More Fluid Motion

Question 1

What is the model to show Poiseuille flow along a pipe?

Answer 1

Water tank with circular cross section outflow pipe of radius a, pipe is length L with pressure P1 inside tank and P2 outside.

Question 2

What is the first step to find the flow through an element Δz in the tank with pipe?

Answer 2

• Cross-sectional area = rΔФΔr
• Need to find net viscous force as before
• v = vz(r) k

Question 3

How can we expand out dvz/dr at r+Δr?

Answer 3

Taylor expansion, so = (dvz/dr at r) + (Δr*d^2vz/dr^2 at r) + …

Question 4

What do we find as the net viscous force?

Answer 4

Fv = μΔzΔФΔr d/dr(rdvz/dr)

Question 5

What do we put into the pressure force equation to find the pressure force in the pipe?

Answer 5

• Area multiplied by the change in pressure

- Fp = rΔФΔr(P(z)-P(z+Δz)) = -rΔФΔrΔz*dP/dz

Question 6

What do we do first with the pressure force equation after finding it?

Answer 6

• Balance pressure force and viscous force and set equal to -Q
• Integrate to get rid of the d/dr and rearrange for dvz/dr

Question 7

What do we do next with the pressure force equation?

Answer 7

• Smooth solution at r=0, so constant A = 0
• Integrate again to find vz, and use boundary condition vz(a) = 0
• Substitute in the equation for Q

Question 8

What do we get as the equation for vz in the end?

Answer 8

vz = (P1-P2)/4μL * (r^2 - a^2)

Question 9

How do we find the flow rate through the pipe?

Answer 9

• Mass flowing through area rΔФΔr is ρvzrΔФΔr

- Double integral from 0 to 2pi and 0 to a of ρ*vz(r) r drdФ

Question 10

What are the two points to consider for a microscopic view of viscosity?

Answer 10

• Average molecular speed v(mol) due to random motion of atoms/molecules
• Speed of centre of mass of fluid element v in direction v

Question 11

What happens if there is a gas at constant temperature, wit v(mol) the same everywhere and v varies?

Answer 11

-Molecules diffuse between layers of different v and exchange momentum: change average speed

Question 12

What is the momentum change on a macroscopic level for the gas?

Answer 12

Inferred as the action of a stress: τ = μ dvx/dy

Question 13

What can viscosity be estimated from?

Answer 13

The molecular dynamics.

Question 14

What happens to μ for gases and liquids as temperature increases?

Answer 14

• Gas: T increases, more collisions and μ increases

- LiquidsL T increases, μ decreases

Question 15

What is the kinematic viscosity?

Answer 15

Dynamic viscosity divided by the density: ν = μ/ρ