Flow patterns behind the free flow front for a Newtonian fluid injected between two infinite parallel plates

Abstract

A complete analytical treatment of the 2-dimensional problem of the injection of a Newtonian fluid between two parallel plates is presented. Explicit formulas are derived for the shape of the free flow front, the streamlines behind the flow front, the velocity, deformation and rotation (orientation) of material elements in the flow front region, and the associated stresses there. The analysis is based on complex function theory, and in this, the flow region, inclusive the unknown free flow front, is mapped onto the interior of the unit circle. The mapping function that determines the shape of the flow front is found by solving a Hilbert problem. It is analytically found in how far the actual flow front differs from a semi-circular shape, and it is concluded that the semi-circular approximation seems acceptable. Deformations of material line or area elements due to the fountain flow in the flow front region are followed in time; large deformation and reorientations of the material elements are observed. Our results are compared with results in literature obtained by numerical simulations and by experimental work, and on the whole good correspondence is found.