- #1
throneoo
- 126
- 2
Homework Statement
a viscous fluid with viscosity η flows through a circular vessel of length L and radius R under a pressure difference of P. Assuming the flow is laminar, calculate:
a)the force on the walls of the vessel.
b)the net force on the vessel.
Homework Equations
Hagen-Poiseuille equation: v(r)=P(R^2-r^2)/4ηL
viscous force on the fluid : F(r)=-η(2*pi*r*L) (dv/dr)
The Attempt at a Solution
a)
on the boundary between the fluid and the wall, the vessel should exert a certain force to drag the fluid , but the viscous force from the fluid would balance it such that the fluid flows at constant velocity ( at rest). Thus the force exerted by the vessel on the fluid =-F(r) . By Newton's 3rd law, the walls experience the reaction force F(r),
which, by differentiation, is P*pi*R^2 at r=R , in the direction of the flow.
b)
I can't think of any other forces except the one mentioned in a) , which would be really weird as the vessel would be accelerating while the fluid flows steadily. It's even worse if the pressure force acts on the vessel too , as the net force will be 2*P*pi*R^2 instead of 0. Perhaps I've messed up the directions in a)