Evgeny S. Asmolov

The inertial lift on a spherical particle in a plane Poiseuille flow at large channel Reynolds number

The inertial migration of a small rigid sphere translating parallel to the walls within a channel flow at large channel Reynolds numbers is investigated. The method of matched asymptotic expansions is used to solve the equations governing the disturbance flow past a particle at small particle Reynolds number and to evaluate the lift. Both neutrally and non-neutrally buoyant particles are considered. The wall-induced inertia is significant in the thin layers near the walls where the lift is close to that calculated for linear shear flow, bounded by a single wall. In the major portion of the flow, excluding near-wall layers, the wall effect can be neglected, and the outer flow past a sphere can be treated as unbounded parabolic shear flow. The effect of the curvature of the unperturbed velocity profile is significant, and the lift differs from the values corresponding to a linear shear flow even at large Reynolds numbers.

STABILITY OF A DUSTY-GAS LAMINAR BOUNDARY LAYER ON A FLAT PLATE

The linear stability of incompressible boundary-layer flow of dusty gas on a semiinnite flat plate is considered. The particles are assumed to be under the action of the Stokes drag only. The problem is reduced to the solution of the modied Orr{Sommerfeld equation (Saman 1962). This is solved numerically using two approaches: directly by orthonormalization method, and by perturbation method at small particle mass content. The stability characteristics are calculated for both monoand polydisperse particles. The dust suppresses the instability waves for a wide range of the particle size. The most ecient suppression takes place when the relaxation length of the particle velocity is close to the wavelength of the Tollmien{Schlicting (TS) wave. The reduction in growth rate per unit dust content is approximately ten times greater than the characteristic value of the growth rate for a clean gas. For monosized dust the complex frequency of the TS wave changes in a discontinuous way. As a result a domain in the space of independent parameters arises where two discrete TS modes exist and a domain where no TS mode may exist. For polydisperse dust with a discrete distribution in particle size the number of breaks in the dependence equals the number of particle sizes. For the continuous distribution in particle size the dependence of the complex-frequency on Reynolds number and wavenumber is continuous. The eigenfunction becomes a non-smooth function of the normal coordinate in this case. Some comments are made about the role of the lift force acting on the particles for the problem in question.

The inertial lift on a small particle in a weak-shear parabolic flow

The lateral migration of a small spherical particle translating within a vertical channel flow with large channel Reynolds numbers Rc is investigated. The weak-shear case is studied when the ratio of the slip velocity to maximum velocity of the channel flow, Vs, is finite, while two other dimensionless groups, particle Reynolds number Rs and Rc−1, are asymptotically small. The disturbance flow at large distances from the sphere is governed by Oseen-like equations. The ratio of Oseen length to channel width is e=ls/l=(Rc|Vs|)−1≪1, i.e., the Oseen region is only a small part of the channel while the major portion of the disturbance flow is inviscid. A solution of the governing equations is constructed in terms of two-dimensional Fourier transform of the disturbance field in a plane parallel to the channel walls. The ordinary differential equation for Fourier transform of lateral velocity Γz(k,z) is solved using the method of matched asymptotic expansions based on e. Several domains in (k,z) space are distin...

Effective slip boundary conditions for arbitrary one­ dimensional surfaces

In many applications it is advantageous to construct effective slip boundary conditions, which could fully characterize flow over patterned surfaces. Here we focus on laminar shear flows over smooth anisotropic surfaces with arbitrary scalar slip , varying in only one direction. We derive general expressions for eigenvalues of the effective slip-length tensor, and show that the transverse component is equal to half of the longitudinal one, with a two times larger local slip, . A remarkable corollary of this relation is that the flow along any direction of the one-dimensional surface can be easily determined, once the longitudinal component of the effective slip tensor is found from the known spatially non-uniform scalar slip.

THE INERTIAL LIFT ON AN OSCILLATING SPHERE IN A LINEAR SHEAR FLOW

Abstract In a shear flow, a small sphere may experience a lift force due to fluid inertia. Most previous workers assumed that the particle was stationary so that they could treat the fluid motion as steady. In spite of this, the results of previous analyses have generally been applied to problems in which particles move in an unsteady fashion. This paper presents the results of singular perturbation calculations of the lift on a sphere in a linear shear flow. The velocity of the sphere oscillates sinusoidally in time. Although the problem is idealized, the results provide some physical understanding of the effects of unsteadiness and the frequency regime in which one may assume quasisteady conditions.

Dynamics of a spherical particle in a laminar boundary layer

The problem of the motion of an individual spherical particle in a laminar boundary layer is considered for small Reynolds numbers determined from the relative velocity and the transverse velocity gradient of the flow undisturbed by the particle. The dependence of the transverse force acting on the particle, which results from the nonuniformity of the free stream, on the distance of the particle from the surface of a flat plate is calculated. It is shown that the direction of the transverse force changes with the distance of the particle from the plate: near the surface the force is positive, i.e., directed away from the plate, and at greater distances negative.

Effective slip-length tensor for a flow over weakly slipping stripes.

We discuss the flow past a flat heterogeneous solid surface decorated by slipping stripes. The spatially varying slip length, b(y), is assumed to be small compared to the scale of the heterogeneities, L, but finite. For such weakly slipping surfaces, earlier analyses have predicted that the effective slip length is simply given by the surface-averaged slip length, which implies that the effective slip-length tensor becomes isotropic. Here we show that a different scenario is expected if the local slip length has steplike jumps at the edges of slipping heterogeneities. In this case, the next-to-leading term in an expansion of the effective slip-length tensor in powers of max[b(y)/L] becomes comparable to the leading-order term, but anisotropic, even at very small b(y)/L. This leads to an anisotropy of the effective slip and to its significant reduction compared to the surface-averaged value. The asymptotic formulas are tested by numerical solutions and are in agreement with results of dissipative particle dynamics simulations.

Effective hydrodynamic boundary conditions for microtextured surfaces

Understanding the influence of topographic heterogeneities on liquid flows has become an important issue with the development of microfluidic systems, and more generally for the manipulation of liquids at the small scale. Most studies of the boundary flow past such surfaces have concerned poorly wetting liquids for which the topography acts to generate superhydrophobic slip. Here we focus on topographically patterned but chemically homogeneous surfaces, and measure a drag force on a sphere approaching a plane decorated with lyophilic microscopic grooves. A significant decrease in the force compared with predicted even for a superhydrophobic surface is observed. To quantify the force we use the effective no-slip boundary condition, which is applied at the imaginary smooth homogeneous isotropic surface located at an intermediate position between the top and bottom of grooves. We relate its location to a surface topology by a simple, but accurate analytical formula. Since grooves represent the most anisotropic surface, our conclusions are valid for any texture, and suggest rules for the rational design of topographically patterned surfaces to generate desired drag.

ASYMPTOTIC MODEL OF THE INERTIAL MIGRATION OF PARTICLES IN A DILUTE SUSPENSION FLOW THROUGH THE ENTRY REGION OF A CHANNEL

The inertial migration of particles in a dilute suspension flow through the entry region of a plane channel (or a circular pipe) is considered. Within the two-fluid approach, an asymptotic one-way coupling model of the dilute suspension flow in the entry region of a channel is constructed. The carrier phase is a viscous incompressible Newtonian fluid, and the dispersed phase consists of identical noncolloidal rigid spheres. In the interphase momentum exchange, we take into account the drag force, the virtual mass force, the Archimedes force, and the inertial lift force with a correction factor due to the wall effect and an arbitrary particle slip velocity. The channel Reynolds number is high and the particle-to-fluid density ratio is of order unity or significantly larger unity. The solution is constructed using the matched asymptotic expansion method. The problem of finding the far-downstream cross-channel profile of particle number concentration is reduced to solving the equations of the two-phase bound...

Flows and mixing in channels with misaligned superhydrophobic walls

Aligned superhydrophobic surfaces with the same texture orientation reduce drag in the channel and generate secondary flows transverse to the direction of the applied pressure gradient. Here we show that a transverse shear can be easily generated by using superhydrophobic channels with misaligned textured surfaces. We propose a general theoretical approach to quantify this transverse flow by introducing the concept of an effective shear tensor. To illustrate its use, we present approximate theoretical solutions and Dissipative Particle Dynamics simulations for striped superhydrophobic channels. Our results demonstrate that the transverse shear leads to complex flow patterns, which provide a new mechanism of a passive vertical mixing at the scale of a texture period. Depending on the value of Reynolds number two different scenarios occur. At relatively low Reynolds number the flow represents a transverse shear superimposed with two corotating vortices. For larger Reynolds number these vortices become isolated, by suppressing fluid transport in the transverse direction.