HEAT TRANSFER BETWEEN A FLUID AND A PLATE : MULTIDIMENSIONAL LAPLACE TRANSFORMATION METHODS

Multidimensional Laplace transformations are used to obtain the surface 
temperature and the surface heat flux of a plate with a fluid flowing across it without solving the complete boundary value problem. It is also shown that the constant initial and boundary values can be relaxed and the method still applies. The solution to the boundary value problem at points away from the surface can be treated similarly

R.G. BUS   Carslaw and Jaeger [4] in finding "exact" or "analytic" solutions.Several related problems and the details of the mathematical model are discussed with care in [1,2].
In this paper we sho that the direct application of the multidimensional Laplace transformation to the boundary value problem, with the assumption of the existence of the transform of the solution as a replacement for the bounded solution condition, leads dlrectly to a 'Wcompatabillty" condition.The transforms of the surface heat flux and temperature can then be computed directly by use of this compatability condition along with the transform of the boundary condition, without first obtaining the temperature function at points within the fluid.These transforms are then inverted.Some simpler applications of a comparability condition were presented by D. Voelker and G. Doetsch [5] a number of years ago.In Sections 2 and 3 we utilize this method to obtain the results of Sucec [1,2] and of Singh, Sharma, and Hisra [3], respectively.In the last section we sho ho the method can be applied as well to silar problems in which arbitrary functions are introduced into the initial conditions and the boundary conditions.
We use reference notations to tabulated pairs of functions which are related -1 by the Laplace transformation such as [2 2.1(22)] to denote Formula 22, Section 2.1 of the inversion part of [2], or [6 A4(5)] to denote Formula 5 of Section A4 of [6].A few useful combinations from tables are recorded in our Appendlx.

2.5) Y
We also replace the condition that e(t,x,y) be bounded, by the assumption that the solution possesses a 3-dLaenslonal Laplace transform, that is 3(e(t,x,y)) f(s,u,v).
(2.6) Properties of the 2-dlmenslonal Laplace transformation are developed in [5], which further includes an extensive inversion table.Initially ve transform vlth respect to the first eo variables and let 2(e(t,x,y)) S(s,u,y). (2.7) If we next transform with respect to the third variable, then, after some simplifica- tions, the problem becomes f(s,u,v)-(v+s') II,x:u,O+) -3'(sV) -1 2(2.8) v (s,,,O+) (,4) S(s,u,0+). (2.9) In (2.8) we observe that the denominator is zero arbitrarily far out in right half planes; thus, f is not analytlc in right hal planes, unless the numerator is also zero for v (u+s) 1/2.Hence, f cannot be a Laplace transform in general.