The influence of electric fields and surface tension on Kelvin–Helmholtz instability in two-dimensional jets

We consider nonlinear aspects of the flow of an inviscid two-dimensional jet into a second immiscible fluid of different density and unbounded extent. Velocity jumps are supported at the interface, and the flow is susceptible to the Kelvin–Helmholtz instability. We investigate theoretically the effects of horizontal electric fields and surface tension on the nonlinear evolution of the jet. This is accomplished by developing a long-wave matched asymptotic analysis that incorporates the influence of the outer regions on the dynamics of the jet. The result is a coupled system of long-wave nonlinear, nonlocal evolution equations governing the interfacial amplitude and corresponding horizontal velocity, for symmetric interfacial deformations. The theory allows for amplitudes that scale with the undisturbed jet thickness and is therefore capable of predicting singular events such as jet pinching. In the absence of surface tension, a sufficiently strong electric field completely stabilizes (linearly) the Kelvin–Helmholtz instability at all wavelengths by the introduction of a dispersive regularization of a nonlocal origin. The dispersion relation has the same functional form as the destabilizing Kelvin–Helmholtz terms, but is of a different sign. If the electric field is weak or absent, then surface tension is included to regularize Kelvin–Helmholtz instability and to provide a well-posed nonlinear problem. We address the nonlinear problems numerically using spectral methods and establish two distinct dynamical behaviors. In cases where the linear theory predicts dispersive regularization (whether surface tension is present or not), then relatively large initial conditions induce a nonlinear flow that is oscillatory in time (in fact quasi-periodic) with a basic oscillation predicted well by linear theory and a second nonlinearly induced lower frequency that is responsible for quasi-periodic modulations of the spatio-temporal dynamics. If the parameters are chosen so that the linear theory predicts a band of long unstable waves (surface tension now ensures that short waves are dispersively regularized), then the flow generically evolves to a finite-time rupture singularity. This has been established numerically for rather general initial conditions.     

The influence of electric fields and surface tension on Kelvin–Helmholtz instability in two-dimensional jets

We consider nonlinear aspects of the flow of an inviscid two-dimensional jet into a second immiscible fluid of different density and unbounded extent. Velocity jumps are supported at the interface, and the flow is susceptible to the Kelvin–Helmholtz instability. We investigate theoretically the effects of horizontal electric fields and surface tension on the nonlinear evolution of the jet. This is accomplished by developing a long-wave matched asymptotic analysis that incorporates the influence of the outer regions on the dynamics of the jet. The result is a coupled system of long-wave nonlinear, nonlocal evolution equations governing the interfacial amplitude and corresponding horizontal velocity, for symmetric interfacial deformations. The theory allows for amplitudes that scale with the undisturbed jet thickness and is therefore capable of predicting singular events such as jet pinching. In the absence of surface tension, a sufficiently strong electric field completely stabilizes (linearly) the Kelvin–Helmholtz instability at all wavelengths by the introduction of a dispersive regularization of a nonlocal origin. The dispersion relation has the same functional form as the destabilizing Kelvin–Helmholtz terms, but is of a different sign. If the electric field is weak or absent, then surface tension is included to regularize Kelvin–Helmholtz instability and to provide a well-posed nonlinear problem. We address the nonlinear problems numerically using spectral methods and establish two distinct dynamical behaviors. In cases where the linear theory predicts dispersive regularization (whether surface tension is present or not), then relatively large initial conditions induce a nonlinear flow that is oscillatory in time (in fact quasi-periodic) with a basic oscillation predicted well by linear theory and a second nonlinearly induced lower frequency that is responsible for quasi-periodic modulations of the spatio-temporal dynamics. If the parameters are chosen so that the linear theory predicts a band of long unstable waves (surface tension now ensures that short waves are dispersively regularized), then the flow generically evolves to a finite-time rupture singularity. This has been established numerically for rather general initial conditions.     

Hamiltonian model for coupled surface and internal waves in the presence of currents

We examine a two dimensional fluid system consisting of a lower medium bounded underneath by a flatbed and an upper medium with a free surface. The two media are separated by a free common interface. The gravity driven surface and internal water waves (at the common interface between the media) in the presence of a depth-dependent current are studied under certain physical assumptions. Both media are considered incompressible and with prescribed vorticities. Using the Hamiltonian approach the Hamiltonian of the system is constructed in terms of ‘wave’ variables and the equations of motion are calculated. The resultant equations of motion are then analysed to show that wave–current interaction is influenced only by the current profile in the ‘strips’ adjacent to the surface and the interface. Small amplitude and long-wave approximations are also presented.     

Surface and internal waves: The two-dimensional problem on forward motion of a body intersecting the interface between two fluids

A linear two-dimensional boundary value problem, that describes steady-state surface and internal waves due to the forward motion of a body in a fluid consisting of two superposed layers with different densities, is considered. The body is fully submerged and intersects the interface between the two layers. Two well-posed formulations of the problem are proposed in which, along with the Laplace equation, boundary conditions, coupling conditions on the interface, and conditions at infinity, a pair of supplementary conditions are imposed at the points where the body contour intersects the interface. In one of the well-posed formulations (where the differences between the horizontal momentum components are given at the intersection points), the existence of the unique solution is proved for all values of the parameters except for a certain (possibly empty) nowhere dense set of values.     

Two-layer Gravity Currents with Topography

Two-dimensional and time-dependent gravity currents involving the initial release of a fixed volume of heavy fluid over a gradually sloping bottom and underlying a layer of lighter fluid are considered. The equations which describe the resulting two-layer flow are derived from the Navier–Stokes equations for a constant density, inviscid, nonrotating fluid, neglecting kinematic viscosity, surface tension, and entrainment between the layers. A new addition to the theory is introduced in the form of a forcing term in the lower layer horizontal momentum equation which is incorporated to produce the characteristic structure typical of such gravity currents in the laboratory. This delaying term is restricted to the front of the gravity current, and as such is shown to be valid under conventional shallow-water scaling assumptions. The hyperbolic character of the equations of motion is shown, a simple numerical test for hyperbolicity is derived from theoretical considerations, and these results are related to the stability Froude number of the flow. Well-posedness of the initial boundary value problem is proven via localization of the equations, and the discussion is extended to a two-point boundary value problem with examples of steady-state and traveling wave solutions given for a bottom surface of constant slope. Numerical results are obtained by using a recently developed finite-difference relaxation scheme for conservation laws, sufficiently modified herein to include spatial variability and forcing terms, which approximates the material interface at the front of the lower fluid layer as a shock. The effects of slope and the delaying force are investigated numerically to determine their theoretical importance, and the range of expected values is compared to published experimental results. Some calculations for the temporal evolution of the flow are produced that display the phenomenon of rear wall separation for nonzero slopes.     

Thin-film approximations of the two-phase Stokes problem

Passing to the limit of small layer thickness in the two-phase Stokes problem we obtain when including only gravity (resp. surface tension) effects a strongly coupled parabolic system of second (resp. fourth) order. In the non-degenerate case we prove that the corresponding evolution problems are locally well-posed. For the gravity driven flow though, we have to assume that the less dense fluid lies on top of the less dense layer. Moreover, we show that the solutions converge exponentially fast towards a flat steady-state, which is uniquely determined by the volume of the two fluids, provided they are initially close to this rest state.     

On the Equation of a Rotating Film

We study positive periodic solutions to a nonautonomous nonlinear third-order ordinary differential equation of the theory of motion of a viscous incompressible fluid with free boundary. This equation describes the steady motions of a thin layer of a fluid film on the surface of a rotating horizontal cylinder in the gravity field. The linear operator on the left-hand side of the equation has a three-dimensional kernel. Moreover, the equation contains two nonnegative parameters proportional to the gravity acceleration and surface tension. Depending on these parameters the problem in question may have either two solutions or no solutions at all. We establish some qualitative properties of solutions to the problem: in particular, their asymptotic behavior at the extremal values of the parameters.     

Existence of undercompressive weak solutions to the Euler equations

We prove the local existence of an undercompressive hydrodynamical shock to the isothermal Euler equations with a non-monotone pressure function. This nonlinear problem will be formulated as an abstract hyperbolic initial boundary value problem. The existence of a weak solution to a linearized version of the problem is shown with the use of Riesz theorem. Using the results of the linear system yields by an iteration scheme (local in time) well-posedness of the nonlinear problem. The system of equations is obtained by modeling the motion of sharp liquid-vapor interfaces including configurational forces as well as surface tension. The considered non-viscous Van der Waals fluid is compressible and allows phase transitions. The propagating phase boundary is controlled by a modified version of the Rankine-Hugoniot jump condition obtained by the Young-Laplace law. Entropy dissipation at the interface is precisely described by a kinetic relation. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modeling of two-phase flows with surface tension by finite pointset method (FPM)

A meshfree method for two-phase immiscible incompressible flows including surface tension is presented. The continuum surface force (CSF) model is used to include the surface tension force. The incompressible Navier–Stokes equation is considered as the mathematical model. Application of implicit projection method results in linear second-order partial differential equations for velocities and pressure. These equations are then solved by the finite pointset method (FPM), which is a meshfree and Lagrangian method. The fluid is represented as finite number of particles and the immiscible fluids are distinguished by the color of each particle. The interface is tracked automatically by advecting the color functions for each particle. Two test cases, Laplace's law and the Rayleigh–Taylor instability in 2D have been presented. The results are found to be consistent with the theoretical results.     

Well-posedness in Sobolev spaces of the full water wave problem in 2-D

. We consider the motion of the interface of 2-D irrotational, incompressible, inviscid water wave, with air above water and surface tension zero. We show that the interface is always not accelerating into the water region, normal to itself, as rapidly as the normal acceleration of gravity, as long as the interface is nonself-intersect. We therefore obtain the existence and uniqueness of solutions of the full water wave problem, locally in time, for any initial interface which is nonself-intersect, including the case that the interface is of multiple heights. Oblatum 1b-II-1996 & XI-1996     

Decay of the 2D Periodic Stationary Viscous Surface Waves

We consider the evolution of viscous fluids in a 2D horizontally periodic slab bounded above by a free top surface and below by a fixed flat bottom. This is a free boundary problem. The dynamics of the fluid are governed by the incompressible stationary Navier–Stokes equations under the influence of gravity and the effect of surface tension. We develop the global theory of solutions in low regularity Sobolev spaces for small data by nonlinear energy estimates.     

On Local Singularities in Ideal Potential Flows with Free Surface

Despite important advances in the mathematical analysis of the Euler equations for water waves, especially over the last two decades, it is not yet known whether local singularities can develop from smooth data in well-posed initial value problems. For ideal free-surface flow with zero surface tension and gravity, the authors review existing works that describe ``splash singularities'', singular hyperbolic solutions related to jet formation and ``flip-through'', and a recent construction of a singular free surface by Zubarev and Karabut that however involves unbounded negative pressure. The authors illustrate some of these phenomena with numerical computations of 2D flow based upon a conformal mapping formulation. Numerical tests with a different kind of initial data suggest the possibility that corner singularities may form in an unstable way from specially prepared initial data.     

Study of vortex breakdown in a stratified fluid

An analysis shows that nonsmooth solutions have to be considered. Weak solutions to the Euler equations describing an incompressible stratified fluid under gravity are defined and studied. The study makes use of a wave energy functional proposed for the nonlinear equations. It is shown that the Euler equations are insufficient for stating a well-posed generalized problem. Additional conditions based on physical considerations are proposed. One condition is energy conservation, and the other is a constraint imposed on the density, which is required for stability. A numerical method is developed that is used to analyze how wave breakdown in a stratified fluid depends on stratification. The numerical results are in satisfactory agreement with experiments.     

Kinetic Equation for Soliton Gas: Integrable Reductions

We study the dynamic behaviour of two viscous fluid films confined between two concentric cylinders rotating at a small relative velocity. It is assumed that the fluids are immiscible and that the volume of the outer fluid film is large compared to the volume of the inner one. Moreover, while the outer fluid is considered to have constant viscosity, the rheological behaviour of the inner thin film is determined by a strain-dependent power-law. Starting from a Navier–Stokes system, we formally derive evolution equations for the interface separating the two fluids. Two competing effects drive the dynamics of the interface, namely the surface tension and the shear stresses induced by the rotation of the cylinders. When the two effects are comparable, the solutions behave, for large times, as in the Newtonian regime. We also study the regime in which the surface tension effects dominate the stresses induced by the rotation of the cylinders. In this case, we prove local existence of positive weak solutions both for shear-thinning and shear-thickening fluids. In the latter case, we show that interfaces which are initially close to a circle converge to a circle in finite time and keep that shape for later times.

     

Well-posedness of the free-surface incompressible Euler equations with or without surface tension

We provide a new method for treating free boundary problems in perfect fluids, and prove local-in-time well-posedness in Sobolev spaces for the free-surface incompressible 3D Euler equations with or without surface tension for arbitrary initial data, and without any irrotationality assumption on the fluid. This is a free boundary problem for the motion of an incompressible perfect liquid in vacuum, wherein the motion of the fluid interacts with the motion of the free-surface at highest-order.

     

Large time existence of small viscous surface waves without surface tension

The motion of a three-dimensional viscous, imcompressible fluid is governed by the Navier-Stokes equations. We study the case where the fluid is in an ocean of infinite extent and finite depth with a free surface on top. This gives rise to a nonlinear free boundary problem. The given data are the initial velocity field and the initial free surface. In general, given smooth data, the solution will develop singularities in finite time; however, the effect of viscosity and surface tension tends to prevent the ingulitrities. It was previously known that when both are present, small, appropriately smooth solutions do not develop singularities; that is, smooth solutions exist globally in time. In this paper, we show that viscosity alone will prevent the formation of singularitics, even without surface tension; i.e., small smooth data which satisfy certain natural compatibility conditions, smooth solutions exist for all time. Uniqueness of the solution for any finite time interval is also proved.     

An Energy-Stable Parametric Finite Element Method for Simulating Solid-State Dewetting Problems in Three Dimensions

We propose an accurate and energy-stable parametric finite element method for solving the sharp-interface continuum model of solid-state dewetting in three-dimensional space. The model describes the motion of the film\slash vapor interface with contact line migration and is governed by the surface diffusion equation with proper boundary conditions at the contact line. We present a weak formulation for the problem, in which the contact angle condition is weakly enforced. By using piecewise linear elements in space and backward Euler method in time, we then discretize the formulation to obtain a parametric finite element approximation, where the interface and its contact line are evolved simultaneously. The resulting numerical method is shown to be well-posed and unconditionally energy-stable. Furthermore, the numerical method is generalized to the case of anisotropic surface energies in the Riemannian metric form. Numerical results are reported to show the convergence and efficiency of the proposed numerical method as well as the anisotropic effects on the morphological evolution of thin films in solid-state dewetting.     

Gravity as a field theory in flat space-time

We propose a formulation of gravity theory in the form of a field theory in a flat space-time with a number of dimensions greater than four. Configurations of the field under consideration describe the splitting of this space-time into a system of mutually noninteracting four-dimensional surfaces. Each of these surfaces can be considered our four-dimensional space-time. If the theory equations of motion are satisfied, then each surface satisfies the Regge-Teitelboim equations, whose solutions, in particular, are solutions of the Einstein equations. Matter fields then satisfy the standard equations, and their excitations propagate only along the surfaces. The formulation of the gravity theory under consideration could be useful in attempts to quantize it.     

Singularity formation in thin jets with surface tension

We derive and study asymptotic models for the dynamics of a thin jet of fluid that is separated from an outer immiscible fluid by fluid interfaces with surface tension. Both fluids are assumed to be incompressible, inviscid, irrotational, and density-matched. One such thin jet model is a coupled system of PDEs with nonlocal terms—Hilbert transforms—that result from expansion of a Biot-Savart integral. In order to make the asymptotic model well-posed, the Hilbert transforms act upon time derivatives of the jet thickness, making the system implicit. Within this thin jet model, we demonstrate numerically the formation of finite-time pinching singularities, where the width of the jet collapses to zero at a point. These singularities are driven by the surface tension and are very similar to those observed previously by Hou, Lowengrub, and Shelley in large-scale simulations of the Kelvin-Helmholtz instability with surface tension and in other related studies. Dropping the nonlocal terms, we also study a much simpler local model. For this local model we can preclude analytically the formation of certain types of singularities, though not those of pinching type. Surprisingly, we find that this local model forms pinching singularities of a very similar type to those of the nonlocal thin jet model. © 1998 John Wiley & Sons, Inc.     

On the well‐posedness of a mathematical model describing water‐mud interaction

In this paper, we consider a mathematical model describing the two‐phase interaction between water and mud in a water canal when the width of the canal is small compared with its depth. The mud is treated as a non‐Newtonian fluid, and the interface between the mud and fluid is allowed to move under the influence of gravity and surface tension. We reduce the mathematical formulation, for small boundary and initial data, to a fully nonlocal and nonlinear problem and prove its local well‐posedness by using abstract parabolic theory. Copyright © 2012 John Wiley & Sons, Ltd.