Intrinsic equations of viscous incompressible steady flows

Summary

The basic equations governing steady incompressible viscous flow in the absence of extraneous forces are obtained in intrinsic form and the properties arrived at are:

• (i) geometric relations existing between curvature and torsions of the curves of congruences, formed by the streamlines, principal normals and binormals to them when the fluid is non-viscous and isovels coincide with them.

• (ii) for an irrotational flow the magnitude of the velocity and curvature of the streamlines are constant along a binormal to the streamline.

• (iii) either the vorticity lies in the normal plane or velocity is constant along the streamlines.

• (iv) when the flow is Beltrami, the streamlines must be normal to a single parameter family of surfaces. The velocities in magnitude and curvature of the streamline are constant in the rectifying plane. The geometric relations existing are deduced.

• (v) for a doubly laminar flow, the vorticity lies in the normal plane and its curvature of the streamlines is constant along its binormal.

• (vi) plane flows are considered, the streamline patterns when isovels coincide with them are determined as either concentric circles or parallel straight lines. This is independent of the viscosity of the fluid. The question declared unsolved by Nemenyi and Prim¹⁰), is solved here.