The role of particle inertia and particle-free-surface collisions in periodic free-surface flows is evaluated in the framework of an analytical flow model for a thermocapillary liquid bridge. Inertia and particle-free-surface collisions lead to particle accumulation, but on different time scales, and can lead to different accumulation patterns. A comparison with experimental results provides strong evidence that the experimentally observed accumulation patterns are due to particle-free-surface collisions.
SUMMARYThe immersed boundary approach for the modeling of complex geometries in incompressible flows is examined critically from the perspective of satisfying boundary conditions and mass conservation. It is shown that the system of discretized equations for mass and momentum can be inconsistent, if the velocity is used in defining the force density to satisfy the boundary conditions. As a result, the velocity is generally not divergence free and the pressure at locations in the vicinity of the immersed boundary is not physical. However, the use of the pseudo-velocities in defining the force density, as frequently done when the governing equations are solved using a fractional step or projection method, combined with the use of the specified velocity on the immersed boundary, is shown to result in a consistent set of equations which allows a divergence-free velocity but, depending on the time step, is shown to have the undesirable effects of inaccurately satisfying the boundary conditions and allowing a significant permeability of the immersed boundary. If the time step is reduced sufficiently, the boundary conditions on the immersed boundary can be satisfied. However, this entails an unacceptable increase in computational expense. Two new methods that satisfy the boundary conditions and allow a divergence-free velocity while avoiding the increased computational expense are presented and shown to be second-order accurate in space. The first new method is based on local time step reduction. This method is suitable for problems where the immersed boundary does not move. For these problems, the first new method is shown to be closely related to the second new method. The second new method uses an optimization scheme to minimize the deviation from the interpolation stencil used to represent the immersed boundary while ensuring a divergence-free velocity. This method performs well for all problems, including those where the immersed boundary moves relative to the grid. Additional results include showing that the force density that is added to satisfy the boundary conditions at the immersed boundary is unbounded as the time step is reduced and that the pressure in the vicinity of the immersed boundary is unphysical, being strongly a function of the time step. A method of computing the total force on an immersed boundary which takes into account
The formation of particle-accumulation structures in the flow in a cylindrical liquid bridge driven by the thermocapillary effect is studied with the aim of determining the physical mechanism which forms the structures. The flow is modeled using the incompressible Navier–Stokes and energy equations with the assumption of constant fluid properties except for surface tension, which is assumed to depend linearly on temperature. Different models for the motion of small non-interacting spherical particles at low concentration are employed, taking into account particle inertia due to density differences between fluid and particles and the restricted particle motion near the boundaries of the flow domain. Attention is focused on differences in formation time between particle-accumulation structures arising as a result of inertial effects only, particle–boundary-interaction effects only, and a combination of the two.
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