text: In this simple model,  can also be derived directly from the expression for fluid flow in the Stokes limit for an incompressible fluid, which is
  for  the fluid flow velocity and  the pressure. We consider an infinite surface in the  plane at , and enforce stick boundary conditions there, i.e., . We take the concentration gradient to be along the  axis, i.e., . Then the only non-zero component of the flow velocity  is along x, , and it depends only on height . So the only non-zero component of the Stokes' equation is
  In diffusioosmosis, in the bulk of the fluid (i.e., outside the interface) the hydrostatic pressure is assumed to be uniform (as we expect any gradients to relax away by fluid flow) and so in bulk
  for the solvent's contribution to the hydrostatic pressure, and  the contribution of the solute, called the osmotic pressure. Thus in the bulk the gradients obey
  As we have assumed the solute is ideal, , and so
  Our solute is excluded from a region of width  (the interfacial region) from the surface, and so in interface , and so there . Assuming continuity of the solvent contribution into the interface we have a gradient of the hydrostatic pressure in the interface
  i.e., in the interface there is a gradient of the hydrostatic pressure equal to the negative of the bulk gradient in the osmotic pressure. It is this gradient in the interface in the hydrostatic pressure  that creates the diffusioosmotic flow. Now that we have , we can substitute into the Stokes equation, and integrate twice, then
  where , ,  and  are integration constants. Far from the surface the flow velocity must be a constant, so . We have imposed zero flow velocity at , so . Then imposing continuity where the interface meets the bulk, i.e., forcing  and  to be continuous at  we determine  and , and so get
  Which gives, as it should, the same expression for the slip velocity, as above. This result is for a specific and very simple model, but it does illustrate general features of diffusioosmoisis: 1) the hydrostatic pressure is, by definition (flow induced by pressure gradients in the bulk is a common but separate physical phenomenon) uniform in the bulk, but there is a gradient in the pressure in the interface, 2) this pressure gradient in the interface causes the velocity to vary in the direction perpendicular to the surface, and this results in a slip velocity, i.e., for the bulk of the fluid to move relative to the surface, 3) away from the interface the velocity is constant, this type of flow is sometimes called plug flow.