Deformation-induced lateral migration of a bubble slowly rising near a vertical plane wall in a stagnant liquid is numerically and theoretically investigated. In particular, our focus is set on a situation with a small clearance $c$ between the bubbl
e interface and the wall. Motivated by the fact that experimentally measured migration velocity (Takemura et al. (2002, J. Fluid Mech. {bf 461}, 277)) is higher than the velocity estimated by the available analytical solution (Magnaudet et al. (2003, J. Fluid Mech. {bf 476}, 115)) using the Fax{e}n mirror image technique for $kappa(=a/(a+c))ll 1$ (here $a$ is the bubble radius), when the clearance parameter $epsilon(=c/a)$ is comparable to or smaller than unit, the numerical analysis based on the boundary-fitted finite-difference approach by solving the Stokes equation is performed to complement the experiment. To improve the understandings of a role of the squeezing flow within the bubble-wall gap, the theoretical analysis based on a soft-lubrication approach (Skotheim & Mahadevan (2004, Phys. Rev. Lett. {bf 92}, 245509)) is also performed. The present analyses demonstrate the migration velocity scales $propto{rm Ca} epsilon^{-1}V_{B1}$ (here, $V_{B1}$ and ${rm Ca}$ denote the rising velocity and the capillary number, respectively) in the limit of $epsilonto 0$.
A new simulation method for solving fluid-structure coupling problems has been developed. All the basic equations are numerically solved on a fixed Cartesian grid using a finite difference scheme. A volume-of-fluid formulation (Hirt and Nichols (1981
, J. Comput. Phys., 39, 201)), which has been widely used for multiphase flow simulations, is applied to describing the multi-component geometry. The temporal change in the solid deformation is described in the Eulerian frame by updating a left Cauchy-Green deformation tensor, which is used to express constitutive equations for nonlinear Mooney-Rivlin materials. In this paper, various verifications and validations of the present full Eulerian method, which solves the fluid and solid motions on a fixed grid, are demonstrated, and the numerical accuracy involved in the fluid-structure coupling problems is examined.
Deformation-induced lateral migration of a bubble slowly rising near a vertical plane wall in a stagnant liquid is numerically and theoretically investigated. In particular, our focus is set on a situation with a short clearance $c$ between the bubbl
e interface and the wall. Motivated by the fact that numerically and experimentally measured migration velocities are considerably higher than the velocity estimated by the available analytical solution using the Fax{e}n mirror image technique for $a/(a+c)ll 1$ (here $a$ is the bubble radius), when the clearance parameter $varepsilon(= c/a)$ is comparable to or smaller than unity, the numerical analysis based on the boundary-fitted finite-difference approach solving the Stokes equation is performed to complement the experiment. The migration velocity is found to be more affected by the high-order deformation modes with decreasing $varepsilon$. The numerical simulations are compared with a theoretical migration velocity obtained from a lubrication study of a nearly spherical drop, which describes the role of the squeezing flow within the bubble-wall gap. The numerical and lubrication analyses consistently demonstrate that when $varepsilonleq 1$, the lubrication effect makes the migration velocity asymptotically $mu V_{B1}^2/(25varepsilon gamma)$ (here, $V_{B1}$, $mu$, and $gamma$ denote the rising velocity, the dynamic viscosity of liquid, and the surface tension, respectively).
