Materials which are heated by the passage of electricity are usually modeled by a nonlinear coupled system of two partial differential equations. The current equation is elliptic, while the temperature equation is parabolic. These equations are coupled one to another through the conductivities and the Joule effect. A computationally attractive discretization method is analyzed and shown to yield optimal error estimates in H ' .

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Singularities are considered in the solution of the laminar boundary-layer equation at a position of separation. The works of Howarth (1938). Goldstein (1948), Stewartson (1958), Terrill (1960) and Akinrelere [(1981), (1982)] are reviewed to fully establish the existence of singularity in the incompressible boundary layer at separation for both the velocity and thermal fields. A flow at a large Reynold's number along an immersed solid surface around which bounda y layer is formed through which the velocity rises rapidly from zero at the surface to its value in the main stream is considered. It is found that whenever separation does occur, the boundary layer equations cease to be valid on the upstream side and also downstream of separation. In this paper, the works of Akinrelere [(1981),(1982)] on the thermal field had been extended to include suction through a porous surface. Following Stewartson (1958) the stream function ψ 1 is expanded in a series of the type ψ 1 = 2 3 2 ξ 3 6 r=0 ξ r f r (η) + 2 3 2 ξ 8 ln ξ[F 5 (η) + ξF 6 (η)] + O(ξ 10 ln ξ), where ξ = x 1 4 1 , η = y 1 /2 1 2 x 1 4 1 and (x 1 , y 1) are non-dimensional distances measured from the separation point. Analytical so1utions for f 1 , f 2 , f 3 , g l , g 2 and g 3 are presented. Results are obtained both for arbitrary Prandtl number σ and σ = 1, with and without suction.