Unsteady boundary layer flow and heat transfer of a viscous fluid over a semi-infinite stationary flat plate: Blasius flow problem

In this paper, a class of similarity solution for the unsteady Navier-Stokes equations which describe the two-dimensional unsteady laminar boundary layer flow of a viscous incompressible fluid over a semi-infinite flat stationary plate surface has been derived. Here, the boundary layer is formed owing to an uniform free stream flow over this plate surface and the unsteadiness in this flow dynamics comes only from the similarity solution which is not yet available in the open literature. For this similarity solution the governing boundary layer equations (momentum and thermal) transform into a set of non-linear ordinary differential equations which conceive two physical parameters, namely, unsteadiness parameter gamma and Prandtl number Pr. The impact of these two parameters on the unsteady boundary layer flows as well as on the heat transfer analysis have been discussed in detailed. The present analysis confirms the non-existence of the boundary layer solutions after a certain negative value of gamma, while it continues for a large positive value of gamma and specifically for the value of gamma = 0, we get the steady boundary layer solution for the Blasius flow problem. Here, we have considered the whole range of the Pr values, which includes both the limiting cases Pr -> 0 and Pr -> infinity, in the discussion of the heat transfer analysis. For these two limiting cases, two approximated separate mathematical formulae for the heat transfer solutions have been derived in terms of the well-known bessel, error and hypergeometric functions, whose results for any given value of gamma within the solution domain are well fitted with the corresponding numerical results.