Strain tensor bulk fluids

The dependence of the viscous stress on the velocity gradient in the fluid is a constitutive law, which is usually called the bulk rheological equation. The general linear relation between the viscous stress tensor, T, and the rate of strain tensor,... 

Here is a generalized (6N dimensional) force-torque vector, U -u (6N dimensional) is the particle translational-angular velocity relative to the bulk fluid flow evaluated at the particle centre, (3x3 matrix) is the traceless symmetric rate of the strain tensor (supposed to be constant in space). The resistance matrices Rfu (6N x6N) and Rfe (6N x 3 x 3) which depend only on the instantaneous relative particle configurations (position and orientation) relate the force-torque exerted by the suspending fluid on the particles to their motion relative to the fluid and to the imposed shear flow, respectively. Note that in ER (MR) fluids torques can be neglected. 

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