The solution of heat-transfer problems by the Wiener—Hopf technique II. Trailing edge of a hot film

Abstract

An incompressible fluid of constant thermal diffusivity k , flows with velocity u = Sy in the x -direction, where S is a scaling factor for the velocity gradient at the wall y = 0. The region — L ≤ x ≤ 0 is occupied by a heated film of temperature T 1 , the rest of the wall being insulated. Far from the film the fluid temperature is T 0 < T 1 . The finite heated film is approximated by a semi-infinite half-plane x < 0 by assuming that the boundary-layer solution is valid somewhere on the finite region upstream of the trailing edge. Exact solutions in terms of Fourier inverse integrals are obtained by using the Wiener-Hopf technique for the dimensionless temperature distribution on the half-plane x > 0 and the heat transfer from the heated film. An asymptotic expansion is made in inverse powers of x and the coefficient of the leading term is used to calculate the exact value of the total heat-transfer as a function of the length L . It is shown that the boundary layer solution differs from the exact solution by a term of order L -1/3 for large L . An expansion in powers of x for the heat transfer upstream of the trailing edge is also found. Application of the theory, together with that of Springer & Pedley (1973), to hot films used in experiments are discussed for the range of values of L(S/K) ½ , up to 20.