Mixed Slip-stick Boundary Conditions Research Articles

Hydrodynamic interactions between aerosol particles in the transition regime

We present a method for calculating the hydrodynamic interactions between particles in the kinetic (or transition regime), characterized by non-negligible particle Knudsen numbers. Such particles are often present in aerosol systems. The method is based on our extended Kirkwood–Riseman theory (Corson et al., Phys. Rev. E, vol. 95 (1), 2017c, 013103), which accounts for interactions between spheres using the velocity field around a translating sphere as a function of Knudsen number. Results for the two-sphere problem at small Knudsen numbers are in good agreement with those obtained using Felderhof’s interaction actions for mixed slip-stick boundary conditions, which are accurate to order $r^{-7}$ (Felderhof, Physica A, vol. 89 (2), 1977, pp. 373–384). The strength of the interactions decreases with increasing Knudsen number. Results for two fractal aggregates demonstrate that one can apply a point force approach for interactions between particles in the transition regime; the interaction tensor is similar to the Oseen tensor for continuum flow. Using this point force approach, we present an analysis for the settling of an unbounded cloud of particles. Our analysis shows that for sufficiently high volume fractions and cloud radii, the cloud behaves as a gas droplet in continuum flow even when the individual particles are small relative to the mean free path of the gas. The method presented here can be applied in a Brownian dynamics simulation analogous to Stokesian dynamics to study the behaviour of a dense aerosol system.

Effect of fluid compressibility on the flow caused by a sudden impulse applied to a sphere immersed in a viscous fluid

The flow of a viscous compressible fluid about a sphere suddenly set in motion by an applied impulse is studied on the basis of linearized hydrodynamic equations. Mixed slip-stick boundary conditions are assumed to hold at the surface of the sphere. An analytic expression for the flow pattern is presented for an idealized fluid in which the damping of sound waves is neglected. The time dependence of the pressure part and the viscous part of the fluid momentum is investigated. It is shown that at long times, one-third of the imparted momentum resides in the pressure part.

Flow caused by a square force pulse applied to a sphere immersed in a viscous incompressible fluid

An analytic solution of the linearized Navier-Stokes equations is presented for the flow of an incompressible viscous fluid around a sphere subjected to a square force pulse. Mixed slip-stick boundary conditions are assumed to hold at the surface of the sphere. The transient flow pattern is expressed in terms of a Stokes stream function. This facilitates visualization of the flow.

Transient flow caused by a sudden impulse or twist applied to a sphere immersed in a viscous incompressible fluid

An analytic solution of the linearized Navier-Stokes equations is presented for the flow of an incompressible viscous fluid about a sphere suddenly set in motion by an impulse or twist. Mixed slip-stick boundary conditions are assumed to hold at the surface of the sphere. The transient flow pattern depends on the mass and moment of inertia of the sphere. The impulsive translational motion of the sphere generates an expanding vortex ring. Conservation of total momentum is discussed and the momentum of the fluid is expressed as a sum of a potential and a viscous contribution. Conservation of total angular momentum is verified in the case of rotational motion.

Image representation of a spherical particle near a hard wall

The motion of a spherical colloidal particle suspended in a moving fluid near a planar hard wall or free surface is considered. The particle types include hard spheres with mixed slip-stick boundary conditions, droplets with high surface tension and porous particles. A general expression for the flow field is obtained in terms of a set of force multipoles induced on the particle and on its mirror image in the bounding surface. An earlier calculation of the friction and mobility functions for the particle is extended to include coupling to an incident linear shear flow. A numerical representation of all these functions is given which is accurate at all separations of particle and boundary. These results are used to study the position and orientation of a hard sphere falling towards a wall under gravity in the presence of a shear flow.

Heat transfer in a slip-flow past a sphere

Convective heat transfer from a sphere in a streaming flow of a viscous incompressible fluid is studied under mixed slip-stick boundary conditions. Using the method of matched asymptotic expansions, analytical expression for the mean Nusselt number Nu is obtained up to O( σ 2Re 2), where Re<1 is the Reynolds number based on uniform streaming and σ of O(1) is the Prandtl number. The results show that for a given Re, heat transfer decreases with increasing slip. When a meridional spin is given to the body, it further decreases the heat transfer, though, quantitatively it is very marginal. This feature is in sharp contrast to the increased heat transfer due to rotation, observed in the absence of slip.

Creeping flow about a slightly deformed sphere

The problem of slow streaming flow of a viscous incompressible fluid past a spheroid which departs but little in shape from a sphere with mixed slip-stick boundary conditions, is investigated. The explicit expression for the stream function is obtained to the first order in the small parameter characterising the deformation. The case of an oblate spheroid is considered as a particular example and the force on this non-spherical body is evaluated. It is found that the parameter λ1, which arises in connection with the boundary condition, has significant effect upon the hydrodynamic force. In fact, it is shown that, the force is a quadratic function of this parameter up to the first order of deformation. Also, it is observed that the drag in the present case is less than that of the Stokes resistance for a slightly oblate spheroid. Some other special cases are also deduced from the present result. A brief discussion of the results to other body shapes is presented.

Circle theorem for stokes flow with mixed slip-stick boundary conditions

Recently, several kinetic descriptions have been utilized for the simulation of gas mixture flows in microgeometries. Although such developments are important, Navier–Stokes computation can find extended use in engineering applications. In the present contribution, a rarefied flow of a gas mixture in the vicinity of a wall is concerned and new velocity-slip and temperature-jump boundary conditions are derived for the whole mixture. Appealing to these new boundary conditions, Navier–Stokes computation of microscale gas mixture flows becomes feasible. Consequently, the proposed conditions in conjunction with the Navier–Stokes equations and an equation for the conservation of species are solved for a He–Ne flow in a microchannel and suitability of the derived boundary conditions is demonstrated in the slip flow regime.

Stokes flow past a sphere with mixed slip-stick boundary conditions

We give a general solution of Stokes equations for an incompressible, viscous flow past a sphere with mixed slip-stick boundary conditions. The Faxén's law for drag and torque on the sphere is also given and illustrated with an example.

Linear response theory of the motion of a spherical particle in an incompressible fluid

We consider a spherical particle immersed in an incompressible fluid described by the linearized Navier-Stokes equations. The particle response to an applied time-dependent force, torque and incident flow is studied for mixed slip-stick boundary conditions. The resulting flow pattern is expressed with the aid of resistance matrices.

Long-time self-diffusion coefficient and zero-frequency viscosity of dilute suspensions of spherical Brownian particles

We study the long-time self-diffusion coefficient and the zero-frequency effective viscosity for a suspension of spherical Brownian particles. The correction to first order in the volume fraction to the self-diffusion coefficient and the correction to second order in the volume fraction to the effective viscosity are given by integral expressions derived by Batchelor. Using exact series expansions of the hydrodynamic interaction functions in powers of the inverse distance between centers of a pair of particles we obtain accurate values for the correction terms. We consider hard spheres with mixed slip-stick boundary conditions as well as spherical liquid droplets with a viscosity different from the bulk.

Short-time diffusion coefficients and high frequency viscosity of dilute suspensions of spherical Brownian particles

We study the short-time collective and self-diffusion coefficients, the rotational self-diffusion coefficient, and the high frequency effective viscosity of a suspension of spherical Brownian particles. At low volume fraction these transport coefficients are given by integrals of hydrodynamic pair interactions. Using exact series expansions in powers of the inverse distance between centers of a pair of particles, we obtain accurate values for the low density transport coefficients. We consider hard spheres with mixed slip–stick boundary conditions, as well as spherical liquid droplets with an internal viscosity different from the bulk. We show that the transport coefficients depend strongly on the particle model.

Hydrodynamic scattering theory of flow about a sphere

We formulate the solution of the linear Navier-Strokes equations for time-dependent incompresible flow about a spherical particle in terms of a scattering formalism. The solution is written as an expansion in terms of incident and scattered waves. The amplitudes of the outgoing waves are related to a set of spherical force multipoles which themselves can be found from the amplitudes of the incident waves with the help of a resistance matrix. The resistance matrix is evaluated explicitly for hard spheres with mixed slip-stick boundary conditions. Transformation equations are given which relate the time-dependent solutions found here to previously studied solutions for the time-independent steady flow case.

Force multipole moments for a spherically symmetric particle in solution

We present a general theorem for the force multipole moments of arbitrary order induced in a spherically symmetric particle immersed in a fluid whose motion satisfies the linear Navier-Stokes equation for steady incompressible viscous flow. The multipole moments are expressed in terms of the unperturbed fluid velocity field. It is shown that for a particle with a finite extension only a few terms give rise to fluid perturbations which are not confined to the interior of the particle. We give explicit results for a polymer satisfying the Debye-Bueche-Brinkman equations and for a hard sphere with mixed slip-stick boundary conditions.

Faxén theorems for spherically symmetric polymers in solution

We derive Faxén theorems for the force, the torque and the symmetric force dipole moment acting on a spherically symmetric polymer suspended in arbitrary flow. The results are immediate generalizations of corresponding theorems for a hard sphere with mixed slip-stick boundary conditions.

Hydrodynamic interaction of two spherically symmetric polymers

We evaluate the complete diffusion matrix describing the translation and rotation of two spherically symmetric polymers in hydrodynamic interaction in a viscous fluid. Explicit results in terms of a series expansion in inverse powers of the distance l between centers is obtained by a method of reflections. Recently derived Faxén theorems allow direct calculation of the diffusion matrix, rather than its inverse the friction matrix. The results can be easily compared with those for hard spheres with mixed slip-stick boundary conditions.

Creeping flow about a sphere

We present the general solution of the linear Navier-Stokes equation for incompressible viscous flow about a sphere with mixed slip-stick boundary conditions. Exphclt results are obtained in cartesian coordinates. The relation wRh Lamb's method is indicated. We derive expressions for a set of fundamental tensors, introduced by Brenner, describing the perturbed flow. Finally the force density induced on the sphere is evaluated. In this paper we give the solution of the linear Navier-Stokes equation for incompressible viscous flow about a sphere with mixed slip-stick boundary conditions for arbitrary incident flow. A method of solution was presented long ago by Lamb l) in terms of solid spherical harmonics, but his solution left something to be desired from a practical point of view. Lamb's solution is described in some detail in the monograph by Happel and Brenner2). The difficulty with Lamb's solution is that application to particular problems often requires writing the harmonics in terms of spherical coordinates, whereby a specific choice of axes, alien to the problem at hand, is made. For many problems an approach using cartesian coordinates is more convenient. In particular for the problem of hydrodynamic interaction of two spheres this approach has recently proved effective34). In the following we persistently use the cartesian point of view and solve the problem by the so-called method of cartesian ansatz. The solution is a direct generalization of earlier results obtained for constant, linear or quadratic incident flow3-7). At the end we show the relation to Lamb's solution. As a benefit we obtain explicit results for a set of tensors introduced by Brenner 8) in his fundamental paper on Stokes resistance. Brenner only gave explicit

Hydrodynamic interaction between two spheres

We evaluate the hydrodynamic interaction tensors for two spheres of unequal size and for general mixed slip-stick boundary conditions. A method of reflections leads to a series expansion for the diffusion tensors in powers of the inverse distance l −1 between sphere centers and explicit results are derived through terms of order l −7. It turns out that the series expansion for the diffusion tensors converges much more rapidly than that for the friction tensors.