Effect of small parameters on the structure of the front formed by unstable viscous-fluid displacement from a Hele-Shaw cell

The process of displacement of a viscous fluid from a Hele-Shaw cell consisting of two plates separated by a small gap is investigated. The front formed when the fluid is displaced from the cell by another, lower-viscosity fluid is unstable. The lower-viscosity fluid breaks through the layer of displaced fluid and forms channels called viscous fingers. As a result, a mixing zone occupied by both displaced and displacing fluids is formed. The structure of the unstable displacement front is investigated when the surface tension forces have no effect on the shape of the fingers. This situation is realized when a water-glycerin mixture is rapidly displaced from the cell by water. Equations taking the inertial and viscous forces acting in the plane of the plates into account are obtained by averaging the Navier-Stokes equations over the cell gap. Using the equations obtained the stability of a plane displacement front traveling in the direction of its normal and the stability of the lateral surfaces of the viscous fingers is investigated when the fluid velocities are parallel to the interface. From the solution for stability of the transverse displacement front it follows that the viscous forces acting in the plane of the plates determine the finger width (when there is no surface tension). Instability also develops in the flow on the longitudinal fluid interface. In this case the destabilizing factor is the inertial forces. Under the action of this instability the fingers, in their turn, lose stability and disintegrate into viscous bubbles.     

Effect of tangential and normal forces on the wave formation in the flow of a liquid film and a gas

The effect of surface forces on nonlinear waves induced by the hydrodynamic instability in the flow of a viscous liquid film along the inner surface of a tube blown with a gas.     

Capillary instabilities in a thin nematic liquid crystalline fiber embedded in a viscous matrix

$m$ to take into account non-axisymmetric modes. Capillary instabilities in nematic fibers reflect the anisotropic nature of liquid crystals, such as the orientation contribution to the surface elasticity and surface bending stresses. Surface gradients of bending stresses provide additional anisotropic contributions to the capillary pressure that may renormalize the classical displacement and curvature forces that exist in any fluid fiber. The exact nature (stabilizing and destabilizing) and magnitude of the renormalization of the displacement and curvature forces depend on the nematic orientation and the anisotropic contribution to the surface energy, and accordingly capillary instabilities may be axisymmetric or non-axisymmetric, with finite or unbounded wavelengths. Thus, the classical fiber-to-droplet transformation is one of several possible instability pathways while others include surface fibrillation. The contribution of the viscosity ratio to the capillary instabilities of a thin nematic fiber in a viscous matrix is analyzed by two parameters, the fiber and matrix Ohnesorge numbers, which represent the ratio between viscous and surface forces in each phase. The capillary instabilities of a thin nematic fiber in a viscous matrix are suppressed by increasing either fiber or matrix Ohnesorge number, but estimated droplet sizes after fiber breakup in axisymmetric instabilities decrease with increasing matrix Ohnesorge number. Received November 26, 2001 / Published online May 21, 2002 Communicated by Epifanio Virga, Pavia     

Displacement of a viscous fluid from a Hele-Shaw cell with a sink

It is found that the radial geometry does not stabilize the evolution of instability in the displacement of a more viscous fluid by a less viscous fluid from a circular Hele-Shaw cell with a sink. A linear analysis shows the absolute instability of the radial displacement front. The appearance of isolated fingers is observed during numerical simulations.     

Linear instability of ultra-thin liquid films flowing down cylindrical fibre

The stability characteristics of an ultra-thin layer of a viscous liquid flowing down a cylindrical fibre are investigated by a linear theory. The film with the thickness less than 100 nm is driven by an external force and under the influence of the van der Waals forces. The results show that, when the relative film thickness decreases, the curvature of the fibre depresses the development of the linear perturbations, whereas the van der Waals forces promote the instabilities. This competition results in a non-monotonous dependence of the growth rate on the relative film thickness. The critical curves are also obtained to describe the transition from the absolute instability to the convective instability, indicating that the van der Waals forces can enlarge the absolutely unstable region. Furthermore, the surface tension can cause the development of the absolute instability, whereas the external force has an opposite effect.     

Linear instability of ultra-thin liquid films flowing down cylindrical fibre

The stability characteristics of an ultra-thin layer of a viscous liquid flowing down a cylindrical fibre are investigated by a linear theory. The film with the thickness less than 100 nm is driven by an external force and under the influence of the van der Waals forces. The results show that, when the relative film thickness decreases, the curvature of the fibre depresses the development of the linear perturbations, whereas the van der Waals forces promote the instabilities. This competition results in a non-monotonous dependence of the growth rate on the relative film thickness. The critical curves are also obtained to describe the transition from the absolute instability to the convective instability, indicating that the van der Waals forces can enlarge the absolutely unstable region. Furthermore, the surface tension can cause the development of the absolute instability, whereas the external force has an opposite effect.     

The instability mechanism of single and multilayer Newtonian and viscoelastic flows down an inclined plane

The instability mechanism of single and multilayer flow of Newtonian and viscoelastic fluids down an inclined plane has been examined based on a rigorous energy analysis as well as careful examination of the eigenfunctions. These analyses demonstrate that the free surface instability in single and multilayer flows in the limit of longwave disturbances (i.e., the most dangerous disturbances) arise due to the perturbation shear stresses at the free surface. Specifically, for viscoelastic flows, the elastic forces are destabilizing and the main driving force for the instability is the coupling between the base flow and the perturbation velocity and stresses and their gradient at the free surface. For Newtonian flows at finite Re, the driving force for the interfacial instability in the limit of longwaves depends on the placement of the less viscous fluid. If the less viscous fluid is adjacent to the solid surface then the main driving force for the instability is interfacial friction, otherwise the bulk contribution of Reynolds stresses drives the instability. For viscoelastic fluids in the limit of vanishingly small Re, the driving force for the instability is the coupling of the base flow and perturbation velocity and stresses and their gradients across the interface. In the limit of shortwaves the interfacial stability mechanism of flow down inclined plane is the same as plane Poiseuille flows (Ganpule and Khomami 1998, 1999a, b). Received: 20 October 2000/Accepted: 11 January 2001     

Displacement of a viscous fluid from an annular Hele–Shaw cell with a sink within the framework of the Brinkman model

The stability of the radial front of viscous fluid displacement from an annular Hele–Shaw cell with a sink of finite radius is analyzed. It is shown that in the absence of the surface tension and at a minimal manifestation of molecular diffusion the role of the stabilizing factor determining the displacement front structure can be played by small viscous forces in the cell plane. The viscous fingers formed in this case turn out to be wider than those in a rectangular cell.     

Free oscillations of a liquid rotating in a cylindrical vessel under conditions of weightlessness

The plane problem of determination of the natural frequencies of small oscillations of a viscous liquid rotating in a partially filled cylindrical vessel under conditions of weight-lessness is examined. If the angular velocity of vessel rotation is sufficiently low, surface forces acting on the liquid-gas boundary prove to be of the same order as the centrifugal forces and significantly affect the oscillatory frequencies. Asymptotic formulas expressing the dependence of the oscillatory frequencies on the parameters of the problem are obtained by the boundary layer method, with the assumption that the ratio of viscous to centrifugal forces is low.Khar'kov. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkostii Gaza, No. 4, pp. 3–9, July–August, 1972.     

Coupled frequencies of a hydroelastic viscous liquid system

The coupled frequencies of a hydroelastic system consisting of an elastic shell and a viscous liquid layer with a free surface have been treated. The system exhibits no z-dependency and may be either an annular liquid layer around an elastic center shell or a liquid layer inside an elastic container. The first case has been evaluated numerically, where the influence of the liquid surface tension parameter, the elasticity parameter of the shell and the thickness of the layer have been determined. In contrast to the hydroelastic system with an ideal liquid, the system with viscous liquid exhibits instability of the liquid surface as well as the shell.     

Radial Viscous Fingering in Case of Poorly Miscible Fluids

The displacement of a viscous fluid from an annular Hele-Shaw cell with a source of finite radius by a less viscous one is investigated. A special case of poorly miscible fluids is considered when corresponding dimensionless criteria—capillary and Peclet numbers—both tend to infinity. Brinkman model which additionally takes into account small viscous forces in a plane of the cell is used to describe the displacement process. Linear analysis shows a stabilizing effect of viscous forces and reveals a geometrical similarity criterion, namely the ratio of the interface’s radius to the gap between the cell’s plates. The displacement patterns, obtained numerically under Brinkman model, are very sensitive to the discovered criterion. The comparison with available experimental data is acceptable.     

The Compressible Viscous Surface-Internal Wave Problem: Nonlinear Rayleigh–Taylor Instability

This paper concerns the dynamics of two layers of compressible, barotropic, viscous fluid lying atop one another. The lower fluid is bounded below by a rigid bottom, and t he upper fluid is bounded above by a trivial fluid of constant pressure. This is a free boundary problem: the interfaces between the fluids and above the upper fluid are free to move. The fluids are acted on by gravity in the bulk, and at the free interfaces we consider both the case of surface tension and the case of no surface forces.We are concerned with the Rayleigh–Taylor instability when the upper fluid is heavier than the lower fluid along the equilibrium interface. When the surface tension at the free internal interface is below the critical value, we prove that the problem is nonlinear unstable.     

Motion of a nonisothermal jet in a field of centrifugal forces

Considerable interest attaches to the study of a jet of viscous liquid in a field of body forces that depend on an axial coordinate. Such flows are realized when slag cotton is obtained by the action on a molten mineral of the centrifugal force of drums rotating in the vertical plane [1]. The behavior of a film of liquid on a rotating cylinder was considered in [2, 3]. The instability of a molten layer and jet separation are explained on the basis of the Taylor mechanism in [4]. In the present paper, a particular solution is given for accelerating nonisothermal jets of a viscous incompressible liquid. This solution is used to explain the dynamics of jet separation from a uniformly rotating drum. The flow stability is analyzed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 27–36, September–October, 1980.We thank A. A. Zaitsev for discussing the results of the work.     

Motion of a cylinder in a viscous electroconductive medium with a magnetic field

The method of force sources is used to consider the planar problem of the motion of a circular cylinder in a viscous electroconductive medium with a magnetic field. The conventional and magnetic Reynolds numbers are assumed to be small. Expressions are obtained for the hydrodynamic reaction forces of the medium, acting on the moving cylinder. It is shown that as a result of the flow anisotropy in the medium, caused by the magnetic field, in addition to the resistance forces on bodies moving at an angle to the field, there are deflecting forces perpendicular to the velocity vector. The velocity field disturbances at great distances from the moving cylinder are determined.The problems of viscous electroconductive flow about solid bodies in the presence of a magnetic field constitute one of the divisions of magnetohydrodynamics. Motion of an electroconductive medium in a magnetic field gives rise to inductive electromagnetic fields and currents which interact with the velocity and pressure hydrodynamic fields in the medium [1, 2]. Under conditions of sufficiently strong interaction, the number of independent flow similarity parameters in MHD is considerably greater than in conventional hydrodynamics. This circumstance complicates the theoretical analysis of MHD flow about bodies, and therefore we must limit ourselves to consideration of individual particular flow cases.Here we consider the linear problem of the motion of an infinite circular cylinder in a viscous incompressible medium with finite electroconductivity located in a uniform magnetic field.There are many studies devoted to the flow of a viscous electroconductive medium with a magnetic field about solid bodies (see, for example, [3–5]). Because of this, some of the results obtained here include previously known results, which will be indicated below. In contrast to the cited studies, the examination is made by the method of force sources, suggested in [6]. This method permits obtaining integral equations for the distribution of the forces acting on the surface of the moving body. Their solution is obtained for small Reynolds and Hartmann numbers. Then the nature of the velocity disturbances at great distances from the body are determined. These results are compared with conventional viscous flow about a cylinder in the Oseen approximation.     

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.     

Dynamics of trains and train-like articulated systems travelling in confined fluid—Part 1: Modelling and basic dynamics

The dynamics and stability of a train of flexibly interconnected rigid cylinders travelling in a confined cylindrical “tunnel” subjected to fluid dynamic forces is studied theoretically. Each cylinder, which is coupled to other cylinders and supported by springs and dampers, has degrees of freedom in the lateral translational and rotational directions. The kinetic, dissipation, and potential energies of the system and the generalized forces associated with the fluid dynamic forces acting on the system, such as inviscid fluid dynamic forces, viscous frictional forces, and form drag, are obtained first. Then the equations of motion are derived in a Lagrangian framework. The principal aim of this study is to investigate the effect of the aerodynamic forces on the dynamics of a high-speed train running in a tunnel, or more generally of a train-like system travelling in a coaxial cylindrical tube. The results of this study show that the system loses stability by flutter and that viscous frictional drag has a considerable effect on stability of the system. In addition, the mechanism of instability of the system is clarified with the aid of a study of the modal shapes and energy considerations.     

Thermocapillary Instability of a Cylindrical Layer in the Presence of a Radial Temperature Gradient

Within a linear formulation, the thermocapillary instability of equilibrium of a cylindrical layer of heat-conducting viscous fluid in the presence of a radial temperature gradient is investigated with respect to arbitrary disturbances. It is shown that the Rayleigh instability mechanism results in the appearance of monotonous disturbances of a new type. For steady disturbances, the neutral curve is split into two separate segments, each corresponding to its own type of disturbances. For a deformable free boundary, new oscillating disturbances in the form of surface waves develop. It is found that, in the case of axial symmetry, the behavior of these disturbances completely coincides with the oscillating disturbance behavior in a plane layer.     

Analysis of a Stochastic Implicit Interface Model for an Immersed Elastic Surface in a Fluctuating Fluid

We present some mathematical analyses of a recently proposed stochastic implicit interface model for an elastic surface immersed in an incompressible viscous fluid subject to fluctuation forces. We derive suitable a priori estimates and establish the well-posedness of pathwise solutions and provide uniform control on the solutions in probability.     

An experimental investigation of initial oscillations in a radial Hele-Shaw cell

Hele-Shaw cell is a laboratory device consisting of two parallel plates of glass separated by a thin gap. In this cell, in the flow of two immiscible fluids, when a fluid of higher viscosity is displaced by a fluid of lower viscosity, the less viscous fluid is observed to form “fingers” into the more viscous one due to the unstable interface. The Saffman-Taylor or viscous finger instability has been examined and modeled for over forty years for the rectilinear Hele-Shaw cell and about half as long for the radial Hele-Shaw cell. In this paper, we study, in detail, the early development of viscous instabilities in a radial Hele-Shaw cell. This source flow configuration has been chosen so that the instability can be monitored precisely. The objective of this study is to examine the onset of fingering, i.e. initial number of fingers that form, and the evolution of interface instability. Our experiments suggest that there may be some order in this formation process and one can model this aspect by considering the unsteady velocity components and predicting temporal changes in wavenumber responsible for the initial number of fingers and may be later accounting for the fingertip oscillations and splitting. We injected a water-based fluid into an oil in a radial Hele-Shaw cell at constant flow rate and recorded the movement of the less viscous droplet as it evolved. The relative curvature changes on the expanding droplet boundary was plotted with the angular positions about the interface and subtracting out the average radius, resulting in a plot of the change in amplitude with respect to time for the interface configuration. Three unstable configured tests at kinematic viscosity contrast (v ^O) of 0.34, 0.68, and 0.94 were run at approximately the same flow rate (2π cm²/s). The droplet exhibited oscillatory movement for these unstable configuration. The amplitude and the rate of oscillations were measured from digitized data. The smaller the viscosity difference, the smaller was the amplitude growth rate and resulted in a longer time to form visible finger initiation. This work was supported by National Science Foundation, grant number EID-9017555. We also like to thank Dr. Len Schwartz, Professor of Mechanical Engineering at the University of Delware for his insight and helpful suggestions.