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On weak solutions of a control-volume model for liquid films flowing down a fibre

Roman M. Taranets, Hangjie Ji, Marina Chugunova

Abstract

This paper presents an analytical investigation of the solutions to a control volume model for liquid films flowing down a vertical fibre. The evolution of the free surface is governed by a coupled system of degenerate nonlinear partial differential equations, which describe the fluid film's radius and axial velocity. We demonstrate the existence of weak solutions to this coupled system by applying a priori estimates derived from energy-entropy functionals. Additionally, we establish the existence of traveling wave solutions for the system. To illustrate our analytical findings, we present numerical studies that showcase the dynamic solutions of the partial differential equations as well as the traveling wave solutions.

On weak solutions of a control-volume model for liquid films flowing down a fibre

Abstract

This paper presents an analytical investigation of the solutions to a control volume model for liquid films flowing down a vertical fibre. The evolution of the free surface is governed by a coupled system of degenerate nonlinear partial differential equations, which describe the fluid film's radius and axial velocity. We demonstrate the existence of weak solutions to this coupled system by applying a priori estimates derived from energy-entropy functionals. Additionally, we establish the existence of traveling wave solutions for the system. To illustrate our analytical findings, we present numerical studies that showcase the dynamic solutions of the partial differential equations as well as the traveling wave solutions.
Paper Structure (9 sections, 7 theorems, 207 equations, 4 figures)

This paper contains 9 sections, 7 theorems, 207 equations, 4 figures.

Table of Contents

  1. Problem statement
  2. Existence of weak solutions
  3. Energy estimate
  4. Entropy estimate
  5. Approximate problem
  6. Compactness and limit processes
  7. Travelling wave solutions
  8. Numerical studies
  9. Conclusions

Key Result

Theorem 1.3

Let the initial data $(h_0,u_0)$ satisfy mass--rr-incond and $T > 0$. Then there exists a weak solution $(h,u)$ in the sense of Definition Def-weak, where $v=h^2-1$. Moreover, the sets $\{ v(.,t) = 0\}$ and $\{ |h_x(.,t)| = \infty\}$ have Lebesgue measure zero for any $t \in [0,T]$ for the plug flow

Figures (4)

  • Figure 1: Schematic plot of a liquid film flowing down a cylindrical fibre. The axial coordinate along the fibre axis is $x$, and the radial distance from that axis is $h(x,t)$. The dimensionless fibre radius $R = 1$.
  • Figure 2: Typical travelling wave profiles (left) $H(\xi)$ and (right) $U(\xi)$ for two plug flow cases ((a) and (b)) and two laminar flow cases ((c) and (d)).
  • Figure 3: Dynamics of plug flow with (top left) $h(x,t)$ and (top right) $u(x,t)$ starting from initial profiles \ref{['eq:ic']} with $h_0 = 2.29$, showing that the PDE solution approaches a travelling wave solution $(H(\xi), U(\xi))$ satisfying equations \ref{['t-1']} -- \ref{['t-2']} with the velocity $s=1.396$. The solutions are shifted so that the maximums are aligned. The corresponding energy (bottom left) satisfies the estimate \ref{['e-3']}, $\mathcal{E}(t) + \mathcal{I}(t) < C_0(t)$, where $\mathcal{I}(t) = c \iint \limits_{Q_t} { v u^2_x \,dx dt} + \iint \limits_{Q_t} { \tfrac{u^2 v }{g(h)} \,dx dt}$. The entropy (bottom right) satisfies the estimate \ref{['n-7']}, $\int_0^L v_x^2/v\ dx < C_3(t)$. The system parameters are $L = 20$, $a=0.2$, $b=10$, $c = 1$ with $g(h) = h^2-1$.
  • Figure 4: Dynamics of laminar flow (top left) $h(x,t)$ and (top right) $u(x,t)$ starting from initial profiles \ref{['eq:ic']} with $h_0 = 2.29$, showing that the PDE solution approaches a travelling wave solution $(H(\xi), U(\xi))$ satisfying equations \ref{['t-1']} - \ref{['t-2']} with the velocity $s=0.1$. The solutions are shifted so that the maximums are aligned. Again, the corresponding energy plot (bottom left) shows that the energy satisfies the estimate \ref{['e-3']}, $\mathcal{E}(t) + \mathcal{I}(t) < C_0(t)$, where $\mathcal{I}(t) = c \iint \limits_{Q_t} { v u^2_x \,dx dt} + \iint \limits_{Q_t} { \tfrac{u^2 v }{g(h)} \,dx dt}$. The entropy (bottom right) satisfies the estimate \ref{['n-7']}, $\int_0^L v_x^2/v\ dx < C_3(t)$. The system parameters are $L = 20$, $a=0.1$, $b=11$, $c = 4$, $g(h) = I(h)/(h^2-1)$.

Theorems & Definitions (18)

  • Remark 1.1
  • Definition 1.2
  • Theorem 1.3
  • Lemma 2.1: Energy inequality
  • proof : Proof of Lemma \ref{['lem-en']}
  • Lemma 2.2: Entropy inequality
  • Remark 2.3
  • proof : Proof of Lemma \ref{['lem-entr']}
  • Remark 2.4
  • Lemma 2.5
  • ...and 8 more