The surface of the reference sphere 'repels' nearby trajectories that spiral into a new stable limit cycle in the shear plane. The topology of the off-plane trajectories is more complicated because the gradient displacement changes sign away from the shear plane.
Introduction
Introduction
Using the method developed in Section 2.2 , a general form of the multiscale hierarchy for a simple shear flow is derived in Section 2.3.2 , and inertial corrections to the Smoluchowski equation are obtained. In Section 2.4 we compare the exact and multiscale solutions for the phase space probability densities for the initial conditions mentioned above and further consider the multiscale form of the hierarchy in the limit where thermal effects are negligible.
Problem formulation
When applied to equation (2.2), the multiple scales method enables the simultaneous evolution of the probability density on a hierarchy of time scales depending on St, the evolution rate on the different time scales being asymptotically separated for small St. having characterized the shape of the solution in velocity space, one can now apply the multiple scales formalism to determine the dependence of the φs on the longer time scales (denoted by tslow above).
Brownian motion in simple shear flow
However, a comparison with the exact solution can be made and serves to verify the applicability of the multiple scaling procedure1. The exact solution of equation (2.11) (where the shear flow constitutes a linear force) is a multivariate Gaussian in phase space (Risken 1989, Miguel & Sancho 1979) with the variance matrix elements being functions of time.
Comparison of exact and multiple scales solutions
We also consider the relaxation of the Brownian particle from a specified initial state in the thermal limit (P e→. It is seen that the multiple power series for this initial state is of the form
Conclusions and discussion
The success of the multi-scale method clearly implies the existence of an equivalent Chapman-Enskog approach to the same problem (see next chapter). 1978 A systematic solution procedure for the Fokker-Planck equation of a Brownian particle in the case of high friction.
Introduction
A Chapman-Enskog-like procedure can be performed for the Fokker-Planck equation, and the resulting expansion then describes the evolution of the probability density of the phase space (for the Brownian particle) on timescales much larger than the inertial relaxation time. (τp); this is done in Chapter 4. However, for the Fokker-Planck equation with a hydrodynamic drag that is linear in velocity, one can make further progress since the equation is linear and the time-dependent equation of leading order reduces to an eigenvalue problem which easily solved. This implies that, unlike the Boltzmann equation, for the Fokker-Planck equation one can formulate a Chapman-Enskog method that takes into account variations in the probability density on the shortest timescales of O(τp)1.
In the following, we outline a similar formulation for the non-equilibrium case when the forcing is due to a simple shear flow; the resulting expansion for the phase space probability density and the corrected Smoluchowski equation obtained are identical to that derived in the previous chapter using the multiple scale formalism. However, the Chapman–Enskog formulation is more general and remains valid even in cases where the drift and diffusion coefficients in the Fokker–Planck equation are configuration dependent.
Equivalence of the Chapman-Enskog and multiple scales formalisms for a single Brownian particle in shear flow
Since each of the Pm,nred's in (3.5) is proportional to −(m+n)t, the action of the time derivative opPm,nred is equivalent to multiplying by the factor −(m+n), that is, for the reduced problem∂/∂t=−(m+n) when acting on Pm,nred. A similar argument suggests the form (3.4) for the action of the time derivative, where the higher order corrections in this case are the operators ∂(i)m,n's fori≥0. The subscriptsm andn indicate that the action of the operator∂m,n(i) (on the Pm,n's) will in general be different for each mand n.
2This requirement is met in a natural way in the multiple scale formalism, since the terms in the expansion of the time derivative are of the form∂/∂ti, where it is still treated as a time-like variable. By a simple rearrangement one finds that the sum on the right-hand side of (3.7) contains terms of the form.
Chapman-Enskog method for a configuration dependent drag force in one dimensiondrag force in one dimension
However, as will be seen below, the requirement of an explicit exponential form for the fast scales (the t1 scale in Chapter 2) in the multiscale formalism limits its applicability to precisely these cases. The Chapman-Enskog expansion developed in the previous section is still valid for this problem since it does not assume any special functional form for fast time scales. The involved relationship between cn and c0 in this case can now be compared to (3.12) and the form (2.9) assumed in the multiscale formalism.
The latter corresponds to zero flow in the steady state and is still given by the solution of the leading-order Smoluchowski equation multiplied by Maxwell's velocity distribution. In the non-equilibrium cases above, the inertial corrections found by the Chapman-Enskog expansion can be important in determining the modified steady-state distributions.
Chapman-Enskog method for inertial suspensions
However, we will restrict our attention to the first effects of particle inertia, i.e. the O(St) inertial modification, while commenting on the general form and the physical relevance of the higher order terms. The Peclet number defined above represents the ratio of the configuration and flow time scales for an isolated Brownian particle. In this section we therefore derive the form of the O(St) correction to the Smoluchowski equation for a constant m.
Here we have explicitly shown the action of the symmetrizing operator for the case of two indices1 andi2. The latter is expected to affect only the forms of the hi(x) factors in the L(k)x operators.
Conclusions
Introduction
The fluid inertia is again negligible and the particle statistics are determined by the Boltzmann equation which takes into account momentum transfer via collisions with solids. In this chapter we consider simple shear flow of dilute suspensions in the limit Re = 0, St 1, and in the absence of Brownian motion (P e. The limit Re = St = 0, in the absence of non-hydrodynamic forces, generates a symmetric microstructure with a Newtonian rheology (Batchor & Green 1972b); we investigate in depth the deviation from this limit for small but finite particle inertia.
In the analysis below we consider only pair-particle interactions, and the results will therefore only be quantitatively accurate for dilute suspensions (φ → 0). A central result of our analysis is that the rheology of a finite St suspension for pairwise interactions in the absence of nonhydrodynamic forces is indeterminate.
Equations for particle trajectories
In this restriction, the structure of the equation governing the pair probability P2 is the same as that for PN derived in Chapter 3 (see Section 4.2). We also show that the asymmetry of finite hydrodynamic interactions St leads to finite (anisotropic) shear-induced self-diffusivity. However, the relative simplicity of the evolving slow scales may have been obscured by the generality of the formalism.
We now investigate the non-Brunian limit of the two-particle Smoluchowski equation for the case of spherical particles in a linear flow. For a statistically homogeneous suspension of spherical particles, only the relative positions of the centers of mass are relevant.
Nature of inertial velocity corrections for small Stokes numbersnumbers
Although not readily apparent, the radial components of the inertial corrections at all higher orders also exhibit the same near-field behavior (see below). We now consider a simplified form of the equation for the relative motion of a pair of particles in a linear flow field to investigate the near-field behavior of the relative velocity for arbitrary St, thereby verifying the aforementioned near-field form of the inertial corrections. Taking the radial component of the above equation (which eliminates the rotating part) and using the expressions for the resistance tensors (Kim & . Karrila 1991), we get.
The solution of the above equation is hindered by the fact that d/dt(Vini) 6= (dVi/dt)ni; the curvature of the particle path causes inertial forces proportional to dni/dt. At smaller spacings, there is a rapid transition from a steep logarithmic decline to a gradual linear variation corresponding to a rapidly decreasing magnitude of the acceleration term.
Relative in-plane trajectories of two spheres in simple shear flowshear flow
Trajectories starting from points asymptotically close to the surface of the reference sphere spiral outward. Finally, in section 4.4.6 we consider the location of the stable limit cycle and its attraction domain in the shear plane. This in turn would imply that the radial coordinates of the two zero-Stokes trajectories differ O(St2).
The thick black arrows indicate the sign of the perturbation vortex in the corresponding regions. Thus, the radial distance of the current trajectory at φ=π/2 for the ‘+’ branch is given by. Also, in the above and subsequent expressions, a neglected term of the general form O(1/rn−scs) is denoted by O(1/rn), since both will contribute terms of O(1 /cn−1) in the final expression for (∆z) in the plane.
In fact, this conclusion can be reached from a comparison of the O(1) and O(St) terms of the governing equation (4.25).
Introduction
It will be seen in section 5.3 that a perturbation scheme for the off-plane trajectories similar in structure to that formulated in Chapter 4 for in-plane trajectories explains the general features of the finite Sttrajek space. Before proceeding with the detailed analysis, however, we discuss the physical mechanisms leading to the behavior shown in Fig. 5.2 and 5.4 are observed. Because of the antisymmetry of simple shear flow and the symmetry about the shear plane, it is sufficient to investigate only a quadrant of the entire trajectory space.
Relative off-plane trajectories
For small values of x−∞ the out-of-plane trajectories still resemble those in the shear plane in that they pass very close to the surface of the reference sphere for small z−∞. A subset of finite Trajectories emerging from the spiral will approach the limit cycle in the shear plane from 'inside'. The long-time behavior of these trajectories asymptotes with that of in-plane trajectories that spiral toward the limit cycle (see Fig. 4.18).
3 To be precise, the neutral plane must correspond to the downstream coordinate out of the plane of the neutral trajectory ie. The above variation of the zero-Stokes trajectory space is also consistent with the dynamical systems perspective presented in Section 4.4.6.
Perturbation analysis for off-plane trajectories
We continue to use the same symbols as before for the r and φ components of the inertial correction. Considering the rhs of the trajectory equations, we note that at φ = π/2 the O(1) terms in the denominator of (5.1) and in both the numerator and denominator of (5.2) are zero. This lack of validity, however, does not affect the values of the leading-order diffusivities (see Section 5.3.4).
The solution ˜φ, similar to the in-plane case, has two unequal branches indicating the asymmetry atO(St). In tables 5.1 and 5.2 we tabulate values of the vortex displacement for open trajectories for two Stokes numbers (St = 0.1 and 0.01).
