Abstract
The main objective of this work is to examine in detail basic unsteady pipe flows and to investigate any new physical phenomena. We take the viscoelastic upper-convected Maxwell fluid as our non-Newtonian model and consider the flow of such a fluid in pipes of uniform circular cross-section in the following three cases: 1. (a) when the pressure gradient varies exponentially with time; 2. (b) when the pressure gradient is pulsating; 3. (c) a starting flow under a constant pressure gradient. In the first problem we looked separately at the pressure gradient rising exponentially with time and falling exponentially with time, i.e. the pressure gradient is proportional to e ± α 2 τ . The behaviour of the flow field depends to a large extent on β where β 2 = α 2(1 ± Hα 2) with H being the quotient of the Weissenberg and Reynolds numbers. In both cases for small | βη|, η being the radial distance from the axis, the velocity profiles are seen to be parabolic. However, for large | βη| the flows are vastly different. In the case of increasing pressure gradient the flow depicts boundary-layer characteristics while for decreasing pressure gradient the velocity depends on the wall distance. The case of a pulsating pressure gradient is investigated in the second problem. Here the pressure gradient is proportional to cos nτ. Again the flow depends to a large extent on a parameter β ( β 2 = i n − n 2 H). For small values of | βη| the velocity profile is parabolic. However, it is found that, unlike Newtonian fluids, the velocity distribution for the upper-convected Maxwell fluid is not in phase with the exciting pressure distribution. In the case of large | βη| the solution displays a boundary-layer characteristic and the phase of the motion far from the wall is shifted by half a period. The final problem examines a flow that is initially at rest and then set in motion by a constant pressure gradient. A closed form solution has been obtained with the aid of a Fourier-Bessel series. The variation of the velocity across the pipe has been sketched and comparison made with the classical solution.
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