Stability of fully developed pipe flow of a shear-thinning fluid that approximates the response of viscoplastic fluids

The stability of steady, fully developed flow in a long cylindrical pipe for a shear-thinning fluid (which approximates a class of viscoplastic materials) is studied using linear stability analysis. The eigenvalues of the frequency of the perturbation of the steady-state solution are obtained using the shooting method. The eigenvalues are negative in the Reynolds number range studied and asymptotically tend to zero as the Reynolds number increases. This shows the pipe flow is stable in the Reynolds number range studied. A qualitatively similar trend is shown by the eigenvalues of a Navier-Stokes fluid of equivalent viscosity. However, the eigenvalues are much lesser than those of the shear-thinning fluid, and this shows that the flow of the Navier-Stokes fluid can be expected to be stable over a much larger Reynolds number range than the shear-thinning fluid.