An Energy Stable Immersed Boundary Method for Deformable Membrane Problem with Non-uniform Density and Viscosity

Abstract

Membrane problems commonly encountered in engineering and biological applications involve large deformations and complex configurations. Immersed boundary (IB) method, formulated by the fluid equations in which the fluid-structure interaction is described in terms of the Dirac function, is one of the most powerful tools to simulate such problems. However, the IB method suffers from severe time step restrictions to maintain stability if the discretization lacks conservation of energy, especially for two-phase flows. In this paper, we develop an energy stable IB method for solving deformable membrane problems with non-uniform density and viscosity. Unlike the classic IB formulation, the evolution of membrane, including elastic tension and bending force, is controlled by its tangent angle and arc length. After minor modifications, it is shown that the model satisfies the continuous energy law. Thus, for the reformulated model, we proposed an implicit unconditionally energy stable scheme, where the energy of the scheme is proved to be dissipative. The resultant system is solved iteratively and the numerical results show that the proposed scheme is energy stable and capable of predicting the dynamics of extensible and inextensible interface problems with non-uniform density and viscosity.

A Appendix

For the term of bending force, we have

$$\begin{aligned}&\Delta t \langle {\varvec{u}}^{n+1},S_h^{n+1}(\textbf{F}_b^{n+1}) \rangle _{\Omega _h} \\&\quad = \sum _\textbf{x} \left( \sum _{k=1}^{M} (\textbf{F}_b)_{k}^{n+1} \delta _h(\textbf{x}-\textbf{X}_k^{n+1})\Delta \alpha \right) \cdot {\varvec{u}}^{n+1}(\textbf{x}) h^2 \Delta t \\&\quad = \sum _{k=1}^M (\textbf{F}_b)_{k}^{n+1} \cdot \left( \sum _\textbf{x} {\varvec{u}}^{n+1}\delta _h(\textbf{x}-\textbf{X}_k^{n+1}) h^2 \right) \Delta \alpha \Delta t \\&\quad = \sum _{k=1}^M c_b\left( D_\alpha \left( \frac{D_\alpha }{s_{\alpha ,k}^{n+1}} \left( \frac{D_\alpha \theta _k^{n+1}}{s_{\alpha ,k}^{n+1}}\right) \right) + \frac{1}{2}\frac{(D_\alpha \theta _k^{n+1})^3}{(s_{\alpha ,k}^{n+1})^2} \right) \varvec{n}_k^{n+1} \cdot (U_k^{n+1}\varvec{n}_k^{n+1} + V_k^{n+1}\varvec{\tau }_k^{n+1}) \Delta \alpha \Delta t \\&\quad = \sum _{k=1}^M c_b\left( D_\alpha \left( \frac{D_\alpha }{s_{\alpha ,k}^{n+1}} \left( \frac{D_\alpha \theta _k^{n+1}}{s_{\alpha ,k}^{n+1}}\right) \right) + \frac{1}{2}\frac{(D_\alpha \theta _k^{n+1})^3}{(s_{\alpha ,k}^{n+1})^2} \right) U_k^{n+1} \Delta \alpha \Delta t \\&\quad = \sum _{k=1}^M c_b\left( -\left( \frac{D_\alpha }{s_{\alpha ,k}^{n+1}} \left( \frac{D_\alpha \theta _k^{n+1}}{s_{\alpha ,k}^{n+1}}\right) \right) D_\alpha U_k^{n+1} + \frac{1}{2}\frac{(D_\alpha \theta _k^{n+1})^3}{(s_{\alpha ,k}^{n+1})^2}U_k^{n+1} \right) \Delta \alpha \Delta t \\&\quad = \sum _{k=1}^M -c_b \left( D_\alpha \left( \frac{D_\alpha \theta _k^{n+1}}{s_{\alpha ,k}^{n+1}}\right) \right) \left( -(\theta _k^{n+1}-\theta _k^n) + \Delta t \frac{V_k^{n+1}}{2} \left( \frac{D_\alpha \theta _{k+1}^{n+1}}{s_{\alpha ,k+1}^{n+1}} + \frac{D_\alpha \theta _{k-1}^{n+1}}{s_{\alpha ,k-1}^{n+1}}\right) \right) \Delta \alpha \\&\qquad + \sum _{k=1}^M \frac{c_b}{2}\frac{(D_\alpha \theta _k^{n+1})^2}{(s_{\alpha ,k}^{n+1})^2}((s_{\alpha ,k}^{n+1} - s_{\alpha ,k}^n) - \Delta t D_\alpha V_k^{n+1}) \Delta \alpha \\&\quad = \sum _{k=1}^M -c_b \left( \frac{D_\alpha \theta _k^{n+1}}{s_{\alpha ,k}^{n+1}}\right) (D_\alpha \theta _k^{n+1}-D_\alpha \theta _k^n)\Delta \alpha - \sum _{k=1}^M \frac{c_b}{2} D_\alpha \left( \frac{D_\alpha \theta _k^{n+1}}{s_{\alpha ,k}^{n+1}}\right) ^2 V_k^{n+1} \Delta t \Delta \alpha \\&\qquad + \sum _{k=1}^M \frac{c_b}{2}\frac{(D_\alpha \theta _k^{n+1})^2}{(s_{\alpha ,k}^{n+1})^2}(s_{\alpha ,k}^{n+1} - s_{\alpha ,k}^n)\Delta \alpha + \sum _{k=1}^M \frac{c_b}{2}D_\alpha \left( \frac{D_\alpha \theta _k^{n+1}}{s_{\alpha ,k}^{n+1}}\right) ^2 V_k^{n+1}\Delta t \Delta \alpha \\&\quad = \sum _{k=1}^M -c_b \left( \frac{(D_\alpha \theta _k^{n+1})^2}{s_{\alpha ,k}^{n+1}} - \frac{D_\alpha \theta _k^{n+1} D_\alpha \theta _k^n}{s_{\alpha ,k}^{n+1}}\right) \Delta \alpha + \sum _{k=1}^M \frac{c_b}{2}\left( \frac{(D_\alpha \theta _k^{n+1})^2}{s_{\alpha ,k}^{n+1}} - \frac{(D_\alpha \theta _k^{n+1})^2 s_{\alpha ,k}^n}{(s_{\alpha ,k}^{n+1})^2}\right) \Delta \alpha \\&\quad = \sum _{k=1}^M -\frac{c_b}{2} \frac{(D_\alpha \theta _k^{n+1})^2}{s_{\alpha ,k}^{n+1}} \Delta \alpha + \sum _{k=1}^M c_b \frac{(D_\alpha \theta _k^n)^2}{s_{\alpha ,k}^n}\frac{D_\alpha \theta _k^{n+1} s_{\alpha ,k}^n}{D_\alpha \theta _k^n s_{\alpha ,k}^{n+1}} \Delta \alpha - \sum _{k=1}^M \frac{c_b}{2} \frac{(D_\alpha \theta _k^n)^2}{s_{\alpha ,k}^n} \frac{(D_\alpha \theta _k^{n+1})^2 (s_{\alpha ,k}^n)^2}{(D_\alpha \theta _k^n)^ 2(s_{\alpha ,k}^{n+1})^2}\Delta \alpha \\&\quad = -B^{n+1} - \sum _{k=1}^M \frac{c_b}{2} \frac{(D_\alpha \theta _k^n)^2}{s_{\alpha ,k}^n} \left( \left( \frac{D_\alpha \theta _k^{n+1} s_{\alpha ,k}^n}{D_\alpha \theta _k^n s_{\alpha ,k}^{n+1}}\right) ^2 - 2\frac{D_\alpha \theta _k^{n+1} s_{\alpha ,k}^n}{D_\alpha \theta _k^n s_{\alpha ,k}^{n+1}} + 1 - 1 \right) \Delta \alpha \\&\quad = -B^{n+1} - \sum _{k=1}^M \frac{c_b}{2} \frac{(D_\alpha \theta _k^n)^2}{s_{\alpha ,k}^n} \left( \frac{D_\alpha \theta _k^{n+1} s_{\alpha ,k}^n}{D_\alpha \theta _k^n s_{\alpha ,k}^{n+1}} - 1 \right) ^2\Delta \alpha + \sum _{k=1}^M \frac{c_b}{2} \frac{(D_\alpha \theta _k^n)^2}{s_{\alpha ,k}^n} \Delta \alpha \\&\quad = -B^{n+1} + B^n - \frac{c_b}{2} \bigg \Vert \sqrt{\frac{(D_\alpha \theta ^n)^2}{s_{\alpha }^n}} \left( \frac{D_\alpha \theta ^{n+1} s_{\alpha }^n}{D_\alpha \theta ^n s_{\alpha }^{n+1}} - 1 \right) \bigg \Vert _{\Gamma _h}. \end{aligned}$$