On a Particular Class of Similar Solutions of the Equations of Motion and Energy of a Viscous Fluid

Abstract

By introducing the similarity concept to the two-dimensional, incompressible Navier-Stokes equations and energy equation, a particular class of solutions is found. Two general types of flows are considered: (1) laminar free convection—i.e., flows which take place due to a body force—and (2) laminar forced convection. For free convection on vertical plates, similar solutions are obtained for two different power-law surface temperature variations, and it is shown that one of these solutions constitutes a new type of boundary problem. Results of numerical integrations of the equations are compared with solutions of the similar boundary-layer equations for free convection, and it is demonstrated that a range of surface temperature variations exists for which the boundary layer equations are no longer valid. For forced convection, it is shown that the use of similarity transformations provides an alternate method of deriving the ordinary differential equations for some well-known solutions, such as Couette and stagnation point flows. Solutions are obtained for radial converging or diverging flows between plane surfaces when the temperatures of the surfaces vary as arbitrary powers of the distance from the origin. Results of numerical integrations of the ordinary differential equations are presented for Prandtl numbers of 0.01 and 1.0 and for linear surface temperature variations. Some rather surprising results are obtained for diverging flows when separation occurs and some revealing comparisons with results from boundary-layer theory are made.