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.     

Kelvin–Helmholtz instability with a free surface

The well-posedness of the hydrostatic equations is linked to long wave stability criteria for parallel shear flows. We revisit the Kelvin--Helmholtz instability with a free surface. In the wall-bounded case, the flow is unstable to all wave lengths. Short wave instabilities are localized and independent of boundary conditions. On the other hand, long waves are shown to be stable if the upper boundary is a free surface and gravity is sufficiently small. We also consider smooth velocity profiles of the base flow rather than a velocity jump. We show that stability of long waves for small gravity generally holds for monotone profiles U(y). On the other hand, this need not be the case if U is not monotone.     

Magnetoviscous potential flow analysis of Kelvin–Helmholtz instability with heat and mass transfer

A linear analysis of the Kelvin–Helmholtz instability of interface between two viscous and magnetic fluids has been carried out where there was heat and mass transfer across the interface while the fluids have been subjected to a constant magnetic field parallel to the streaming direction. The viscous potential flow theory has been used for the investigation. A dispersion relation has been obtained and a stability criterion is given by a critical value of relative velocity as well as the critical value of magnetic field. The resulting plots show the effect of various physical parameters such as wave number, viscosity ratio, ratio of magnetic permeabilities and heat transfer coefficient. It has been observed that heat and mass transfer has a destabilizing effect whereas the horizontal magnetic field stabilizes the system.     

A note on “Quasi-analytical solution of two-dimensional Helmholtz equation”

The recent paper of Van Hirtum in this journal repeats a number of misconceptions about the use of conformal mappings in solving the two-dimensional Helmholtz equation. These are discussed, as is the fact that the numerical approach presented does not lead to accurate results. In general conformal mapping is not useful in solving Helmholtz’s equation. Other, accurate, techniques are briefly reviewed.     

Analysis of electroosmotic flow with periodic electric and pressure fields via the lattice Poisson–Boltzmann method

This paper analyzes the electroosmotic flow fields in heterogeneous microchannels by applying the lattice Poisson–Boltzmann equation. The influences of surface potential, ionic molar concentration, channel height, and driving force fields on fluid velocity are discussed in detail. A scheme for producing vortexes in a straight channel by adjusting the heterogeneous surface potentials and phase angles of the periodic driving force fields is introduced. By distributing the heterogeneous surface potentials at particular positions, we can create vortexes near walls or in the center of the channel. The size, strength, and rotational direction of vortexes are further variable by introducing appropriate phase angles for a single driving force field or for the phase differences between combined driving force fields, such as electric/pressure fields. These obstacle-like vortexes perturb fluids and hinder flow, and thus, may be useful for enhancing micromixer performance.     

The influence of a DC electric field on the von Kármán vortex street in the wake of a confined cylinder

We investigate the influence of a DC electric field on the flow around and in the wake of a confined cylinder by means of numerical simulations. Our results indicate that even very small electrical perturbations have significant impact on the settling time of the lift coefficient. Moreover, the oscillations of the lift coefficient of pure pressure-driven and pressure-driven flow with induced electrical field are in anti-phase. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Landau levels on the hyperbolic plane in the presence of Aharonov–Bohm fields

We consider the magnetic Schrödinger operators on the Poincaré upper half plane with constant Gaussian curvature ?1. We assume the magnetic field is given by the sum of a constant field and the Dirac δ measures placed on some lattice. We give a sufficient condition for each Landau level to be an infinitely degenerated eigenvalue. We also prove the lowest Landau level is not an eigenvalue if the above condition fails. In particular, the infinite degeneracy of the lowest Landau level is equivalent to the infiniteness of the zero-modes of the two-dimensional Pauli operator.     

Qualitative behaviour of solutions for the two-phase Navier–Stokes equations with surface tension

The two-phase free boundary value problem for the isothermal Navier–Stokes system is studied for general bounded geometries in absence of phase transitions, external forces and boundary contacts. It is shown that the problem is well-posed in an $L_p$ -setting, and that it generates a local semiflow on the induced state manifold. If the phases are connected, the set of equilibria of the system forms a $(n+1)$ -dimensional manifold, each equilibrium is stable, and it is shown that global solutions which do not develop singularities converge to an equilibrium as time goes to infinity. The latter is proved by means of the energy functional combined with the generalized principle of linearized stability.     

The Popular lectures and addresses of William Thomson,Baron Kelvin of Largs (1824–1907)

William Thomson, Lord Kelvin, was the best-known scientist of his day, an eminent natural philosopher, engineer, and mathematician. His Popular lectures and addresses cover many aspects of science in a non-technical way. After a general introduction on popular scientific writing, Kelvin's varied topics are listed, and just one is described in more detail: his lecture ‘On ship waves’. This paper was first presented at a colloquium at the University of Glasgow, Glasgow on Literature and Mathematics in May 2011.     

Theoretical analysis of Rayleigh–Taylor instability on a spherical droplet in a gas stream

A linear analysis of the Rayleigh–Taylor (R–T) instability on a spherical viscous liquid droplet in a gas stream is presented. Different from the most previous studies in which the external acceleration is usually assumed to be radial, the present study considers a unidirectional acceleration acting on a spherical droplet with arbitrary initial disturbances and therefore can provide insights into the influence of R–T instability on the atomization of spherical droplets. A general recursion relation coupling different spherical modes is derived and two physically prevalent limiting cases are discussed. In the limiting case of inviscid droplet, the critical Bond numbers to excite the instability and the growth rates for a given Bond number are obtained by solving two eigenvalue problems. In the limiting case of large droplet acceleration, different spherical modes are asymptotically decoupled and an explicit dispersion relation is derived. For given Bond number and Ohnesorge numbers, the critical size of stable droplet, the most-unstable mode and its corresponding growth rate are determined theoretically.     

Influence of finite-density fluctuations on the development of the Rayleigh–Taylor instability in a porous medium

Theoretical and Mathematical Physics - We numerically simulate the Rayleigh–Taylor convection in a porous medium in the presence of initial density fluctuations at the interface between two...     

The influence of perceived stock value price histories in the mean–variance-instability model

The model introduced by H. Talpaz, A. Harpaz and J.B. Penson (1984. European Journal of Operational Research 14, 262–269) extends the mean–variance model introducing the concept of instability. In this way it is possible to see an investor's attitude towards predicted instability. In this paper we show how optimisation procedures based on penalty (or preferred) weighted instability matrices can be interpreted in terms of real time utility functions which depend on an `actual' and a `remembered' time series due to fading memory. This approach justifies some bounded normalised functions used to represent the investors preference between the irregular frequency fluctuations.     

Hydrodynamic Coherence and Vortex Solutions of the Euler–Helmholtz Equation

The form of the general solution of the steady-state Euler–Helmholtz equation (reducible to the Joyce–Montgomery one) in arbitrary domains on the plane is considered. This equation describes the dynamics of vortex hydrodynamic structures.     

A third-order Taylor–Galerkin model for the simulation of two-dimensional unsteady free surface flows

A Taylor–Galerkin third-order method is presented to integrate the equations describing two-dimensional, unsteady flows having a free surface. The discretization in time, with Taylor series expansions, is based on a fractional step, while the spatial approximation is obtained by the conventional Galerkin finite element method. Results are presented, in terms of the water profile history and flow velocity field, for two simulations: a partial and sudden breakage of a dam in a horizontal and frictionless river-bed, and the flow through a channel contraction.     

A Davey–Stewartson description of two-dimensional solitons in nonlocal media

Novel soliton solutions of a two-dimensional (2D) nonlocal nonlinear Schrödinger (NLS) system are revealed by asymptotically reducing the system to a completely integrable Davey–Stewartson (DS) set of equations. In so doing, the reductive perturbation method in addition to a multiple scales scheme are utilized to derive both the DS-I and DS-II systems, depending on the strength of the nonlocality, which in turn, may be regarded here as a measure of the surface tension. As such, two different soliton solutions are obtained: the breather and dromion solutions in the case of DS-I (weak nonlocality), as well as lump solutions in the case of DS-II (strong nonlocality). Besides their immediate mathematical importance, our results find a wide range of applications due the high applicability of the relative nonlocal NLS (optics, plasmas, liquid crystals, and thermal media in the strong nonlocality regime, etc.) and hence these structures can also be realized experimentally in various physical setting.