Viscous flow between two rotating nonconcentric cylinders for small eccentricity

The flow of a viscous and incompressible fluid between two rotating nonconcentric cylinders is investigated. An approximate solution of the Navier-Stokes equations is obtained by a perturbation method for the case of small eccentricity. A second solution of the basic flow is obtained by imposing the additional geometric restriction of small gap between the two cylinders and employing the asymptotic expansion of Bessel functions by Meissel's series. This second solution is also obtained by formulating a small gap boundary value problem. The transverse velocity profiles are presented for the case when the eccentricity and gap are small and the outer cylinder is stationary.     

Fluid-dynamic interaction between two spheres

Experiment of fluid-dynamic interaction between two spheres was conducted to obtain basic information concerning the two-phase flow, especially in dense phase. Two or three spheres were set up in a water tunnel in the longitudinal or transverse direction with Reynolds numbers less than 10³. The flow behind the sphere was visualized by the use of condense milk and change in vortex structure due to the interaction was observed in detail. Additionally, drag force on the sphere was measured by a pendulum method which was developed to detect small drag, and the range of distance in which the drag is affected by the interaction was shown.     

Motion of a viscous liquid between two rotating spheres

An analysis is presented of the steady-state asymmetric motion of an incompressible viscous liquid between two concentric spheres rotating with constant angular velocities about various axes passing through their common center. The reaction force of the liquid on the inner sphere is determined; this force reduces to a resistive torque.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 179–180, May–June, 1974.     

A capillary force model for interactions between two spheres

Based on the method of energy principle, an analytical approach for computing the capillary force for sphere/sphere geometry is presented in this paper. In modeling the capillary force, we consider spheres with both equal and non-equal radii, for both symmetric and asymmetric configurations at liquid/solid interfaces. We use numerical analysis to investigate the validity and efficiency of the derived model. The effect of various parameters including humidity, distance between two spheres, radii of spheres and contact angles on the meniscus force are investigated. Finally the results obtained from the model are compared with experimental measurements, and the accuracy and precision of the presented approach is verified.     

The flow of viscoplastic material between two concentric spheres

The motion of a viscoplastic medium between two concentric spheres is considered upon rotation of one sphere with constant angular velocity. This problem is solved by an heuristic iterative method. The boundary of the stagnation zones is found and its specific shape is shown. The flow characteristics versus the parameter of the medium are obtained. Voronezh State Engineering Academy, Voronezh 394017. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 1, pp. 133–139, January–February, 1999.     

Unsteady flow in an annulus between two concentric rotating spheres

An analysis is presented for the unsteady laminar flow of an incompressible Newtonian fluid in an annulus between two concentric spheres rotating about a common axis of symmetry. A solution of the Navier-Stokes equations is obtained by employing an iterative technique. The solution is valid for small values of Reynolds numbers and acceleration parameters of the spheres. In applying the results of this analysis to a rotationally accelerating sphere, a virtual moment of inertia is introduced to account for the local inertia of the fluid.     

Steady rotatory flow of Bingham material between two concentric spheres

Summary Steady flow of Bingham material between two concentric rotating spheres has been investigated taking into account the constitutive equations given by Oldroyd. Motions in the elastic as well as in the flow region have been discussed. A critical value of the Bingham number has been found below which the flow takes place in the whole region and above which the elastic and flow regions occur side by side.Nomenclature r, , space co-ordinates - u _r ,u ,u velocity components - density - modulus of rigidity (constant) - bulk modulus - e _ii , the dilatation - ₁ coefficient of viscosity (constant) - e _ik strain tensor - d _ik rate of strain tensor - p _ik stress tensor primes denote deviatoric components of tensors, e.g. - p _ik p _ik +p _k ⁱ , p=–1/3p _ii - yield value (constant) - D/Dt materiaal derivative with regard to time following the particle - R ₁ radius of the inner sphere - R ₂ radius of the outer sphere - R radius of the yield surface (spherical) - ₁ velocity of the inner sphere - B Bingham number     

Unsteady flow in an annulus between two concentric rotating spheres

An analysis is presented for the unsteady laminar flow of an incompressible Newtonian fluid in an annulus between two concentric spheres rotating about a common axis of symmetry. A solution of the Navier-Stokes equations is obtained by employing an iterative technique. The solution is valid for small values of Reynolds numbers and acceleration parameters of the spheres. In applying the results of this analysis to a rotationally accelerating sphere, a virtual moment of intertia is introduced to account for the local inertia of the fluid.Nomenclature R _i radius of the inner sphere - R _o radius of the outer sphere - radial coordinate - r dimensionless radial coordinate, - meridional coordinate - azimuthal coordinate - time - t dimensionless time, - Re _i instantaneous Reynolds number of the inner sphere, _i R _k ² / - Re _o instantaneous Reynolds number of the outer sphere, _o R _o ² / - radial velocity component - V _r dimensionless radial velocity component, - meridional velocity component - V dimensionless meridional velocity component, - azimuthal velocity component - V dimensionless azimuthal velocity component, - viscous torque - T dimensionless viscous torque, - viscous torque at surface of inner sphere - T _i dimensionless viscous torque at surface of inner sphere, - viscous torque at surface of outer sphere - T _o dimensionless viscous torque at surface of outer sphere, - externally applied torque on inner sphere - T _p,i dimensionless applied torque on inner sphere, - moment of inertia of inner sphere - Z _i dimensionless moment of inertia of inner sphere, - virtual moment of inertia of inner sphere - Z _i,v dimensionless virtual moment of inertia of inner sphere, - virtual moment of inertia of outer sphere - _i instantaneous angular velocity of the inner sphere - _o instantaneous angular velocity of the outer sphere - density of fluid - viscosity of fluid - kinematic viscosity of fluid,/ - radius ratio,R _i/R _o - swirl function, - dimensionless swirl function, - stream function - dimensionless stream function, - _i acceleration parameter for the inner sphere, - _o acceleration parameter for the outer sphere, - shear stress - _r dimensionless shear stress,      

Treatment of numerical diffusion in strong convective flows

A three-dimensional extension of the QUICK scheme adapted for the finite volume method and non-uniform grids is presented to handle convection-diffusion problems for high Peclet numbers and steep gradients. The algorithm is based on three-dimensional quadratic interpolation functions in which the transverse curvature terms are maintained and the diagonal dominance of the coefficient matrix is preserved. All formulae are explicitly given in an appendix. Results obtained with the classical upwind (UDS), the simplified QUICK (transverse terms neglected) and the present full QUICK schemes are given for two benchmark problems, one two-dimensional, steady state and the other three-dimensional, unsteady state. Both QUICK schemes are shown to give superior solutions compared with the UDS in terms of accuracy and efficiency. The full QUICK scheme performs better than the simplified QUICK, giving even for coarse grids acceptable results closer to the analytical solutions, while the computational time is not affected much.     

Natural frequencies of submerged piezoceramic hollow spheres

An exact 3D analysis of free vibration of a piezoceramic hollow sphere submerged in a compressible fluid is presented in this paper. A separation method is adopted to simplify the basic equations for spherically isotropic piezoelasticity. It is shown that there are two independent classes of vibration. The first one is independent of the fluid medium as well as the electric field, while the second is associated with both the fluid parameter and the piezoelectric effect. Exact frequency equations are derived and numerical results are obtained. The project supported by the National Natural Science Foundation of China (No. 19872060)     

Natural convective heat transfer in water enclosed between pairs of differentially heated vertical plates

This paper presents the results of an experimental study of buoyancy-driven convective heat transfer between three parallel vertical plates, symmetrically spaced with water as the intervening medium. The centre plate was electrically heated, while the other side plates were water-cooled forming two successive parallel vertical channels of dimensions 20 cm × 3.5 cm × 35 cm (length W, gap L, height H) each. Top, bottom and sides of the channels were open to water in the chamber which is the novel aspect of this study. Plate surface temperature and bath temperature at different levels of height from the bottom of channel were measured by K-type thermocouples. Experimental data have been correlated as under:

The instantaneous slow flow of a visco-elastic fluid between two concentric spheres

Summary A theoretical analysis is made of the flow of an incompressible viscoelastic fluid contained between two concentric spheres when the outer sphere is moved instantaneously in a given direction, whilst the inner sphere remains at rest. The solution is developed by successive approximations, the first corresponding to the instantaneous slow flow of a Newtonian viscous fluid. By allowing the radius of the outer sphere to approach infinity, the result obtained can be used to give an approximate solution to the equations of motion of a visco-elastic fluid flowing slowly past a fixed sphere.     

Unsteady convective diffusion in a rotating parallel plate channel

The paper presents an exact analysis of the dispersion of a passive contaminant in a viscous fluid flowing in a parallel plate channel driven by a uniform pressure gradient. The channel rotates about an axis perpendicular to its walls with a uniform angular velocity resulting in a secondary flow. Using a generalized dispersion model which is valid for all time, we evaluate the longitudinal dispersion coefficientsK _i (i=1, 2, ...) as functions of time. It is shown thatK ₁=0 andK ₃,K ₄, ... decay rapidly in comparison withK ₂. ButK ₂ decreases with increasing (the dimensionless rotation parameter) for values of upto approximately =2.2. ThereafterK ₂ increases with further increase in and its value gets saturated for large values of (say, 500) and does not change any further with increase in . A physical explanation of this anomalous behaviour ofK ₂ is given.

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Instationäre konvektive Diffusion in einem rotierenden Parallelplattenkanal

Zusammenfassung In dieser Untersuchung wird eine exakte Analyse der Ausbreitung eines passiven Kontaminierungsstoffes in einer zähen Flüssigkeit gegeben, die, befördert durch einen gleichförmigen Druckgradienten, in einem Parallelplattenkanal strömt. Der Kanal rotiert mit gleichförmiger Winkelgeschwindigkeit um eine zu seinen Wänden senkrechte Achse, wodurch sich eine Sekundärströmung ausbildet. Unter Verwendung eines generalisierten, für alle Zeiten gültigen Dispersionsmodells werden die longitudinalen DispersionskoeffizientenK _i (i=1, 2, ...) als Funktionen der Zeit ermittelt. Es wird gezeigt, daßK ₁=0 gilt und dieK ₃,K ₄, ... gegenüberK ₂ schnell abnehmen.K ₂ nimmt ab, wenn , der dimensionslose Rotationsparameter, bis etwa zum Wert 2,2 ansteigt. Danach wächstK ₂ mit bis auf einem Endwert an, der etwa ab =500 erreicht wird. Dieses anomale Verhalten vonK ₂ findet eine physikalische Erklärung.

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List of symbols C solute concentration - D molecular diffusivity - K _i longitudinal dispersion coefficients - 2L depth of the channel - P ₀ dimensionless pressure gradient along main flow - Pe Péclet number - q velocity vector - Q _x,Q _y mass flux along the main flow and the secondary flow directions - dimensionless average velocity along the main flow direction - (x, y, z) Cartesian co-ordinates Greek symbols dimensionless rotation parameter - the inclination of side walls withx-axis - kinematic viscosity - fluid density - dimensionless time - angular velocity of the channel - dimensionless distance along the main flow direction - dimensionless distance along the vertical direction - dimensionless solute concentration - integral of the dispersion coefficientK ₂() over a time interval     

Interaction between two spheres placed in tandem arrangement in steady and pulsating flow

The interaction among two spheres in tandem formation are studied for a Reynolds number of 300 using both steady and pulsating inflow conditions. The purpose is to further investigate the force characteristics as well as the shedding patterns of the two spheres as the separation distance is changed from 1.5 to 12 sphere diameters. The method used for the simulations is the volume of solid (VOS) method, an approach based on the volume of fluid (VOF) method. Comparisons with other computational methods have shown VOS to accurately resolve the flow field around solid spheres. The results show that the separation distance plays a significant role in changing the flow patterns and shedding frequencies at moderate separation distances, whereas effect on drag is observed even at a separation distance of 12 diameters.     

Liquid bridge force between two unequal-sized spheres or a sphere and a plane

Liquid bridge force acting between wet particles is an important property in particle characterization. This paper deals with liquid bridge force between either two unequal-sized spherical particles or a sphere and a flat plate under conditions where gravitational effect arising from bridge distortion is negligible. In order to calculate the force of the liquid bridge efficiently and accurately, expressions of liquid configuration and liquid bridge force were derived by building a mechanical model, which assumes the liquid bridge to be circular in shape between either two unequal-sized spheres or a sphere and a plane. To assess the accuracy of the numerical results of the calculated liquid bridge forces, they were compared to the published experimental data.     

Experimental study of mutual interference between two spheres placed on a plane boundary

This paper describes an experimental study of the mutual interference between two spheres placed on a plane boundary. The experiment was carried out in an N. P. L. type wind-tunnel having a working section of 500×500×2000 mm³ in size at a Reynolds number of 4.74×10⁴. The surface-pressure distributions of two spheres were measured for the various relative positions of two spheres and the drag, side-force, and lift coefficients were determined from surface-pressure distributions. The separation of the flow and the formation of vortices were observed by the method of visualization. The distributions of velocities, and turbulent intensities of the flow past two spheres were measured. The experimental results for two spheres were compared with those of a single sphere.List of symbols C _D drag coefficient - C _L lift coefficient - C _p surface-pressure coefficient of sphere=(P-P )(qU ² ) - C _s coefficients of side force - D diameter of sphere [mm] - P static pressure [Pa] - P static pressure in free stream [Pa] - Re Reynolds number= DU/v - S spacing between the centers of two adjoining spheres in plane view [mm] - U time-mean velocity in X-direction [m/s] - [m/s] free stream velocity [m/s] - u, v, w X, Y and Z-components of velocity fluctuation [m/s] - X, Y, Z coordinate axes with origin at the bottom center of test sphere, X, Y, Z axis being taken in the streamwise, lateral and vertical directions respectively [mm] (Fig. 1) - latitude angle [°] - longitude angle [°] - angle between the line connected with the centers of two spheres and wind direction [°] (Fig. 2) - kinematic viscosity of air [m²/s] - density of air [N/m³] This paper was presented at the 10th Symposium on Turbulence, University of Missouri-Rolla, Sept. 22–24, 1986