Numerical simulation of deformation of a droplet in a stationary electric field using DG

Numerical simulation of deformation of a droplet in a stationary electric field is performed in the present research. The droplet is suspended in another immiscible fluid with the same density and viscosity but a different dielectric property (permittivity). By applying the electric field, the fluids are polarized that gives rise to mechanical forces and deformation. A two-way coupling occurs because of the forces exerted from the electric field on the droplet and the deformation of the droplet which changes the geometry for the electric field calculations. The droplet continues to deform until a force balance between the electric force, pressure and the surface tension is achieved and the droplet becomes a spheroid. An electromechanical approach is adopted to solve the above mentioned problem, which includes solving the governing equations of both the electric and fluid fields, computing the coupling forces and capturing the movement of the interface of the droplet and the surrounding fluid. A one-fluid approach is followed, which enables us to solve one set of the governing equations for both the droplet and the surrounding fluid. The interface is represented as the zero iso-value of a level set function and an advection equation is solved to find the movement of the interface. A diffuse interface model is used to regularize the jump in the fluid and electric properties. The governing equations of the electric and fluid fields and the level set advection equation are discretized using the Discontinuous Galerkin Finite Element method (DG) in the BoSSS code for solving conservation laws. The electric field is computed from the electric potential by considering the electrostatic equations. To find the electric potential, a Laplace equation is solved which has a jump in the permittivity at the interface. The Laplace equation is discretized using the interior penalty method (IP) which we modified for the case of high jumps in the permittivity. Assuming that the fluids are linear dielectric materials, the electric force is the dielectrophoretic force which is computed from the Kortweg-Helmholtz formula. This force is added as a body force to the incompressible Navier-Stokes equations, which are the governing equations for the fluid flow. Considering that there is no jump in the fluid properties, a single phase solver of the Navier-Stokes equations including the surface tension at the interface is developed. The surface tension force is added as a body force to the Navier-Stokes equations using the continuum surface force model (CSF). This model is known for producing a spurious velocity field. To decrease the spurious velocities, the surface tension term is calculated by using high degree polynomials for a precise calculation of the normal vector and curvature. To solve the incompressible Navier-Stokes equations using the DG method, a projection scheme with a consistent Neumann pressure boundary condition is employed and the same polynomial order for the velocity and pressure (equal-order method) is applied. Using the above-mentioned pressure boundary condition leads to an optimal convergence rate of k + 1 in the L2-norm for the pressure, which is not reported from other DG solvers. However, using the DG method, we have observed that discontinuities in the solutions at the cell boundaries can affect the solution accuracy and even cause a numerical instability. These accuracy and stability issues occur when the derivatives of the solution are computed. Therefore a flux-based method for calculation of the derivatives of the flow variables was adopted. As the results showed considerably improved accuracy and stability characteristics, we used the proposed method also in solving the above mentioned coupled problem

Levitation of non-magnetizable droplet inside ferrofluid

The central theme of this work is that a stable levitation of a denser non-magnetizable liquid droplet, against gravity, inside a relatively lighter ferrofluid -- a system barely considered in ferrohydrodynamics -- is possible, and exhibits unique interfacial features; the stability of the levitation trajectory, however, is subject to an appropriate magnetic field modulation. We explore the shapes and the temporal dynamics of a plane non-magnetizable droplet levitating inside ferrofluid against gravity due to a spatially complex, but systematically generated, magnetic field in two dimensions. The effect of the viscosity ratio, the stability of the levitation path and the possibility of existence of multiple-stable equilibrium states is investigated. We find, for certain conditions on the viscosity ratio, that there can be developments of cusps and singularities at the droplet surface; this phenomenon we also observe experimentally and compared with the simulations. Our simulations closely replicate the singular projection on the surface of the levitating droplet. Finally, we present an dynamical model for the vertical trajectory of the droplet. This model reveals a condition for the onset of levitation and the relation for the equilibrium levitation height. The linearization of the model around the steady state captures that the nature of the equilibrium point goes under a transition from being a spiral to a node depending upon the control parameters, which essentially means that the temporal route to the equilibrium can be either monotonic or undulating. The analytical model for the droplet trajectory is in close agreement with the detailed simulations. (See draft for full abstract).Comment: This article has been published in a revised form in Journal of Fluid Mechanics http://dx.doi.org/10.1017/jfm.2018.733. Copyright: copyright holde

A dipolar droplet bound in a trapped Bose-Einstein condensate

We study the statics and dynamics of a dipolar Bose-Einstein condensate (BEC) droplet bound by inter-species contact interaction in a trapped non-dipolar BEC. Our findings are demonstrated in terms of stability plots of a dipolar 164Dy droplet bound in a trapped non-dipolar 87Rb BEC with a variable number of 164Dy atoms and the inter-species scattering length. A trapped non-dipolar BEC of a fixed number of atoms can only bind a dipolar droplet containing atoms less than a critical number for the inter-species scattering length between two critical values. The shape and size (statics) as well as the small breathing oscillation (dynamics) of the dipolar BEC droplet are studied using the numerical and variational solutions of a mean-field model. We also suggest an experimental procedure for achieving such a 164Dy droplet by relaxing the trap on the 164Dy BEC in a trapped binary 87Rb-164Dy mixture

Oscillations of weakly viscous conducting liquid drops in a strong magnetic field

We analyse small-amplitude oscillations of a weakly viscous electrically conducting liquid drop in a strong uniform DC magnetic field. An asymptotic solution is obtained showing that the magnetic field does not affect the shape eigenmodes, which remain the spherical harmonics as in the non-magnetic case. Strong magnetic field, however, constrains the liquid flow associated with the oscillations and, thus, reduces the oscillation frequencies by increasing effective inertia of the liquid. In such a field, liquid oscillates in a two-dimensional (2D) way as solid columns aligned with the field. Two types of oscillations are possible: longitudinal and transversal to the field. Such oscillations are weakly damped by a strong magnetic field - the stronger the field, the weaker the damping, except for the axisymmetric transversal and inherently 2D modes. The former are overdamped because of being incompatible with the incompressibility constraint, whereas the latter are not affected at all because of being naturally invariant along the field. Since the magnetic damping for all other modes decreases inversely with the square of the field strength, viscous damping may become important in a sufficiently strong magnetic field. The viscous damping is found analytically by a simple energy dissipation approach which is shown for the longitudinal modes to be equivalent to a much more complicated eigenvalue perturbation technique. This study provides a theoretical basis for the development of new measurement methods of surface tension, viscosity and the electrical conductivity of liquid metals using the oscillating drop technique in a strong superimposed DC magnetic field.Comment: 17 pages, 3 figures, substantially revised (to appear in J. Fluid Mech.