Dynamics of incompressible fluids with incompatible distortion rates

Abstract

Listen

Translate article

We consider the implication of allowing the distortion rate in a fluid to be the sum of two smooth components, both incompatible in the sense that they are not gradients of vector fields. Whereas one of these components embodies a smooth distribution of slippage, the other describing the repair needed to ensure that the distortion rate is the gradient of the velocity field. Our considerations lead to a generalization of the Navier–Stokes equations for an incompressible fluid. A tensor field which characterizes incompatibility is introduced and a fundamental equation for its evolution is proposed. Properties of these equations are exemplified by revisiting the classical problem of pressure-driven flow in a plane, rectangular channel. For a sufficiently large instantaneously applied pressure drop, a novel action due to the diffusivity κ associated with the transport of incompatibility is illustrated in addition to the normal diffusive and dissipative affect of viscosity ν. The steady state of the velocity and incompatibility is governed by a system of two ordinary differential equations which are fully analyzed and the resulting fields are determined and discussed. The transient problem is solved numerically and the graphical results show how these fields are structured in time by diffusion and dissipation in the channel during the transition from rest to steady state. A precursor laminar flow persists until a particular time at which the wall shear stress becomes equal to an a priori given cut-off value that initiates incompatibility at the walls of the channel. This incompatibility diffuses inward and interacts synergistically with the viscous action present in the channel to flatten the velocity profile so as to take on a ‘plug flow-like’ appearance, and to redistribute the viscous dissipation and the dissipation due to incompatibility into boundary layers at the channel walls.