In a chemical coating process a belt is moving with a constant speed of V=2 m/s. Belt is drawing up a liquid film with a thickness h=2mm. The flow of the liquid is assumed to be steady, laminar and fully-developed. The belt is b=1 m wide in the z direction. The liquid film is open to the atmosphere and the friction between the liquid and the air could be neglected.
a. Using Navier-Stokes equations show that the pressure field within the liquid is constant i.e. P=Patm and the velocity profile is u=[g/(2n)]y2-(gh/n)y+V. Do it step by step showing every assumption and simplification.
b. If shear stress on the belt surface tw is a function of r,h,g, find the dimensionless groups that represent this problem using uckingham-P theorem, i.e find P1 =f(P2,P3,..).
c. Using results in part a and b, find the function f in part b.
d. What is the power used by the belt to draw up the liquid?
e. What is the volume flow rate of the liquid Q(m3/s)?