Laminar Flow : shear force on walls

[throneoo]

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)

 

[Chestermiller]

Your result for part (a) is correct. For part (b) the only axial force that the fluid exerts on the vessel is the shear force on the wall from part (a). But the pipe obviously isn't accelerating. So there must be another force acting. You need to have an axial tensile force within the pipe metal at the beginning of the pipe to hold the pipe in place.

Chet