The influence of relative velocity on the eddy structure between two spheres in Stokes flow

Davis et al. (1976) have shown that if two solid spheres move together in an axisymmetric Stokes flow, then provided they are sufficiently close, a body of fluid becomes trapped between the spheres. Here it is shown how the small eddy motions induced in this trapped fluid are significantly disrupted when one sphere moves relative to the other.     

Stokes flow before plane boundary with mixed stick-slip boundary conditions

A general theorem for the Stokes flow over a plane boundary with mixed stick-slip boundary conditions is established. This is done by using a representation for the velocity and pressure fields in the three-dimensional Stokes flow in terms of a biharmonic function and a harmonic function. The earlier theorem for the Stokes flow due to fundamental singularities before a no-slip plane boundary is shown to be a special case of the present theorem. Furthermore, in terms of the Stokes stream function, a corollary of the theorem is also derived, providing a solution to the problem of the axisymmetric Stokes flow along a rigid plane with stick-slip boundary conditions. The formulae for the drag and torque exerted by the fluid on the boundary are established. An illustrative example is given.     

Viscosity of a Suspension with a Cubic Array of Spheres in a Shear Flow

Viscous fluid flow past an infinite periodic array of rigid spheres of the same radius is considered. A solution of the Stokes equations periodic in three variables is obtained for viscous incompressible flow with a linear velocity profile. The solution takes into account the hydrodynamic interaction of an infinite number of particles in the array. An expression for the effective viscosity of a suspension with a cubic array of particles is obtained.     

Slow motion of a permeable sphere in a viscous fluid

Flow of a viscous fluid past a permeable sphere is investigated in the Stokes approximation. An example of such a flow is flow past a perforated or meshed spherical surface. The elements of the sphere contain rigid impermeable sections and openings through which the fluid can flow. The interaction of the sphere with the flow is described by two drag coefficients, which established the connection between the flow velocity of the fluid at the sphere and the stress tensor on it. The dependence of the flow pattern and also the drag and flow rate of the fluid on these coefficients is investigated. In special cases, the obtained solution describes flow past solid and liquid spheres.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 165–167, September–October, 1982.     

Thrust augmentation of flapping airfoils in low Reynolds number flow using a flexible membrane

The unsteady aerodynamic thrust and aeroelastic response of a two-dimensional membrane airfoil under prescribed harmonic motion are investigated computationally with a high-order Navier–Stokes solver coupled to a nonlinear membrane structural model. The effects of membrane prestress and elasticity are examined parametrically for selected plunge and pitch–plunge motions at a chord-based Reynolds number of 2500. The importance of inertial membrane loads resulting from the prescribed flapping is also assessed for pure plunging motions. This study compares the period-averaged aerodynamic loads of flexible versus rigid membrane airfoils and highlights the vortex structures and salient fluid–membrane interactions that enable more efficient flapping thrust production in low Reynolds number flows.     

Squeeze flow of a second-order fluid between two parallel disks or two spheres

The normal viscous force of squeeze flow between two arbitrary rigid spheres with an interstitial second-order fluid was studied for modeling wet granular materials using the discrete element method. Based on the Reynolds‘ lubrication theory, the small parameter method was introduced to approximately analyze velocity field and stress distribution between the two disks. Then a similar procedure was carried out for analyzing the normal interaction between two nearly touching, arbitrary rigid spheres to obtain the pressure distribution and the resulting squeeze force. It has been proved that the solutions can be reduced to the case of a Newtonian fluid when the non-Newtonian terms are neelected.     

A strongly coupled,embedded-boundary method for fluid–structure interactions of elastically mounted rigid bodies

In the present paper, an embedded-boundary formulation that is applicable to fluid–structure interaction problems is presented. The Navier–Stokes equations for incompressible flow are solved on a Cartesian grid which is not aligned with the boundaries of a body that undergoes large-angle/large-displacement rigid body motions through the fixed grid. A strong-coupling scheme is adopted, where the fluid and the structure are treated as elements of a single dynamical system, and all of the governing equations are integrated simultaneously and interactively in the time domain. A demonstration of the accuracy and efficiency of the method is given for a variety of fluid–structure interaction problems.     

Optical measurements of pore geometry and fluid velocity in a bed of irregularly packed spheres

Imaging methods are proposed for the characterisation of liquid flows through transparent porous media of matched refractive index. The methods are based on the analysis of laser-illuminated slices, and specialized for the case in which the porous medium is composed of irregularly packed spheres. They include algorithms for the reconstruction of the three-dimensional (3D) sphere arrangement based on a laser scan of the packed bed, particle tracking velocimetry applied to the motions of micro-tracers in a laser-illuminated plane, and techniques for the co-registration of geometry and velocity measurements acquired from different slices. The methods are applied to a cylindrical flow cell filled with mono-sized spheres and operated at Reynolds number Re = 28. The data produced include the full 3D geometry of the packed spheres assembly, the 2D fluid velocity field in the axial centre-plane of the flow cell, and the corresponding porosity and velocity distributions.     

Three-dimensional tracking of the long time trajectories of suspended particles in a lid-driven cavity flow

Stereo imaging methods are used to measure the positions of solid spherical particles suspended in a viscous liquid and enclosed in a transparent cubic cavity. The liquid and particle motions are driven at the top lid by a conveyor belt operated at constant speed. Based on sequences of stereo views of the full cavity, the particles are tracked continuously along their three-dimensional orbits. The corresponding position histories are treated as noisy stochastic data and processed using Kalman filters to fill data gaps and attenuate the effect of measurement errors. The lid-driven viscous flow is characterised by an intricate internal structure which is mirrored in the particle paths. The tracks of the solid particles align with long exposure images of laser-illuminated micro-particles in selected transverse planes. Nevertheless, their long time trajectories appear to cluster along preferential pathways of the internal circulation pattern.     

Direct numerical simulation of particle dispersion in a turbulent jet considering inter-particle collisions

To investigate the behaviour of inter-particle collision and its effects on particle dispersion, direct numerical simulation of a three-dimensional two-phase turbulent jet was conducted. The finite volume method and the fractional-step projection algorithm were used to solve the governing equations of the gas phase fluid and the Lagrangian method was applied to trace the particles. The deterministic hard-sphere model was used to describe the inter-particle collision. In order to allow an analysis of inter-particle collisions independent of the effect of particles on the flow, two-way coupling was neglected. The inter-particle collision occurs frequently in the local regions with higher particle concentration of the flow field. Under the influence of the local accumulation and the turbulent transport effects, the variation of the average inter-particle collision number with the Stokes number takes on a complex non-linear relationship. The particle distribution is more uniform as a result of inter-particle collisions, and the lateral and the spanwise dispersion of the particles considering inter-particle collision also increase. Furthermore, for the case of particles with the Rosin–Rammler distribution (the medial particle size is set d₅₀ = 36.7 μm), the collision number is significantly larger than that of the particles at the Stokes number of 10, and their effects on calculated results are also more significant.     

Numerical simulation of the hydrodynamic interaction between a sedimenting particle and a neutrally buoyant particle

A new method for the simulation of the translational and rotational motions of a system containing a sedimenting particle interacting with a neutrally buoyant particle has been developed. The method is based on coupling the quasi-static Stokes equations for the fluid with the rigid body equations of motion for the particles. The Stokes equations are solved at each time step with the boundary element method. The stresses are then integrated over the surface of each particle to determine the resultant forces and moments. These forces and moments are inserted into the rigid body equations of motion to determine the translational and rotational motions of the particles. Unlike many other simulation techniques, no restrictions are placed on the shape of the particles. Superparametric boundary elements are employed to achieve accurate geometric representations of the particles. The simulation method is able to predict the local fluid velocity, resolve the forces and moments exerted on the particles, and track the particle trajectories and orientations.     

The Motion of a Fluid-Rigid Ball System at the Zero Limit of the Rigid Ball Radius

We study the limiting motion of a system of rigid ball moving in a Navier–Stokes fluid in ${\mathbb{R}^3}$ as the radius of the ball goes to zero. Recently, Dashti and Robinson solved this problem in the two-dimensional case, in the absence of rotation of the ball (Dashti and Robinson in Arch Rational Mech Anal 200:285–312, 2011). This restriction was caused by the difficulty in obtaining appropriate uniform bounds on the second order derivatives of the fluid velocity when the rigid body can rotate. In this paper, we show how to obtain the required uniform bounds on the velocity fields in the three- dimensional case. These estimates then allow us to pass to the zero limit of the ball radius and show that the solution of the coupled system converges to the solution of the Navier–Stokes equations describing the motion of only fluid in the whole space. The trajectory of the centre of the ball converges to a fluid particle trajectory, which justifies the use of rigid tracers for finding Lagrangian paths of fluid flow.     

Two phase mixed convection Al2O3–water nanofluid flow in an annulus

Heat transfer enhancement of a mixed convection laminar Al₂O₃–water nanofluid flow in an annulus with constant heat flux boundary condition has been studied employing two phase mixture model and effective expressions of nanofluid properties. The fluid flow properties are assumed constant except for the density in the body force, which varies linearly with the temperature (Boussinesq’s hypothesis), thus the fluid flow characteristics are affected by the buoyancy force. The Brownian motions of nanoparticles have been considered to determine the effective thermal conductivity and the effective dynamic viscosity of Al₂O₃–water nanofluid, which depend on temperature. Three-dimensional Navier–Stokes, energy and volume fraction equations have been discretized using the finite volume method while the SIMPELC algorithm has been introduced to couple the velocity–pressure. Numerical simulations have been presented for the nanoparticles volume fraction (?) between 0 and 0.05 and different values of the Grashof and Reynolds numbers. The calculated results show that at a given Re and Gr, increasing nanoparticles volume fraction increases the Nusselt number at the inner and outer walls while it does not have any significant effect on the friction factor. Both the Nusselt number and the friction coefficient at the inner wall are more than their corresponding values at the outer wall.     

Squeeze flow of interstitial Herschel–Bulkley fluid between two rigid spheres

Interaction between two spheres with an interstitial fluid is crucial in discrete element modeling for simulating the behaviors of ‘wet’ particulate materials. The normal viscous force of squeeze flow between two arbitrary rigid spheres with an interstitial Herschel–Bulkley fluid was studied on the basis of Reynolds’ lubrication theory, resulting in analytical integral expressions of pressure distribution and the viscous force between the two spheres. According to the variation of shear stress, the fluid was divided into yielding and unyielding regions, followed by a discussion on the thickness of the two regions. The result of this paper could be reduced to either the power-law fluid or the Bingham fluid case.     

Transient rolling friction model for discrete element simulations of sphere assemblies

The rolling resistance between a pair of contacting particles can be modeled with two mechanisms. The first mechanism, already widely addressed in the DEM literature, involves a contact moment between the particles. The second mechanism involves a reduction of the tangential contact force, but without a contact moment. This type of rotational resistance, termed creep-friction, is the subject of the paper. Within the creep-friction literature, the term “creep” does not mean a viscous mechanism, but rather connotes a slight slip that accompanies rolling. Two extremes of particle motions bound the range of creep-friction behaviors: a pure tangential translation is modeled as a Cattaneo–Mindlin interaction, whereas prolonged steady-state rolling corresponds to the traditional wheel–rail problem described by Carter, Poritsky, and others. DEM simulations, however, are dominated by the transient creep-friction rolling conditions that lie between these two extremes. A simplified model is proposed for the three-dimensional transient creep-friction rolling of two spheres. The model is an extension of the work of Dahlberg and Alfredsson, who studied the two-dimensional interactions of disks. The proposed model is applied to two different systems: a pair of spheres and a large dense assembly of spheres. Although creep-friction can reduce the tangential contact force that would otherwise be predicted with Cattaneo–Mindlin theory, a significant force reduction occurs only when the rate of rolling is much greater than the rate of translational sliding and only after a sustained period of rolling. When applied to the deviatoric loading of an assembly of spheres, the proposed creep-friction model has minimal effect on macroscopic strength or stiffness. At the micro-scale of individual contacts, creep-friction does have a modest influence on the incremental contact behavior, although the aggregate effect on the assembly's behavior is minimal.     

Method of fundamental solutions for multidimensional Stokes equations by the dual-potential formulation

A simple and accurate boundary-type meshless method of fundamental solutions (MFS) is applied to solve both 2D and 3D Stokes flows based on the dual-potential formulation of velocity potential and stream function vector. Using the dual-potential concept, the solutions of both 2D and 3D Stokes flows are obtained by combining the much simpler fundamental solutions of Laplace (potential) and bi-harmonic equations without using the complicated singular fundamental solutions such as Stokeslets and their derivatives as well as source doublet hypersingularity. The developed algorithm is used to test five numerical experiments for 2D flows: (1) circular cavity, (2) wave-shaped bottom cavity and (3) circular cavity with eccentric rotating cylinder; and for 3D flows: (4) a uniform flow passing a sphere and (5) a uniform flow passing a pair of spheres. Good results are obtained as comparing with solutions of analytical and numerical methods such as FEM, BEM and other meshfree schemes.     

Numerical modeling of flow past a volumeless and thin rigid body using direct forcing immersed boundary method

The new capability has been added as the numerical method for modeling volumeless and thin rigid bodies to the direct forcing immersed boundary (DFIB) method. The DFIB approach is based on adding a virtual force to the Navier–Stokes equations of incompressible flow to account for the interaction between the fluid and structures. The volume of a solid function (VOS) identifies the stationary or moving solid structures in a given fluid domain. A new VOS-based algorithm was developed to identify thin, rigid structure boundary points in fluid flow and ensure that the fluid cannot cross through the boundary of a thin rigid structure while moving or stationary. The DFIB method was first validated in a three-dimensional (3D) turbulent flow over a circular cylinder. The large-eddy simulation simulated the turbulent flow scales. The proposed algorithm was tested using a 3D turbulent flow past a stationary and rotating Savonius wind turbine that functions as a thin, rigid body. The validation results showed that the selected DFIB approach, combined with the novel algorithm, could simulate a thin, volumeless, rigid structure that is stationary and rotating in incompressible turbulent flows. The current method is also applicable for two-way fluid-structure interaction problems.     

Spiral thermocapillary flows in two-layer systems

Parallel thermocapillary flows in infinite layers can be calculated easily. However, in reality thermocapillary flows occur in channels or closed cavities, and they are three-dimensional. In the present paper, a two-layer fluid system filling a channel with a rectangular cross section is considered. A numerical investigation of three-dimensional spiral thermocapillary flows generated by a temperature gradient imposed along the channel has been performed. Both the case of a zero longitudinal pressure gradient (a through flow in the channel) and that of a zero longitudinal fluid flux (a flow in the closed cavity) are investigated. Steady and oscillatory thermocapillary motions, and transitions between them are studied.     

A New Application of the Differential Quadrature Element-Incremental Method in Moving-Boundary Problems

Upper bounds on the permeability of random porous media are presented, which improve significantly on existing bounds. The derived bounds rely on a variational formulation of the upscaling problem from a viscous flow at the pore scale, described by Stokes equation, to a Darcy formulation at the macroscopic scale. A systematic strategy to derive upper bounds based on trial force fields is proposed. Earlier results based on uniform void or interface force fields are presented within this unified framework, together with a new proposal of surface force field and a combination of them. The obtained bounds feature detailed statistical information on the pore morphology, including two- and three-point correlation functions of the pore phase, the solid–fluid interface and its local orientation. The required spatial correlation functions are explicitly derived for the Boolean model of spheres, in which the solid phase is modelled as the union of penetrable spheres. Existing and new bounds are evaluated for this model and compared to full-field simulations on representative volume elements. For the first time, bounds allow to retrieve the correct order of magnitude of permeability for a wide range of porosity and even improve on some estimates. However, none of the bounds reproduces the non-analytic behaviour of the permeability–porosity curve at low solid concentration.