Torque-free Particles Research Articles

Shear-induced Electrokinetic Lift at Large Péclet Numbers

We analyze the problem of shear-induced electrokinetic lift on a particle freely suspended near a solid wall, subject to a homogeneous (simple) shear. To this end, we apply the large-Peclet-number generic scheme recently developed by Yariv et al. (J. Fluid Mech. , Vol. 685, 2011, p. 306). For a force- and torque-free particle, the driving flow comprises three components, respectively describing (i) a particle translating parallel to the wall; (ii) a particle rotating with an angular velocity vector normal to the plane of shear; and (iiii) a stationary particle in a shear flow. Symmetry arguments reveal that the electro-viscous lift, normal to the wall, is contributed by Maxwell stresses accompanying the induced electric field, while electro-viscous drag and torque corrections, parallel to the wall, are contributed by the Newtonian stresses accompanying the induced flow. We focus upon the near-contact limit, where all electro-viscous contributions are dominated by the intense electric field in the narrow gap between the particle and the wall. This field is determined by the gap-region pressure distributions associated with the translational and rotational components of the driving Stokes flow, with the shear-component contribution directly affecting only higher-order terms. Owing to the similarity of the corresponding pressure distributions, the induced electric field for equal particle–wall zeta potentials is proportional to the sum of translation and rotation speeds. The electro-viscous loads result in induced particle velocities, normal and tangential to the wall, inversely proportional to the second power of particle–wall separation.

Irreversible Electrokinetic Repulsion at Zero-Reynolds-Number Sedimentation

Stokes-flow reversibility is violated in electrolyte solutions by a streaming-potential mechanism, where nonuniform convective currents within Debye layers surrounding charged particles induce electric fields in the electroneutral Ohmic bulk. We demonstrate the irreversibility consequences of this phenomenon for the problem of particle-pair sedimentation, where the two particles experience a repulsive force driven by bulk Maxwell stresses. At small separations the force scales inversely with the third power of separation distance. This singular behavior is associated with the counterrotation of the two torque-free particles, which leads through a lubrication mechanism to an intense electric field in the narrow gap between them. At large separations the force follows an inverse dependence upon the fourth power of separation, now associated with rectilinear particle motion.

Heat transfer from a neutrally buoyant sphere in a second-order fluid

In the absence of inertial effects, the heat or mass transfer from torque-free neutrally buoyant spheres in planar shearing flows of Newtonian fluids is diffusion limited at large Pe; here, the Peclet number ( Pe) is a dimensionless measure of the relative importance of convective and diffusive transfer mechanisms. In the inertialess Newtonian limit, the linearity and reversibility of the governing Stokes equations of motion leads to the existence of a region of closed streamlines around the freely-rotating particle that precludes convective enhancement in these flows. Non-Newtonian stresses act to break this symmetry, and the rate of heat/mass transfer from the particle is significantly increased for large Pe. It is possible to analytically determine the transport rate in the limit of weak non-Newtonian effects. For a torque-free particle in a planar linear flow of a second-order fluid, the dimensionless rate of heat or mass transfer, characterized by the Nusselt number, is found to be N u = 0.478 [ P e D e ( 1 + λ ) 2 ( 1 + ϵ ) ] 1 / 3 in the limit P e D e ≫ 1 and D e ≪ 1 , where ϵ is a dimensionless property of the fluid, and λ is a parameter that depends on the relative magnitudes of extension and vorticity in the ambient flow. In simple shear flow, corresponding to λ = 0 , the Nusselt number may be alternatively written as 6.01 ( P e D e ) 1 / 3 [ ( ( 1 / 2 ) + ( Ψ 2 / Ψ 1 ) ) / ( 1 + ( Ψ 2 / Ψ 1 ) ) ] 1 / 3 ; here, Ψ 1 and Ψ 2 are the first and second normal stress coefficients. The Deborah number ( De) is the ratio of the intrinsic relaxation time of the fluid to the macroscopic flow time scale in all above cases, and serves as a dimensionless measure of the relative magnitudes of the non-Newtonian (elastic) and Newtonian stresses; for simple shear flow, one may define D e = ( Ψ 1 + Ψ 2 ) γ ˙ / η in terms of the normal stress coefficients, η being the solvent viscosity.

Inertial effects on the transfer of heat or mass from neutrally buoyant spheres in a steady linear velocity field

Microscale inertia is found to break the degenerate closed-streamline configuration that occurs in a shearing flow past a neutrally buoyant torque-free spherical particle in the inertialess limit. The broken symmetry at small but finite Re allows heat or mass to be convected away in an efficient manner in sharp contrast to the inertialess diffusion-limited scenario. Inertial forces scale with the particle Reynolds number, defined as Re=γ̇a2∕ν, where a is the radius of the particle, γ̇ is the characteristic magnitude of the velocity gradient, and ν is the kinematic viscosity of the suspending fluid. The dimensionless heat or mass transfer rate is then given by Nu=C(RePe)1∕3+O(1) when Re≪1 and RePe≫1, the constant C being a function of the flow in the vicinity of the particle. Here, Nu is the Nusselt number defined as Q∕(4πkaΔF), where Q is the dimensional heat/mass flux, k the appropriate transport coefficient, and ΔF the driving force viz. the temperature or concentration difference between the particle and the ambient fluid; for pure diffusion, Nu=1. The Peclét number (Pe) is a dimensionless measure of the relative dominance of the convective and diffusive transfer mechanisms. It is shown that C equals 0.325(1+λ)2∕3 for a two-dimensional linear flow, where λ measures the relative magnitudes of extension and vorticity. For simple shear (λ=0), knowledge of the inertial velocity field to O(Re3∕2) enables one to determine the next term in the asymptotic expansion for Nu; one finds Nu=(RePe)1∕3(0.325−0.0414Re1∕2)+O(1) in the limit 1≫Re≫Pe−2∕5. It is argued that the convective enhancement at finite Re via symmetry-breaking streamline bifurcations will occur in generic shearing flows with nonlinear velocity profiles; the degenerate Stokes streamline pattern around a neutrally buoyant torque-free particle in a quadratic flow serves to reinforce this assertion. The above mechanism represents a possible means for heat or mass transfer enhancement from the dispersed phase in multiphase systems. Implications for particles in turbulent flows are also discussed.

Interception of two spheroidal particles in shear flow

A numerical method is developed for simulating the motion of particles with arbitrary shape in an effectively infinite or bounded viscous flow. The particle translational and angular velocities are computed directly by solving an integral equation of the second kind arising from the double-layer representation for Stokes flow, subject to a specified force and torque. The deflated integral equation is solved by the method successive substitutions using a spectral boundary-element method. In the case of force- and torque-free particles, the contribution of the particles to the effective viscosity of the suspension is expressed in terms of the distribution density of the double-layer potential over the particle surfaces. The method is applied to investigate the interception of two spheroidal particles in simple shear flow, with emphasis on the net particle displacement and shift in the phase of rotation after separation, and on the transient signature of the interception on the effective viscosity of the suspension. The net particle displacement normal to the streamlines of the shear flow is found to have both positive and negative values depending on the relative particle configuration before interception. For moderate particle separations, the phase of rotation is shifted only slightly with respect to the Jeffery orbit executed by a particle in isolation.

The motion of axisymmetric dipolar particles in homogeneous shear flow

The respective effects of an external field and departure of suspended particles from spherical shape on the rotary motion of axisymmetric dipolar particles placed in a homogeneous shear flow are studied. The analysis shows that, owing to cumulative effects, even a weak external field or a small deviation from spherical shape can significantly modify the resulting motion relative to that found in the corresponding classical problems of torque-free particles or dipolar spheres in homogeneous shear. Thus, unlike the latter problems, there are in the present problem cases when all particles approach a single limit cycle; in other cases multiple stable equilibria simultaneously coexist and the orientation space is appropriately divided into corresponding domains of attraction; in some situations possessing appropriate symmetry properties, particles may, depending upon their respective initial orientations, either move along a family of periodic orbits or else converge to a stable equilibrium orientation.

MOTION OF A RIGID CYLINDER BETWEEN PARALLEL PLATES IN STOKES FLOW: PART 2: POISEUILLE AND COUETTE FLOW.

A numerical solution is presented for the motion of a neutrally buoyant circular cylinder in Poiseuille and Couette flows between two plane parallel boundaries. The force and torque on a stationary particle are calculated for a wide range of particle sizes and positions across the channel. The resistance matrix calculated in Ref. [1] (henceforth referred to as Part 1) is utilized to find the translational and angular velocity for a drag- and torque-free particle. The results are compared with analytical perturbation solutions for a small cylindrical particle situated on the channel centerline, and for the motion of a spherical particle in a circular tube or between plane parallel boundaries. It is found the behavior of flow around a cylindrical particle in a channel is qualitatively similar to the behavior of flow around a spherical particle in a tube, while the flow around a spherical particle in a channel frequently exhibits different trends from the above two cases.

A micromechanical derivation of Fick's law for interfacial diffusion of surfactant molecules

A theory is presented for the diffusion of surfactant molecules, modeled as noninteracting Brownian spheres in proximity to, or straddling, the interface between two immiscible fluids. The spheres are assumed to be physicochemically inert except for the existence of short-range forces, which may be attractive and/or repulsive with respect to the interface. The number density of such particles is assumed small. Two alternative views of the diffusion process are presented; one, termed microscopic, in which variations in concentration are resolvable down to the scale l imposed by the short-range attractive or repulsive forces between the particles and the interface, and the other, termed macroscopic, in which variations are only discernible at the much larger scale, L, of gradients in the bulk surfactant concentration distribution, which are assumed to exist parallel to the interface. It is demonstrated that a rigorous theory can be developed, at the microscale, for prediction of the concentration profiles and diffusive flux of surfactant molecules parallel to the interface using only the well-established Stokes-Einstein theory of bulk-phase Brownian diffusion, and low Reynolds number hydrodynamics for the motion of a torque-free particle in close proximity, or straddling, a fluid-fluid interface. On the other hand, a macroscopic description of the same phenomena requires the introduction of new concepts, such as “adsorption” and “surface diffusion,” which are specifically associated with the interface, as seen from the macroscale of O( L). In the present paper, constitutive relationships for these macrosurface processes are derived in a rigorous manner from the more fundamental and complete microscopic description of the system. It is shown that Fick's law is applicable to surface diffusion, with the driving gradient based on a surface-excess concentration for surfactant particles. Furthermore, the surface diffusion coefficient is found to depend upon the Stokesian hydrodynamic resistance of a torque-free sphere translating parallel to the interface, as a function of the distance of its center from the interface, and upon the “adsorptive-potential” energy function which tends to cause the particles to accumulate there. Numerical values await the solution of the low Reynolds number hydrodynamics problem thereby posed, as well as the acquisition of knowledge relating to the potential energy function. Two examples of such potential energy functions are discussed for Brownian particles, one based upon a difference in area-specific, surface-free energies (i.e., solid/liquid interfacial tensions) for the two fluids, and the other on a simple density differential between the particle and the two fluids. Simple geometric models of surfactant molecules are also discussed in the context of potential energy functions for interaction between the molecules and the interface. The possibility of using such models in the analysis of other equilibrium and surface transport phenomena is pointed out.