Ellipsoid flowed around by a harmonic vector field

We consider the screening of an external magnetic field in which a superconducting ellipsoid is inserted and a change in the velocity distribution in an ideal liquid flowing around an ellipsoid inserted in it. In both cases, the solution is given by a harmonic vector field parallel to the surface near the ellipsoid.     

A numerical study of flows driven by a rotating magnetic field in a square container

The influence of the geometry on the magnetically driven flow is studied by means of numerical simulations. Low–frequency, low–induction and low–interaction conditions are assumed. The rotating magnetic field (RMF) gives rise to a time–independent magnetic body force, computed via the electrical potential equation and Ohm's law and a time–dependent part that is neglected due to the low interaction parameter. Flow results of the cylindrical and square container are compared with respect to the magnetic body force, time–averaged velocity fields, first flow instabilities and Reynolds stress tensors. The dependency of the maximal velocity magnitude and the intensity of the magnetic induction is identical in axisymmetric and non–axisymmetric containers and in good agreement with Davidson's theory. However, significant differences are recognized, for instance, in the distribution of the Reynolds stress tensors. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Calculating the electric charges that shield from an external coaxial electric field on the surface of a conducting axially symmetric body

A method of solving a one-dimensional integral equation for finding charges on the surface of a conducting axially symmetric body is given. For the case of an ellipsoid of rotation in an electric field with polynomial values on the axis of symmetry, an exact solution is obtained. The axis of symmetry of the body and the axis of the external field coincide. A numerical algorithm based on a combination of a projective method and a method of iterative regularization for solving a Fredholm equation of the first kind is proposed. The projectors are chosen as B-splines. The charges calculated for an ellipsoid of rotation are close to the analytical ones.     

The steady motions of a heavy ellipsoid filled with liquid on a plane with friction

The motion of a heavy uniform thin-walled ellipsoid of revolution, completely filled with an ideal incompressible liquid, performing uniform vortex motion is investigated. It is assumed that the ellipsoid is situated on a horizontal plane, from the side of which a normal reaction and a force of viscous sliding friction act on it. The equations of motion of the system, suitable both in the general case and in limiting cases of zero ellipsoid mass or zero liquid mass, are set up. Steady and periodic motions of the ellipsoid with the liquid are obtained. The conditions for uniform rotations of the ellipsoid about a vertically situated axis of symmetry to be stable are obtained.     

On a periodic motion of a rigid body carrying a material point in the presence of impacts with a horizontal plane

A material system consisting of an outer rigid body (a shell) and an inner body (a material point) is considered. The system moves in a uniform field of gravity over a fixed absolutely smooth horizontal plane. The central ellipsoid of inertia of the shell is an ellipsoid of rotation. The material point moves according to the harmonic law along a straight-line segment rigidly attached to the shell and lying on its axis of dynamical symmetry. During its motion, the shell may collide with the plane. The coefficient of restitution for an impact is supposed to be arbitrary. The periodic motion of the shell is found when its symmetry axis is situated along a fixed vertical, and the shell rotates around this vertical with an arbitrary constant angular velocity. The conditions for existence of this periodic motion are obtained, and its linear stability is studied.     

On the development of a wake vortex in inviscid flow

The evolution of an initial perturbation in an axisymmetric subsonic normal inviscid gas flow through a pipe is directly simulated. The basic (unperturbed) flow has a zero radial velocity component, while its axial velocity component (along the axis of symmetry) increases or decreases linearly with the radius. The perturbation is specified as a swirl (rotation about the axis) with a positive or negative velocity vanishing on the central axis and the lateral surface. Irrespective of its direction, the swirl gives rise to a steady-state vortex carried by the flow. It shape is spherical (contiguous to the rotation axis) or circular (sliding along the impermeable lateral surface).     

The stability of the steady motion of a gyrostat with a liquid in a cavity

A symmetrical rigid body with a spherical base, carrying a rotor and having a cavity in the shape of an ellipsoid of revolution, completely filled with an ideal incompressible liquid in uniform vortex motion, is moving along an absolutely rough plane. It is shown that this system admits of an energy integral, Jellett's integral, the integral of constant vorticity and a geometric integral. The construction of a Lyapunov function as a linear combination of first integrals [1] yields the sufficient conditions for the rotation of the gyrostat about the vertically positioned axis of symmetry to be stable. The conditions for the gyrostat's rotation to be unstable are found. It is shown that the rotor may prove to have either a stabilizing or destabilizing effect on the system and that the gyrostat admits of motions of the type of regular precession. The sufficient conditions for the stability of these motions are obtained.     

The stability of Couette flow in a toroidal magnetic field

The stability of the hydromagnetic Couette flow is investigated when a constant current is applied along the axis of the cylinders. It is shown that if the resulting toroidal magnetic field depends only on this current, no linear instability to axisymmetric disturbances is possible.     

Numerical and asymptotic study of non‐axisymmetric magnetohydrodynamic boundary layer stagnation‐point flows

Both numerical and asymptotic analyses are performed to study the similarity solutions of three‐dimensional boundary‐layer viscous stagnation point flow in the presence of a uniform magnetic field. The three‐dimensional boundary‐layer is analyzed in a non‐axisymmetric stagnation point flow, in which the flow is developed because of influence of both applied magnetic field and external mainstream flow. Two approaches for the governing equations are employed: the Keller‐box numerical simulations solving full nonlinear coupled system and a corresponding linearized system that is obtained under a far‐field behavior and in the limit of large shear‐to‐strain‐rate parameter (λ). From these two approaches, the flow phenomena reveals a rich structure of new family of solutions for various values of the magnetic number and λ. The various results for the wall stresses and the displacement thicknesses are presented along with some velocity profiles in both directions. The analysis discovered that the flow separation occurs in the secondary flow direction in the absence of magnetic field, and the flow separation disappears when the applied magnetic field is increased. The flow field is divided into a near‐field (due to viscous forces) and far‐field (due to mainstream flows), and the velocity profiles form because of an interaction between two regions. The magnetic field plays an important role in reducing the thickness of the boundary‐layer. A physical explanation for all observed phenomena is discussed. Copyright © 2017 John Wiley & Sons, Ltd.     

Exact and approximate expressions for resistance coefficients of a colloidal sphere approaching a solid surface at intermediate Reynolds numbers

A mathematical model has been developed to describe the force of liquid flow acting on a colloidal spherical particle as it approaches a solid surface at intermediate-Reynolds-number-flow regime. The model has incorporated bispherical coordinates to determine a stream function for the flow disturbed by the sphere. The stream function was then used to derive the flow force on the particle as a function of the inter-surface separation distance. The force equation was related to the modified Stokes equation to obtain an exact analytical expression for the correction factor to the Stokes law. Finally, a rational approximation is presented, which is in good agreement with the exact numerical result, and can be readily applied to more general particle–surface interactions involving short-range hydrodynamics associated with colloidal particles in the near vicinity of a large solid collector surface at intermediate Reynolds number of the supporting flow.     

Analytic construction of the near field in a transonic flow near a thin axisymmetric body

A solution of the axisymmetric problem of unsteady transonic flow around thin bodies of revolution is proposed in the form of a double series expansion in powers of the distance to the axis of symmetry and its logarithm in a neighborhood of a given point at the symmetry axis. Chains of recurrence equations are obtained for the coefficients of the series. The convergence of the constructed series is proved by the method of special majorants. The theorem of existence and uniqueness of the solution to the boundary-value problem for a nonlinear partial differential equation with a singularity at the symmetry axis is obtained in the asymptotic model of unsteady transonic flow under consideration. Thereby the application of the proposed series is justified to the problems of unsteady transonic flow around thin axisymmetric bodies with a drift of the nonpenetration condition onto the symmetry axis. Hence, these series can be used in numerical-analytical methods and model computations.     

Heat transfer in MHD viscoelastic boundary layer flow over a stretching sheet with non-uniform heat source/sink

This paper presents the study of momentum and heat transfer characteristics in a hydromagnetic flow of viscoelastic liquid over a stretching sheet with non-uniform heat source, where the flow is generated due to a linear stretching of the sheet and influenced by uniform magnetic field applied vertically. Here an analysis has been carried out to study the effect of magnetic field on the visco-elastic liquid flow and heat transfer over a stretching sheet with non-uniform heat source. The non-linear boundary layer equation for momentum is converted into ordinary differential equation by means of similarity transformation and is solved exactly. Heat transfer differential equation is also solved analytically. The effect of magnetic field on velocity, skin friction and temperature profiles are presented graphically and discussed.     

Hydromagnetic instability of two coaxial streaming cylinders with double perturbed interfaces

The magnetohydrodynamic (MHD) stability of a double interface perturbed streaming liquid cylinder coaxial with a streaming fluid mantle acting upon capillary, inertial, pressure gradient and electromagnetic forces has been developed. The problem is formulated, solved and the stability criterion of the model is estabilished. The latter is discussed analytically and the results are confirmed numerically and interpreted physically. Some reported works are recovered as limiting cases. The capillary force is stabilizing or not according to restrictions. The magnetic field has a strong stabilizing influence. The radii (liquid–fluid) ratio plays an important role in increasing the MHD stabilizing domains. The density of liquid–fluid ratio has a little stabilizing effect. The streaming has a destabilizing influence for all kinds of (non-) axisymmetric perturbation modes. However, if the magnetic field strength is so strong such that the Alfvén wave velocity is greater than the streaming velocity, then the destabilizing character due to capillary force or/and streaming is completely suppressed and stability sets in. In the absence of the magnetic field and we neglect the fluid inertial force, the present results are in good agreement with the experimental results of (Kendall J.M. Phys Fluids 1986;29:2086).     

Heat transfer enhancement for boundary layer flow over a wedge by the use of electric fields

The effect of an electric field is investigated for heat transfer properties in a laminar, incompressible, non-isothermal boundary layer gas flow over a wedge. The governing boundary layer equations are reduced to an ordinary differential equation system using similarity transformations. The reduced equations are solved numerically for different values of electric and flow field parameters characterizing the ratio of electric force to fluid inertia force, Joule heating and ion kinetic work. For specific electric field function forms, leading to similarity solutions, velocity boundary layers are observed to become thinner and heat transfer properties are shown to be enhanced near the wall. The level of enhancement is controlled by the electric body force with additional effects of Joule heating and ion kinetic work on the bulk flow. The effects of low and high Prandtl numbers are also demonstrated. Heat transfer enhancement is observed to increase with increasing Prandtl number.     

Open-surface MHD flow over a curved wall in the 3-D thin-shear-layer approximation

3-D thin-shear-layer equations for flows of conducting fluids in a magnetic field have been derived in orthogonal body-oriented coordinates and then applied to the analysis of MHD open-surface flows over a curved wall. Unlike the classic boundary-layer-type equations, present ones permit information to be propagated upstream through the induced magnetic field. Another departure from the classic theory is that the normal momentum equation keeps the balance between the pressure gradient term, and those related to gravity, centrifugal forces, and Lorentz force. Thus, the normal pressure variations are allowed. The model describes basic 3-D effects due to the wall curvature and spatial variations of the applied magnetic field. As a particular case, equations for flows with rotational symmetry have been derived. Numerical calculations were performed for open-surface flows over a body of revolution under conditions relevant to a fusion reactor (Hartmann number is 8500). Some specific flow patterns, such as flow thickening and spiral-type flows, have been observed and discussed. A special attention has been paid to the analysis of the magnetic propulsion as a tool for the active flow control by applying an electric current. It has been shown that depending on the applied current, the axial pressure gradient can act as an adverse pressure gradient or propulsion force.     

Large deformations of bodies of revolution made of elastic homogeneous and fiber-reinforced materials 1. Torsion of toroidal bodies

Equations of a mathematical model for bodies of revolution made of elastic homogeneous and fiber-reinforced materials and subjected to large deformations are presented. The volume content of reinforcing fibers is assumed low, and their interaction through the matrix is neglected. The axial lines of the fibers can lie both on surfaces of revolution whose symmetry axes coincide with the axis of the body of revolution and along trajectories directed outside the surfaces. The equations are obtained for the macroscopically axisymmetric problem statement where the parameters of macroscopic deformation of the body vary in its meridional planes, but are constant in the circumferential directions orthogonal to them. The equations also describe the torsion of bodies of revolution and their deformation behavior under the action of inertia forces in rotation around the symmetry axis. The results of a numerical investigation into the large deformations of toroidal bodies made of elastic homogeneous and unidirectionally reinforced materials under torsion caused by a relative rotation of their butt-end sections around the symmetry axis are presented.     

3-D axisymmetric subsonic flows with nonzero swirl for the compressible Euler–Poisson system

We address the structural stability of 3-D axisymmetric subsonic flows with nonzero swirl for the steady compressible Euler–Poisson system in a cylinder supplemented with non-small boundary data. A special Helmholtz decomposition of the velocity field is introduced for 3-D axisymmetric flow with a nonzero swirl (=?angular momentum density) component. With the newly introduced decomposition, a quasilinear elliptic system of second order is derived from the elliptic modes in Euler–Poisson system for subsonic flows. Due to the nonzero swirl, the main difficulties lie in the solvability of a singular elliptic equation which concerns the angular component of the vorticity in its cylindrical representation, and in analysis of streamlines near the axis $r=0$.     

Flow engendered by the time-varying motion of a porous plate in the presence of a magnetic field

The problem of flow of an electrically conducting viscous liquid due to the time-varying motion of an infinite porous plate has been studied. There is a uniform magnetic field imposed transversely to the plate and the magnetic lines of force are taken to be fixed relative to the fluid. Exact solutions for the velocity field and the skin-friction have been obtained and some particular cases have been discussed. The effect of suction parameter and magnetic field strength on the flow characteristics have been depicted through several graphs and tables.     

Distortion of liquid metal flow in a square duct due to the influence of a magnetic point dipole

We consider liquid metal flow in a square duct with electrically insulating walls under the influence of a magnetic point dipole using three-dimensional direct numerical simulations with a finite-difference method. The dipole acts as a magnetic obstacle. The Lorentz force on the magnet is sensitive to the velocity distribution that is influenced by the magnetic field. The flow transformation by an inhomogeneous local magnetic field is essential for obtaining velocity information from the measured forces. In this paper we present a numerical simulation of a spatially developing flow in a duct with laminar inflow and periodic boundary conditions. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Lorentz force flow meter for application at blast furnaces: design and calibration

A reference Lorentz force flow meter (LFF) has been developed to measure molten steel mass flow at the end of the runner of an experimental blast furnace. It works according to the principles of Lorentz force velocimetry [1] in which a static magnetic field interacts with a liquid metal stream. The magnetic field lines are generated by an arrangement of permanent magnets and penetrate the entire cross-section of the flow generating eddy currents and a total Lorentz force inside the melt. This force is proportional to the mass flow of the liquid metal and owing Newton's third law, there is a counter force of the same magnitude acting on the magnet system which is connected to a load cell. For accurate flow rate measurements, a “dry and wet calibration” of the LFF needs to be performed [2]. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)