Interdisciplinary Applied Mathematics

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The total    force dipole    strength    is the    sum    of    the    contributions    due    to the


torque and due to stresslet, and is of the form


T?n _ T?n{s) I 1 , rpn


+2eb’fcJfc>


where ТЩ is the torque acting on the fluid due to either an external torque on the particle    or    the    effects    of moment    of    intertia    of the    particle.    Often


this term is zero or negligible. Also, F^A is the stresslet and is adjusted so that the condition


En = Eij0(x — Y,ap)d3x


is zero for each particle, where


F.. = I i I duA


2 dxj dxi J


In the context of Stokes’s flow for an isolated particle, the torque needed to maintain the rotation of a sphere with angular velocity П is


T = 8na3


For a sphere placed in an external flow UTO that has a uniform rate of strain Ef°, the symmetric force dipole needed to neutralize this is


Щ = f


Combining the above equations, we obtain


Flj = pf vVp(3Qij + 5E*j).


The torque term is where eijk is the tensorial index and Qpk is the angular velocity of particle к. The stresslet contribution is defined implicitly by assuming that the average rate of strain is zero, since the particles are not allowed to deform. To this end, we obtain E*4 from the convolution


ij


Ej = E™0(x — Yn,aD)d3x,


where


1 fdUT    dU?°


2 у dxj    dxi )


is the strain tensor of the undisturbed flow field.


The details of the implementation were presented by (Lomholt, 2000), who employed an iterative procedure to impose the constraint of zero average strain rate. A more direct approach would be to consider the mobility matrix, which is the inverse of an influence matrix, and can be constructed from the linear response to multiunit pulses. This, however, may be computationally expensive, but hybrid approaches can be followed that combine efficiency and accuracy. A robust and simple to implement approach is the penalty method developed in (Liu, 2004).

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