Rotational and axial flow of pseudoplastic fluids

Abstract

Effects of a controllable axial flow on the stability of rotational flow of pseudoplastic fluids in the gap between concentric cylinders are studied. It is assumed that the outer cylinder is fixed while the inner one has simultaneous and independent rotational and translational motions. The fluid follows the Carreau model and mixed boundary conditions are imposed. The four-dimensional low-order dynamical system resulting from Galerkin projection of the conservation of mass and momentum equations includes highly non-linear terms in the velocity components originating from the shear-dependent viscosity. In the absence of the axial flow as the pseudoplasticity effect increases, the critical Taylor number at which the rotational flow loses its stability to the vortex structure decreases. The emergence of the vortices corresponds to the onset of a supercritical bifurcation which is also seen in the flow of a Newtonian fluid in rotation. The existence of an axial flow, induced by the translational motion of the inner cylinder, appears to further advance the emergence of the vortex flow. Complete flow analysis together with viscosity maps are given for the entire flow field.

Appendix

The constants in Eqs. (11a)–(11d) are given in this appendix:

$$\begin{aligned} & A_{1} = A_{5} = A_{6} = B_{1} =-1,\qquad A_{2} = k^{2} \tau ^{2}, \qquad A_{3} = \frac{16k^{2} \tau ^{2}}{3 \pi ^{2}}, \qquad A _{4} = A_{8} =1 \\ & A_{7} =-\frac{32\sqrt{2}}{9} \frac{k\tau }{\pi^{3}}, \qquad A_{9} =-4 \pi ^{2} \tau , \qquad A_{10} =- \pi^{2} \tau , \qquad A_{11} = \frac{\sqrt{2} k\tau }{\pi } \\ & B_{2} =- \frac{32k\pi \tau ^{3}}{3 \sqrt{2}}, \qquad B_{3} =- \frac{k^{4}\tau ^{2}}{4} -\frac{5 k^{2} \pi ^{2} \tau ^{2}}{2} - \frac{\pi ^{4} \tau^{2}}{4} \\ & B_{4} =- \frac{64 \sqrt{2} k^{5} \tau ^{3}}{9 \pi ^{3}} + \frac{32 \sqrt{2} k^{3} \tau ^{3}}{3\pi } - \frac{32 \sqrt{2} k \pi \tau }{9} \\ & B_{5} = \frac{\tau }{8} \biggl( \frac{-9 k^{6}}{\pi ^{2}} +13 k^{4} -43 k ^{2} \pi ^{2} - \pi ^{4} \biggr), \qquad B_{6} = \frac{256k\pi \tau ^{3}}{3 \sqrt{2}} \\ & B_{7} =2 k^{6} \tau ^{3} -10 k^{4} \tau ^{3} \pi ^{2} -10 k^{2} \tau ^{3} \pi ^{4} +2 \tau ^{3} \pi ^{6}, \qquad B_{8} =-\frac{512}{15 \sqrt{2}} \tau ^{2} k\pi \\ & B_{9} =-2k^{4} \tau ^{2} -4 k^{2} \tau ^{2} \pi ^{2} -2 \tau ^{2} \pi ^{4}, \qquad B_{10} =-k^{2} \tau -3\tau \pi ^{2} \\ & B_{11} =-\biggl(\frac{k ^{2} \tau }{4} + \frac{9\tau \pi ^{2}}{4} \biggr),\qquad B_{12} =- \frac{1}{16} \biggl( \frac{18 k^{4} \tau }{\pi ^{2}} +4\tau k^{2} +18 \tau \pi ^{2} \biggr) \\ & B_{13} =-2\tau\bigl( k^{2} -3 \pi ^{2} \bigr), \qquad B_{14} =-2\tau k^{2} -6\tau \pi ^{2}, \qquad B_{15} =38 \pi^{2} \\ & B_{16} = \frac{k^{2} \tau }{\pi } -3\pi \tau , \qquad B_{17} =B_{20} =-12 \pi ^{2} \tau , \qquad B_{18} =-6 \pi^{2} \tau \\ & B_{19} =-2k\tau -6 \pi ^{2} \tau,\qquad B_{21} =-3 \pi ^{2} \tau ,\qquad B_{22} =- \frac{3}{4} \pi ^{2} \tau \\ & B_{23} =- \frac{\tau }{4} \bigl( k^{2} +9 \pi^{2} \bigr), \qquad B_{24} =-6\tau \pi , \qquad B_{25} =-6\tau\pi ^{2}. \end{aligned}$$