A ferrofluid drop located on a horizontal plane and covering a vertical current-carrying conductor under the action of a magnetic field generated by the conductor and gravity is considered. The numerical study is based on a mathematical model of equilibrium states, consisting of equations for the free-surface shape of the ferrofluid drop and equations for the concentration of interacting magnetic nanoparticles inside the ferrofluid. The computational algorithm is organized as an iterative coupling between the finite-difference approximation of the free-surface and the Newton's method for the concentration equations formulated on an a-posteriori adaptive mesh, representing the discrete free-surface. Numerical results predict that the interactions between magnetic nanoparticles have little effect on the equilibrium free-surface shapes in weak magnetic fields but lead to a stronger shape elongation in moderate magnetic fields when compared to simulations with diffusion of non-interacting particles. Simulations have shown a sharp transition from drop-like shapes to sheath-like coverage of the conductor for a ferrofluid drop of large volume at different contact angles.

Objectives. A variational-difference method for numerical simulation of equilibrium capillary surfaces based on the minimization of the energy functional is proposed. As a test task a well-known axisymmetric hydrostatic problem on equilibrium shapes of a drop adjacent to a horizontal rotating plane under gravity is considered. The mathematical model of the problem is built on the basis of the variational principle: the shape of the drop satisfies the minimum total energy for a given volume. The problem of the functional minimization is reduced to a system of nonlinear equations using the finite element method. To solve the system a Newton's iterative method is applied. Methods. The variational-difference approach (the finite element method) is used. The finite linear functions are chosen as basic functions. Results. Equilibrium shapes of a drop on a rotating plane are constructed by the finite element method in a wide range of defining parameters: Bond number, rotational Weber number and wetting angle. The influence of these parameters on the shape of a drop is investigated. The numerical results are matched with the results obtained using the iterative-difference approach over the entire range of physical stability with respect to axisymmetric perturbations. Conclusion. The finite element method responds to the loss of stability of a drop with respect to axisymmetric perturbations. Therefore it can be used to study the stability of the equilibrium of axisymmetric capillary surfaces.

Shielding properties of a cylindrical thick-walled ferrofluid layer that protects against externally applied uniform magnetic fields are numerically investigated. We take into account the diffusion of magnetic nanoparticles in the ferrofluid with magnetic dipole-dipole, steric and hydrodynamic interactions between particles. Permeability of the ferrofluid is considered to be dependent on the magnetic-field strength and the particle concentration. A combined method of finite differences and boundary elements is applied to solve a nonlinear transmission problem of magnetostatics in the whole space, augmented by nonlinear algebraic equations based on the mass transfer equation for magnetic nanoparticles in ferrofluids. Numerical experiments revealed that the diffusion of particles has negligible influence on the shielding properties at weak and strong intensities of the applied magnetic field when comparing with the results of computations for a uniform particle distribution.

A coupled method of finite differences and boundary elements is applied to solve a nonlinear transmission problem of magnetostatics. The problem describes an interaction of a uniform magnetic field with a cylindrical ferrofluid layer. Ferrofluid magnetisations, based on expansions over the Langevin law, are considered to model ferrofluids with a different concentration of ferroparticles. The shielding effectiveness factor of the cylindrical thick-walled ferrofluid layer is calculated depending on intensities of the uniform magnetic field and on thickness of the ferrofluid layer.

A stable finite-difference scheme is constructed on a minimum stencil of a uniform mesh for a two-dimensional steady-state convection – diffusion equation of a general form; the scheme is theoretically studied and tested. It satisfies the maximum principle and has the fourth order of approximation. The scheme monotonicity is controlled by two regularization parameters introduced into the difference operator. The scheme is focused on solving applied convection – diffusion problems with a developed boundary layer, including gravitational convection, thermomagnetic convection, and diffusion of particles in a magnetic fluid. The scheme is tested on the well-known problem of a high-intensive gravitational convection in a horizontal channel of a square cross-section with a uniform heating from the side. A detailed comparison is performed with the monotone Samarskii scheme of the second order approximation on the sequences of square meshes with the number of partitions from 10 to 1000 on each side of the square domain and over the entire range of the Rayleigh numbers, corresponding to the laminar convection mode. A significant advantage of the fourth order scheme in the convergence rate is shown for the decreasing mesh step.

This paper considers the numerical solution of boundary integral equations for an exterior transmission problem in a three-dimensional axisymmetric domain. The resulting potential problem is formulated in a meridian plane as the second kind integral equation for a boundary potential and the first kind integral equation for a boundary flux. The numerical method is an axisymmetric collocation with equal order approximations of the boundary unknowns on a polygonal boundary. The complete elliptic integrals of the kernels are approximated by polynomials. An asymptotic kernels behavior is analyzed for accurate numerical evaluation of integrals. A piecewise-constant midpoint collocation and a piecewise-linear nodal collocation on a circular arc and on its polygonal interpolation are used for test computations on uniform meshes. We analyze empirically the influence of the polygonal boundary interpolation to the accuracy and the convergence of the presented method. We have found that the polygonal boundary interpolation does not change the convergence behavior on the smooth boundary for the piecewise-constant and the piecewise-linear collocation.

Three mathematical models on static equilibrium distribution of ferromagnetic particles in a ferrofluid, which has a free surface, are described for dilute, weakly concentrated and moderately concentrated ferrofluids, respectively. The models differ in description of the mass transfer of particles and fluid magnetization. Numerical computations are performed concerned with the problem of ferrofluid layer instability in an applied uniform magnetic field (the Rosensweig instability) to resolve axisymmetric equilibrium states relative to the particles' distribution, to the magnetic field structure and to the free surface shape of the fluid. It is observed that the onset of ferrofluid layer instability does not depend on the diffusion of particles (interacting or not). However, diffusion causes an elongation of peaks in a long-term range (several days) with the highest concentration in the peak region.

We derive four different numerical schemes to compute the shape of axisymmetric drops at the tip of a capillary for a given drop volume and a given Bond number. Small drop volumes show a convex shape, however increasing the drop volume a point of inflection appears and at a critical drop volume (depending from the Bond number) the drop detaches from its capillary. One of the four schemes produces drop shapes completely different from the others. We show that these shapes are unstable by solving numerically the free boundary value problem for the transient Navier-Stokes equations. (C) 2015 Elsevier B.V. All rights reserved.

We derive a mathematical model for studying the stability of magnetic fluid seal under the action of external pressure drop in the static case. We propose a numerical algorithm for computing the free surface shape under the influence of diffusion of magnetic particles. We include also the case in which the particle concentration achieves its maximum corresponding to the dense packing of the particles. Numerical experiments for various sets of parameters give interesting insights in the dependence of the burst pressure on different parameters. (C) 2011 Elsevier B.V. All rights reserved.

The present study is devoted to the development of an ADI approach to simulate two-dimensional time-dependent diffusion process of ferromagnetic particles in magnetic fluids. Specific features of the problem are the Neumann boundary conditions. We construct an ADI scheme of formally second order accuracy approximation in time and space. It is proved that the scheme is absolutely stable and it has the accuracy in the energy norm of the second order in time and the order 3/2 in space. The numerical results of a test problem indicate that the convergence rate in space is of the second order as well.

The present study is devoted to the classical problem on stability of a magnetic fluid layer under the influence of gravity and a uniform magnetic field. A periodical peak‐shaped stable structure is formed on the fluid surface when the applied magnetic field exceeds a critical value. The mathematical model describes a single peak in the pattern assuming axial symmetry of the peak shape. The field configuration in the whole space, the magnetic particle concentration inside the fluid and the free surface structure are unknown quantities in this model. The unknown free surface is treated explicitly, using a parametric representation with respect to the arc length. The nonlinear problem is discretized by means of a finite element method for the Maxwell's equations and a finite‐difference method for the free surface equations. Numerical modelling allows to get over‐critical equilibrium free surface shapes in a wide range of applied field intensities. Our numerical results show a significant influence of the particle diffusion on the overcritical shapes.

In a magnetic field longitudinal to a cylindrical capillary, a magnetic fluid drop elongates until surface instability occurs. Two different types of instabilities arise as the magnetic field intensity increases. For fluids with small contact angles, the instability is caused by the drop breakage along the capillary axis with a subsequent fluid spreading over the capillary wall. For fluids with large contact angles, the instability is due to fluid separation from the capillary wall with a formation of a freely suspended drop inside the capillary. The main objective of the present work is a numerical study of the instability phenomena under a longitudinal uniform magnetic field. The problem is considered in a wide range of contact angles.

New aspects related to the redistribution of magnetic particles concentration in a magnetic fluid caused by magnetophoresis and Brownian diffusion in a nonuniform magnetic field are considered. These aspects deal with the influence of these processes on the pressure redistribution and levitation of bodies in a magnetic fluid. It is shown that due to these processes the pressure force acting on bodies changes significantly with time and can be reduced dozens of percent if compared to a homogenous fluid.

A mathematical model and a computational method for studying the influence of the particle diffusion on equilibrium shapes of a magnetic liquid is developed. It is then applied on the ferrohydrostatic problem of doubly connected equilibrium shapes of a magnetic fluid located on a horizontal plate around a vertical cylindrical conductor with a direct current. Numerical simulations show the limits of the uniform concentration approximation.

A mathematical model for the diffusion process of ferromagnetic particles in a magnetic fluid is described. The unique solvability of the steady‐state particle concentration problem is investigated and an analytical expression for its solution is found. In case that the fluid is under the action of a high‐gradient magnetic field a Stefan‐type diffusion problem can arise. An algorithm for solving the Stefan‐type steady‐state problem is developed.

In the present study, the influence of Brownian diffusion of magnetic particles in a magnetic fluid on a force, which acts upon a fluid volume under an external inhomogenous magnetic field to be induced by a pair of flat poles, is considered in plane geometry. It is shown that the re-distribution of magnetic particle concentrations causes this force to undergo considerable changes with time.

Introduction. As a magnetic fluid represents a colloid of a solid ferromagnetic, it is natural that the processes of magnetophoresis and Brownian diffusion proceed in it [1]. A more detailed description of these processes is given in [2], especially as applied to magnetic concentration convection. They take on great significance when magnetic fluids are used for designing high-precision sensor facilities and their stable parameters should be provided, on the one hand, and when magnetic fields have large gradients, on the other [3, 4]. The most essential manifestation of the processes of magnetodiffusion is revealed in magnetic fluid seals [5, 6]. More frequently, there is a tendency to diminish the influence of these processes in a magnetic fluid by choosing moderate values of magnetic field gradients and also their short residence in the static state in these fields. Nevertheless, these processes always occur in magnetic fluids and their detailed study is of undoubted interest. When the intensity of a magnetic field is non-linearly distributed in a magnetic fluid, magnetic particles are concentrated in those regions, where the magnetic field intensity is larger. Accordingly, the fluid magnetization increases in these regions. As a rule, these regions are characterized by the highest gradients of the magnetic field intensity. Since a magnetic force acting upon a magnetic fluid volume is defined as the product of fluid magnetization by field gradient, its nonlinear growth with time must be observed. For short, this force will be further referred to as the magnetic weight of a fluid.