= Solution
The first term is the local viscous force per unit length from <resistive-force theory>. With <unit tangent vector> $\mathbf t=\mathbf r_s$, the drag tensor gives $-c_\parallel\mathbf v_\parallel-c_\perp\mathbf v_\perp$, where $\mathbf v=\mathbf r_t$ is velocity relative to the fluid.
The second term is the bending force of an <inextensible filament>. Varying its energy $E_b=(B/2)\int|\mathbf r_{ss}|^2ds$ gives bulk force density $-B\mathbf r_{ssss}$. The final term, $\partial_s(T\mathbf r_s)$, is the force density from <filament tension>; $T$ is the <Lagrange multiplier> enforcing $|\mathbf r_s|=1$. At low <Reynolds number>, these forces balance without a filament acceleration term.
With mass, length, and time dimensions $M,\mathsf L,\mathsf T$, the coefficients have
$$
\boxed{[c_\perp]=[c_\parallel]=M\mathsf L^{-1}\mathsf T^{-1},\qquad [B]=M\mathsf L^3\mathsf T^{-2},\qquad [T]=M\mathsf L\mathsf T^{-2}.}
$$
Thus the drag coefficients have the dimensions of <dynamic viscosity>, the <filament bending modulus> has dimensions force times length squared, and tension has dimensions force.
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