Generalized integral transform solution for boundary layer flow over a sphere

The generalized integral transform technique is applied to the boundary layer equations for flow over a sphere in their primitive variables. Even though a diffusion-based eigenvalue problem is used, the velocity profile, shear stress and separation point have been calculated with high accuracy. Low-order approximations are shown to be accurate near the surface and the predictions of the separation point is very good. Comparison with finite difference results shows the better convergence behaviour of the integral transform method.     

Visco-elastic boundary layer flow past a stretching plate with suction and heat transfer

In this paper the study of visco-elastic (Walters' liquid B model) flow past a stretching plate with suction is considered. Exact solutions of the boundary layer equations of motion and energy are obtained. The expressions for the coefficient of skin friction and of boundary layer thickness are obtained.     

A new exact solution for the flow of a Maxwell fluid past an infinite plate

A new exact solution corresponding to the flow of a Maxwell fluid over a suddenly moved flat plate is determined. This solution is in all accordance with a previous one and for λ→0 it goes to the well-known solution for Navier-Stokes fluids.     

Vortex stretching in a laminar boundary layer flow

 A new technique to produce controlled stretched vortices is presented. The initial vorticity comes from a laminar boundary layer flow and the stretching is parallel to the initial vorticity. This low velocity flow enables direct observations of the formation and destabilization of vortices. Visualizations are combined with quasi-instantaneous measurements of a full velocity profile obtained with an ultra-sonic pulsed Doppler velocimeter. Several modes of destabilization are observed and include pairing of two vortices, hairpin deformation, and vortex breakdown into a coil shape. Received: 3 April 1996/ Accepted: 4 October 1996     

Accurate solutions for viscoelastic boundary layer flow and heat transfer over stretching sheet

In this article, we present accurate analytical solutions for boundary layer flow and heat transfer of an incompressible and electrically conducting viscoelastic fluid over a linearly stretching surface subject to a transverse uniform magnetic field using the homotopy analysis method （HAM） for two general types of non-isothermal boundary conditions. In addition, we demonstrate that the previously reported analytical solutions for the temperature field given in terms of Kummer＇s function do not converge at the boundary. We provide a graphical and numerical demonstration of the convergence of the HAM solutions and tabulate the effects of various parameters on the skin friction coefficient and wall heat transfer.     

Mixed convection boundary layer flow over a horizontal plate with thermal radiation

The effects of thermal radiation and thermal buoyancy on the steady, laminar boundary layer flow over a horizontal plate is investigated. The plate temperature is assumed to be inversely proportional to the square root of the distance from the leading edge. The set of similarity equations is solved numerically, and the solutions are given for some values of the radiation and buoyancy parameters for Prandtl number unity. It is found that dual solutions exist for negative values of the buoyancy parameter, up to certain critical values. Beyond these values, the solution does no longer exist. Moreover, it is found that there is no local heat transfer at the surface except in the singular point at the leading edge. The radiation parameter is found to increase the local Stanton number.     

Viscoelastic MHD flow boundary layer over a stretching surface with viscous and ohmic dissipations

In this study the momentum and heat transfer characteristics in an incompressible electrically conducting viscoelastic boundary layer fluid flow over a linear stretching sheet are considered. Highly non-linear momentum and thermal boundary layer equations are reduced to set of nonlinear ordinary differential equations by appropriate transformation.     

Analytic solution for axisymmetric flow over a nonlinearly stretching sheet

An analysis is performed for the boundary-layer flow of a viscous fluid over a nonlinear axisymmetric stretching sheet. By introducing new nonlinear similarity transformations, the partial differential equations governing the flow are reduced to an ordinary differential equation. The resulting ordinary differential equation is solved using the homotopy analysis method (HAM). Analytic solution is given in the form of an infinite series. Convergence of the obtained series solution is explicitly established. The solution for an axisymmetric linear stretching sheet is obtained as a special case.     

MHD boundary layer flow and heat transfer over a stretching sheet with induced magnetic field

In this paper, the problem of steady magnetohydrodynamic boundary layer flow and heat transfer of a viscous and electrically conducting fluid over a stretching sheet is studied. The effect of the induced magnetic field is taken into account. The transformed ordinary differential equations are solved numerically using the finite-difference scheme known as the Keller-box method. Numerical results are obtained for various values of the magnetic parameter, the reciprocal magnetic Prandtl number and the Prandtl number. The effects of these parameters on the flow and heat transfer characteristics are determined and discussed in detail. When the magnetic field is absent, the closed analytical results for the skin friction are compared with the exact numerical results. Also the numerical results for the heat flux from the stretching surface are compared with the results reported by other authors when the magnetic field is absent. It is found that very good agreement exists.     

Radiation effects on the thermal boundary layer flow over a moving plate with convective boundary condition

The steady laminar boundary layer flow over a moving plate in a moving fluid with convective surface boundary condition and in the presence of thermal radiation is investigated in this paper. Under certain conditions, the present problem reduces to the classical Blasius and Sakiadis problems. The effects of radiation and convective parameters on the thermal field are thoroughly examined and discussed. Dual solutions are found to exist when the plate and the fluid move in the opposite directions.     

Analysis of a laminar boundary layer flow over a flat plate with injection or suction

An analysis is performed to study a laminar boundary layer flow over a porous flat plate with injection or suction imposed at the wall. The basic equations of this problem are reduced to a system of nonlinear ordinary differential equations by means of appropriate transformations. These equations are solved analytically by the optimal homotopy asymptotic method (OHAM), and the solutions are compared with the numerical solution (NS). The effect of uniform suction/injection on the heat transfer and velocity profile is discussed. A constant surface temperature in thermal boundary conditions is used for the horizontal flat plate.     

Finite difference solution of unsteady natural convection boundary layer flow over an inclined plate with variable surface temperature

The unsteady natural convection boundary layer flow over a semi-infinite inclined plate is considered with the wall temperatureT _w ,(x) (=T +ax ⁿ )varying as the power of the axial coordinate. The governing equations are solved by an implicit finite difference scheme of Crank-Nicolson type. Numerical results are obtained for different values of Prandtl number, Grashof number and n for different angles of inclination. The steadystate velocity and temperature profiles, local and average skin frictions and Nusselt numbers are shown graphically. The effects of the angle of inclination and exponent n on velocity and temperature profiles, skin friction and Nusselt number have been discussed. The velocity, temperature and Nusselt number of the present study are compared with the available results and a good agreement is found to exist between the two.

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Differenzlösung für nichtstationäre natürliche Grenzschicht-Konvektionsströmung an einer geneigten Platte mit veränderlicher Oberflächentemperatur

Zusammenfassung Die nichtstationäre natürliche Grenzschicht-Konvektionsströmung an einer halbunendlichen geneigten Platte wird unter Zugrundelegung der GesetzmäßigkeitT _w ,(x) (=T +ax ⁿ für die Wandtemperatur als Funktion der Achsialkoordinate untersucht, und zwar mit Hilfe eines impliziten Differenzverfahrens vom Crank-Nicolson Typ und bei Variation der Prandtl- und Grashof-Zahlen, des Exponenten n und des Neigungswinkels. Graphisch dargestellt sind die Geschwindigkeits-und Temperaturprofile im stationären Zustand, die örtlichen und gemittelten Reibungsbeiwerte und der Nusselt-Zahlen. Der Einfluß des Neigungswinkels und des Exponenten n auf diese Größe wird diskutiert. Im Vergleich mit den Ergebnissen aus anderen Arbeiten konnte gute Übereinstimmung festgestellt werden.

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Nomenclature g acceleration due to gravity - Gr _L Grashof number at x=L - L length of the plate - n exponent in the power law variation of the wall temperature - Nu _x local Nusselt number - Nu _X dimensionless local Nusselt number - average Nusselt number - dimensionless average Nusselt number - p pressure - Pr Prandtl number - t time - t dimensionless time - T temperature - Tw temperature on the plate - T dimensionless temperature - u x-velocity component - U dimensionlessX-velocity component - v y-velocity component - V dimensionlessY-velocity component - x spatial coordinate along the plate - X dimensionless spatial coordinate along the plate - y spatial coordinate normal to the plate - Y dimensionless spatial coordinate normal to the plate Greek symbols thermal diffusivity - ß volumetric coefficient of thermal expansion - t dimensionless time-step - X dimensionless finite difference grid spacing in theX-direction - Y dimensionless finite difference grid spacing in theY-direction - angle of inclination of plate with horizontal - kinematic viscosity - density - _x local skin friction - _X dimensionless local skin friction - average skin friction - dimensionless average skin friction     

Similarity solution of MHD boundary layer flow with diffusion and chemical reaction over a porous flat plate with suction/blowing

Similarity analysis of diffusion of chemically reactive solute distribution in MHD boundary layer flow of an electrically conducting incompressible fluid over a porous flat plate is presented. The reaction rate of the solute is considered inversely proportional along the plate. Adopting the similarity transformation technique the governing equations are converted into the self-similar ordinary differential equations which are solved by shooting procedure using Runge-Kutta method. For increase of the Schmidt number the solute boundary layer thickness is reduced. Most importantly, the effects of reaction rate and order of reaction on concentration field are of conflicting natures, due to increasing reaction rate parameter the concentration decreases, but for the increase in order of reaction it increases. In presence of chemical reaction, the concentration profiles attain negative value when Schmidt number is large.     

A new branch of solutions of boundary-layer flows over a permeable stretching plate

The steady-state boundary-layer flows over a permeable stretching sheet are investigated by an analytic method for strongly non-linear problems, namely the homotopy analysis method (HAM). Two branches of solutions are obtained. One of them agrees well with the known numerical solutions. The other is new and has never been reported in general cases. The entrainment velocity of the new branch of solutions is always smaller than that of the known ones. For permeable stretching sheet with sufficiently large suction of mass flux, the difference between the shear stresses and velocity profiles of two branches of solutions is obvious: the shear stress of the new branch of solutions is considerably larger than that of the known ones. However, for impermeable sheet and permeable sheet with injection or small suction of mass flux, the shear stress and the velocity profile of two branches of solutions are rather close: in some cases the difference is so small that the new branch of solutions might be neglected even by numerical techniques. This reveals the reason why the new branch of solutions has not been reported. This work also illustrates that, for some non-linear problems having multiple solutions, analytic techniques are sometimes more effective than numerical methods.     

Unsteady boundary layer flow due to a stretching surface in a rotating fluid

The induced unsteady flow due to a stretching surface in a rotating fluid, where the unsteadiness is caused by the suddenly stretched surface is studied in this paper. After a similarity transformation, the unsteady Navier–Stokes equations have been solved numerically using the Keller-box method. Also, the perturbation solution for small times as well as the asymptotic solution for large times, when the flow becomes steady, has been obtained. It is found that there is a smooth transition from the small time solution to the large time or steady state solution.     

A new approach on MHD natural convection boundary layer flow past a flat plate of finite dimensions

A new approach on MHD natural convection boundary layer flow from a finite flat plate of arbitrary inclination in a rotating environment, is presented. This problem plays a significant role on boundary layer flow control. It is shown that taking into account the pressure rise region at the leading edge of the plate leads to avoid separation and the back flow is reduced by the strong magnetic field. It is also shown that the frictional drag at the leading edge of the plate is reduced when the inclination angle α=π/4. In the case of isothermal flat plate, the bulk temperature becomes identical for any value of Gr (Grashof number) when the value of M ² (Hartmann number) and K ² (rotation parameter) are kept fixed.