8.2.1 Gosling and Singer

[Mol.Phys.26(1973)1475]
An external force $\vec{F}=\{F_{x},0,0\}$ is introduced in the equations of motion according to

$$F_{i,x}=F_{0}\sin k z_{i} \;\;\;\;{\rm with}\;\;\;\; k\equiv 2 \pi \frac{m}{L}; \;\; m=1,2,\dots$$ (8.15)

(Note: $\vec{A}_{q}=\{\sin kz, 0, 0 \}$ and $\vec{A}_{p}=0$ which fulfills 8.12.)

This will eventually lead to an average velocity profile which theoretically should look like

$$<v_{i,x}>=\frac{n}{k^{2}\eta} F_{0} \sin kz_{i}=v_{0} \sin kz_{i}$$ (8.16)

where $n \equiv N/V$. Fitting the actual velocity profile to this expression we find $v_{0}$ and from that the $k$-dependent viscosity:

$$\eta(k)=\frac{nF_{0}}{v_{0}k^{2}}$$ (8.17)

Repeating the calculation for several values of $k$ and extrapolating to $k \rightarrow 0$ we find an estimate for $\eta \equiv \eta(k=0)$

Note: The applied force does work against the viscous forces, thus inducing an undesirable temperature rise.