Straight-rod resistance in a linear flow ($C=4\pi \mu /|\log ϵ|$)

For a straight rod $\mathbf{X}=s\mathbf{p}$, $-L<s<L$, leading slender-body theory uses $\mathbf{f}=C(I-\mathbf{pp}/2)[\Delta \mathbf{U}+s(\Delta \mathbf{\Omega }\times \mathbf{p}-E\mathbf{p})]$. Integration gives $\mathbf{F}=2CL(I-\mathbf{pp}/2)\Delta \mathbf{U}$ and $\mathbf{G}=(2C{L}^3/3)[(I-\mathbf{pp})\Delta \mathbf{\Omega }-\mathbf{p}\times E\mathbf{p}]$. Axial spin is outside this centreline approximation.

The orientation of a force-free, torque-free infinitely slender straight rod follows background rotation plus the transverse component of strain. The last term preserves $|\mathbf{p}|=1$. Finite aspect ratio changes the strain coefficient and permits periodic tumbling in a simple shear flow.

A force-free, torque-free straight rod contributes a particle stresslet tensor $(C{L}^3/3)(\mathbf{p}\cdot E\mathbf{p})\mathbf{pp}$. Its extra viscous dissipation is $(C{L}^3/3)(\mathbf{p}\cdot E\mathbf{p}{)}^2$. In a simple shear flow, $\dot{\theta }=-\gamma {\sin }^2\theta$; the dissipation is greatest along the extensional or compressional axes and vanishes at flow and gradient alignment.