Analyze the problem of electro-osmotic flow with an additional imposed pressure gradient. Show that the volumetric flow rate can be expressed as

\[Q=L_{11} \Delta \phi+L_{12} \Delta P\]

where \(\Delta \phi\) is an applied potential gradient in the \(x\)-direction equal to \(E_{x} / L\).

Also show that the current can be expressed as

\[I=L_{21} \Delta \phi+L_{22} \Delta P\]

Derive expressions for the coefficients \(L_{11}\) etc. Verify that \(L_{12}=L_{21}\), a type of Onsager reciprocity relation (Onsager, 1931).