On the Stability of Flow of a Thermally Stratified Fluid under the Action of Gravity

Abstract

The equation for the small disturbances from the plane‐parallel flow of a thermally stratified fluid under the influence of gravity acting perpendicular to the plane of stratification is derived. It was found necessary to include not only viscosity but also heat conductivity to preclude the resulting differential equation from having a singularity. Asymptotic solutions of the sixth‐order differential equation thus derived are obtained. They show the presence of a Stokes point. The limiting form of the differential equation near the Stokes point is next obtained and an exact solution of this equation is derived by means of a Laplace transformation. In the general case the integrand of the Laplace transformation involves Whittaker's confluent hypergeometric functions. In the special case of a Prandtl number of 1, the integrand is considerably simpler and for this case asymptotic representations of the solutions on both sides of the Stokes point have been derived from the Laplace transformation solution by the method of steepest descent. The connection formulas between the solutions are the same as that previously derived by Tollmien and Lin for the case when stratification, gravity, and heat conduction are neglected.