An analytic solution of unsteady boundary-layer flows caused by an impulsively stretching plate

The development of velocity and temperature fields of an incompressible viscous electrically conducting fluid, caused by an impulsive stretching of the surface in two lateral directions and by suddenly increasing the surface temperature from that of the surrounding fluid, is studied. The partial differential equations governing the unsteady laminar boundary-layer flow are solved numerically using an implicit finite difference scheme. For some particular cases, closed form solutions are obtained, and for large values of the independent variable asymptotic solutions are found. The surface shear stresses inx-andy-directions and the surface heat transfer increase with the magnetic field and the stretching ratio, and there is a smooth transition from the short-time solution to the long-time solution.

Unsteady three-dimensional MHD-boundary-layer flow due to the impulsive motion of a stretching surface

An analytic technique, namely, the homotopy analysis method, is applied to give series solution of the unsteady boundary-layer flows over an impermeable stretching plate. Different from all previous perturbation solutions, our series solutions are convergent in the whole time region 0 ≤ r < +∞. To the best of our knowledge, such kind of series solution has never been reported for this problem. Besides, two kinds of new similarity transformations about dimensionless time are proposed. Using these two different similarity transformations, we obtain the same convergent solution valid in the whole time region 0 < r < +∞. Furthermore, it is shown that a nonlinear initial/boundary-value problem can be replaced by an infinite number of linear boundary-value subproblems.

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Series Solutions of Unsteady Boundary‐Layer Flows over a Stretching Flat Plate

Homotopy analysis of unsteady boundary layer flow adjacent to permeable stretching surface in a porous medium

Finite difference solution of the transient free-convection flow of an incompressible viscous fluid past an impulsively started semi-infinite vertical plate with heat and mass transfer is presented here. The steady state velocity, temperature and concentration profiles are shown graphically. Velocity profiles are compared with exact solution. It has been observed that there is a rise in the velocity due to the presence of a mass diffusion. An increase in Schmidt number, leads to a fall in the velocity. Stability and the convergence of the finite difference scheme are established.

Unsteady flow past an impulsively started vertical plate with heat and mass transfer

SUMMARY The manner in which the variable x, measuring distance from the leading edge, first enters into the solution of the title problem is elucidated. The structure of the corresponding eigenfunctions is different in each of four regions of the boundary layer and the corresponding contribution to the velocity is weaker than any finite power of — x + Ut, where t measures time and the variable x first enters at x = Ut. A comparison between the predicted variations of displacement thickness and of the skin friction with the computed values obtained by Dennis (2) is favourable. IN part I of this paper Stewartson (6) considered the development of the unsteady boundary layer on a semi-infinite flat plate immersed in an infinite viscous fluid. The plate is given, impulsively, a velocity U at time t = 0 in a direction parallel to its length and thereafter moves in the same direction with the same velocity. It was found that the structure of the boundary layer at a fixed distance x from the leading edge is different, according as T (= Utjx) Jjl. If T 1, x enters the structure, which approaches the steady Blasius form asr->-oo. Physically, the significance of T = 1 is due to the wave-like character of the governing equation when regarded as a function of x and t. This means that the disturbance caused by the presence of the leading edge travels through the boundary layer with the maximum local velocity at any station of x, i.e. the mainstream velocity. Hence, the effect of the leading edge is felt when Ut > x, i.e. T > 1. Two interesting questions which naturally arise relate to the mathematical mechanism by which the variable x enters the solution at T = 1+ and the variable t disappears as T->-OO. An answer to the second question was indicated in the paper but very little progress was made with the first question. Since then the problem has aroused considerable interest and of the subsequent contributions we mention especially the first full numerical solution by Hall (3), the numerical solution of the similarity form of the governing equation by Dennis (2) and the completion of the mathematical description of the solution as T -> oo by Watson. This study has just been

K. Stewartson

Papers

On the impulsive motion of a flat plate in a viscous fluid. ii