Integral Boundary-Layer Solution for Laminar Flow along a Flat Plate

Viscous Flows ◽  
1988 ◽  
pp. 271-278 ◽  
Author(s):  
Stuart Winston Churchill
1983 ◽  
Vol 48 (6) ◽  
pp. 1571-1578 ◽  
Author(s):  
Ondřej Wein

Theory has been formulated of a convective rotating spherical electrode in the creeping flow regime (Re → 0). The currently available boundary layer solution for Pe → ∞ has been confronted with an improved similarity description applicable in the whole range of the Peclet number.


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.


1975 ◽  
Vol 97 (3) ◽  
pp. 482-484 ◽  
Author(s):  
C. B. Watkins

Numerical solutions are described for the unsteady thermal boundary layer in incompressible laminar flow over a semi-infinite flat plate set impulsively into motion, with the simultaneous imposition of a constant temperature difference between the plate and the fluid. Results are presented for several Prandtl numbers.


Starting from the differential equation of mass transfer in laminar flow and the appropriate boundary condition, expressions are derived for the rate of mass transfer from ( a ) a flat plate in a longitudinal fluid stream, ( b ) a vertical flat plate by natural convection, ( c ) the forward stagnation point of a sphere in a fluid stream. Only outward mass transfer is considered; this corresponds to blowing outwards from the plate at a rate inversely proportional to the boundary-layer thickness. The Kármán-Pohlhausen-Kroujiline method is used. Where appropriate the Prandtl or Schmidt number has been taken as 0⋅71. The calculations are valid for all mass-transfer processes for which a single diffusion coefficient can be ascribed to the diffusing property, but are particularly relevant to the combustion of liquid fuels, for which the outward mass-transfer rates are so high that important deviations occur from boundary-layer profiles without mass transfer. Despite the great temperature variations present in boundary layers with combustion, mean values for the fluid properties are assumed. In the case of natural convection, it is assumed that the body forces on the fluid in the boundary layer are everywhere zero; this leads to a less serious over-estimate of the buoyancy than the usual assumptions which are valid only for small temperature differences.


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