlinear problems may be resolved in a simple way. The present work shows that the homotopy analysis method is an effective tool for solving nonlinear problems with multiple solutions.     

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.     

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.     

Parabolic flow over a flat plate with wave disturbance in the main stream

In this paper the effects of unsteady flow past a flat plate, which is at rest, are studied when the steady parabolic flow outside the boundary layer is in a wave disturbance. Solutions are developed for large and small times. The behaviour of the amplitudes and phase leads of skin friction components along chordwise and spanwise directions are studied in detail. Two cases, one when the main stream lines are straight and yawed, and the other when the main stream lines are parabolic and concave with respect to the points on the chordwise direction are studied separately and the results are interpreted graphically.     

On the behavior of a slightly rarefied gas mixture over plane boundaries

Summary The behavior of a slightly rarefied gas mixture bounded by plane boundaries is investigated on the basis of the linearized Boltzmann equation ofB-G-K type for gas mixtures under the diffusive boundary condition. A useful result of the present analysis is that the macroscopic equations and the appropriate boundary conditions in terms of slip and jump are obtained together with the Knudsen-layer corrections near the boundaries. This system of equations makes possible the treatment at fluid dynamic level for various problems of gas mixtures with plane geometry which require kinetic theory consideration. As an application of this system, some basic flow problems of a slightly rarefied gas mixture, namely, Couette flow, thermal slip flow and diffusion slip flow between two plates are taken up. The total velocity distributions of these concrete problems are explicitly obtained for the first time, and their dependence on the properties and concentration of the component gases in the mixture are clarified in some detail.

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Zusammenfassung Das Verhalten einer verdünnten Gasmischung, bei kleiner aber nicht vernachlässigbarer Knudsen Zahl, zwischen zwei parallelen Platten wird analytisch untersucht. Die linearisierten Boltzmann Gleichungen desB-G-K Typs für Gasmischungen mit diffusiven Randbedingungen werden angewendet. Aus der vorliegenden Untersuchung resultieren brauchbare makroskopische Gleichungen mit den zugehörigen Randbedingungen, — mit Gleitgeschwindigkeit und Temperatursprung formuliert, — sowie die Korrekturen für die wandnahe Knudsen-Schicht. Verschiedene Strömungen von Gasmischungen, bei denen gas-kinetische Effekte eine Rolle spielen, entlang ebener Begrenzungen können mit den Gleichungssystemen auf strömungsmechanischem Niveau behandelt werden. Das System wird auf die Couette Strömung und die thermal-slip und diffusion-slip Strömungen angewendet. Zum ersten Mal werden die Geschwindigkeitsprofile dieser elementaren Strömungen explizit berechnet. Der Einfluß der Gaseigenschaften und Konzentrationen auf diese Profile werden weitgehend erklärt.

     

Numerical study of forced and free convective boundary layer flow of a magnetic fluid over a flat plate under the action of a localized magnetic field

The two-dimensional, steady, laminar, forced and free convective boundary layer flow of a magnetic fluid over a semi-infinite vertical plate, under the action of a localized magnetic field, is numerically studied. The magnetic fluid is considered to be water-based with temperature dependent viscosity and thermal conductivity. The study of the boundary layer is separated into two cases. In case I the boundary layer is studied near the leading edge, where it is dominated by the large viscous forces, whereas in case II the boundary layer is studied far from the leading edge of the plate where the effects of buoyancy forces increase. The numerical solution, for these two different cases, is obtained by an efficient numerical technique based on the common finite difference method. Numerical calculations are carried out for the value of Prandl number Pr =  49.832 (water-based magnetic fluid) and for different values of the dimensionless parameters entering into the problem and especially for the magnetic parameter Mn, the viscosity/temperature parameter Θ _r and the thermal/conductivity parameter S*. The analysis of the obtained results show that the flow field is influenced by the application of the magnetic field as well as by the variation of the viscosity and the thermal conductivity of the fluid with temperature. It is hoped that they could be interesting for engineering applications.     

Numerical study of forced and free convective boundary layer flow of a magnetic fluid over a flat plate under the action of a localized magnetic field

The two-dimensional, steady, laminar, forced and free convective boundary layer flow of a magnetic fluid over a semi-infinite vertical plate, under the action of a localized magnetic field, is numerically studied. The magnetic fluid is considered to be water-based with temperature dependent viscosity and thermal conductivity. The study of the boundary layer is separated into two cases. In case I the boundary layer is studied near the leading edge, where it is dominated by the large viscous forces, whereas in case II the boundary layer is studied far from the leading edge of the plate where the effects of buoyancy forces increase. The numerical solution, for these two different cases, is obtained by an efficient numerical technique based on the common finite difference method. Numerical calculations are carried out for the value of Prandl number Pr =  49.832 (water-based magnetic fluid) and for different values of the dimensionless parameters entering into the problem and especially for the magnetic parameter Mn, the viscosity/temperature parameter Θ _r and the thermal/conductivity parameter S*. The analysis of the obtained results show that the flow field is influenced by the application of the magnetic field as well as by the variation of the viscosity and the thermal conductivity of the fluid with temperature. It is hoped that they could be interesting for engineering applications.     

Solutions of thermal boundary layer equations when temperature gradient at the moving flat plate in parabolic flow is prescribed

Solutions of steady as well as unsteady three-dimensional incompressible thermal boundary layer equations are studied when the temperature gradient at the moving flat plate in parabolic flow is prescribed. A general analysis is made and different cases are studied by giving values to β and Cx which determine the gradient and curvature of the outer flow steam lines. The components of velocity in boundary layer are discussed by Sarma and Gupta and those results are used to analyse the thermal boundary layer equations. The response of temperature of the plate are studied for large and small times and curves are drawn representing the variation of temperature with time for various cases. The limiting time is also calculated.     

On the viscous dissipation in the boundary layer of a high Prandtl number fluid in laminar flow over a flat plate

Summary An asymptotic expansion for large Prandtl number fluids has been obtained for the recovery factor in a laminar flow over flat plate. The expression underestimates the recovery factor by no more than 1% as long as the Prandtl number is greater than seven.

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Zusammenfassung Die adiabatische Oberflächentemperatur (recovery factor) für ein Fluid mit hoher Prandtl-Zahl wurde für laminare Strömung über eine ebene Platte mit Hilfe einer asymptotischen Entwicklung bestimmt. Der Ausdruck unterschätzt das genaue Resultat um weniger als 1 % für Prandtl-Zahlen höher als 7.

     

Steady boundary layer flow and reactive mass transfer past an exponentially stretching surface in an exponentially moving free stream

An analysis is made to study the steady two-dimensional boundary layer flow and reactive mass transfer past an exponentially stretching sheet in an exponentially moving free stream. The reaction rate of solute and the wall concentration distribution are taken variable. The governing equations are transformed and then solved numerically. The study reveals that the momentum boundary layer thickness is considerably smaller than that of stagnation point flow over stretching sheet. Due to increase of Schmidt number and reaction rate parameter the mass transfer considerably enhances. Importantly, for solute distribution, in addition to mass transfer, mass absorption occurs in certain situations.     

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.     

Magnetohydrodynamic flow of a rarefied gas near an accelerated porous plate

An investigation is made of the flow of an electrically conducting rarefied gas due to the time-varying motion of an infinite porous plate, the gas being permeated by a transverse magnetic field. The suction is taken to be a constant and the magnetic lines of force are taken to be fixed relative to the fluid. The effects of magnetic field, rarefaction parameter, suction parameter are shown by means of some tables. The expressions of the skin friction for the two particular cases have also been obtained.     

On the flow of an unsteady,compressible boundary layer over a semi-infinite flat plate whose temperature is varied with time

Zusammenfassung Es wird die Grenzschicht untersucht, die sich in laminarer, kompressibler Strömung an einer ebenen Platte unter der Annahme einstellt, dass sich diese nach einer Seite ins Unendliche ausdehnt und von der Ruhe aus gleichförmig beschleunigt wird. Die Temperatur der Platte wird in vorgeschriebener Weise mit der Zeit geändert, und der Effekt dieser Änderung kurz nach dem Beginn der Bewegung wird für Punkte berechnet, die weit vom Plattenrand entfernt sind.

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The computer time for this project was supported byNational Aeronautics and Space Administration Grant NsG-398.