Steady slow rotation of a sphere in a visco-elastic liquid

Summary In this paper, we obtain the flow due to slow steady rotation of a sphere in a visco-elastic liquid characterized by the constitutive relation given by Rivlin. The non-Newtonian effects are strongly dependent on a non-dimensional parameter K independent of the angular velocity of the sphere. If 1<K3, we notice four vortices symmetrically placed around the sphere. When K lies outside this range, the direction of the flow pattern is the same as that in the Newtonian case but displaced towards the sphere as K decreases. Also the expression for the couple on the sphere has been obtained which depends on K.     

Thermal convection at a melting benzene surface

Summary Heat transfer by thermal free convection at the surface of a sphere has been studied experimentally by melting a sphere of solid benzene in a large quantity of liquid benzene of homogeneous temperature. The influence of cold liquid produced by the melting process is taken into account to yield results that are representative for the pure effect of heat transfer without melting. In the general formula for heat transfer by thermal convection, =C(GrPr)^1/4, we found C=0.525.     

Numerical analysis of the rotating viscous flow approaching a solid sphere

A numerical simulation is performed to investigate the flow induced by a sphere moving along the axis of a rotating cylindrical container filled with the viscous fluid. Three‐dimensional incompressible Navier–Stokes equations are solved using a finite element method. The objective of this study is to examine the feature of waves generated by the Coriolis force at moderate Rossby numbers and that to what extent the Taylor–Proudman theorem is valid for the viscous rotating flow at small Rossby number and large Reynolds number. Calculations have been undertaken at the Rossby numbers (R_o) of 1 and 0.02 and the Reynolds numbers (R_e) of 200 and 500. When R_o=O(1), inertia waves are exhibited in the rotating flow past a sphere. The effects of the Reynolds number and the ratio of the radius of the sphere and that of the rotating cylinder on the flow structure are examined. When R_o ? 1, as predicted by the Taylor–Proudman theorem for inviscid flow, the so‐called ‘Taylor column’ is also generated in the viscous fluid flow after an evolutionary course of vortical flow structures. The initial evolution and final formation of the ‘Taylor column’ are exhibited. According to the present calculation, it has been verified that major theoretical statement about the rotating flow of the inviscid fluid may still approximately predict the rotating flow structure of the viscous fluid in a certain regime of the Reynolds number. Copyright © 2004 John Wiley & Sons, Ltd.     

Numerical study of transient laminar natural convection heat transfer over a sphere subjected to a constant heat flux

Transient laminar natural convection over a sphere which is subjected to a constant heat flux has been studied numerically for high Grashof numbers (10⁵ ≤ Gr ≤ 10⁹) and a wide range of Prandtl numbers (Pr = 0.02, 0.7, 7, and 100). A plume with a mushroom-shaped cap forms above the sphere and drifts upward continuously with time. The size and the level of temperature of the transient cap and plume stem decrease with increasing Gr and Pr. Flow separation and an associated vortex may appear in the wake of the sphere depending on the magnitude of Gr and Pr. A recirculation vortex which appears and grows until “steady state” is attained was found only for the very high Grashof numbers (10⁵ ≤ Gr ≤ 10⁹) and the lowest Prandtl number considered (Pr = 0.02). The appearance and subsequent disappearance of a vortex was observed for Gr = 10⁹ and Pr = 0.7. Over the lower hemisphere, the thickness of both the hydrodynamic (δ_H) and the thermal (δ_T) boundary layers remain nearly constant and the sphere surface is nearly isothermal. The surface temperature presents a local maximum in the wake of the sphere whenever a vortex is established in the wake of the sphere. The surface pressure recovery in the wake of the sphere increases with decreasing Pr and with increasing Gr. For very small Pr, unlike forced convection, the ratio δ_T/δ_H remains close to unity. The results are in good agreement with experimental data and in excellent agreement with numerical results available in the literature. A correlation has also been presented for the overall Nusselt number as a function of Gr and Pr.     

The effect of a small isolated roughness element on the forces on a sphere in uniform flow

The effect of an isolated roughness element on the forces on a sphere was examined for a Reynolds number range of 5 × 10⁴ < Re < 5 × 10⁵ using a novel sting-mounted sphere apparatus. The roughness element was a circular cylinder, and its width and height was varied to be 1, 2, and 4% of the sphere diameter. At subcritical Re, a lateral force is produced in the direction of the roughness, while at supercritical Re, the force is in the opposite direction. This is caused by asymmetric boundary layer separation, as shown using particle image velocimetry. At supercritical Re, a roughness element that is only 1% the sphere diameter produces a lift to drag ratio of almost one. It was found that the isolated roughness element has the largest effect on the lateral forces when it is located between a streamwise angle of about 40° and 80°. In addition to the mean forces, the unsteady forces were also measured. It was found that at subcritical Re, vortex shedding is aligned to the plane of the roughness element. In addition, the probability distribution of the forces was nearly Gaussian for subcritical Re, but for supercritical Re, the skewness and kurtosis deviate from Gaussian, and the details are dependent on the roughness size. A simple model developed for the vortical structure formed behind the roughness element can be extended to explain aspects of nominally smooth sphere flow, in which external disturbances perturb the sphere boundary layer in an azimuthally local sense. These results also form the basis of comparison for an investigation into the effectiveness of a moving isolated roughness element for manipulating sphere flow.     

Impact of a sphere submerged in an ideal liquid of finite depth

We examine the problem of horizontal and vertical hydrodynamic impact of a sphere submerged in a liquid layer of finite depth. Two limiting cases, corresponding to Froude numbers of zero (problem 1) and infinity (problem 2) are analyzed for horizontal impact.The condition /z=0 is satisfied at the free surface for problem 1 (the problem is hydrodynamically equivalent to the problem of longitudinal flow past a sphere positioned arbitrarily in the layer).The condition=0 is satisfied at the free surface for problem 2. This problem has been examined by Blokh [1] and Sabaneev [2] for the case without a bottom.Also examined is the problem of vertical impact of a sphere submerged in a liquid layer of finite depth.These problems reduce to the solution of an infinite, completely regular system of algebraic equations where the system admits of expansion of the solution in powers of a small parameter; a numerical study of the problem is presented.The influence of the bottom on the basic impact characteristics is studied. The additional-mass coefficients for impact and the pressure distribution over the surface of the sphere are found. The limiting problems corresponding to infinitely deep liquid are examined.     

Heat transfer by laminar flow from a rotating sphere

Summary The heat transfer problem for the flow of an incompressible viscous, heat-conducting fluid, due to uniform rotation about a diameter of a sphere, which is kept at a constant temperature, has been solved with viscous dissipation included. Due to inflow at the poles the cooler liquid is drawn from infinity towards the rotating sphere and this causes a lowering of the temperature there. After flowing in the boundary layer of the sphere the liquid gets heated up and causes a rise in temperature near the equator. Numerical results are given in case of water (Prandtl number σ=5), and it is found that the isothermals are surfaces of revolution flattened at the poles and elongated near the equator. The thermal and the velocity boundary layers turn out to be of the same order of magnitude.     

New analysis of contact melting of phase change material around a hot sphere

The process of contact melting of the solid phase change material (PCM) around a hot sphere, which is driven by the temperature difference between the PCM and the sphere, is analyzed in this paper. Considering the difference of the normal angle between the sphere surface and the solid–liquid interface of the melting PCM, the fundamental equations of the melting process are derived with the film theory. The new film thickness and pressure distribution inside the liquid film and the variation law of the normal angle of the solid–liquid interface and the melting velocity of the sphere are also obtained. It is found that (1) while normal angle at sphere surface φ is within a certain value φ₀, which is related to Ste number and the outside force F, it has no obvious effect on the pressure distribution inside the liquid film and the numerical results by the present model are in accordance with the analytical results in the published literature, (2) the film thickness at φ = ±90° is constringent to a certain value and not the infinity, (3) the analytical results can be employed approximately to analyze the contact melting process except for the film thickness at φ = ±90°.     

On the viscosity of suspensions of solid spheres

A cell theory is used to derive the dependence of the zero-shear-rate viscosity on volume concentration for a suspension of uniform, solid, neutrally buoyant spheres. This result reduces to Einstein's solution at infinite dilution and to Frankel and Acrivos's expression in the limit as the concentration approaches its maximum value. Good agreement is found between the solution and the available data for the entire concentration range, provided that the maximum concentration is determined from the viscosity data themselves.Nomenclature a radius of sphere - d the distance separating the sphere surfaces measured parallel to the line connecting the sphere centers - E energy dissipation rate in one-half the liquid volume separating the spheres - E _cell total energy dissipation rate in the cell - E _homogeneous energy dissipation rate in the cell of a hypothetical one phase fluid - E _interaction energy dissipation rate in the cell due to sphere interactions - E _sphere energy dissipation rate in the cell due to the sphere at the cell center - F force on one sphere - h minimum separation distance between two spheres - J (1/2)d = one-half the distance separating the sphere surfaces measured parallel to the line connecting the sphere centers - p pressure - W velocity of one sphere in squeezing flow between two spheres relative to the midpoint of the line connecting the sphere centers - _i unit vectors in thei-th direction - elongation rate - viscosity of the suspending fluid - _r */ = relative viscosity - * viscosity of the suspension - the total stress tensor - the part of the total stress tensor that vanishes at equilibrium - volume fraction of spheres     

Molecular orientation in non-Newtonian flow of dilute polymer solutions around spheres

A novel approach is presented to study the benchmark problem of flow around spheres in model dilute solutions of monodisperse samples of atactic polystyrene in di-octyl phthalate. Spheres are held stationary on flexible cantilevers of known spring-constant, k, while the polymer solutions are pumped past at controlled flow rates, allowing access to a wide range of Deborah number. In this way the non-Newtonian forces experienced by the spheres can be measured as a function of Deborah number, while detailed observations and measurements of birefringence are made, enabling assessment of macromolecular strain and orientation. In addition the flow field around a sphere has been measured in an a-PS solution. Experiments have been performed on a single sphere and on two spheres axially aligned in the direction of flow. The extensional flow around the downstream stagnation point of the single sphere is found to play a pivotal role in the development of molecular strain and stress, resulting in flow modification and subsequent non-Newtonian behaviour. The flow birefringence in the wake is found to modify severely the flow around a second, downstream, sphere, affecting the non-Newtonian forces encountered by the second sphere. This provides an explanation for the time interval dependent terminal velocity often observed when two spheres follow the same path through viscoelastic liquids.     

Analytical and numerical analysis for the FGM thick sphere under combined pressure and temperature loading

Thermo-mechanical analysis of functionally graded hollow sphere subjected to mechanical loads and one-dimensional steady-state thermal stresses is carried out in this study. The material properties are assumed to vary non-linearly in the radial direction, and the Poisson’s ratio is assumed constant. The temperature distribution is assumed to be a function of radius, with general thermal and mechanical boundary conditions on the inside and outside surfaces of the sphere. In the analysis presented here, the effect of non-homogeneity in FGM thick sphere was implemented by choosing a dimensionless parameter, named β _i (i = 1, . . . , 3), which could be assigned an arbitrary value affecting the stresses in the sphere. It is observed that the solution process for β _i (i = 3) = −1 are different from those obtained for other values of β _i (i = 1, . . . , 3). Cases of pressure, temperature, and combined loadings were considered separately. It is concluded that by changing the value of β _i (i = 1 . . . 3), the properties of FGM can be so modified that the lowest stress levels are reached. The present results agree well with existing results. Using FEM simulations, the analytical findings for FGM spheres under the influence of internal pressure and temperature gradient were compared to finite element results.     

Interaction of a shock with a sphere suspended in a vertical shock tube

Shock wave interaction with a sphere is one of the benchmark tests in shock dynamics. However, unlike wind tunnel experiments, unsteady drag force on a sphere installed in a shock tube have not been measured quantitatively. This paper presents an experimental and numerical study of the unsteady drag force acting on a 80 mm diameter sphere which was vertically suspended in a 300 mm x 300 mm vertical shock tube and loaded with a planar shock wave of M _s = 1.22 in air. The drag force history on the sphere was measured by an accelerometer installed in it. Accelerometer output signals were subjected to deconvolution data processing, producing a drag history comparable to that obtained by solving numerically the Navier-Stokes equations. A good agreement was obtained between the measured and computed drag force histories. In order to interpret the interaction of shock wave over the sphere, high speed video recordings and double exposure holographic interferometric observations were also conducted. It was found that the maximum drag force appeared not at the time instant when the shock arrived at the equator of the sphere, but at some earlier time before the transition of the reflected shock wave from regular to Mach reflection took place. A negative value of the drag force was observed, even though for a very short duration of time, when the Mach stem of the transmitted shock wave relfected and focused at the rear stagnation point of the sphere.Received: 31 March 2003, Accepted: 7 July 2003, Published online: 2 September 2003     

The Remarkable Nature of Radially Symmetric Deformation of Spherically Uniform Linear Anisotropic Elastic Solids

Antman and Negron-Marrero [1] have shown the remarkable nature of a sphere of nonlinear elastic material subjected to a uniform pressure at the surface of the sphere. When the applied pressure exceeds a critical value the stress at the center r=0 of the sphere is infinite. Instead of nonlinear elastic material, we consider in this paper a spherically uniform linear anisotropic elastic material. It means that the stress-strain law referred to a spherical coordinate system is the same for any material point. We show that the same remarkable nature appears here. What distinguishes the present case from that considered in [1] is that the existence of the infinite stress at r=0 is independent of the magnitude of the applied traction σ0 at the surface of the sphere. It depends only on one nondimensional material parameter κ. For a certain range of κ a cavitation (if σ0>0) or a blackhole (if σ0<0) occurs at the center of the sphere. What is more remarkable is that, even though the deformation is radially symmetric, the material at any point need not be transversely isotropic with the radial direction being the axis of symmetry as assumed in [1]. We show that the material can be triclinic, i.e., it need not possess a plane of material symmetry. Triclinic materials that have as few as two independent elastic constants are presented. Also presented are conditions for the materials that are capable of a radially symmetric deformation to possess one or more symmetry planes. This revised version was published online in August 2006 with corrections to the Cover Date.     

Some further observations on the creeping motion of spheres through Ellis model fluids

Using the formulation of Hopke and Slattery, upper and lower bounds on the drag coefficient of a sphere moving slowly in Ellis model fluids have been calculated, over wide range of conditions, and compared with the suitable experimental data available in the literature. C _D drag coefficient - d diameter of sphere - El Ellis number - Re Reynolds number based on zero-shear viscosity - V terminal falling velocity of a sphere - X drag correction factor - Ellis model parameter - ₀ zero-shear viscosity - _1/2 Ellis model parameter     

The stokes flow of the rotating double spheres and multiple spheres

In this paper the expressions for a kind of new rotlets in Stokes flow are derived. By means of superposition of this new rotlets, the drag moment for rotating double spheres and multiple spheres along the smae axis are presented. It has been demonstrated that drag moment for each sphere is the linear function of its angular velocity.     

Effect of unsteady disturbances on the flow in a forward separation zone

The effect of low-frequency disturbances of the three-dimensional separation zone formed in supersonic flow over a sphere with a capped spike on the flow in the forward separation zone has been systematically analyzed on the basis of a large series of experiments. The separation zone was disturbed by rotating the spike about its own axis at various angular velocities. The investigation was carried out using motion-picture records of the flow pattern around the model and the pressure and heat flux distributions on the surface of the sphere.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.2, pp. 185–188, March–April, 1992.     

Stress decomposition analysis in hard and soft sphere suspensions: double peaks in the elastic stress of hard sphere suspensions and its characteristic and structural origin

Brownian dynamics simulations under large amplitude oscillatory shear flow at an intermediate volume fraction in both hard and soft sphere systems have been carried out. Elastic and viscous stresses for the two systems are calculated by using the stress decomposition method. Careful investigation of “double peaks,” which are experimentally observed only in the elastic stress of hard sphere systems, has been conducted. When comparing hard and soft sphere systems in simulation, double peaks are observed only in hard sphere systems, within a specific strain amplitude range. The structures of hard sphere systems at different strain amplitudes, where double peaks appear and not, are compared. Excess entropy concept is adopted to evaluate the extent of particle alignment during the cycle. According to the structural analyses, double peaks are created when the structural difference between maximum-ordered and minimum-ordered states is large during 1 cycle.     

Void nucleation in tensile dead-loading of a composite incompressible nonlinearly elastic sphere

In this paper, the effect of material inhomogeneity on void formation and growth in incompressible nonlinearly elastic solids is examined. A bifurcation problem is considered for a solid composite sphere composed of two neo-Hookean materials perfectly bonded across a spherical interface. Under a uniform radial tensile dead-load, a branch of radially symmetric configurations involving a traction-free internal cavity bifurcates from the underformed configuration. Such a configuration is the only stable solution for sufficiently large loads. In contrast to the situation for a homogeneous neo-Hookean sphere, bifurcation here may occur either locally to the right orto the left. In the latter case, the cavity has finite radius on first appearance. This discontinuous change in stable equilibrium configurations is reminiscent of the snap-through buckling phenomenon observed in certain structural mechanics problems.Since this paper was written, the authors have carried out further analysis of the class of problems of concern here [11]. In particular the stress distribution in the composite neo-Hookean sphere has been described in [11].Paper presented at the 17th International Congress of Theoretical and Applied Mechanics, Grenoble, France, August 1988.     

The arbitrarily and the periodically laminated elastic hollow sphere: exact solutions and homogenization

Summary The aim of this paper is (1) to develop a rational method for the analysis of an arbitrarily laminated elastic, isotropic or transversely isotropic hollow sphere under internal and/or external pressure, (2) to solve the problem of a periodically layered sphere consisting of many equal groups of n different thin layers. The transfer matrix method is used, and exact closed-form solutions are worked out, supplemented by a numerical example. It turns out that by means of the proposed homogenization an originally (periodically) inhomogeneous isotropic sphere is replaced by a homogeneous anisotropic one belonging to the type of spherical symmetric anisotropy. Received 20 October 1997; accepted for publication 15 December 1997