On laminar flow through a channel or tube with injection: application of method of averages

The equations for the two-dimensional incompressible laminar flow through a channel with a uniform normal fluid velocity vw at the walls have been recently [1] reduced to an ordinary nonlinear differential equation with mixed boundary conditions and with a normal-velocity parameter which, in practice, may be large, but whose reciprocal is the coefficient of the highest-order derivative in this equation.A small-perturbation solution of this equation was obtained in [1], valid for small normal velocities at the walls.The corresponding problem for flow through a circular tube has also been treated quite recently [2].Here, the ordinary differential equation which was obtained was again solved by a small-perturbation procedure for small values of the normal fluid velocity at the wall.In addition, however, an asymptotic solution, to first powers of the reciprocal of the normal-velocity Reynolds Number, and hence valid for large values of the injection velocity at the wall, was developed.The purpose of this paper is to present a simple approximate closed-form solution for the flow through a channel and through a circular tube with porous walls valid for the entire range of normal fluid injection velocities from zero to indefinitely large.The method of analysis will be based on the method of averages.Although this method is fairly well known in curve-fitting, it will be seen that it is particularly fruitful here in solving the ordinary differential equations under the given boundary conditions.The method of averages serves here, in fact, as a relatively simple alternative to the method of least squares (see [3]).The method of averages, moreover, will be applied here in conjunction with auxiliary boundary conditions derived from the governing ordinary differential equation, and hence it is to some extent** analogous to the relatively well-known integral methods, such as the Karman-Pohlhausen method, of solving the the 'partial differential equations of the laminar boundary layer.The solutions thus obtained will be shown to reduce exactly to the small-perturbation solutions for small values of the injection (or suction) velocity at the wall, and to reduce approximately to the exact asymptotic solutions for infinite values of the injection velocity.The problem of normal fluid injection is of practical interest in connection with the transpirationor sweat-cooling of heated surfaces such as turbine blades, rocket walls, or wing surfaces in high-speed flight.