Laminar flow analysis in a pipe with locally pressure-dependent leakage through the wall

The paper analyzes the problem of the leaky pipe (or a porous-walled pipe), namely the laminar flow of a pure fluid that takes place in a pipe, the wall of which is composed of a porous material. This configuration is inspired by some watering systems or by the cross-flow (or tangential) filtration configuration for membrane separation or capillary flow. It assumes that the leakage through the wall (or permeate) results from the pressure difference between both sides of the pipe wall, and is here modeled by the Starling–Darcy law.The inner pressure along the pipe behaves accordingly with two competitive features: the viscous pressure drop competing against the pressure increase due to pipe axial flow deceleration. It is long known that both features compensate at a critical value, Rtiso, of the transverse Reynolds number RtRt (based on transpiration velocity); this corresponds to the only situation where the pressure remains uniform along the channel. The case with uniform leakage–known as Berman flow–possesses a similarity solution due to Yuan and Finkelstein (1956) [2] for the pipe configuration. The paper is aimed at extending the latter study to a non-uniform leakage depending linearly on local pressure.First, the similarity solution is revisited. Its expansion in a series of RtRt allows us to propose a hierarchy of new ordinary differential equations (ODEs), that extend–to small or moderate RtRt–the linear ODE proposed for the limit case Rt=0Rt=0 by Regirer (1960) [25]. As by-products, we propose approximate analytical solutions that solve the problem of the leaking pipe with increasing accuracy in the weakly non-linear case (WNL) (i.e. for small and moderate RtRt). Finally, the validity of ODEs and WNL solutions is numerically checked with respect to flow simulations in the Prandtl approximation.