The Reynolds equation for lubrication theory, A more general scenario for the lubrication approximation is the problem where the fluid is contained in a gap of height h, which is assumed to be a general function of x, i

The Reynolds equation for lubrication theory, A more general scenario for the lubrication approximation is the problem where the fluid is contained in a gap of height h, which is assumed to be a general function of x, i.e., h = h(x). The lower surface is assumed flat and moves with a velocity v. The top surface can also move in the y-direction, and hence the gap height is changing at a rate of dh/dt. This is similar to the squeezing flow in the previous problem. The velocity profile locally is the superposition of the Poiseuille flow and the Couette flow: vx = 1/ 2μ (y2 − hy) dp /dx + V ( 1 − y h ) Develop an expression for the pressure distribution in the system using the lubrication approximation. For this the continuity equation is to be used in an integral sense rather than in a differential form: