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Vasudevaiah, Macherla; Rajagopal, Kumbakonam R.

On fully developed flows of fluids with a pressure dependent viscosity in a pipe. (English) Zbl 1099.76019

Appl. Math., Praha 50, No. 4, 341-353 (2005).
Summary: G. G. Stokes recognized that the viscosity of a fluid can depend on the normal stress, and that in certain flows such as flows in a pipe or in channels under normal conditions, this dependence can be neglected. However, there are many other flows, which have technological significance, where the dependence of viscosity on the pressure cannot be neglected. Numerous experimental studies have unequivocally shown that the viscosity depends on the pressure, and that this dependence can be quite strong, depending on the flow conditions. However, there have been few analytical studies that address the flows of such fluids despite their relevance to technological applications such as elastohydrodynamics. Here we study the flow of such fluids in a pipe under sufficiently high pressures wherein the viscosity depends on pressure, and establish an explicit exact solution for the problem. Unlike the classical Navier-Stokes solution, we find that the solutions can exhibit a structure that varies all the way from a plug-like flow to a sharp profile that is essentially two intersecting lines (like a rotated V). We also show that unlike in the case of Navier-Stokes fluid, the pressure depends both on the radial and the axial coordinates of the pipe, logarithmically in the radial coordinate and exponentially in the axial coordinate. Exact solutions such as those established in this paper serve a dual purpose, not only do they offer solutions that are transparent and provide the solution to a specific but simple boundary value problems, but they can be used also to test complex numerical schemes used to study technologically significant problems.

Cited in 18 Documents

MSC:

76D99 Incompressible viscous fluids

Keywords:

Poiseuille flow; plug-like flow
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References:

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