Spheroidal Container Research Articles

Slow viscous stream over a non-Newtonian fluid sphere in an axisymmetric deformed spherical vessel

The creeping motion of a non-Newtonian (Reiner-Rivlin) liquid sphere at the instant it passes the center of an approximate spherical container is discussed. The flow in the spheroidal container is governed by the Stokes equation, while for the flow inside the Reiner-Rivlin liquid sphere, the expression for the stream function is obtained by expressing it in the power series of a parameter S , characterizing the cross-viscosity. Both the flow fields are then determined explicitly by matching the boundary conditions at the interface of Newtonian fluid and non-Newtonian fluid, and also the condition of imperviousness and no-slip on the outer surface. As an application, we have considered an oblate spheroidal container. The drag and wall effects on the liquid spherical body are evaluated. Their variations with regard to the separation parameter l , viscosity ratio \( \lambda\), cross-viscosity S, and deformation parameter \( \varepsilon\) are studied and demonstrated graphically. Several renowned cases are derived from the present analysis. It is observed that the drag not only varies with \( \varepsilon\), but as l increases, the rate of change in behavior of drag force also increases. The influences of these parameters on the wall effects has also been studied and presented in a table.

Slow Steady Rotation of an Approximate Sphere in an Approximate Spherical Container with Slip Surfaces

The slow steady rotation of an approximate sphere (whose shape deviates slightly from that of a sphere) in a concentric approximate spherical container is studied analytically. The slip boundary condition is applied at the particle and container surfaces. An exact solution is obtained to the first order in the small parameter characterizing the deformation. The hydrodynamic couple and coupling coefficient exerted on the particle in presence of the cavity is calculated. As an application, the case of flow past spheroid in a spheroidal container is considered and the couple experienced by spheroid is evaluated. In the limiting cases, the hydrodynamic couple acting on the sphere in an unbounded medium is obtained in case of no slip condition. It is noticed that under the Stokesian assumption, the deformation in the body has significant influence on the couple experienced.

On precessing flow in an oblate spheroid of arbitrary eccentricity

AbstractWe consider a homogeneous fluid of viscosity$\nu $confined within an oblate spheroidal cavity of arbitrary eccentricity$\mathcal{E}$marked by the equatorial radius$d$and the polar radius$d \sqrt{1-\mathcal{E}^2}$with$0<\mathcal{E}<1$. The spheroidal container rotates rapidly with an angular velocity${\boldsymbol{\Omega}}_0 $about its symmetry axis and precesses slowly with an angular velocity${\boldsymbol{\Omega}}_p$about an axis that is fixed in space. It is through both topographical and viscous effects that the spheroidal container and the viscous fluid are coupled together, driving precessing flow against viscous dissipation. The precessionally driven flow is characterized by three dimensionless parameters: the shape parameter$\mathcal{E}$, the Ekman number${\mathit{Ek}}=\nu /(d^2 \delimiter "026A30C {\boldsymbol{\Omega}}_0\delimiter "026A30C )$and the Poincaré number${\mathit{Po}}=\pm \delimiter "026A30C {\boldsymbol{\Omega}}_p\delimiter "026A30C / \delimiter "026A30C \boldsymbol{\Omega}_0\delimiter "026A30C $. We derive a time-dependent asymptotic solution for the weakly precessing flow in the mantle frame of reference satisfying the no-slip boundary condition and valid for a spheroidal cavity of arbitrary eccentricity at${\mathit{Ek}}\ll 1$. No prior assumptions about the spatial–temporal structure of the precessing flow are made in the asymptotic analysis. We also carry out direct numerical simulation for both the weakly and the strongly precessing flow in the same frame of reference using a finite-element method that is particularly suitable for non-spherical geometry. A satisfactory agreement between the asymptotic solution and direct numerical simulation is achieved for sufficiently small Ekman and Poincaré numbers. When the nonlinear effect is weak with$\delimiter "026A30C {\mathit{Po}}\delimiter "026A30C \ll 1$, the precessing flow in an oblate spheroid is characterized by an azimuthally travelling wave without having a mean azimuthal flow. Stronger nonlinear effects with increasing$\delimiter "026A30C {\mathit{Po}}\delimiter "026A30C $produce a large-amplitude, time-independent mean azimuthal flow that is always westward in the mantle frame of reference. Implications of the precessionally driven flow for the westward motion observed in the Earth’s fluid core are also discussed.

Axi-symmetric motion of a porous approximate sphere in an approximate spherical container

The creeping motion of a porous approximate sphere at the instant it passes the center of an approximate spherical container with Ochoa-Tapia and Whitaker’s stress jump boundary condition has been investigated analytically. The Brinkman’s model for the flow inside the porous approximate sphere and the Stokes equation for the flow in an approximate spherical container were used to study the motion. The stream function (and thus the velocity) and pressure (both for the flow inside the porous approximate sphere and inside an approximate spherical container) are calculated. The drag force experienced by the porous approximate spherical particle and wall correction factor are determined in closed forms. The special cases of porous sphere in a spherical container and oblate spheroid in an oblate spheroidal container are obtained from the present analysis. It is observed that drag not only changes with the permeability of the porous region, but as the stress jump coefficient increases, the rate of change in behavior of drag increases.

Stokes flow between two confocal rotating spheroids with slip

The steady axisymmetric flow problem of a viscous fluid confined between two confocal spheroids that are rotating about their axis of revolution with different angular velocities is considered. A linear slip, of Basset type, boundary condition on both surfaces of the spheroidal particle and the container is used. Under the Stokesian assumption, a general solution is constructed from the superposition of basic solutions in prolate and oblate spheroidal coordinates. The boundary conditions on the particle’s surface and spheroidal container are satisfied by a collocation technique. The torque exerted on the spheroidal particle by the fluid is evaluated with good convergence for various values of the slip parameters, the relative angular velocity and aspect ratios of the spheroids. The limiting case of no-slip is in good agreement with the available values in the literature.

Translation and rotation of a porous spheroid in a spheroidal container

The flow problem of an incompressible axisymmetrical quasisteady translation and steady rotation of a porous spheroid in a concentric spheroidal container are studied analytically. The same small departure from a sphere is considered for each spheroidal surface. In the limit of small Reynolds number, the Brinkman equation for the flow inside the porous region and the Stokes equation for the outside region in their stream functions formulations and velocity components, which are proportional to the translational and angular velocities, respectively, are used. Explicit expressions are obtained for both inside and outside flow fields to the first order in a small parameter characterizing the deformation of the spheroidal surface from the spherical shape. The hydrodynamic drag force and couple exerted on the porous spheroid are obtained for the special cases of prolate and oblate spheroids in closed forms. The dependence of the normalized wall-corrected translational and rotational mobilities on permeability for a porous spheroid in an unbounded medium and for a solid spheroid in a cell on the particle volume fraction is discussed numerically and graphically for various values of the deformation parameter. In the limiting cases, the analytical solutions describing the drag force and torque or mobilities for a porous spheroid in the spheroidal vessel reduce to those for a solid sphere and for a porous sphere in a spherical cell.

Precessing rotating flows with additional shear: Stability analysis

We consider unbounded precessing rotating flows in which vertical or horizontal shear is induced by the interaction between the solid-body rotation (with angular velocity Omega(0)) and the additional "precessing" Coriolis force (with angular velocity -epsilonOmega(0)), normal to it. A "weak" shear flow, with rate 2epsilon of the same order of the Poincaré "small" ratio epsilon , is needed for balancing the gyroscopic torque, so that the whole flow satisfies Euler's equations in the precessing frame (the so-called admissibility conditions). The base flow case with vertical shear (its cross-gradient direction is aligned with the main angular velocity) corresponds to Mahalov's [Phys. Fluids A 5, 891 (1993)] precessing infinite cylinder base flow (ignoring boundary conditions), while the base flow case with horizontal shear (its cross-gradient direction is normal to both main and precessing angular velocities) corresponds to the unbounded precessing rotating shear flow considered by Kerswell [Geophys. Astrophys. Fluid Dyn. 72, 107 (1993)]. We show that both these base flows satisfy the admissibility conditions and can support disturbances in terms of advected Fourier modes. Because the admissibility conditions cannot select one case with respect to the other, a more physical derivation is sought: Both flows are deduced from Poincaré's [Bull. Astron. 27, 321 (1910)] basic state of a precessing spheroidal container, in the limit of small epsilon . A Rapid distortion theory (RDT) type of stability analysis is then performed for the previously mentioned disturbances, for both base flows. The stability analysis of the Kerswell base flow, using Floquet's theory, is recovered, and its counterpart for the Mahalov base flow is presented. Typical growth rates are found to be the same for both flows at very small epsilon , but significant differences are obtained regarding growth rates and widths of instability bands, if larger epsilon values, up to 0.2, are considered. Finally, both flow cases are briefly discussed in view of a subsequent nonlinear study using pseudospectral direct numerical simulations, which is a natural continuation of RDT.

Numerical study of viscous modes in a rotating spheroid

The motion of an incompressible viscous rotating fluid in an oblate spheroidal container is studied by direct numerical simulation in an appropriate spheroidal coordinate system and in the linear approximation. The behaviour of a few eigenmodes is investigated as a function of the eccentricity , where the decay rate deviates strongly from the predicted theoretical variation. This resonance is discussed in relation to the accidental coincidence between the spin-over frequency and two other frequencies corresponding to the (8,1,5) and (14,1,9) inertial eigenmodes.

Numerical study of a rotating fluid in a spheroidal container

The motion of an incompressible, viscous rotating fluid contained in a spheroidal container is studied by a direct numerical simulation in an oblate spheroidal coordinate system. An appropriate formalism is first derived which allows us to expand any scalar field in spherical harmonics and to decompose any vector field into its sphero-poloidal and sphero-toroidal scalar parts. The spinover mode is then considered, by linearizing the equation of motion for the fluid. Boundary conditions specific of the spheroidal geometry are explicited. Finite difference method for the sphero-radial component and spherical harmonics expansion for angular components are considered. A few numerical results are given about the viscous effects on the spinover mode, for different eccentricities of the container, in good agreement with the analytical solution given in the literature.

Wall effects on a spherical particle

The motion of a solid spherical particle at the instant it passes the centre of a spheroidal container has been investigated using the no-slip condition at the surfaces. The shape of this spheroidal container is assumed to deviate slightly from that of a sphere at the surface. An expression for the stream function describing the flow field has been obtained to the first order in the parameter characterizing the deviation and this in turn has been utilized to evaluate the drag on the particle. Wall effects are then examined and special cases deduced.

Transition of the electronic response from molecular-like to jellium-like in cold, small sodium clusters

Absolute photoabsorption cross sections for Nan+ (2≤n≤21) were measured in the visible energy range. The cluster ions were produced in a gas aggregation source and thus have a canonical distribution of internal energy corresponding to a temperature of ∼ 105 K. The spectra for n≤9 and 11 exhibit between two and six absorption lines, and are in qualitative agreement with ab inito quantum chemical calculations. For n=15 and 21, the position of the resonances can be explained as excitations of a nearly free electron gas in a spheroidal container. An evolution is thus observed from molecular-like transitions to a giant collective resonance of the electron cloud. The integrated oscillator strength is 0.95 per 3s-electron in the energy range covered for n≥4, showing that the main excitations of the valence electrons have been found.

Temperature Dependence of the Optical Response of Small, Open Shell Sodium Clusters.

The optical absorption of small, open sodium cluster ions ( ${\mathrm{Na}}_{n}^{\phantom{\rule{0ex}{0ex}}+},n\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}4,7,\mathrm{and}11$) exhibit an unexpectedly large temperature dependence. Clearly separated absorption lines observed for cold clusters $(T\ensuremath{\sim}35$ K) are interpreted as transitions between electronic states of the ${\mathrm{Na}}_{n}^{\phantom{\rule{0ex}{0ex}}+}$ molecule. At the highest temperature $(T>380$ K), however, the clusters are liquid, and broad absorption peaks are observed whose energy positions can be explained for $n>7$ by the model of nearly free electrons oscillating in a spheroidal container.

The instability of precessing flow

Abstract An explanation is put forward for the instability observed within a precessing, rotating spheroidal container. The constant vorticity solution for the flow suggested by Poincare is found to be inertially unstable through the parametric coupling of two inertial waves by the underlying constant strain field. Such resonant couplings are due either to the elliptical or shearing strains present which elliptically distort the circular streamlines and shear their centres respectively. For the precessing Earth's outer core, the shearing of the streamlines and the ensuing shearing instability are the dominant features. The instability of some exact, linear solutions for finite precessional rates is established and used to corroborate the asymptotic analysis. A complementary unbounded analysis of a precessing, rotating fluid is also presented and used to deduce a likely upperbound on the growth rate of a small disturbance. Connection is made with past experimental studies.

Liquid oscillations in a prolate spheroidal container

Free and forced oscillations of a frictionless liquid with a free surface in prolate spheroidal container geometries are investigated analytically. The container geometries treated here include all types of annular sectorial and truncated spheroids for which the natural frequencies of the sloshing liquid were determined. For a partially filled simple prolate spheroidal container the natural frequencies were evaluated numerically for different slenderness ratios and compared for a particular container with experimental results. The agreement of test data with the theoretical results is quite good. Equivalent cylindrical containers which exhibit equal liquid surface area and equal volume as the liquid in a spheroid show also good results and could be employed for design work of space vehicles.