Analysis of a Two-Fluid Taylor–Couette Flow with One Non-Newtonian Fluid

Abstract

We study the dynamic behaviour of two viscous fluid films confined between two concentric cylinders rotating at a small relative velocity. It is assumed that the fluids are immiscible and that the volume of the outer fluid film is large compared to the volume of the inner one. Moreover, while the outer fluid is considered to have constant viscosity, the rheological behaviour of the inner thin film is determined by a strain-dependent power-law. Starting from a Navier–Stokes system, we formally derive evolution equations for the interface separating the two fluids. Two competing effects drive the dynamics of the interface, namely the surface tension and the shear stresses induced by the rotation of the cylinders. When the two effects are comparable, the solutions behave, for large times, as in the Newtonian regime. We also study the regime in which the surface tension effects dominate the stresses induced by the rotation of the cylinders. In this case, we prove local existence of positive weak solutions both for shear-thinning and shear-thickening fluids. In the latter case, we show that interfaces which are initially close to a circle converge to a circle in finite time and keep that shape for later times.

Long-time asymptotics and regularity estimates for weak solutions to a doubly degenerate thin-film equation in the Taylor-Couette setting

“…In particular, the effects of surface tension, the shear forces induced by the rotation of the cylinder and the characteristic stresses of the non-Newtonian have a strong impact on the form of the resulting evolution equation for the interface. In [22] a more evolved equation is derived for the situation in which all three effects -surface tension, shear induced by the rotation of the cylinders and the characteristic stress of the non-Newtonian fluid -are comparable. In the present paper, we study the equation (1.1) which is obtained in the following asymptotic regime…”

“…Conversely, if > 1, the viscosity decreases with increasing shear rate and the fluids are called shear-thinning fluids. The Newtonian case has been treated in [25] and the shear-thickening case has been studied in [22]. More details on these two cases are provided in Subsection 1.2.…”

“…when the non-Newtonian fluid located next to the interior cylinder is shear-thickening. This problem is studied in [22], where the following results are proved.…”

“…A more detailed review on the above mentioned works is provided in the introduction of our previous paper [22]. 1.4.…”

“…Derivation of the equation from the two-phase Taylor-Couette setting. We briefly recall from our previous work [22] how problem (1.1) can be derived as a model equation for the interface separating two viscous, non-Newtonian immiscible fluids that are confined between two concentric cylinders rotating at a small relative velocity. For a detailed derivation of the model we refer the reader to [22,Section 2].…”