Repulsive interactions between magnetic bubbles: Consequences for bubble devices

Experiments and calculations are reported about some aspects of the repulsive interactions of cylindrical magnetic bubble domains. Experiments were done on hexagonal lattices of bubbles in which, for one case, the stable positions of the bubbles were determined by a Permalloy overlay. The calculations refer to different structures and Show that from a point of view of bubble interactions the maximum bubble density for a square lattice is almost the same as for a hexagonal lattice. It is also shown that a preferred thickness of the bubble material sheet exists.     

Parameters of bubbles and drops moving in a liquid

The author uses wave theory to examine the processes of formation of bubbles and drops moving in a liquid.Translated from Inzhenerno-Fizicheskoi Zhurnal, Vol. 59, No. 1, pp. 109–115, July, 1990.     

A fixed domain method for diffusion with a moving boundary

Summary The one-dimensional diffusion equation for a region with one fixed boundary and one unknown moving boundary is transformed to a non-linear equation on a fixed region by using the moving boundary position as the time variable. The boundary velocity becomes a second dependent variable, with dependence only on the new time variable. An implicit finite difference scheme, marching in time, is applied to a problem with known analytic solution to demonstrate the computing speed and accuracy of this approach, and also to a problem solved previously by variable time step methods. This transformation reduces any parabolic or elliptic system of equations on a domain with moving boundary, or with unknown free surface in two space variables, to a non-linear fixed domain system which has advantages for computation.     

Dissolution of diagnostic gas bubbles in transformer oil

This work is devoted to the problem of studying the propagation processes of the main diagnostic gases in oil-filled electric equipment. By experimentally and numerically simulating bubble dissolution, the diffusion coefficients of three gases have been determined: hydrogen, methane, and ethane in transformer oil. The value of the diffusion coefficient of hydrogen, 3.3 × 10⁻⁹ m²/s is not abnormally large, which has important practical value for diagnostics of oil-filled electric equipment.     

Modelling and estimation for a moving boundary problem

The displacement of a slightly compressible liquid by another in a porous medium has been modelled. This problem, which involves a moving boundary, has been numerically solved for the one-dimensional case by using the Galerkin method. State and parameter estimation have been carried out using the extended Kalman filter.     

Coupling of polynomial approximations with application to a boundary meshless method

New bridging techniques are introduced to match high degree polynomials. This permits piecewise resolutions of elliptic partial differential equations (PDEs) in the framework of a boundary meshless method introduced recently. This new meshless method relies on the computation of Taylor series approximations deduced from the PDE, the shape functions being high degree polynomials. In this way, the PDE is solved quasi‐exactly inside the subdomains so that only discretization of the boundary and the interfaces are needed, which leads to small size matricial problems. The bridging techniques are based on the introduction of Lagrange multipliers and a set of collocation points on the boundary and the interfaces. Several numerical applications establish that the method is robust and permits an exponential convergence with the degree. Copyright © 2013 John Wiley & Sons, Ltd.     

Natural element approximations involving bubbles for treating mechanical models in incompressible media

In this paper, a new approach is proposed to address issues associated with incompressibility in the context of the meshfree natural element method (NEM). The NEM possesses attractive features such as interpolant shape functions or auto‐adaptive domain of influence, which alleviates some of the most common difficulties in meshless methods. Nevertheless, the shape functions can only reproduce linear polynomials, and in contrast to moving least squares methods, it is not easy to define interpolations with arbitrary approximation consistency. In order to treat mechanical models involving incompressible media in the framework of mixed formulations, the associated functional approximations must satisfy the well‐known inf–sup, or LBB condition. In the proposed approach, additional degrees of freedom are associated with some topological entities of the underlying Delaunay tessellation, i.e. edges, triangles and tetrahedrons. The associated shape functions are computed from the product of the NEM shape functions related to the original nodes. Different combinations can be used to construct new families of NEM approximations. As these new approximations functions are not related to any node, as they vanish at the nodes, from now on we refer these shape functions as bubbles. The shape functions can be corrected enforcing different reproducing conditions, when they are used as weights in the moving least square (MLS) framework. In this manner, the effects of the obtained higher approximation consistency can be evaluated. In this work, we restrict our attention to the 2D case, and the following constructions will be considered: (a) bubble functions associated with the Delaunay triangles, called b1‐NEM and (b) bubble functions associated with the Delaunay edges, called b2‐NEM. We prove that all these approximation schemes allow direct enforcement of essential boundary conditions. The bubble‐NEM schemes are then used to approximate the displacements in the linear elasticity mixed formulation, the pressure being approximated by the standard NEM. The numerical LBB test is passed for all the bubble‐NEM approximations, and pressure oscillations are removed in the incompressible limit. Copyright © 2005 John Wiley & Sons, Ltd.     

Pressure distribution around a single bubble moving in a fluidized bed

Results are presented on the pressure distribution within a bubble and around it, the bubble being in a fluidized bed.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol.21, No. 6, pp. 998–1004, December, 1971.     

A numerical simulation for two-dimensional moving boundary problems with a mushy zone

For materials such as alloy, organic phase-change materials and many others, the change of phases may take place over a temperature range. This leads to phase-change problems with the mushy zone in which the solid and liquid phases coexist. The present study introduces a numerical method combining the Laplace transform technique and the control volume method to solve two-dimensional phase-change problems with the mushy zone. The hybrid numerical method involves the control volume formulation for the space domain and the Laplace transform technique for the time domain. The Taylor's series approximation is applied to linearize nonlinear terms in the governing equation. The transfinite mapping method is used to generate control-volume meshes in each region. The growth of the mushy zone is unknown a priori and is predicted by using the least-square iteration scheme. It will be found that the present hybrid numerical method can be efficiently applied to solve two-dimensional phase-change problems with a mushy zone.     

Solution of a drying problem for a colloidal porous material with a moving evaporation boundary

Closed solutions are derived for linear differential equations for one-dimensional drying with an evaporation boundary moving in accordance witht.Translated from Inzhenerno-Fizieheskii Zhurnal, Vol. 25, No. 5, pp. 871–876, November, 1973.     

Some moving boundary problems for a generalised diffusion equation

Diffusion controlled growth of planar, cylindrical or spherical particles is considered where the composition satisfies a generalised diffusion equation. Asymptotic methods are used to determine composition gradients near the growing particles in the cases of “fast” or “slow” growth. It is shown that in these limits under certain conditions a constant average diffusion coefficient can be defined for which the growth rate calculated from the differential equation with this constant diffusion coefficient agrees with that with the full variable coefficient. Similar considerations are also made of the case of composition dependent diffusion and bulk flow for growing spherical particles.     

Transient heat conduction in hollow spheres with a moving inner boundary

The finite integral transform method is used to obtain the solution of unsteady heat conduction problems for a hollow sphere with a moving internal boundary and various boundary conditions at the outer surface. For the solution of the problems of interest integral transform formulas are presented with kernels (16), (20), and (24) and the corresponding inversion formulas (18), (22), (26), (29) and characteristic equations (17), (21), (25), (28), (31), (33).Nomenclature a, thermal diffusivity and conductivity - t temperature of phase transformation - density - heat transfer coefficient - Q total quantity of heat passing through inner boundary - F latent heat of phase transformation - Fo(1,)=a/R ₁ ² , Fo(i,)=/r _i ² , Fo(i, _i)=a _i/r _i ² Fourier numbers - Bi₂=R₂/ Biot number     

Nonlinear oscillations of a bubble under various boundary conditions

The method of many scales is used to examine the nonlinear oscillations of a spherical gas bubble that occur under the action of a periodically changing external pressure in a spherical volume of inviscid, incompressible liquid and in a liquid flow. The influence of the finite dimensions of the volume and liquid flow velocity on the conditions for the existence of certain types of the equilibrium state of an oscillating bubble is analyzed. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 70, No. 2, pp. 211–215, March–April, 1997.     

Modelling and state estimation for a two-dimensional moving boundary problem

The five-spot waterflooding problem has been modelled as a two-dimensional moving boundary problem with a sharp interface separating the water and oil regions. The Galerkin method was used to solve for the shape and movement of the interface, as well as for the pressures in the reservoir. Having obtained a working model, the extended Kalman filter was then used to estimate the interfacial position when using corrupted pressure measurements from a single sensor in the field.     

Unsteady heat conduction of a thin rod with a uniformly moving fusion boundary or a thermally insulated boundary

A solution is obtained to the problem of unsteady heat conduction of a semi-infinite thin rod, thermally insulated along its generators, or a plate, with a uniformly moving fusion boundary, or a thermally insulated boundary. The results are presented of a numerical calculation for a copper rod with various rates of motion of the boundary.     

On the velocity of a moving phase boundary in solids

Summary A general relationship between the driving force and the velocity of a moving phase boundary in thermoelastic solids is established on the basis of non-equilibrium jump relations at the discontinuity. The non-equilibrium jump relations are formulated in terms of contact quantities and local equilibrium fields. The contact quantities are introduced following ideas of the thermodynamics of discrete systems. It is shown that under certain simplifications the derived relationship can be reduced to a known kinetic relation.     

A moving boundary circulation model for regions with large tidal flats

A numerical model is developed for computing two-dimensional circulation in coastal regions dominated by large tidal flat motion. This model is based on the shallow water momentum and continuity equations being operated upon by the Galerkin finite element method. The particular advantage of the model lies in its ability to accommodate a changing domain boundary. This is accomplished by adopting finite elements that change shape so as to consistently move with the water's edge. Application of the model to the Kuwait Bay is described, and the associated results on circulation are given.     

Soap bubbles in analytical chemistry. Conductometric determination of sub-parts per million levels of sulfur dioxide with a soap bubble

Soap bubbles provide a fascinating tool that is little used analytically. With a very low liquid volume to surface area ratio, a soap bubble can potentially provide a very useful interface for preconcentration where mass transfer to an interfacial surface is important. Here we use an automated system to create bubbles of uniform size and film thickness. We utilize purified Triton-X 100, a nonionic surfactant, to make soap bubbles. We use such bubbles as a gas-sampling interface. Incorporating hydrogen peroxide into the bubble provides a system where electrical conductance increases as the bubble is exposed to low concentrations of sulfur dioxide gas. We theoretically derive the conductance of a hollow conducting spherical thin film with spherical cap electrodes. We measure the film thickness by incorporating a dye in the bubble making solution and laser transmission photometry and find that it agrees well with the geometrically computed thickness. With the conductance of the bubble-making soap solution being measured by conventional methods, we show that the measured values of the bubble conductance with known bubble and electrode dimensions closely correspond to the theoretically computed value. Finally, we demonstrate that sub-ppm levels of SO(2) can readily be detected by the conductivity change of a hydrogen peroxide-doped soap bubble, measured in situ, when the gas flows around the bubble.     

Heat transfer during filtration in a heterogeneous medium with input conditions at a moving boundary

Theoretical results on the heat transfer in combustion-product filtration in the rock-collapse zone behind a borehole-damaged coal bed are outlined. A sufficiently simple engineering solution of the problem is obtained by means of an equivalent heat-conduction equation, and the limiting temperatures corresponding to different parameter values are analyzed.Translated from Inzhenerno-fizicheskii Zhurnal, Vol. 61, No. 2, pp. 313–318, August, 1991.