Frictional heating in plane Couette flow

Abstract

When a liquid whose viscosity decreases as its temperature increases is made to undergo a simple shearing flow the wall speed, or the centreline temperature, is a double valued function of the wall stress. This is called the base solution curve and the turning point is called the nose of the curve. The question we address is this: is there a point of neutral stability on this curve? The answer turns out to depend on whether the wall speed or the wall stress is the control variable. We introduce an eigenvalue problem which explains the shape of the base curve. It leads to a useful definition of the nose and to a rule which forecasts when a nose ought to arise. It then helps us determine where the neutral points lie. The result is this: if the wall speed is the control variable there are no points of neutral stability; if the wall stress is the control variable the nose of the curve is a point of neutral stability. This supports our conviction that in a physical experiment the wall speed must be the control variable, it cannot be the wall stress. Because the wall stress plays the same role here as does the Frank–Kamenetskii number in thermal ignition we conclude that thermal ignition is not a good model of fluid frictional heating.