Indirect boundary element method for unsteady linearized flow over prolate and oblate spheroids and hemispheroidal protuberances

The indirect boundary element method was used to study the hydrodynamics of oscillatory viscous flow over prolate and oblate spheroids, and over hemispheroidal bodies hinged to a plate. Analytic techniques, such as spheroidal coordinates, method of images, and series representations, were used to make the numerical methods more efficient. A novel method for computing the hydrodynamic torque was used, since for oscillatory flow the torque cannot be computed directly from the weightings. Instead, a Green's function for torque was derived to compute the torque indirectly from the weightings. For full spheroids, the method was checked by comparing the results to exact solutions at low and high frequencies, and to results computed using the singularity method. For hemispheroids hinged to a plate, the method for low frequencies was checked by comparing the results to previous results, and to exact solutions at high frequencies. Copyright © 2004 John Wiley & Sons, Ltd.     

Rotary oscillations of axi‐symmetric bodies in an axi‐symmetric viscous flow with slip: Numerical solutions for sphere and spheroids

Rotary oscillations of several axi‐symmetric bodies in axi‐symmetric viscous flows with slip are investigated. A numerical method based on the Green's function technique is used wherein the relevant Helmholtz equation, as obtained from the unsteady Stokes equation, is converted into a surface integral equation. The technique is benchmarked against a known analytical solution, and accurate numerical results for local stress and torque on spheres and spheroids as function of the frequency parameter and the slip coefficients are obtained. It is found that in all cases, slip reduces stress and torque, and increasingly so with the increasing frequency parameter. The method discussed here can be potentially extended to the realistic case of an oscillating disk viscometer. Copyright © 2003 John Wiley & Sons, Ltd.     

The orientation dynamics of small prolate and oblate spheroids in linear shear flows

The present work deals with the stable orientation of oblate and prolate spheroids in general steady linear flows and with the mode of convergence to these stable orientations. The orientation dynamics is governed by the Jeffery equation. The stable orientations are either fixed points or limit cycles in the orientation space. The type of stable orientation depends on whether the eigenvalues of the linear part of Jeffery equation are real or complex. We define prolate and oblate spheroids to be equivalent if the aspect ratio of one is the reciprocal of the other. We show that, in a given flow, equivalent oblate and prolate spheroids possess the same number of fixed points and limit cycles of which only one is stable. If they possess only fixed points, then their corresponding stable fixed points are orthogonal. If they possess one fixed point and one limit cycle each, then the stable fixed point of one is orthogonal to the plane of the limit cycle of the other. The rate of convergence to these attractors is important to consideration of the orientations in time-space varying flow fields. We show that non-normal growth (NNG) of the distance to these attractors may delay the convergence by several characteristic shear time scales. We derive conditions for occurrence of NNG and explicit expressions for the maximal duration of the growth. We consider a specific case of which the vorticity is a stable orientation of prolate spheroids. We analyze the conditions that imply monotonic or, conversely, non-monotonic convergence to this orientation due to NNG. We thereby find the corresponding conditions for convergence of the equivalent oblate spheroids to their attractors, normal to the vorticity. We show that the convergence is monotonic if the vorticity is parallel to the strain tensor’s largest eigenvector, but that NNG occurs if the vorticity is parallel to the strain tensor’s intermediate eigenvector. The NNG duration decreases with increasing vorticity-strain ratio and with the strain intermediate eigenvalue approaching the largest eigenvalue.     

The three-dimensional fundamental solution to Stokes flow in the oblate spheroidal coordinates with applications to multiples spheroid problems

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Creeping flow past and within a permeable spheroid

Flow past and within an isolated permeable spheroid directed along its axis of symmetry is studied. The flow velocity field is solved using the Stokes creeping flow equations governing the fluid motion outside the spheroid, and the Darcy equation within the spheroid. Expressions for the hydrodynamic resistance experienced by oblate and prolate spheroids are derived and analyzed. The limiting cases of permeable circular disks and elongated rods are examined. It is shown that the spheroid’s resistance varies significantly with its aspect ratio and permeability, expressed via the Brinkman parameter.     

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.     

Pathlines around freely rotating spheroids in simple shear flow

The pathlines around oblate and prolate spheroids freely rotating in shear flow according to Jeffery's equations have been calculated numerically. When the spheroid is aligned with the vorticity axis, open and closed pathlines exist separated by a surface of limiting pathlines. This is very similar to pathlines around spheres and similarly aligned (infinite) cylinders. For spheroids with an arbitrary orientation, four kinds of pathline exist: (i) closed pathlines: (ii) open (single pass) pathlines; (iii) transient orbits; and (iv) permanent non-closed orbits. In general the permanent (closed and non-closed) orbits are separated from the open pathlines by a region occupied by transient orbits.The relevance of pathlines around spheroids to problems of heat and mass transfer and particle deposition in flowing sols is discussed.     

Fluid Flows past Spheroids at Moderate Reynolds Numbers

Axisymmetric viscous fluid flows past spheroids are considered. The time-independent complete Navier-Stokes equations written in a spherical coordinate system are used for describing the flow. The problem is solved by the stabilization method on the basis of a variable direction scheme. The input coordinate system is transformed in order to construct a regular computational grid. As a result of the numerical investigation, the stream patterns of flow past elongated and oblate spheroids are obtained for various values of the determining parameters. Numerical values of the dimensions of the circulation zone and the drag coefficient are given for various values of the spheroid semi-axis ratio in the domain of moderate Reynolds numbers.     

Unsteady weakly perturbed motion of a body of revolution in a liquid with jet separation

The unsteady weakly perturbed motion of a body in a liquid with jet separation has been investigated on various occasions in the twodimensional formulation [1–3]. The present paper gives a generalization of the formulation of this two-dimensional problem to the threedimensional case of flow past a body of revolution in accordance with Kirchhoff's scheme. A method is proposed for solving the obtained boundary-value problem using a Green's function. This function is constructed in a special system of curvilinear coordinates. To obtain an effective solution, a Laplace transformation is used. Expressions are given for the Laplace transforms of the vectors of the force and torque acting on the body in the unsteady motion.     

On the stability of distorted laminar flow(III)—The nonlinear stability analysis of distorted parallel shear flow

Based on the hydrodynamic stability theory of distorted laminar flow and the kind of distortion profiles on the mean velocity in parallel shear flow given in paper [1], this paper investigates the nonlinear stability behaviour of parallel shear flow, carries on stability calculation taking account of the perturbations of background turbulence noise under certain assumption, and obtains some results in accordance qualitatively with those of experiment for plane Poiseuille flow and pipe Poiseuille flow.The author thanks Prof. Zhou Heng sincerely for his kind offer of his computer program of the artificial neutrality method on the stability in subcritical range of plane Poiseuille flow.     

Evaluation of constitutive models for voided solids

In this work four constitutive models for voided solids are reviewed and evaluated; namely, Gurson's, Green's, Liao's et al. and Shima-Oyane's models. Two modified forms of Gurson's model and a non-quadratic form of Green's model are developed for normal anisotropy prevailing in the analysis of sheet metal forming. A simplified version — more amenable to analytical derivations — of Gurson's yield function and its modified forms are proposed for plane stress conditions. The associated flow rules are presented and the laws governing void growth with accumulated strain are derived using the above models. Their predictions are compared with experiments. As an application, the flow curves of voided materials of known initial porosities are predicted and compared with experiments.     

Interparticle force and torque on rigid spheroidal particles in acoustophoresis

The movement of the particles in acoustophoresis is driven by the acoustic radiation force acting on the particles. Particles with positive contrast factor tend to agglomerate once they are pushed by the primary force to the vicinity of the pressure node. The main driving force of this agglomeration is the interparticle force. In this study, the boundary element method is used to calculate the interparticle force and torque acting on a pair of spheroidal particles. The numerical results show that the interparticle force is dominant over the primary force when the spheroids are near the pressure nodal plane, similar to the case of two spheres. On contrary, the interparticle torque is insignificant compared to the primary torque, even when the spheroids are close to each other. The results also provide a preliminary study about how biological cells, which are mostly not spherical in shape, agglomerate and orient themselves in the vicinity of the pressure node.     

Simulation of the rheological properties of suspensions of oblate spheroidal particles in a Newtonian fluid

A simulation algorithm was developed to predict the rheological properties of oblate spheroidal suspensions. The motion of each particle is described by Jeffery’s solution, which is then modified by the interactions between the particles. The interactions are considered to be short range and are described by results from lubrication theory and by approximating locally the spheroid surface by an equivalent spherical surface. The simulation is first tested on a sphere suspension, results are compared with known experimental and numerical data, and good agreement is found. Results are then presented for suspensions of oblate spheroids of two mean aspect ratios of 0.3 and 0.2. Results for the relative viscosity η _r, normal stress differences N ₁ and N ₂ are reported and compared with the few available results on oblate particle suspensions in a hydrodynamic regime. Evolution of the orientation of the particles is also observed, and a clear alignment with the flow is found to occur after a transient period. A change of sign of N ₁ from negative to positive as the particle concentration is increased is observed. This phenomenon is more significant as the particle aspect ratio increases. It is believed to arise from a change in the suspension microstructure as the particle alignment increases.     

Hydrodynamic force and torque models for a particle moving near a wall at finite particle Reynolds numbers

This research work is aimed at proposing models for the hydrodynamic force and torque experienced by a spherical particle moving near a solid wall in a viscous fluid at finite particle Reynolds numbers. Conventional lubrication theory was developed based on the theory of Stokes flow around the particle at vanishing particle Reynolds number. In order to account for the effects of finite particle Reynolds number on the models for hydrodynamic force and torque near a wall, we use four types of simple motions at different particle Reynolds numbers. Using the lattice Boltzmann method and considering the moving boundary conditions, we fully resolve the flow field near the particle and obtain the models for hydrodynamic force and torque as functions of particle Reynolds number and the dimensionless gap between the particle and the wall. The resolution is up to 50 grids per particle diameter. After comparing numerical results of the coefficients with conventional results based on Stokes flow, we propose new models for hydrodynamic force and torque at different particle Reynolds numbers. It is shown that the particle Reynolds number has a significant impact on the models for hydrodynamic force and torque. Furthermore, the models are validated against general motions of a particle and available modeling results from literature. The proposed models could be used as sub-grid scale models where the flows between particle and wall can not be fully resolved, or be used in Lagrangian simulations of particle-laden flows when particles are close to a wall instead of the currently used models for an isolated particle.     

Fundamental solutions in perturbed shear flow

In this paper infinite plane Couette flow in a viscous incompressible fluid is considered subject to general three-dimensional perturbations and the equations of motion are linearized. Furthermore, initial-value problems are posed and a set of closed-form solutions are obtained for a variety of conditions, such as the system under the influence of: (i) a mass source; (ii) an external force; or (iii) initial vorticity. The result is a knowledge of both the early transient dynamics and the near spatial field behavior, as well as the state after a long time and the far field behavior. It is shown that the solutions can be considered as fundamental (in the sense that source-sink solutions are regarded fundamental for irrotational motion) and therefore are useful in analyzing other boundary-value, initial-value problems where the basic flow can be synthesized from piece-wise linear (constant shear) variations. To this end a generalized Green's function for the system is determined.     

ON THE STABILITY OF DISTORTED LAMINAR FLOW(Ⅱ)——THE LINEAR STABILITY ANALYSIS OF DISTORTED PARALLEL SHEAR FLOW

Based on the hydrodynamic stability theory of distorted laminar flow and the kind of distortion profiles on the mean velocity in parallel shear flow given in paper [1], this paper investigates the linear stability behaviour of parallel shear flow, presents unstable results of plane Couette flow and pipe Poiseuille flow to two-dimensional or axisymmetric disturbances for the first time, and obtains neutral curves of these two motions under certain definition.     

Stokes flow in a cylindrical tube containing a line of spheroidal particles

Viscous flow in a circular cylindrical tube containing an infinite line of rigid spheroidal particles equally spaced along the axis of the tube is considered for (a) uniform axial translation of the spheroids (b) flow past a line of stationary spheriods and (c) flow of the suspending fluid and spheroids under an imposed pressure gradient. The fluid is assumed to be incompressible and Newtonian. The Reynolds number is assumed to be small and the equations of creeping flow are used. Two types of solutions are developed: (i) an exact solution in the form of an infinite series which is valid for ratios of the spheroid diameter to the tube diameter up to 0.80, (ii) an approximate solution using lubrication theory which is valid for spheroids which nearly fill the tube. The drag on each spheroid and the pressure drop are computed for all cases. Both prolate and oblate spheroids are considered. The results show that the drag and pressure drop depend on the spheroidal diameter perpendicular to the axis of tube primarily and the effects of the spheroidal thickness and spacing are secondary. The results are of interest in connection with mechanics of capillary blood flow, sedimentation, fluidized beds, and fluid-solid transport.     

Forced convective mass transfer studies from spheroids

In order to obtain information on the effect of shape on mass transfer, overall mass transfer rates were measured from naphthalene spheroids suspended in a wind tunnel (Schmidt number 2.4). Spheroidal shapes which included spheres, oblate spheroids and spheroids with composite halves were employed for the study. The ratio of the minor to major axes of the spheroids ranged from 1∶1 to 1∶4. The data obtained from one-hundred and fifty six experimental runs were best correlated by the use of Pasternak and Gauvin's characteristic dimension defined as total surface area of the body divided by maximum perimeter normal to flow. The correlations for the ranges 200 < Re < 2000 and 2000 < Re < 32000 are as follows. $$\begin{gathered} Sh = 0.62 (Re)^{0.5} (Sc)^{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3}} \hfill \\ Sh = 0.26 (Re)^{0.6} (Sc)^{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3}} \hfill \\ \end{gathered}$$ which correlated the data with standard deviation of 3.75% and 3.50% respectively.     

Rheology of a dilute suspension of axisymmetric Brownian particles

Explicit results are presented for the complete rheological properties of dilute suspensions of rigid, axisymmetric Brownian particles possessing fore-aft symmetry, when suspended in a Newtonian liquid subjected to a general three-dimensional shearing flow, either steady or unsteady. It is demonstrated that these rheological properties can be expressed in terms of five fundamental material constants (exclusive of the solvent viscosity), which depend only upon the sizes and shapes of the suspended particles. Expressions are presented for these scalar constants for a number of solids of revolution, including spheroids, dumbbells of arbitrary aspect ratio and long slender bodies. These are employed to calculate rheological properties for a variety of different shear flows, including uniaxial and biaxial extensional flows, simple shear flows, and general two-dimensional shear flows. It is demonstrated that the rheological properties appropriate to a general two-dimensional shear flow can be deduced immediately from those for a simple shear flow. This observation greatly extends the utility of much of the prior Couette flow literature, especially the extensive numerical calculations of Scheraga et al. (1951, 1955).The commonality of many disparate results dispersed and diffused in earlier publications is emphasized, and presented from a unified hydrodynamic viewpoint.     

On the Green's Functions and Boundary Integral Formulation of Elastic Planes with Cutouts

ABSTRACT

The problem of an infinite elastic plane that contains a hole of arbitrary shape and is subjected to a concentrated unit load is considered. The Green's function (influence function) for the problem is formulated by means of two complex potential functions. This is accomplished by mapping the region that is exterior to the hole onto a unit circle. A class of closed contour hole shapes is analyzed. Green's functions for an elliptical hole and a class of triangular holes are determined. Green's functions for a class of rectangular holes are also discussed. In order to determine stress and displacement fields for the finite plane problem, Green's function is employed and an indirect boundary integral equation is formulated, with the integrand of the integral equation incorporating the effect of the hole. The contour of the hole is no longer considered a part of the boundary and only the contour of the region that is exterior to the hole is subdivided into boundary elements. Examples for elliptical and triangular holes are solved.