Low-Reynolds-number motion of a deformable drop between two parallel plane walls

The motion of a three-dimensional deformable drop between two parallel plane walls in a low-Reynolds-number Poiseuille flow is examined using a boundary-integral algorithm that employs the Green’s function for the domain between two infinite plane walls, which incorporates the wall effects without discretization of the walls. We have developed an economical calculation scheme that allows long-time dynamical simulations, so that both transient and steady-state shapes and velocities are obtained. Results are presented for neutrally buoyant drops having various viscosity, size, deformability, and channel position. For nearly spherical drops, the decrease in translational velocity relative to the undisturbed fluid velocity at the drop center increases with drop size, proximity of the drop to one or both walls, and drop-to-medium viscosity ratio. When deformable drops are initially placed off the centerline of flow, lateral migration towards the channel center is observed, where the drops obtain steady shapes and translational velocities for subcritical capillary numbers. With increasing capillary number, the drops become more deformed and have larger steady velocities due to larger drop-to-wall clearances. Non-monotonic behavior for the lateral migration velocities with increasing viscosity ratio is observed. Simulation results for large drops with non-deformed spherical diameters exceeding the channel height are also presented.     

Stationary flow of a conducting liquid in a rectangular channel with two nonconducting walls and two walls having an arbitrary conductivity parallel to an external magnetic field

The flow of a conducting liquid in a channel of rectangular cross section with two walls (parallel to the external magnetic field) having an arbitrary conductivity, the other two being insulators, is considered. The solution of the problem is presented in the form of infinite series. The relationships obtained are used for numerical calculations of the velocity distribution and the distribution of the induced magnetic field over the cross section for several modes of flow.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkostt i Gaza, No. 5, pp. 46–52, September–October, 1970.     

Instability of the motion of a liquid between two rotating spherical surfaces

A study has been made of the stability of the steady-state motion of a viscous incompressible liquid, arising in a thin spherical layer, when both spheres are rotating in the same direction at different angular velocities. For a ratio of the radii of the spheres r₂/r₁=1.10, 1.07, a stability curve is obtained which is analogous to the stability curve for the motion of a liquid between rotating cylinders.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 155–156, July–August, 1970.     

Instability of a plane interface between two immiscible conducting viscous fluids in a normal electrostatic field

The dispersion relation for motions of a charged plane interface between two viscous incompressible immiscible conducting fluids is analyzed numerically for finite values of all the parameters involved. It is shown that in addition to the well-known aperiodic (Tonkes-Frenkel’ type) instability for certain values of the physical parameters an oscillatory instability with periodically growing amplitude may be realized in the system. Yaroslavl’. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, pp. 116–123, November–December, 1998.     

Instability and breakup of capillary liquid jets in a parallel airstream

The problem of the development and interaction of nonlinear two-dimensional perturbations in a rotating capillary jet is solved. The main attention is devoted to the study of the nonuniform breakup of the jet with allowance for the influence of the parallel airstream and the rotation. The solution is found by Galerkin's method [1–3]. The nonlinear development and interaction of a large number of perturbations is considered. A significant influence of long-wavelength modulation on the nature of drop formation is established. It is shown that an increase in the velocity of the parallel stream leads to a decrease in the relative size of the satellite (for the characteristic wavelengths). It is also shown that the rotation extends the region of unstable wave numbers in the complete range of flow velocities and air densities.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 124–128, May–June, 1981.I am sincerely grateful to G. I. Petrov, V. Ya. Shkadov, and S. Ya. Gertsenshtein for constant interest in the work.     

Instability of Bingham fluids in Taylor-Dean flow between two concentric cylinders at arbitrary gap spacings

Stability of Bingham fluids is investigated numerically in azimuthal pressure-driven flow between two infinitely long concentric cylinders. An infinitesimal perturbation is introduced to the basic flow and its time evolution is monitored using normal mode linear stability analysis. An eigenvalue problem is obtained which is solved numerically using pseudo-spectral collocation method. Numerical results are obtained for two different cases: (i) the inner cylinder is rotating at constant velocity while the outer cylinder is fixed (i.e., the Taylor-Dean flow) and (ii) both cylinders are fixed (i.e., the Dean flow). The results show that the yield stress always has a stabilizing effect on the Taylor-Dean flow. But, for the Dean flow the effect of the yield stress is predicted to be stabilizing or destabilizing depending on the magnitude of the Bingham number and also the gap size.     

Instability of a liquid surface with laser heating

Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 28–34, May–June, 1991.     

Calculation of a supersonic jet emerging from a hole with planar walls

The coordinates of the sonic line are derived by means of Frankl's solution [1], while the supersonic part of the jet is considered by the method of characteristics. The numerical solution has been used to calculate the family of rarefaction waves and a family of nozzles having a corner point and a curvilinear transition surface. These calculations show that, when there is a corner point, the shape of the sonic line has hardly any effect on the velocity distribution along the symmetry axis. It is also shown that a positive pressure gradient arises on the surface of the nozzle beyond the corner point if that point lies upstream from the limiting characteristic of the first family.     

Forced convection — axial conduction between parallel walls with unequal heat fluxes

Laminar flow heat transfer is computed for a situation in which a fluid moves along a parallel plate channel with unequal wall heat fluxes (one wall is insulated). The fluid enters the heating section through an upstream region which is perfectly insulated. This situation serves to describe an upper bound for the commonly encountered case of double pipe heat exchangers with identical thermal conditions. A control volume approach has been employed for the numerical work enabling a fast calculation for the thermally developing regime in the parallel plate channel. The merits of the adopted procedure are assessed by comparison with other results available in the literature for the one-region and for the two-region problem.     

Direct numerical simulation of a liquid sheet in a compressible gas stream in axisymmetric and planar configurations

A thin liquid sheet present in the shear layer of a compressible gas jet is investigated using an Eulerian approach with mixed-fluid treatment for the governing equations describing the gas–liquid two-phase flow system, where the gas is treated as fully compressible and the liquid as incompressible. The effects of different topological configurations, surface tension, gas pressure and liquid sheet thickness on the flow development of the gas–liquid two-phase flow system have been examined by direct solution of the compressible Navier–Stokes equations using highly accurate numerical schemes. The interface dynamics are captured using volume of fluid and continuum surface force models. The simulations show that the dispersion of the liquid sheet is dominated by vortical structures formed at the jet shear layer due to the Kelvin–Helmholtz instability. The axisymmetric case is less vortical than its planar counterpart that exhibits formation of larger vortical structures and larger liquid dispersion. It has been identified that the vorticity development and the liquid dispersion in a planar configuration are increased at the absence of surface tension, which when present, tends to oppose the development of the Kelvin–Helmholtz instability. An opposite trend was observed for an axisymmetric configuration where surface tension tends to promote the development of vorticity. An increase in vorticity development and liquid dispersion was observed for increased liquid sheet thickness, while a decreasing trend was observed for higher gas pressure. Therefore surface tension, liquid sheet thickness and gas pressure factors all affect the flow vorticity which consequently affects the dispersion of the liquid.      

Slow motion of a conducting fluid between two rotating walls

Summary An investigation is presented of the flow of a viscous, conducting fluid between two plane walls, which rotate around a common axis towards each other. The flow is considered to be under the influence of a magnetic field, set up by a line-current along the axis of rotation.Under the assumption of low hydrodynamic and magnetic Reynolds-number, expressions are given for the velocity and the pressure gradient.At present at Brussels: Ecole Royale Militaire, Physique des Plasmas.     

Instability of a compound liquid jet

In the linear Rayleigh theory [1] the degree of stability of a jet is determined by the viscosity and inertia characteristics of the fluids and the interphase surface tension. The stability of a jet in an infinite medium increases with increase in the viscosity of both the jet and the medium [2, 3]. The presence of two interfaces is responsible for various features of the development of instability in a liquid layer on the surface of a cylinder, and in particular a layer on the inner surface of a cylinder is more unstable than one on the outer surface [4]. In [5, 6] the breakup of a hollow jet in an external medium was investigated. In this paper we examine, in the linear approximation, the stability of a compound jet of nonmiscible liquids with respect to small axisynmetric perturbations of the interfaces. The instability characteristics are given for jets with inviscid and very viscous outer shells. The conditions governing the suppression of rapidly growing instabilities of the inner part (core) of the jet by a viscous shell are determined.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 3–8, July–August, 1985.     

Finite-part integral and boundary element method for interaction between two parallel planar cracks in three-dimensional finite bodies

By using the Somigliana representation and the concepts of finite-part integrals, a set of hypersingular integral equations of the interaction between two parallel planar cracks in a three-dimensional finite body subjected to arbitrary loads is derived, and then its numerical method is proposed by the finite-part integral method combined with the boundary element method. According to the analytic theory of hypersingular integral equations, the square root models of displacement discontinuities in the elements near the crack front are applied, and thus the computational precision is raised. Based on this, the stress intensity factors can be directly calculated. Finally, the stress intensity factors of several typical interaction problems are calculated.     

Motion of a circular cylinder in a viscous liquid between parallel plates

The two-dimensional motion of a cylinder in a viscous fluid between two parallel walls of a vertical channel is studied. It is found that when the cylinder moves very closely along one of the channel walls, it always rotates in the direction opposite to that of contact rolling along the nearest wall. When the cylinder is away from the walls, its rotation depends on the Reynolds number of the flow. In this study two numerical methods were used. One is for the unsteady motion of a sedimenting cylinder initially released from a position close to one of the channel walls, where the Navier-Stokes equations are solved for the fluid and Newton's equations of motion are solved for the rigid cylinder. The other method is for the steady flow in which a cylinder is fixed in a uniform flow field where the channel walls are sliding past the cylinder at the speed of the approaching flow, or equivalently a cylinder is moving with a constant velocity in a quiescent fluid. The flow field, the drag, the side force (lift), and the torque experienced by the cylinder are studied in detail. The effects of the cylinder location in the channel, the size of the channel relative to the cylinder diameter, and the Reynolds number of the flow are examined. In the limit when the cylinder is translating very closely along one of the walls, the flow in the gap between the cylinder and the wall is solved analytically using lubrication theory, and the numerical solution in the other region is used to piece together the whole flow field.This research was supported by NSF DMR91-20668 through the Laboratory for Research on the Structure of Matter at the University of Pennsylvania and from the Research Foundation of the University of Pennsylvania.