Natural convection in liquids

Abstract

Lorenz (1881) gave a solution to the problem of natural convection set up near a heated vertical plate, in which he assumed that the surfaces of equal fluid velocity and equal fluid temperature are parallel to the plate. Experiments show, however, that this is not the case. Schmidt and Beckmann (1930) suggested applying the approximations of the boundary layer theory, and Pohlhausen (1930) showed how the partial differential equations thus obtained could be transformed into equations with a single independent variable. Five boundary conditions required to be satisfied in Pohlhausen’s problem, three at the plate and two at infinity. Owing to the slow convergence of the terms, Pohlhausen found it impracticable to solve the equations in series for these boundary conditions. He therefore used measurements by Schmidt and Beckmann of the normal gradients of temperature and velocity in air very close to a heated plate, to obtain two more boundary conditions at the plate. Starting with the five given boundary conditions at the plate he obtained a solution in series of the equations for air, which was found to satisfy the conditions at infinity. Pohlhausen considered air only. His method cannot be used for other fluids because the gradients of velocity and temperature near the plate have not been measured except for air; and it would not be safe to assume that the boundary conditions based on experiments in air are valid for other fluids.