1Department of Mechanical Engineering
New Jersey Institute of Technology Newark, NJ 07102
2Department of Aerospace Engineering and
University of Minnesota Minneapolis, MN 55455
A finite element code based on the level set and distributed Lagrange multiplier (DLM) methods is developed for simulating the motion of rigid spherical particles on two-fluid interfaces. The interface position is tracked using a level set function. The contact angle on the particle surface is assumed to be constant, even when the contact line on the particle surface moves. The fluid-particle system is treated implicitly using a combined weak formulation in which the forces and moments between the particles and fluid cancel. The governing Navier-Stokes equations are solved everywhere, including inside the particles. The flow inside the particles is forced to be a rigid body motion by a distribution of Lagrange multipliers. The Marchuk-Yanenko operator-splitting technique is used to decouple the difficulties associated with the nonlinear convection term, the incompressibility constraint, the rigid body motion constraint inside the particles and the interface motion problem.
The code is verified by performing a convergence study to show that the numerical results are independent of the mesh and time step sizes. Simulations show that a spherical particle released on a two-fluid interface can remain suspended even when it is heavier than both liquids, provided the vertical component of interfacial force balances its buoyant weight. This can happen when the sphere is hydrophobic with the lower fluid and hydrophilic with the upper fluid, as in this case the interfacial force can act against gravity. The numerically determined floating height and the interface shape are in good agreement with the analytical results for static equilibrium. Our simulations also show that when two or more particles are released along the interface they, as in experiments, form clusters due the attractive capillary force which arises due to the asymmetric interface deformation around the particles. Specifically, the heavy particles lower the interface creating depression. The particles then experience a lateral attractive force due to gravity and capillarity. The gravity force causes the particle to fall into the depression; this lowers and tilts the contact line. The lower contact line is associated with a smaller vertical component of the capillary force for a fixed angle of contact and the rotation of the contact line produces imbalance in the horizontal component. The former increases the depth to which the particles sink in the lower fluid.
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