On permanent nonlinear waves in a rotating fluid

The governing equation for long nonlinear gravity waves in a rotating fluid changes with the value of the Coriolis parameter f. (1) When f is large, i.e. in the strong rotation case, in an infinite ocean, there are only Sverdrup waves; in a semi-infinite ocean or in a channel, there are either solitary Kelvin waves, for which the governing equation is a KdV equation, or Poincaré waves, which can be obtained by superposition of two Sverdrup waves. (2) When f is small, i.e. in the weak rotation case, in an infinite ocean there are solitary or cnoidal waves governed by the Ostrovskiy equation, and we provide an explicit solution for both solitary and cnoidal Ostrovskiy progressive waves; and in a semi-infinite ocean or a channel, there are Sverdrup waves, which are governed either by Ostrovskiy equations or by the Grimshaw-Melville equation. (3) When f is very small, i.e. in the very weak rotation case, in an infinite ocean, or in a channel, there are solitary waves with a horizontal crest, but with a velocity component or a pressure gradient, which are governed by KdV equations as in the non-rotating case. Physically, that means that the most determining factor is the ratio of the Rossby radius of deformation over a characteristic length of the wave.     

Hydromagnetic Stokes flow in a rotating fluid with suspended small particles

An initial value investigation is made of the motion of an incompressible, viscous conducting fluid with embedded small spherical particles bounded by an infinite rigid non-conducting plate. Both the plate and the fluid are in a state of solid body rotation with constant angular velocity about an axis normal to the plate. The flow is generated in the fluid-particle system due to non-torsional oscillations of a given frequency superimposed on the plate in the presence of a transverse magnetic field. The operational method is used to derive exact solutions for the fluid and the particle velocities, and the wall shear stress. The small and the large time behaviour of the solutions is discussed in some detail. The ultimate steady-state solutions and the structure of the associated boundary layers are determined with physical implications. It is shown that rotation and magnetic field affect the motion of the fluid relatively earlier than that of the particles when the time is small. The motion for large times is set up through inertial oscillations of frequency equal to twice the angular velocity of rotation. The ultimate boundary layers are established through inertial oscillations. The shear stress at the plate is calculated for all values of the frequency parameter. The small and large-time behaviour of the shear stress is discussed. The exact solutions for the velocity of fluid and the wall shear stress are evaluated numerically for the case of an impulsively moved plate. It is found that the drag and the lateral stress on the plate fluctuate during the non-equilibrium process of relaxation if the rotation is large. The present analysis is very general in the sense that many known results in various configurations are found to follow as special cases.     

Stokes flow over a cavity on a superhydrophobic surface containing a gas bubble

Within the approximation of Stokes hydrodynamics, several problems of a steady-state flow over a two-dimensional cavity containing a gas bubble are solved using the method of boundary integral equations. In contrast to previous publications, the method developed makes it possible to study the situation in which the cavity is only partially filled with gas, and the edges of a curved phase interface do not coincide with the cavity corners. Using periodic boundary conditions for the velocity, the flows with pure-shear and parabolic velocity profiles, and also the flow over a group of cavities were considered. The aim of the study was to calculate the effective (average) slip velocity over a microcavity, as applied to flows near textured superhydrophobic surfaces. A parametric numerical study of the effective velocity slip as a function of the radius of curvature of the interface and the position of the interface relative to the cavity boundaries was performed. The accuracy of the method is validated by the calculations of a number of limiting flows over a cavity, for which a quantitative agreement with the results known in the literature is demonstrated.     

Internal and inertial waves in a viscous rotating stratified fluid

The effects of viscosity on the propagation of a St. Andrew's cross wave which is generated by a simple-harmonic localized disturbance in a rotating stratified fluid are considered. A similarity solution of the linearised equations shows that the velocities decay and that the wave width increases away from the disturbance. Previous solutions in a stratified non-rotating fluid are recovered by letting the rotation tend to zero. The solutions are also valid in the limit of a homogeneous rotating fluid. Further solutions for waves in a realistic ocean and in an isothermal atmosphere on a rotating Earth are also included.     

Experimental investigation of nonaxisymmetric inertia waves in a rotating fluid

The properties of nonaxisymmetric waves in a fluid rotating as a rigid body in a cylindrical vessel has been studied experimentally. The method of resonance generation of the required mode is used, as this does not lead to restructuring of the basic flow. It is shown that resonance generation of the natural mode is an essentially unsteady process, whose initial stage conforms closely to the linear theory. It is established that after reaching a critical amplitude the wave ceases to grow and its original structure decays nonlinearly.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 176–180, January–February, 1987.     

Stretching surface in rotating viscoelastic fluid

The boundary layer flow over a stretching surface in a rotating viscoelastic fluid is considered. By applying a similarity transformation, the governing partial differ- ential equations are converted into a system of nonlinear ordinary differential equations before being solved numerically by the Keller-box method. The effects of the viscoelastic and rotation parameters on the skin friction coefficients and the velocity profiles are thor- oughly examined. The analysis reveals that the skin friction coefficients and the velocity in the x-direction increase as the viscoelastic parameter and the rotation parameter in- crease. Moreover, the velocity in the y-direction decreases as the viscoelastic parameter and the rotation parameter increase.     

Shallow-water waves on a rotating ocean due to oscillatory surface stresses

The initial-value problem of shallow-water waves due to an oscillatory surface stress distribution on a homogeneous rotating ocean is solved by the method of integral transforms. For the wave integral, an asymptotic analysis is given which is uniform across the line produced by the coalescing of the pole and the stationary point of the wave spectrum; the result, unlike previous findings, is non-singular when the circular frequency of oscillation equals the Coriolis parameter. Some limiting cases of interest are deduced and the asymptotic envelope of the progressive waves at the surface is illustrated graphically.     

Nonlinear capillary waves on a viscous fluid jet surface

Capillary instability of a fluid jet is one of the classical problems of hydrodynamics [1]. Studying it is of practical interest, particularly for the optimization of the ignition of a liquid propellant and the development of granulating apparatus in the chemical industry [2]. Until recently, the main attention has been paid to analyzing linear problems. Dispersion equations have been obtained for small perturbations of a jet surface with the viscosity of the external medium taken into account [3]. The construction of a theory of finite-amplitude waves on an ideal fluid jet surface was started in [4, 5]. Up to now this theory has achieved substantial results, as can be assessed by the successful numerical modeling of the dissociation of an inviscid fluid jet into drops [6] (see [7, 8] also). This paper is devoted to a discussion of the nonlinear development stage of viscous fluid jet instability under conditions allowing the influence of the surrounding medium and the gravity field to be neglected.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 179–182, March–April, 1977.The author is grateful to B. M. Konyukhov and G. D. Kuvatov for suggesting this problem and performing the experiment and to M. I. Rabinovich for useful discussions.     

Ray method for surface waves on a ferromagnetic fluid

The ray method, or geometrical optics method, is used to study surface waves on a ferromagnetic fluid of variable depth. Both time-dependent and time-reduced cases are considered on the basis of the shallow-water approximation. An application of the time-reduced ray method leads to the discovery of edge waves on a ferromagnetic fluid over a uniformly sloping bottom.     

Three-dimensional waves on the surface of a viscous fluid

The study of the effect of viscosity on the propagation of surface waves has traditionally been confined to the consideration of plane (two-dimensional) waves [1, 2]. So far the effect associated with taking the transverse modulation of the wave profile into account have not been studied. In what follows, a solution is constructed and analyzed for linear three-dimensional periodic waves in an infinitely deep fluid. Their distinguishing property is the presence of a vorticity in the direction of propagation. An expression for the average (over the wavelength) velocity of horizontal particle drift is found in the quadratic approximation in the small wave steepness.     

Gravitational waves on a bounded area of a fluid surface

Ufa State Technical Aviation University, Ufa 450000. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 37, No. 2, pp. 83–89, March–April, 1996.     

Fluid flow in a rotating cylindrical container with a rotating disk at the fluid surface

Fluid flow in a rotating cylindrical container of radius R_w and height H with a co-axially rotating disk of radius R_d at the fluid surface is numerically investigated. The container and the disk rotate with angular velocities Ω_w and Ω_d, respectively. We solve the axisymmetric Navier-Stokes equations using a finite-volume method. The effects of the relative directions and magnitudes of the disk and container rotations are studied. The calculations are carried out with various ratios of Ω_w and Ω_d for H/R_w = 2 and R_d/R_w = 0.7. Streamlines and velocity vectors in the meridional plane and azimuthal velocities are obtained. The flow fields in the meridional plane are discussed with relation to azimuthal velocities in the interior of the container. The numerical results are also compared with experimental data.     

Hall current effects on waves in an electrically conducting rotating fluid

The waves generated by an obstacle moving with constant velocity U along the axis of an inviscid, incompressible, infinitely conducting fluid with Hall current effects, rotating with an angular velocity are studied by Lighthill's technique. Three cases arise according as U>A ₁, U=A ₁ or U ₁, where A ₁ is the modified Alfven velocity due to Hall current term. In the cases U> and <A ₁ the wave-number surface consists of two distinct spheres and four coincident planes. The waves corresponding to the points on the outer sphere trail behind and those on the inner sphere travel ahead of the disturbance if U>A ₁ and vice-versa if U ₁. When U=A ₁ the wave-number surface consists of two coincident spheres and four coincident planes and the spherical waves are found both ahead and behind. By drawing appropriate normals to the four planes, it is seen that the formation of the Taylor column ahead of the disturbance is possible in all the cases.     

Flow of a yield stress fluid over a rotating surface

We study the flow of yield stress fluids over a rotating surface when both the viscoelastic solid behavior below a critical deformation (γ _c) and liquid properties beyond γ _c can play a significant role. We review the detailed characteristics of the flow in the solid regime in the specific case of a pure elongational strain (large height to radius ratio). We, in particular, show that there exists a critical rotation velocity (ω _c) associated with the transition from the solid to the liquid regime. We then consider the specific case of lubricational regime (small height to radius ratio) in the liquid regime. In that case we describe the different possible evolutions of the equilibrium shape of the material as a function of the rotation velocity (ω), from which we extrapolate the transient shape evolutions as ω increases. We show that for a sufficiently large rotation velocity the sample separates into two parts, one remaining at rest around the rotation axis, the other going on moving radially. These predictions are then compared with systematic spin-coating tests under increasing rotation velocity ramps followed by a plateau at ω _f with typical yield stress fluids. It appears that there exists a critical velocity below which the material undergoes a limited elongation and beyond which it starts to spread significantly over the solid surface. For a larger ω _f value the sample forms a thick peripheral roll, leaving behind it a thin layer of fluid at rest relatively to the disc. These characteristics are in qualitative agreement with the theoretical predictions. Beyond a sufficiently large ω _f value this roll eventually spreads radially in the form of thin fingers. Moreover, in agreement with the theory in the lubricational regime, the different curves of deformation vs ω fall along a master curve when the rotation velocity is scaled by ω _c for different accelerations, different sample radii, or different material yield stress. The final thickness of the deposit seems to be mainly governed by the displacement of the roll, the characteristics of which take their origin in the initial stage of the spreading, including the solid–liquid transition.     

Azimuthal wave motions on the surface of a rotating fluid cylinder

Wave motions in a fluid cylinder rotating about the axis are investigated within the framework of the linear theory. The cylinder is assumed to be fairly long. This makes it possible to restrict attention to the study of the plane oscillation pattern. The fluid is assumed to be ideal and incompressible. The models in which the fluid particles are confined by gravitational (body) or/and capillary forces (surface stress forces) are considered. A mode analysis is carried out and the dispersion relations are constructed. Traveling and steady-state waves on the surface of the fluid cylinder are investigated; qualitative effects ("wave inertia") are established. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 128–133, May–June, 1998. The work was carried out with financial support from the Russian Foundation for Basic Research (project No. 96-01-00221). An erratum to this article is available at .     

Nonlinear surface waves on a ferromagnetic fluid of variable depth

A nonlinear ray method is used to study surface waves on a ferromagnetic fluid of variable depth subject to a horizontal magnetic field, and an equation of the KdV type with variable coefficients is derived. An approximate solution of the equation representing a three-dimensional soliton with varying amplitude and phase is constructed and numerical results are presented.     

Theory of weak turbulence of waves on a fluid surface

The nonlinear interaction of waves in a fluid of finite depth is discussed. Forbidden decay processes in the gravitational portion of the spectrum are eliminated from the Hamiltonian by means of a canonical transformation. This provides an opportunity to obtain a kinetic equation which takes into account scattering of capillary waves by gravitational waves, in addition to decays in the subsystem of gravitational waves. The distribution N_k P^1/2h^1/4k^–4 is obtained for capillary waves in shallow water with constant flow of energy P with respect to the spectrum in the space of the wave numbers k. The interaction of the gravitational and capillary turbulence spectra is discussed. An induced distribution of gravitational waves is found which results from their interaction with capillary waves. It is an increasing function of the wave numbers q in the region bounded by the capillary constant k_o, N_q q^9/4 (q < k_o). The coupling of spectra in the gravitational and capillary regions and the conversion from slightly turbulent distributions to universal distributions are discussed.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 97–106, November–December, 1974.     

Threshold of parametric excitation of waves on a fluid surface

Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 61–67, July–August, 1988.