The validity of an operational solution of Laplace's equation

A physical problem that has received considerable attention i s that of leakage in mine ventilation. Most underground methods of mining leave porous material between roadways^ with the result that through leakage^ the quantity of air reaching the working face is usually less than that entering the mine. Investigations into this problem have assumed that air flows through the barrier in a direction perpendicular to the roadways. This is clearly wrong unless flow is impervious in any other direction. In the present study^ the more general consideration of air movement in any direction is made. It is shown that the pressure of air at points in the barrier is governed by Laplace's equation and from other fluid flow relationships, certain non-linear boundary conditions apply. An operational method is used to find a solution but an approximation must be introduced for this to be achieved. The author wishes to establish the accuracy of this solution. For the special case of streamlined flow in the airways, an analytical solution is able to be determined. However, to do so depends on the validity of a modified Dini expansion of a function. This is established so that the operational and analytical solutions can be compared.