Surface shapes of fluids in rotating vessels

The shape of the phase boundary of two fluids in systems with a rotational symmetry, and particularly the level of the liquid in a rotating vessel, is studied. The knowledge of the shape of the surface of such a liquid is necessary for centrifugal moulding of optical surfaces. The dimensionless surface equations are formulated, and the solutions of these equations, taking into account the boundary conditions, are discussed.     

Perturbation theory for viscoelastic fluids between eccentric rotating cylinders

The circumferential and radial profiles of velocity, pressure and stress are derived for the flow of model viscoelastic liquids between two slightly eccentric cylinders with the inner one rotating. Singular perturbation methods are used to derive expansions valid for small gaps between the cylinders, but for all Deborah numbers. Results for Newtonian, second-order, Criminale-Ericksen-Filbey, upper-convected Maxwell, and White-Metzner constitutive equation separate the effects of elasticity, memory, and shear thinning on the development of the large stress gradients that hinder numerical solutions with these models in more complicated geometries. The effect of the constitutive equation on the critical Deborah number for flow separation is presented.     

Inhomogeneous plane waves in viscous fluids

The propagation of elliptically polarised inhomogeneous plane waves in a linearly viscous fluid is considered. The angular frequency and the slowness vector are both assumed to be complex. Use is made throughout of Gibbs bivectors (complex vectors). It is seen that there are two types of solutions—the zero pressure solution, for which the increment in pressure due to the propagation of the wave is zero, and a universal solution which is independent of the viscosity.Since the waves are attenuated in time, the usual mean energy flux vector is not a suitable way of measuring energy flux. A new energy flux vector, appropriate to these waves is defined, and results relating it with energy dissipation and energy density are obtained. These results are related to a result derived directly from the balance of energy equation.     

Non-Newtonian fluids v frictional resistance of discs and cones rotating in power-law non-Newtonian fluids

Summary Relations have been derived for the frictional resistance of finite discs and cones rotating in Ostwald-de Waele (power-law) type non-Newtonian fluids. The obtained equations can be formulated as dimensionless relations between the dimensionless moment coefficient and the generalized Reynolds number; the flow-behaviour index n enters the equations as a parameter. The relations derived for cones contain the apex angle 2₀ as an additional parameter in the form of A=sin ₀. The validity of the theoretically derived relations has been verified by measurements of the torque of discs and cones for a number of pseudoplastic power-law fluids.Nomenclature A sin ₀ parameter - b exponent in regression equation (16) - C coefficient in regression equation (16) - c _Mi dimensionless moment coefficient, for bodies wetted on one side (i=1) and for completely wetted bodies (i=2), equations (8) and (9b) - d diameter of turntable - F, G velocity functions of exact solution, equation (4) - K consistency coefficient of non-Newtonian fluids - M _Ki torque of rotating bodies, i=1 for bodies wetted on one side, i=2 for completely wetted bodies - n flow-behaviour index of non-Newtonian fluids - N=K/ kinematic consistency coefficient - P tangential force - r(y) perpendicular distance of point on cone surface from axis - R radius of disc or of base of cone - modified Reynolds number defined by equation (14) - Re _ow generalized Reynolds number defined by equation (10) - S, S area - u, v components of velocity vector - x, y, z coordinates according to fig. 1 - ₀ half the apex angle of cone - coefficient of frictional resistance defined by equation (11) - thickness of boundary layer - independent variable in exact solution, defined by equation (5) - density of fluid - _zx, _zy tangential stresses - angular velocity of rotation Indices T theoretical value - E experimental value - 0 refers to surface of rotating body     

Solitary waves in stratified fluids and their interaction

A systematic procedure is proposed for obtaining solutions for solitary waves in stratified fluids. The stratification of the fluid is assumed to be exponential or linear. Its comparison with existing results for an exponentially stratified fluid shows agreement, and it is found that for the odd series of solutions the direction of displacement of the streamlines from their asymptotic levels is reversed when the stratification is changed from exponential to linear. Finally the interaction of solitary waves is considered, and the Korteweg-de Vries equation and the Boussinesq equation are derived. Thus the known solutions of these equations can be relied upon to provide the answers to the interaction problem.     

Gyroscopic waves in a rotating liquid layer

The dispersion characteristics of gyroscopic waves in an incompressible liquid layer in a cavity of a rapidly rotating cylinder are studied. It is shown that in a viscous incompressible liquid layer, an inertial wave can be represented as the sum of six helical harmonics. The effects of the liquid viscosity and the ratio of the wave frequency to the angular velocity of rotation of the cylinder on the real and imaginary parts of the wavenumber are studied. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 2, pp. 15–20, March–April, 2008.     

Spiral waves in rotating twisted elastic pipes

The problem of propagation of bending waves in rotating pipes prestressed by longitudinal force and torque is stated and solved. Such waves are shown to be spiral ones. It is established that four waves exist for every wavelength, two of which are right-handed spirals and the other two are left-handed. These waves propagate with different velocities in different directions __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 3, pp. 125–134, March 2008.     

Inhomogeneous plane waves in compressible viscous fluids

The propagation of inhomogeneous plane waves in a compressible viscous fluid is considered. The frequency and the slowness vector are both allowed to be complex. There are seen to be two types of solutions: (a) two transverse waves, which involve no density or pressure fluctuations, (b) a longitudinal wave, which involves no fluctuations in vorticity. For each type, a propagation condition is obtained giving the (complex) squared length of the slowness vector as a function of frequency. Each depends also on the viscosities. It is seen how to recover the incompressible case as the limit in which the inviscid acoustic wave speed tends to infinity. Each wave is shown to be linearly stable for real frequencies. These waves are attenuated in space and time but nevertheless it is possible to define constant weighted mean values (over a cycle of the propagating part of the wave) of the energy density, energy flux and dissipation. The energy-dissipation equation and the propagation conditions are used to derive relationships between these constant weighted means, some of which are generalizations to compressible fluids of previously known results for incompressible fluids. Explicit expressions in terms of frequency are given for the weighted means.     

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.     

Surface waves in a rotating anisotropic elastic half-space

The Stroh formalism for surface waves in an anisotropic elastic half-space is extended to the case when the half-space rotates about an axis with a constant rotation rate. The sextic equation for the Stroh eigenvalues, the eigenvectors, the orthogonality and closure relations are obtained. The Barnett–Lothe tensors are no longer real, but two of them are Hermitian. Taziev’s equation is generalized and used to derive the polarization vector and the secular equation without computing the Stroh eigenvalues and eigenvectors. An alternative derivation using the method of first integrals by Mozhaev and Destrade yields new invariants that relate the displacement and stress and are independent of the depth from the free surface. Explicit expression of the polarization vector and the secular equation for monoclinic materials with the symmetry plane at x₃ = 0 with the rotation about the x₃-axis obtained by Destrade is re-examined, and new results are presented. Also presented is the one-component surface wave in the rotating half-space.     

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.     

Dynamics of vortex rings in viscous fluids

As part of a long range study of vortex rings, their dynamics, interactions with boundaries and with each other, we present the results of experiments on thin core rings generated by a piston gun in water. We characterize the dynamics of these rings by means of the traditional equations for such rings in an inviscid fluid suitably modifying them to be applicable to a viscous fluid. We develop expressions for the radius, core size, circulation, and bubble dimensions of these rings.     

Schlieren measurements of internal waves in non-Boussinesq fluids

Previous experiments that have examined the generation of internal gravity waves by a monochromatic source have been restricted to small amplitude forcing in Boussinesq stratified fluids. Here we present measurements of internal waves generated by a circular cylinder oscillating at large amplitude in a non-Boussinesq fluid. The ‘synthetic schlieren’ optical measurement technique (Sutherland et al. in J Fluid Mech 390:93–126, 1999) is extended to stratifications in which the index of refraction of the fluid may vary nonlinearly with density. The method is applied to examine disturbances in approximately uniformly stratified ambient fluids consisting either of sodium chloride (NaCl) or sodium iodide (NaI) solutions whose concentrations increase to near-saturation at the bottom of the tank. In particular, we report upon the first extensive measurements of the optical properties of NaI solutions as they depend upon concentration and density. Applying the results to experiments, we find that large amplitude forcing generates a patch of oscillatory turbulence surrounding the cylinder, thereby increasing the effective cylinder size and decreasing the amplitude of the waves in comparison with the predictions of linear theory. We parameterize the influence of the turbulent boundary layer in terms of an effective cylinder radius and forcing amplitude.     

Viscosity effect on free surface waves in fluids

The frequencies and damping coefficients of gravitational-capillary waves are found for a wide range of controlling dimensionless parameters. The transition to the limiting cases of deep water and an ideal fluid is analyzed. In the parameter plane, the boundary between the regions of oscillatory and aperiodic perturbations is determined and the region of weak damping is indicated. The equilibrium state of thin liquid films with account for the Van der Waals forces is considered and the dispersion equation for the capillary-Van der Waals surface waves is obtained. For a suitably chosen frequency scale, this equation is the same as that for gravitational-capillary waves. The physical conditions making it possible to observe capillary and Van der Waals waves in thin fluid layers are estimated. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 4, pp. 156–164, July–August, 2000.     

Travelling waves of density for a fourth-gradient model of fluids

In mean-field theory, the non-local state of fluid molecules can be taken into account using a statistical method. The molecular model combined with a density expansion in Taylor series of the fourth order yields an internal energy value relevant to the fourth-gradient model, and the equation of isothermal motions takes then density’s spatial derivatives into account for waves travelling in both liquid and vapour phases. At equilibrium, the equation of the density profile across interfaces is more precise than the Cahn and Hilliard equation, and near the fluid’s critical point, the density profile verifies an Extended Fisher–Kolmogorov equation, allowing kinks, which converges towards the Cahn–Hillard equation when approaching the critical point. Nonetheless, we also get pulse waves oscillating and generating critical opalescence.     

Flow in near-critical fluids induced by shock and expansion waves

Unsteady shock and expansion waves are proposed as means to produce flows near the liquid-vapor critical-point without imposing pressure gradients. By choosing appropriate initial conditions and wave speeds, near-critical post-wave conditions can be obtained. The post-shock conditions are shown to be stable with respect to perturbations in the pre-shock conditions. The initial conditions are sufficiently far from the critical-point to allow fast thermal equilibration, permitting the use of larger fluid volumes. Example calculations for the cases of an impulsively accelerated piston, of a shock tube, and of a Ludwieg-like tube are presented yielding flows up to 20 m/s in sulfur hexafluoride (SF₆), where the limit is due to the region of validity of the equation of state. The proposed setup also allows one to study shock wave propagation into near-critical fluids.Received: 13 August 2003, Revised: 7 October 2004, Published online: 4 February 2005[/PUBLISHED]PACS: 47.40.Nm, 47.50. + d, 47.55.Kf, 64.70.Fx, 64.60.HtCorrespondence to: S. Schlamp