Effect of surface slip on Stokes flow past a spherical particle in infinite fluid and near a plane wall

Abstract

The motion of a spherical particle in infinite linear flow and near a plane wall, subject to the slip boundary condition on both the particle surface and the wall, is studied in the limit of zero Reynolds number. In the case of infinite flow, an exact solution is derived using the singularity representation, and analytical expressions for the force, torque, and stresslet are derived in terms of slip coefficients generalizing the Stokes-Basset-Einstein law. The slip velocity reduces the drag force, torque, and the effective viscosity of a dilute suspension. In the case of wall-bounded flow, advantage is taken of the axial symmetry of the boundaries of the flow with respect to the axis that is normal to the wall and passes through the particle center to formulate the problem in terms of a system of one-dimensional integral equations for the first sine and cosine Fourier coefficients of the unknown traction and velocity along the boundary contour in a meridional plane. Numerical solutions furnish accurate predictions for (a) the force and torque exerted on a particle translating parallel to the wall in a quiescent fluid, (b) the force and torque exerted on a particle rotating about an axis that is parallel to the wall in a quiescent fluid, and (c) the translational and angular velocities of a freely suspended particle in simple shear flow parallel to the wall. For certain combinations of the wall and particle slip coefficients, a particle moving under the influence of a tangential force translates parallel to the wall without rotation, and a particle moving under the influence of a tangential torque rotates about an axis that is parallel to the wall without translation. For a particle convected in simple shear flow, minimum translational velocity is observed for no-slip surfaces. However, allowing for slip may either increase or decrease the particle angular velocity, and the dependence on the wall and particle slip coefficients is not necessarily monotonic.

Motion of a solid particle in a shear flow along a porous slab

“…For a given sphere location d * , these velocities increase with K * for prescribed λ; whereas, for a given permeability K * , note that, as λ increases, then u l 1 increases, but ω l 2 may either increase (for K * = 10 −4 ) or decrease (for K * = 10 −2 ). For a low permeability, the results versus λ are in good agreement with those for an impermeable slip wall obtained by bispherical coordinates in Loussaief (2008) and by BIM in Luo & Pozrikidis (2008). For the quadratic shear flow, u q 1 and ω q 2 exhibit the same behaviour as u l 1 and ω l 2 , except that, for large enough d * , ω q 2 admits a minimum value when λ is small enough and K * large (see the curve for K * = 10 −2 and λ = 0.1/3).…”

“…Note that, as expected, for K * = 10 −2 the coefficient f 33 decreases as the slip length λ increases, since the shear rate of the flow taking place in the slab-sphere gap is smaller when λ is bigger and thus the lubrication force is then smaller. Using the bipolar coordinates method, Michalopoulou et al (1992) considered the axisymmetric translation of a solid sphere towards another motionless and porous Davis, Kezirian & Brenner (1994) using the bipolar coordinates method ( for λ = 0.1, and ♦ for λ = 0.01) and by Luo & Pozrikidis (2008) using the boundary integral equations technique ( for λ = 0.1), and the results obtained for a solid wall and a no-slip particle by Luo & Pozrikidis (2008) (⋆ for λ = 0) and by Chaoui & Feuillebois (2003) …”

“…This means that it is easier to move (translate or rotate) the sphere parallel with the slab than normal to it. For the weak permeability K * = 10 −4 , the porous slab behaves as an impermeable boundary of solid nature if λ 0.01 (see the comparisons with Chaoui & Feuillebois (2003) and Luo & Pozrikidis (2008) for λ = 0) or of slipping nature for λ large enough (see the comparisons with Davis et al (1994) and Luo & Pozrikidis (2008) for λ = 0.1). The coefficients f 12 and c 21 , displayed in figure 5, exhibit the same behaviours versus (d * , K * , λ) as the coefficients f 11 and c 22 but are of weaker magnitudes.…”

Slip at the surface of an oscillating spheroidal particle in a micropolar fluid

“…Basset [12] derived expressions E4 for the force exerted by the surrounding fluid on a translating rigid sphere with a slip boundary condition at its surface (for example a settling aerosol sphere). The hydrodynamic effects of homogeneous and inhomogeneous slip boundary conditions for Newtonian fluids have been discussed extensively in the literature [13,14,15,16]. Usually, slip exists to a degree between the fluid and the surface of the solid.…”

Boundary integral method for Stokes flow with linear slip flow conditions in curved surfaces

“…Luo and Pozrikidis [19] study the motion of spherical particles in infinite fluid and near a plane wall subjected to slip boundary conditions. The boundary integral formulation presented in this work takes advantage of the axial symmetry of the boundaries with respect to the axis that is normal to the wall and passes through the particle center, reducing the solution to a system of one-dimensional integral equations.…”