Growth rates for the linearized motion of fluid interfaces away from equilibrium

We consider the motion of a two-dimensional interface separating an inviscid, incompressible, irrotational fluid, influenced by gravity, from a region of zero density. We show that under certain conditions the equations of motion, linearized about a presumed time-dependent solution, are wellposed; that is, linear disturbances have a bounded rate of growth. If surface tension is neglected, the linear equations are well-posed provided the underlying exact motion satisfies a condition on the acceleration of the interface relative to gravity, similar to the criterion formulated by G. I. Taylor. If surface tension is included, the linear equations are well-posed without qualifications, whether the fluid is above or below the interface. An interesting qualitative structure is found for the linear equations. A Lagrangian approach is used, like that of numerical work such as [3], except that the interface is assumed horizontal at infinity. Certain integral equations which occur, involving double layer potentials, are shown to be solvable in the present case. © 1993 John Wiley & Sons, Inc.     

对冲圆射流生成径向扩张液膜研究

采用两股互相冲击的圆射流可以形成环形的液体薄膜,液膜在径向扩展到一定的临界半径距离会破碎．数值模拟了液膜在周围气体中形成和破碎的非定常过程．考虑了液体和气体都是不可压缩Newton流体的轴对称问题．液体和气体的界面采用Level set函数来跟踪,Navier-Stokes 控制方程和物理边界条件采用有限差分格式离散求解．计算结果给出了环形液体薄膜形成并在其环形边缘处破碎,并缓慢运动的过程．液膜的厚度随着液膜在轴向的扩展会逐渐变薄,因此定义的局部Weber数会在径向逐渐减小,这里的局部Weber数定义为ρu2h/σ,其中ρ和σ分别为液体的密度和界面的张力,u和h分别为在径向某个位置的液膜的平均径向速度和半液膜厚度．数值结果表明就像实验中所观察到的那样,液膜径向扩展的过程的确会在局部Weber数趋向于1的时候终结而停止扩张．根据空间-时间线性稳定性理论,液膜的破碎最初是由正弦模式在临界局部Weber数Wec=1引起的,在临界局部Weber数小于1时会发生绝对不稳定性．在线性理论中另一个独立的模式,所谓的余弦模式,则增长比正弦模式要慢,从而会推测到正弦模式主导破碎的结论．然而,这里的数值结果却表明,余弦模式在界面波的非线性发展阶段实质的超越了正弦模式的增长,并对液膜的最终阶段的破碎起主导作用．这验证了线性理论只能够对触发时扰动波的性质进行预测,而对失稳后情况和结果的预测则不一定正确．     

Lie group analysis for the effect of temperature-dependent fluid viscosity with thermophoresis and chemical reaction on MHD free convective heat and mass transfer over a porous stretching surface in the presence of heat source/sink

This paper concerns with a steady two-dimensional flow of an electrically conducting incompressible fluid over a vertical stretching sheet. The flow is permeated by a uniform transverse magnetic field. The fluid viscosity is assumed to vary as a linear function of temperature. A scaling group of transformations is applied to the governing equations. The system remains invariant due to some relations among the parameters of the transformations. After finding three absolute invariants a third-order ordinary differential equation corresponding to the momentum equation and two second-order ordinary differential equation corresponding to energy and diffusion equations are derived. The equations along with the boundary conditions are solved numerically. It is found that the decrease in the temperature-dependent fluid viscosity makes the velocity to decrease with the increasing distance of the stretching sheet. At a particular point of the sheet the fluid velocity decreases with the decreasing viscosity but the temperature increases in this case. It is found that with the increase of magnetic field intensity the fluid velocity decreases but the temperature increases at a particular point of the heated stretching surface. Impact of thermophoresis particle deposition with chemical reaction in the presence of heat source/sink plays an important role on the concentration boundary layer. The results thus obtained are presented graphically and discussed.     

Remarks on the derivation of the hydrostatic Euler equations

The motion of an inviscid incompressible fluid between two horizontal plates is studied in the limit when the plates are infinitesimally close. The convergence of the solutions of the Euler equations to those of their formal ‘hydrostatic’ limit can be established in the case when the initial velocity field satisfies a local Rayleigh conditions. This result, originally obtained by Grenier through weighted energy estimates based on Arnold's stability analysis of the Euler equations, is proven here by a more straightforward method even closer to Arnold's method.     

Propagation of harmonic waves in prestressed fiber viscoelastic thick tubes filled with a viscous dusty fluid

In this work, propagation of harmonic waves in initially stressed cylindrical viscoelastic thick tubes filled with a Newtonian fluid is studied. The tube, subjected to a static inner pressure P_i and a positive axial stretch λ, will be considered as an incompressible viscoelastic and fibrous material. The fluid is assumed as an incompressible, viscous and dusty fluid. The field equations for the fluid are obtained in the cylindrical coordinates. The governing differential equations of the tube’s viscoelastic material are obtained also in the cylindrical coordinates utilizing the theory of small deformations superimposed on large initial static deformations. For the axially symmetric motion the field equations are solved by assuming harmonic wave solutions. A closed form solution can be obtained for equations governing the fluid body, but due to the variability of the coefficients of resulting differential equations of the solid body, such a closed form solution is not possible to obtain. For that reason, equations for the solid body and the boundary conditions are treated numerically by the finite-difference method to obtain the effects of the thickness of the tube on the wave characteristics. Dispersion relation is obtained using the long wave approximation and, the wave velocities and the transmission coefficients are computed.     

Vibration of fluid-conveying nanotubes subjected to magnetic field based on the thin-walled Timoshenko beam theory

In this article, the flutter vibrations of fluid-conveying thin-walled nanotubes subjected to magnetic field is investigated. For modeling fluid structure interaction, the nonlocal strain gradient thin-walled Timoshenko beam model, Knudsen number and magnetic nanoflow are assumed. The Knudsen number is considered to analyze the slip boundary conditions between the fluid-flow and the nanotube's wall, and the average velocity correction parameter is utilized to earn the modified flow velocity of nano-flow. Based on the extended Hamilton's principle, the size-dependent governing equations and associated boundary conditions are derived. The coupled equations of motion are transformed to a general eigenvalue problem by applying extended Galerkin technique under the cantilever end conditions. The influences of nonlocal parameter, strain gradient length scale, magnetic nanoflow, longitudinal magnetic field, Knudsen number on the eigenvalues and critical flutter velocity of the nanotubes are studied.     

Dynamics of a rotor partially filled with a viscous incompressible fluid

A rotor partially filled with a viscous incompressible fluid is modeled as planar system. Its structural part, i. e. the rotor, is assumed to be rigid, circular, elastically supported and running with a prescribed time-dependent angular velocity. Both parts, structure and fluid, interact via the no-slip condition and the pressure. The point of departure for the mathematical formulation of the fluid filling is the Navier-Stokes equation, which is complemented by an additional equation for the evolution of its free inner boundary. Further, rotor and fluid are subjected to volume forces, namely gravitation. Trial functions are chosen for the fluid velocity field, the pressure field and the moving boundary, which fulfill the incompressibility constraint as well as the boundary conditions. Inserting these trial functions into the partial differential equations of the fluid motion, and applying the method of weighted residuals yields equations with time derivatives only. Finally, in combination with the rotor equations, a nonlinear system of 12 differential-algebraic equations results, which sufficiently describes solutions near the circular symmetric state and which may indicate the loss of its stability. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Radiation effects on the MHD flow near the stagnation point of a stretching sheet: revisited

This paper considers the effects of radiation on the flow near the two-dimensional stagnation point of a stretching sheet immersed in a viscous and incompressible electrically conducting fluid in the presence of an applied constant magnetic field. The external velocity and the stretching velocity of the sheet are assumed to vary linearly with the distance from the stagnation point. The governing partial differential equations are transformed into a system of ordinary differential equations using a similarity transformation, before being solved numerically by the Keller-box method. The features of the heat transfer characteristics for different values of the governing parameters are analyzed and discussed. The results indicate that the heat transfer rate at the surface decreases in the presence of radiation.     

Radiation effects on the MHD flow near the stagnation point of a stretching sheet: revisited

This paper considers the effects of radiation on the flow near the two-dimensional stagnation point of a stretching sheet immersed in a viscous and incompressible electrically conducting fluid in the presence of an applied constant magnetic field. The external velocity and the stretching velocity of the sheet are assumed to vary linearly with the distance from the stagnation point. The governing partial differential equations are transformed into a system of ordinary differential equations using a similarity transformation, before being solved numerically by the Keller-box method. The features of the heat transfer characteristics for different values of the governing parameters are analyzed and discussed. The results indicate that the heat transfer rate at the surface decreases in the presence of radiation.     

Love waves in a fluid-saturated porous layer under a rigid boundary and lying over an elastic half-space under gravity

In this paper, mathematical modeling of the propagation of Love waves in a fluid-saturated porous layer under a rigid boundary and lying over an elastic half-space under gravity has been considered. The equations of motion have been formulated separately for different media under suitable boundary conditions at the interface of porous layer, elastic half-space under gravity and rigid layer. Following Biot, the frequency equation has been derived which contain Whittaker’s function and its derivative that have been expanded asymptotically up to second term (for approximate result) for large argument due to small values of Biot’s gravity parameter (varying from 0 to 1). The effect of porosity and gravity of the layers in the propagation of Love waves has been studied. The effect of hydrostatic initial stress generated due to gravity in the half-space has also been shown in the phase velocity of Love waves. The phase velocity of Love waves for first two modes has been presented graphically. Frequency equations have also been derived for some particular cases, which are in perfect agreement with standard results. Subsequently the lower and upper bounds of Love wave speed have also been discussed.     

The Modulation of an Internal Gravity-Wave Packet,and the Resonance with the Mean Motion

An inviscid, incompressible, stably stratified fluid occupies a horizontal channel, along which an internal gravity-wave packet is propagating. The wave induced mean motions are calculated, and the equations describing the evolution of the wave amplitude derived. When the group velocity of the wave packet coincides with a long-wave speed there is a resonance, and the equations describing this resonance are derived.     

Near-resonace transverse oscillations in an elastic layer. Steady-state solutions

The solutions of the equations of the non-linear evolution of transverse oscillations in a layer of an incompressible elastic medium under conditions close to resonance conditions are investigated qualitatively and using analytical methods. The oscillations are created by a small periodic motion of one of the boundaries in its plane, with a period that is close to the period of the natural oscillations of the layer. It is assumed that the medium can possess slight anisotropy and that the amplitude of the oscillations which arise is small. Previously obtained differential equations are used, which describe the slow evolution of the wave pattern of non-linear transverse waves. Two possible formulations of problems for these equations are considered. In the first formulation, it is determined what the external action must be in order that the non-linear evolution of oscillations or periodic oscillations occurs according to a (previously specified) desired law. In the second formulation it is assumed that the periodic motion of one of the boundaries is given. It is shown that a steady-state solution, that does not vary from period to period, can be represented by a continuous solution and also by a solution which contains discontinuities in the strain and velocity components. The mechanism of the overturn of a non-linear wave during its evolution and the formation of a discontinuity are qualitatively described.     

Natural frequencies of rectangular Mindlin plates coupled with stationary fluid

The present study is concerned with the free vibration analysis of a horizontal rectangular plate, either immersed in fluid or floating on its free surface. The governing equations for a moderately thick rectangular plate are analytically derived based on the Mindlin plate theory (MPT), whereas the velocity potential function and Bernoulli’s equation are employed to obtain the fluid pressure applied on the free surface of the plate. The simplifying hypothesis that the wet and dry mode shapes are the same, is not assumed in this paper. In this work, an exact-closed form characteristics equation is used for the plate subjected to a combination of six different boundary conditions. Two opposite sides are simply supported and any of the other two edges can be free, simply supported or clamped. To demonstrate the accuracy of the present analytical solution, a comparison is made with the published experimental and numerical results in the literature, showing an excellent agreement. Then, natural frequencies of the plate are presented in tabular and graphical forms for different fluid levels, fluid densities, aspect ratios, thickness to length ratios and boundary conditions. Finally, some 3-D mode shapes of the rectangular Mindlin plates in contact with fluid are illustrated.     

A Hybrid Immersed Interface Method for Driven Stokes Flow in an Elastic Tube

We present a hybrid numerical method for simulating fluid flow through a compliant, closed tube, driven by an internal source and sink. Fluid is assumed to be highly viscous with its motion described by Stokes flow. Model geometry is assumed to be axisymmetric, and the governing equations are implemented in axisymmetric cylindrical coordinates, which capture 3D flow dynamics with only 2D computations. We solve the model equations using a hybrid approach: we decompose the pressure and velocity fields into parts due to the surface forcings and due to the source and sink, with each part handled separately by means of an appropriate method. Because the singularly-supported surface forcings yield an unsmooth solution, that part of the solution is computed using the immersed interface method. Jump conditions are derived for the axisymmetric cylindrical coordinates. The velocity due to the source and sink is calculated along the tubular surface using boundary integrals. Numerical results are presented that indicate second-order accuracy of the method.     

Dynamics of water evaporation fronts

The evolution and shapes of water evaporation fronts caused by long-wave instability of vertical flows with a phase transition in extended two-dimensional horizontal porous domains are analyzed numerically. The plane surface of the phase transition loses stability when the wave number becomes infinite or zero. In the latter case, the transition to instability is accompanied with reversible bifurcations in a subcritical neighborhood of the instability threshold and by the formation of secondary (not necessarily horizontal homogeneous) flows. An example of motion in a porous medium is considered concerning the instability of a water layer lying above a mixture of air and vapor filling a porous layer under isothermal conditions in the presence of capillary forces acting on the phase transition interface.     

Singular integrals with generalized Cauchy kernels for the velocity field in boundary value problems of filtration in an anisotropic inhomogeneous porous layer

We study two-dimensional stationary and nonstationary boundary value problems of fluid filtration in an anisotropic inhomogeneous porous layer whose conductivity is modeled by a not necessarily symmetric tensor. For the velocity field, we introduce generalized singular Cauchy and Cauchy type integrals whose kernels are expressed via the leading solutions of the main equations and have a hydrodynamic interpretation. We obtain the limit values of a Cauchy type generalized integral (Sokhotskii-Plemelj generalized formulas). This permits one to develop a method for solving boundary value problems for the filtration velocity field. The idea of the method and its efficiency are illustrated for the boundary value problem of filtration in adjacent layers of distinct conductivities and the problem of the evolution of liquid interface.     

Electrohydrodynamic linear stability of finitely conducting flows through porous fluids with mass and heat transfer

In this work, a linear stability analysis is used to investigate a capillary surface waves between two horizontal finite fluid layers. The system is acted upon by a vertical periodic electric field. The problem examines few representatives of porous media. It is also includes finite conductivity, mass and heat transfer. It is assumed that the basic flow is two-dimensional streaming flow. A general dispersion relation governing the linear stability is derived. In contrast with our previous work [23], the present problem shows that the stability criterion depends on the mass and heat transfer parameter. The present study recovers some special cases upon appropriate data choices. The presence of finite conductivity’s together with the dielectric permeability’s make the uniform electric field plays a dual role in the stability criterion. This shows some analogy with the nonlinear stability theory. In addition, the mass and heat transfer parameter as well as the Darcy’s coefficients play a stabilizing role in the stability picture. In case of the Rayleigh–Taylor instability, by means of the Whittaker technique, the parametric excitation of the electrohydrodynamic surface waves is obtained. The transition curve equations are calculated up to the fourth order for a small dimensionless parameter. The analytical results are numerically confirmed.     

Existence and qualitative theory for stratified solitary water waves

This paper considers two-dimensional gravity solitary waves moving through a body of density stratified water lying below vacuum. The fluid domain is assumed to lie above an impenetrable flat ocean bed, while the interface between the water and vacuum is a free boundary where the pressure is constant. We prove that, for any smooth choice of upstream velocity field and density function, there exists a continuous curve of such solutions that includes large-amplitude surface waves. Furthermore, following this solution curve, one encounters waves that come arbitrarily close to possessing points of horizontal stagnation.We also provide a number of results characterizing the qualitative features of solitary stratified waves. In part, these include bounds on the wave speed from above and below, some of which are new even for constant density flow; an a priori bound on the velocity field and lower bound on the pressure; a proof of the nonexistence of monotone bores in this physical regime; and a theorem ensuring that all supercritical solitary waves of elevation have an axis of even symmetry.     

Annular liquid jets and other axisymmetric free-surface flows at high Reynolds numbers

A perturbation method based on a long wavelength approximation is used to obtain the leading order equations governing the fluid dynamics of laminar, annular, round and compound liquid jets and liquid films on convex and concave cylindrical surfaces. An approximate, integral balance method is also used to determine the inviscid core and the thickness of the boundary layers of annular liquid jets near the nozzle exit. The steady state equations are transformed into parabolic ones by means of the von Mises transformation and solved in an adaptive, staggered grid to determine the axial velocity distribution and the location of the free surfaces. It is shown that, for free surface flows subject to inertia, gravity and surface tension, there is a contraction near the nozzle which increases as the Reynolds and Froude numbers are decreased, and is nearly independent of the Weber number for Weber numbers larger than about one hundred. It is also shown that this contraction depends on the flow considered, and is larger for films on convex surfaces. It is also shown that, for round jets, the acceleration of the jet's free surface is larger than that of the jet's centerline, although, sufficiently far from the nozzle exit, the axial velocity is uniform across the jet.     

Deformation of a drop in a viscous stream and conditions of existence of its equilibrium form

The difference method is used for obtaining a solution of the problem of unsteady motion of a drop in a stream, taking into account its deformation under conditions of axial symmetry. The fluid inside and outside the drop is assumed viscous and incompressible. The stable forms of drop are represented for various Reynolds and Weber numbers of external stream. By analyzing the conditions for normal stresses at the drop boundary, the critical Weber number was obtained, which establishes the conditions of existence of equilibrium form of the drop.