Tensor mobility

The form of the effective mobility tensor remains unchanged as in Eq. (125), which imphes that the fluid flow does not affect the mobility terms. This is reasonable for an uncharged medium, where there is no interaction between the electric field and the convective flow field. However, the hydrodynamic term, Eq. (128), is affected by the electric field, since electroconvective flux at the boundary between the two phases causes solute to transport from one phase to the other, which can change the mean effective velocity through the system. One can also note that even if no electric field is applied, the mean velocity is affected by the diffusive transport into the stationary phase. Paine et al. [285] developed expressions to show that reversible adsorption and heterogeneous reaction affected the effective dispersion terms for flow in a capillary tube the present problem shows how partitioning, driven both by electrophoresis and diffusion, into the second phase will affect the overall dispersion and mean velocity terms. 

Hydrodynamic and frictional effects may be described by a Cartesian mobility tensor which is generally a function of all of the system coordinates. In models of systems of beads (i.e., localized centers of hydrodynamic resistance) with hydrodynamic interactions, is normally taken to be of the form... 

To obtain a more compact expession for the Cartesian drift velocity, it is useful to generalize the underlying diffusion equation in the /-dimensional constraint surface to a diffusion equation in the unconstrained 3N dimensional space. To define a mobility tensor throughout the unconstrained space, we adopt Eq. (2.133) as the definition of the constrained Cartesian mobility everywhere. To allow Eqs. (2.133) and (2.134) to be evaluated away from the constraint surface, we must also define n = 0c /0R everywhere, and specify definitions of the... 

Other definitions may be constructed by the following generalization of the relationship between the dynamical reciprocal vectors and the mobility tensor Given any invertible symmetric covariant Cartesian tensor S v with an inverse we may take... 

In the case of a singular mobility tensor, we will replace condition (266) by the weaker condition... 

The discrete Markov process used to define a kinetic SDE in Eq. (2.315) or (2.318) can be directly implemented as a numerical algorithm for the integration of a set of SDEs. The resulting simulation algorithm would require the evaluation of neither derivatives of the mobility nor any corrective pseudoforce. It would, however, require an efficient method of calculating the elements of the mobility tensor and derivatives of U and in in the chosen system of generalized coordinates. 

Here, is the mobility tensor in the chosen system of coordinates, which is a constrained mobility for a constrained system and an unconstrained mobility for an unconstrained system. As discussed in Section VII, in the case of a constrained system, Eq. (2.344) may be applied either to the drift velocities for the / soft coordinates, for which is a nonsingular / x / matrix, or to the drift velocities for a set of 3N unconstrained generalized or Cartesian coordinates, for a probability distribution (X) that is dynamically constrained to the constraint surface, for which is a singular 3N x 3N matrix. The equilibrium distribution is. (X) oc for unconstrained systems and... 

In models of beads with full hydrodynamic interactions, for which the mobility tensor is represented by a dense matrix, the Cholesky decomposition of H requires 3N) /6 operations. Eor large N, this appears to be the most expensive operation in the entire algorithm. The only other unavoidable 0 N ) operation is the LU decomposition of the K x K matrix W that is required to solve for the K constraint forces, which requires /3 operations, or roughly... 

As in Section H of this appendix, we restrict ourselves in this section to the case of a nonsingular mobility tensor. By combining Eq. (2.323) for the drift velocites in the original coordinate system with the Ito transformation rule for drift velocities, we find that the variables X, ..., X experience drift velocities... 

Hint Rotational Brownian diffusivity is the manifestation of random walks of the orientation of the rod. By analogy with translational diffusion, the rotational diffusivity D,- = kTMaG, where M eis the mobility tensor relating angular velocity and torque. 

Fluid velocity, y component, normal to channel walls or plate surface Mobility of particle in solution Specific volume Mobility tensor, Eq. (5.1.3b)... 

For an approximate estimate of the mobilities, either a constant, isotropic relaxation time r or a constant, isotropic mean free path X can be assumed. In these two approximations, the components pty of the mobility tensor are found from the Einstein relation (Eq. (8.22)) to be... 

The tensor Rij is called the translational tensor, or the resistance tensor. Its components depend on particle s size and shape and have the dimensionality of length. They can be interpreted as equivalent radii of the body. The tensor is called the friction tensor, and the values Vy are known as mobilities. They are similar to mobilities introduced in Section 4.5. Therefore the tensor with components Vij is called the mobility tensor. 

In the Rouse model, the hydrodynamic interaction and the excluded volume interaction among beads are disregarded and the mobility tensor and the interaction potential are given by... 

In equation (80), r = r is the distance between the particle centres and 1 is the unit tensor. The mobility tensor by contains four independent coefficients. All, Ai2, Bn, and B12, each of which depends on r and a2/ai. The behaviour of these coefficients has been discussed by Batchelor in relation to the estimation of first-order corrections to the sedimentation velocity and diffusion coefficient. The mean settling velocity is... 

The mobility tensor by refers to an isolated pair of particles. According to Batchelor, the determination of the hydrodynamic functions occurring in the expressions for the mobility of one particle in the presence of two other particles would be a formidable task . Since there is little evidence that hydrodynamic forces are to any extent additive, a rigorous treatment of the dynamics of concentrated dispersions is not yet in sight. 

Here is the mobility tensor satisfying f = X- Again, this diffusion coefficient contains two contributions, one depending on the temperature T of the flmd and another one independent from it and proportional to the velocity gradient of the imposed external flow. Therefore, the first contribution is related to the usual thermal Brownian motion whereas the last one is related to non-thermal effects. Notice also that the Smoluchowski equation (26) also contains the usual convective term V (pvo). 

Figure 2.10 Sketch of a tensorial hydrodynamic slip. The normal traction fn exerted by the fluid on an anisotropic surface produces an effective slip velocity Au = Mf in a different direction. At the molecular level, the Interfacial mobility tensor M is related to the trajectories of diffusing particles In the interfacial region, such as the one shown (reprinted with permission from Ref 28, copyright 2008, Cambridge University Press].

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