The secondary flow induced around a sphere in an oscillating stream of elastico-viscous liquid

The present paper is devoted to the theoretical study of the secondary flow induced around a sphere in an oscillating stream of an elastico-viscous liquid. The boundary layer equations are derived following Wang's method and solved by the method of successive approximations. The effect of elasticity of the liquid is to produce a reverse flow in the region close to the surface of the sphere and to shift the entire flow pattern towards the main flow. The resistance on the surface of the sphere and the steady secondary inflow increase with the elasticity of the liquid.Nomenclature a radius of the sphere - b ^ik contravariant components of a tensor - e contravariant components of the rate of strain tensor - F() see (47) - G total nondimensional resistance on the surface of the sphere - g _ik covariant components of the metric tensor - f, g, h secondary flow components introduced in (34) - k ₀ measure of relaxation time minus retardation time (elastico-viscous parameter) - K =k ₀ ²/V ₀ ² , nondimensional parameter characterizing the elasticity of the liquid - n measure of the ratio of the boundary layer thickness and the oscillation amplitude - N, T defined in (44) - p arbitrary isotropic pressure - p _ik covariant components of the stress tensor - p ^ik contravariant components of the stress tensor associated with the change of shape of the material - R =V ₀ a/v, the Reynolds number - S =a/V ₀, the Strouhall number - r, , spherical polar coordinates - u, v, w r, , component of velocity - t time - V(, t) potential velocity distribution around the sphere - V ₀ characteristic velocity - u, v, t, y, P nondimensional quantities defined in (15) - reciprocal of s - density - defined in (32) - defined in (42) - ₀ limiting viscosity for very small changes in deformation velocity - complex conjugate of - oscillation frequency - = ₀/, the kinematic coefficient of viscosity - , defined in (52) - (, y) stream function defined in (45) - =(NT/2n)^1/2 y - /t convective time derivative (1) _ik      

MHD viscous flow between torsionally oscillating disks

The unsteady motion of an incompressible, viscous, stratified fluid between two parallel infinite disks maintained at different temperatures is studied under the influence of a uniform transverse magnetic field. The whole system is under rigid rotation in the initial state and perturbations are created by the small amplitude torsional oscillations of the disks. The time required for the transient velocity and temperature to decay is found for various ranges of the values of the forcing frequency of the disks. The steady state velocity and temperature distributions represent boundary layers on the disks and an interior flow. The interplay between the Hartmann number and the Ekman number in determining the boundary layers on the disks is discussed.     

The unsteady motion of a lamina in an elastico-viscous liquid

Summary Consideration is given to the unsteady motion of a lamina moving under a constant force in an idealised elastico-viscous liquid contained between walls parallel to the lamina.It is shown that the velocity of the lamina is strongly dependent upon the elasticity of the liquid.Both the time at which the velocity of the lamina attains an effective steady value and the corresponding distance travelled by the lamina depend in a marked way upon an elastic parameter.The authors are indebted to Dr.K. Walters for helpful comments and suggestions.     

The unsteady motion of a sphere in an elastico-viscous liquid

Summary Consideration is given to the unsteady motion of a sphere moving under a constant force in elastico-viscous liquid, the sphere being initially at rest. The problem is solved by the familiarLaplace-transform method. It is shown that the presence of elasticity in the liquid increases the velocity of the sphere. This increase is very pronounced for small values of the time, but is not very significant for large values of the time. It is concluded that the time taken to reach the terminal velocity is unaffected by the presence of elasticity in the liquid.Formulae are given which may be of use to experimentalists who use the falling-ball method to determine the zero shear-rate viscosity.     

Start-up of an elastico-viscous liquid draining from a vertical surface

The free drainage of thin films of non-Newtonian liquids from vertical surfaces occurs in a variety of industrial processes. Consideration is given in this paper to the effect of liquid elasticity on the much neglected “start-up” phase of the flow. The analysis utilizes the solution derived in 1970 by Waters and King for the generation of plane Poiseuille flow, to show that the presence of elasticity has a profound effect on the initial and the subsequent drainage profile.     

Pulsatile flow of an elastico-viscous liquid in a channel of elliptic cross-section

Summary The pulsatile flow of an elastico-viscous liquid through an elliptic tube is considered in this note. The problem is solved as a boundary and initial value problem by using the Laplace Transform method. Explicit expressions for velocity distribution are obtained in the two limiting cases of small and large values of time. The effect of elasticity on the flow is discussed analytically.

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Zusammenfassung In dieser Mitteilung wird die pulsierende Strömung einer viskoelastischen Flüssigkeit durch ein elliptisches Rohr betrachtet. Das Problem wird als Anfangs- und Randwertproblem mit Hilfe der Laplace-Transformation gelöst. Explizite Ausdrücke der Geschwindigkeitsverteilung werden für die beiden Grenzfälle kurzer und langer Zeiten angegeben. Die Wirkung der Elastizität auf die Strömung wird analysiert.

     

Drop in an oscillating liquid

At the present time there is a complete lack of studies devoted to the forced motion of a drop located in an oscillating liquid. However, this problem is of considerable interest. For example, it represents simulation of hydrodynamic processes occurring during the irradiation of drops of one liquid located in another liquid by longwave sound. The stationary flows occurring in this case may have a significant influence on the heat- and mass-transfer processes. In the present article we investigate the velocity field in the interior and exterior of a drop executing forced oscillatory motion as a result of its interaction with the ambient liquid. At a sufficiently large distance from the drop the ambient liquid oscillates in a specified way, where s/R ? 1 (s is the amplitude of displacement of the liquid particles, and R is the radius of the drop).     

Exact solutions for the flow of second grade fluid in annulus between torsionally oscillating cylinders

The velocity field and the associated shear stress corresponding to the torsional oscillatory flow of a second grade fluid, between two infinite coaxial circular cylinders, are determined by means of the Laplace and Hankel transforms. At time t = 0, the fluid and both the cylinders are at rest and at t = 0 + , cylinders suddenly begin to oscillate around their common axis in a simple harmonic way having angular frequencies ω 1 and ω 2 . The obtained solutions satisfy the governing differential equation and all imposed initial and boundary conditions. The solutions for the motion between the cylinders, when one of them is at rest, can be obtained from our general solutions. Furthermore, the corresponding solutions for Newtonian fluid are also obtained as limiting cases of our general solutions.     

Unsteady flow in the Ekman layer of an elastico-viscous liquid

The transient flow in the Ekman layer of an elastico-viscous liquid near a flat plate is discussed. Initially the fluid and the plate were rotating together and the plate then suddently starts moving with a uniform velocity in its own plane relative to the rotating frame of reference. It is shown that the ultimate steady state is reached through decay of inertial oscillations whose frequency decreases with increase in the elastic parameter.     

The generation of a draining liquid film by the vertical withdrawal of a flat plate from an elastico-viscous liquid

The analysis of Waters and Keeley [1] for the start-up of the drainage of an elastico-viscous liquid from a vertical surface is extended to include the effect of the vertical surface moving vertically with an arbitrary velocity. This is used to examine the effect of elasticity on the interesting drainage problems associated with the sudden withdrawal of a vertical plate from an elastico-viscous liquid.     

Flow interference between a stationary cylinder and an elastically mounted cylinder arranged in proximity

The flow interference between two circular cylinders, one stationary and the other free to oscillate in the transverse direction, are studied numerically at Re=150. The incompressible Navier–Stokes equation in two space dimensions, an assumption that is expected to be valid at the considered Re, is solved by the characteristic-based-split (CBS) finite element method using the T4/C3 MINI triangular element. The center-to-center spacing between the two cylinders is fixed at 4D, where D is the cylinder diameter. The angle of incident flow, α, with respect to the line through the two cylinder centers, varies within the range from α=0° to 90°. For the elastically mounted cylinder, the reduced mass considered is M_r=2.0; the structural damping coefficient is assigned to be zero, which encourages high amplitude oscillations. For each α, the computations are conducted for a wide range of reduced velocities, U_r. The flow interference is examined by scrutinizing (i) the frequency characteristics of the vortex shedding and oscillation; (ii) the dynamic response of the oscillating cylinder, including the amplitude of displacement, the drag and lift force characteristics and the phase relationship between the lift and the displacement series; and (iii) the flow response in terms of the instantaneous vorticity field. It was found that the flow interference type is significantly affected by the angle of the incident flow. As the cylinder is oscillated outside of the region of the wake behind the stationary cylinder (α≥30°), it behaves similarly to its isolated counterpart. In contrast, if the cylinder is partially or entirely submerged within the upstream wake (α<30°), then both the flow and body responses are substantially modified due to the vigorous interaction between the upstream wake and the oscillating cylinder; the response therefore belongs to the wake-induced regime. The U_r range associated with the higher amplitude response is significantly shifted toward a higher U_r. The maximum vibration amplitude builds up to a significantly higher level, even increasing the U_r far beyond the resonance regime. In general, the wake flow associated with the wake-induced vibration (WIV) regime appears to be more unperiodic than does that corresponding to the vortex-induced oscillation regime. It is also revealed that both the vortex-cylinder and the shear layer-cylinder interaction mechanisms are responsible for the characteristics of the responses of oscillating cylinder. The larger momentum required for the higher oscillation amplitude is obtained from the duration of the energy transfer from the fluid to the cylinder, which is ascribed to the phase lag between the lift force and the cylinder displacement.     

Distribution of the friction heat between two stationary pins and a rotating disc

A closed-form model for the computation of heat fluxes distribution between two motionless cylindrical pins and a fast rotating disc is investigated. Based on a finite Fourier transform technique and an asymptotic analysis, a series solution for the disc surface temperature is obtained. The solution for the pin is trivial. By matching the average temperatures over the contact areas the coefficients of the heat distribution between a pin and a disc are found. It is demonstrated that the present analysis covers the previously known case of one pin as well as new results are obtained. Received on 4 November 1996     

Stability of a swarm of air bubbles in an oscillating liquid

The phenomenon of the formation of a swarm of air bubbles in a liquid executing vertical oscillations within a vertical tube is described. The explanation for the mechanism of air entry into the liquid and the behavior of the individual minute bubbles was given in [1], In the present paper we examine a swarm of bubbles—an air cushion; the stability of this swarm is studied. The results of experimental studies and theoretical calculations are compared.     

Motion of solid particles in an oscillating liquid

The problem of motion of two “free” solid particles in a uniformly oscillating liquid is formulated and solved. In particular, it is found that the particles are capable of performing (side by side with oscillations) an average, monotonous displacement in the liquid as a whole.     

Waves in a heavy two-layer liquid excited by an oscillating sphere

An approximate solution of an initial-boundary-value problem appropriate for the semiaxist>0 (t is time) is constructed for a system of integrodifferential equations which describes the waves excited in an initially stationary unbounded heavy two-layer fluid by a vertically oscillating sphere located at a distance from the interface that is significantly greater than its radius. The shape of the steady-state wave is found by passing to the limit as time increases indefinitely. The wave resistance experienced by the sphere during the transient process and in the steady-state regime is studied as a function of frequency. Nizhnii Novgorod. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 120–133, March–April, 1998.     

The interaction between a single particle and an oscillating fluid

The motion of single particles in a sinusoidally oscillating flow field as the simplest case of a turbulent motion of an eddy is investigated theoretically and experimentally. Since complete knowledge about the interaction between the particles and the surrounding fluid is lacking, three models are evaluated for the interpretation of the sphere's behaviour. The different phenomena of the particle behaviour in an oscillating flow field are described, and the validity of the model which agrees most closely with the experimental results is limited.