Exact Solution of Boundary Layer Flow Due to a Quadratically Shrinking Sheet

Using a Newtonian cooling liquid, the shrinking sheet problem is explored for the closed form solution. The velocity with which the sheet shrinks is assumed to vary as a quadratic fucntion of the distance from the slit. The shrinking sheet is assumed to be permeable and the boundary layer approximation is used. The basic boundary layer equation that governs conservation of linear momentum is a non-linear partial differential equation and the same is converted into a set of nonlinear ordinary differential equations using suitable similarity transformation. The analytical procedure for solving the resultant system of ordinary differential equations is discussed in detail. The mathematical expressions, involving shrinking parameters, for the stream function and the velocity components are obtained. The linear shrinking sheet problem is shown to be the limiting case of present study. The shrinking sheet problem is shown to be qualitatively much different than the stretching sheet problem though technically it looks like a cosmetic change in one of the boundary conditions.