Also, for the y- and z-directions,∂p∂y=0and∂p∂z=0.The equations from the y- and z-directions imply the pressure only depends on x.But pressure doesnot depend on x at the air/water interface, and pressure is independent of y and z, so the pressure isconstant everywhere in this problem.The flow is driven purely by gravity.We haved2vxIdy2=−ρwgμwso integrating twice givesvxIy( )=−ρwgμwy22+C1y+C2.The same analysis applies in the air, but the density in the air is negligible (this assumption isconsistent with the problem statement, in which pressure variations in the air are neglected).Flow inthe air is driven by friction (i.e., the no-slip surface) between the water and air.In the air we havevxIIy( )=C3y+C4.The no-slip condition at y=0 yields C2=0, so in the watervxIy( )=−ρwgμwy22+C1y0≤y≤hI.At the air/wall interface the no-slip condition isvxII=0aty=hI+hIIso0=C3hI+hII()+C4orvxIIy( )=C41−yhI+hII⎛⎝⎜⎞⎠⎟forhI≤y≤hI+hII.Two constants remain and for those we use the conditions at the air/water interface. We require that theshear stress be continuous, so