Interdisciplinary Applied Mathematics

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хг ‘ хг i


where FB includes all body forces, e.g., gravity or electromagnetic forces, and FiH    is    the    force    from    the fluid    on    the particle.    Since flow inertia    is


already included in FCM, it has to be subtracted from the particle inertia. The force monopole strength is therefore


+ fB


For example, if gravity is the only body force, then


FiB


(pP    Pf )Vp gi,


where Vpn is the volume occupied by particle n.


The last term in the FCM governing equations (14.15) is associated with the dipole contribution (Kim and Karrila, 1991). It is caused by two different effects:


an antisymmetric part, which is due to torque exerted on the fluid by particle n, and


a symmetric part, the stresslet, which corresponds to the rate of strain tensor.


The formulation of the force dipole effect ensures that the angular mo-mentum/kinetic budget is consistently related to the work done by any torque and that the stresslet term does not impart net work on the flow.


The envelope    0(x,aD)    employed    for    the    dipole    contribution    is    also    a


Gaussian envelope but with a different width, characterized by the length scale ap, i.e.,


0(x,aD) = A(x, ad).


This length scale is determined by matching the particle angular velocity found from the convolution


Qi = ^ J u>i(x)G(x — YaD)d3x,


with the angular velocity of a fixed sphere in an unbounded quiescent flow of vorticity w*; see (Kim and Karrila, 1991), and also (Lomholt, 2000). This matching results in


a


<?d =

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