8.3 Unbounded Stokes flows [Stokes flow top]

Unbounded flows (i.e., flows of fluid around immersed solid objects) are also characterized by the Reynolds number, but in this case the characteristic velocity U and length scale ℓ now come from the velocity and size of the object. Unlike bounded flows, where numerical approaches are the norm, unbounded flows are commonly treated analytically, and solution of model problems leads to results that are of immense value in common systems. For example, a microfluidic device may be used to process blood or a cellular suspension. In this case, calculating all details of the flow would be difficult and largely unnecessary, since we can encapsulate the effects of particles in a simple way that can be described analytically. The analytical solutions for flow over a sphere or for the flow response to a point force explain in great part the dynamics (e.g., diffusivity) of macromolecules and particles. Given a flow over a body, the viscous stress, integrated over the body surface, is a measure of the drag force on the object as it moves in the fluid. This drag force is used to calculate particle settling times, electrophoretic and dielectrophoretic particle velocities, and the accuracy of particle image velocimetry measurements in microsystems. We start with discussion of Stokes flow over a sphere in an infinite domain, which can be solved directly.

8.3.1 Stokes flow over a sphere in an infinite domain

Consider axisymmetric flow at velocity U over a sphereof radius a at low Re (Figure 8.2). Here, the relevant Reynolds number is customarily defined as Re = ρUd∕η, where d = 2a is the sphere diameter. The governing equations are the Stokes equations, and the boundary conditions are that the velocity is zero at r = a and the velocity is equal to U as r →∞.

The solution for this flow can be obtained in a number of ways, the simplest of which is to assume that the result can be written in terms of a power series in a∕r. The solution (Exercise 8.2) is

and

The variation of the pressure from the freestream value (Δp) is given by

The viscous forces on the sphere can be integrated to get the total drag force on the sphere:

Tese steady results are applicable for time-varying U owing to the instantaneity of the Stokes equations. For a particle of finite size and Reynolds number, we can confirm this approximation by evaluating the characteristic time for Stokes particles to equilibrate with their fluid surroundings as compared to the experimental time scales. Thus we can assume that, for time scales long as compared to the particle lag timeτ_p = , the system can be assumed in quasi-steady state, i.e., the magnitude of the unsteady term in the equation is small, even when the flow itself is unsteady.

The Stokes flow solution for flow over a sphere has three terms, which relate to the multipolar solutions discussed in Appendix F. The constant term refers to the uniform freestream velocity–this is the flow that would be observed if the particle were absent. The termproportional to a∕r is the Stokeslet term—it corresponds to the response of the flow caused by a point force of 6πηUa applied to the fluid at the center of the sphere. This component of the flowfield is shown in Figure 8.3. The Stokeslet term describes the viscous response of the fluid to the no-slip condition at the particle surface, and this term contains all of the vorticity caused by the viscous action of the particle. The termproportional to a³∕r³ is the stresslet term—it is identical to Equation F.6 as described in the appendix. This term contributes an irrotational flow—it is unrelated to the viscous force of the sphere, and is caused by the finite size of the particle. This component of the flowfield is shown in Figure 8.4. Since the stresslet term decays proportional to r^-3 while the Stokeslet term decays proportional to r^-1, the primary long-range effect of the particle is induced by the Stokeslet. Thus, the net force on the fluid induced by the sphere is required to prescribe the flow far from a sphere, rather than the particle size or velocity alone. Far from a sphere moving in a Stokes flow, the flow does not distinguish between the effects of one particle that has velocity U and radius 2a and another that has velocity 2U and radius a, since these two spheres have the same drag force. Close to these spheres, of course, the two flows are different, as distinguished by the different stresslet terms.

It is customary to define a dragcoefficient, which normalizes the drag in Equation 8.15 by the dynamic pressure of the freestream fluid (in a coordinate reference frame where the particle is motionless) multiplied by the cross-sectional area of the sphere:

(8.26)

where A_p = πa² is the cross-sectional area of the sphere, and Re_d = is the Reynolds number based on the particle diameter.

To predict particle dynamics in microsystems, we require the drag force reported in Equation 8.15 but not other details of the flow. Since microparticles achieve equilibrium quickly, a force applied to a spherical microparticle induces particle motion at the velocity such that the drag force and the motive force are equal and opposite. Thus the steady-state velocity of a microparticle with radius a in Stokes flow with an applied force  is given by

(8.27)

As is the case for all steady-state flows (cf. steady Poiseuille flow), Equation 8.27 relates force to velocity.

Given a system with small but finite Reynolds number, the instantaneity of the particle response can be quantified by calculating theStokes number Sk, which is the ratio of the particle lag time to the characteristic time over which the flow changes. The characteristic flow time can come from the characteristic time of an unsteady boundary condition, or from the ratio U∕ℓ of the characteristic velocity and length scale from a steady boundary condition in a nonuniform flow. Choosing the latter, we have

(8.28)

Particles with Stokes number Sk ≪ 1 can be assumed to be always in steady-state with a local velocity field given by the idealized solution derived earlier.

8.3.2 General solution for Stokes flow over a sphere in an infinite domain

The above solution applies for a stationary sphere with a uniform flow of velocity U. In microsystems, it is typically more appropriate to consider the fluid as being nominally quiescent and the particle to be in motion. Further, particle motion is rarely aligned with the coordinate system in use, or the presence of multiple particles may make it impossible to define a coordinate system in line with the motion of all particles. To this end, we benefit from writing a general solution. We write this with a hydrodynamic interaction tensor which, when dotted against the force applied to the particle, gives the velocity field:

(8.29)

As we have shown earlier, the force  applied to a particle induces a particle velocity of ∕6πηa. The same force induces a fluid velocity field of ⋅. For a sphere, the hydrodynamic interaction tensor  is given by

(8.30)

In this equation,  denotes the distance vector from the sphere center to the fluid location and Δr is the magnitude of that distance.  is the identity tensor. The  terms lead to dyads and thus  is a dyadic tensor. The general solution for the velocity can be found by evaluating the dot product, and noting that ⋅ = (⋅):

(8.31)

The first term in this equation is the Stokeslet term, while the second term is the stresslet term (see the appendix, Section F.2 for details). The stresslet term becomes unimportant as Δr becomes large, and this hydrodynamic tensor is often simplified to the Oseen-Burgers tensor:

(8.32)

By neglecting the stresslet term, the Oseen-Burgers tensor no longer describes the flow caused by a sphere, but rather the flow caused by a point force appliedat Δr = 0. At large Δr∕a, the two tensors give the same result.

Now we consider the same rigid spherical particle of radius a moving because of a force , and we consider the pressure change caused by the particle motion and attendant flow. This pressure change Δp can be written as

(8.33)

Where  is the (first-order) pressure interaction tensor, given by

(8.34)

The resulting pressure change owing to a particle of radius a moving with velocity is

(8.35)

In the following sections, we report how the drag, the drag coefficient, or the particle velocity changes when the shape of the particle changes or it is in proximity to other objects.

8.3.3 Flow over prolate ellipsoids

For ellipsoidal particles, the flow is a function of a particle’s orientation and axis lengths, and thus the drag force is a function of these parameters as well. The force dependence is expressed through changes in both A_p, the cross-sectional area perpendicular to the flow, and C_D, the drag coefficient based on an effective particle diameter. For an ellipsoid with axes a₁, a₂, and a₃ (a₁ > a₂ > a₃), the effective particle diameter is the diameter of a sphere of equivalent volume (i.e., 2).

For the special case of a prolate ellipsoid (a₁ > a₂ = a₃) — useful because of its similarity to rod-shaped particles, for flows along the long axis induce a drag given by

(8.40)

wheretheeccentricity, is given by e = .

8.3.4 Stokes flow over particles in finite domains

As a moving particle approaches a wall, the fluid velocity field resulting from the moving particle is retarded because of the no-slip boundary condition at the wall. For a given force, the presence of the wall therefore reduces the particle velocity as it approaches the wall. For a particle of radius a located a distance d from the wall, the force-velocity relation normal to the wall can be approximated by

(8.41)

and the force-velocity relation tangent to the wall can be approximated by

(8.42)

In both cases, the effects of the wall are small at d = 10a.

8.3.5 Stokes flow over multiple particles

At low interparticle spacing distances, caused by high particle concentrations or by particle localization due to hydrodynamic and/or electrokinetic forces, the drag force on the particles and therefore the velocity of particles is impacted by particle-particle interactions. As was the case with flow near walls, the isolated sphere relation is accurate as long as particle-particle separations exceed ten times the particle diameter. The hydrodynamic interaction tensor can be used to evaluate the forces of particles on each other and thus predict the forces and particle velocities.

Some examples of our research where Stokes flow analysis is relevant include our circulating tumor cell capture microchips and our dielectrophoretic manipulation of microparticles.

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