Hinch

E.J. Hinch and A. Acrivos Long Slender Drops in a Simple Shear Flow. J. Fluid. Mech. 98, 305 (1980). 

Because appears contracted with in the equation of motion, the hard components of have no dynamical effect, and are arbitrary. The values of the soft components of F depend on the form chosen for the generalized projection tensor, and reduce to the metric pseudoforce found by Fixman and Hinch in the case of geometric projection. 

The present analysis builds directly on three previous analyses of SDEs for constrained systems by Fixman [9], Hinch [10], and Ottinger [11]. Fixman and Hinch both considered an interpretation of the inertialess Langevin equation as a limit of an ordinary differential equation with a finite, continuous random force. Both authors found that, to obtain the correct drift velocity and equilibrium distribution, it was necessary to supplement forces arising from derivatives of C/eff = U — kT n by an additional corrective pseudoforce, but obtained inconsistent results for the form of the required correction force. Ottinger [11] based his analysis on an Ito interpretation of SDEs for both generalized and Cartesian coordinates, and thereby obtained results that... 

The results given above are essentially identical to those obtained by Hinch [10] by a similar method, except for the fact that Hinch did not retain any of the terms involving the force bias (tIv)o which he presumably assumed to vanish. An apparent contradiction in Hinch s results may be resolved by correcting his neglect of this bias. In a traditional interpretation of the Langevin equation as a limit of an underlying ODE, the bead velocities are rigorously independent of the hard components of the random forces, since the random forces in Eq. (2.291) appear contracted with K , which has nonzero components only in the soft subspace. Physically, the hard components of the random forces are instantaneously canceled by the constraint forces, and thus can have no effect... 

The first analysis of the constrained Langevin equation was given by Fixman [9], who worked primarily in generalized coordinates. In order to resolve a discrepancy between Fixman s results and those of Hinch [10], we now follow Fixman by considering a Langevin equation for the soft coordinates, ...,... 

This equation agrees with Fixman s Eq. (3.26), except for the fact that Fixman, like Hinch, implicitly assumed that ( )Q = 0 for the Cartesian random forces, and so did not retain the first term on the RHS of Eq. (2.306). Fixman correctly emphasized, however, that the transformation from Cartesian to generalized... 

Geometrically projected random forces, which were introduced by Hinch [10], have a variance given by the geometrically projected friction tensor... 

In either interpretation of the Langevin equation, the form of the required pseudoforce depends on the values of the mixed components of Zpy, and thus on the statistical properties of the hard components of the random forces. The definition of a pseudoforce given here is a generalization of the metric force found by both Fixman [9] and Hinch [10]. An apparent discrepancy between the results of Fixman, who considered the case of unprojected random forces, and those of Hinch, who was able to reproducd Fixman s expression for the pseudoforce only in the case of projected random forces, is traced here to an error in Fixman s use of differential geometry. 

In the traditional interpretation of the Fangevin equation for a constrained system, the overall drift velocity is insensitive to the presence or absence of hard components of the random forces, since these components are instantaneously canceled in the underlying ODF by constraint forces. This insensitivity to the presence of hard forces is obtained, however, only if both the projected divergence of the mobility and the force bias are retained in the expression for the drift velocity. The drift velocity for a kinetic interpretation of a constrained Langevin equation does not contain a force bias, and does depend on statistical properties of the hard random force components. Both Fixman and Hinch nominally considered the traditional interpretation of the Langevin equation for the Cartesian bead coordinates as a limit of an ordinary differential equation. Both authors, however, neglected the possible existence of a bias in the Cartesian random forces. As a result, both obtained a drift velocity that (after correcting the error in Fixman s expression for the pseudoforce) is actually the appropriate expression for a kinetic interpretation. 

The stochastic stress o in Grassia and Hinch s algorithm is given by a discrete approximation to the time average of Eq. (2.378) over one timestep, as... 

Equation (A.99) may, however, also be obtained by explicitly using projected random forces, as Hinch did in order to reproduce Frxman s result. The use of the projected friction tensor for Z v in Eq. (2.306), rather than the unprojected tensor has the same effect in that equation as did Fixman s neglect of the hard components of because the RHS of Eq. (2.306) depends only on the dot... 

T.A-Erikson R.J.Hinch, Jr, "Investigation of Crystallographic Properties of Primary Explosives , Armour Res Foundation, IIT Quarterly Rept No 1 (1958), [Ord Proj TB3-0115A, Contract DA-11-022-501-ORD-2731] 36)W.C.McCrone Assoc Inc, "The Crystallography of Explosives , Final Rept (1962) (Contract DA-11-022-ORD-4090) 37)R.C.Evans,... 

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