Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Christina Lienstromberg, Juan J. L. Velázquez

Abstract

We study the dynamic behaviour of solutions to a fourth-order quasilinear degenerate parabolic equation for large times arising in fluid dynamical applications. The degeneracy occurs both with respect to the unknown and with respect to the sum of its first and third spatial derivative. The modelling equation appears as a thin-film limit for the interface separating two immiscible viscous fluid films confined between two cylinders rotating at small relative angular velocity. More precisely, the fluid occupying the layer next to the outer cylinder is considered to be Newtonian, i.e. it has constant viscosity, while we assume that the layer next to the inner cylinder is filled by a shear-thinning power-law fluid. Using energy methods, Fourier analysis and suitable regularity estimates for higher-order parabolic equations, we prove global existence of positive weak solutions in the case of low initial energy. Moreover, these global solutions are polynomially stable, in the sense that interfaces which are initially close to a circle, converge at rate $1/t^{1/β}$ for some $β> 0$ to a circle, as time tends to infinity. In addition, we provide regularity estimates for general nonlinear degenerate parabolic equations of fourth order.

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

Abstract

We study the dynamic behaviour of solutions to a fourth-order quasilinear degenerate parabolic equation for large times arising in fluid dynamical applications. The degeneracy occurs both with respect to the unknown and with respect to the sum of its first and third spatial derivative. The modelling equation appears as a thin-film limit for the interface separating two immiscible viscous fluid films confined between two cylinders rotating at small relative angular velocity. More precisely, the fluid occupying the layer next to the outer cylinder is considered to be Newtonian, i.e. it has constant viscosity, while we assume that the layer next to the inner cylinder is filled by a shear-thinning power-law fluid. Using energy methods, Fourier analysis and suitable regularity estimates for higher-order parabolic equations, we prove global existence of positive weak solutions in the case of low initial energy. Moreover, these global solutions are polynomially stable, in the sense that interfaces which are initially close to a circle, converge at rate for some to a circle, as time tends to infinity. In addition, we provide regularity estimates for general nonlinear degenerate parabolic equations of fourth order.
Paper Structure (16 sections, 28 theorems, 281 equations, 2 figures)

This paper contains 16 sections, 28 theorems, 281 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Known results and the Newtonian case $\alpha = 1$
  3. Known results in the shear-thickening case $0 < \alpha < 1$
  4. The shear-thinning case $\alpha > 1$ -- main results of the paper
  5. Derivation of the equation from the two-phase Taylor--Couette setting
  6. Outline of the paper
  7. Global weak solutions and asymptotic behaviour for $\alpha > 1$ -- The main results.
  8. The regularised problem -- Local existence of solutions and polynomial stability
  9. Definition of the regularised problem and main result on local existence and asymptotic behaviour
  10. The energy functional and convergence to a circle
  11. Regularity estimates for quasilinear problems of fourth order
  12. Weighted interpolation inequalities.
  13. Regularity estimates
  14. Optimal decay estimates for the derivatives of the Fourier modes $h_{\pm 1}$ of solutions to \ref{['eq:PDE_regularised']}
  15. The limit $\sigma \to 0^+$. Solutions to the original problem \ref{['eq:PDE_alpha>1']}.
  16. ...and 1 more sections

Key Result

Lemma 2.2

Given $h_0 \in H^1(S^1)$, let be a weak solution to eq:PDE_alpha>1 on $[0,T]$. Then $h$ satisfies

Figures (2)

  • Figure 1: Two-phase flow of liquids confined between two rotating cylinders
  • Figure 2: Problem setting with thin-film next to the inner cylinder in non-dimensional variables

Theorems & Definitions (63)

  • Definition 2.1
  • Lemma 2.2: Conservation of mass
  • proof
  • Remark 2.3
  • Theorem 2.4
  • Theorem 3.1: Local existence for \ref{['eq:PDE_regularised']}
  • Remark 3.2
  • proof
  • Lemma 3.3: Conservation of mass for \ref{['eq:PDE_regularised']}
  • proof
  • ...and 53 more