Reynold's Number & Continuity Equation

Question 1

What are the dimensions of μ?

Answer 1

M*L^-1 * T^-1

Question 2

What are the dimensions of ρ0v0d0, where ρ is the mass density, v is the speed and d is the scale-length?

Answer 2

M*L^-1 * T^-1

Question 3

How do we make a dimensionless number using these dimensions?

Answer 3

ρ0v0d0/μ = v0*d0/v

Question 4

What is the Reynolds number?

Answer 4

Dimensionless number Re = ρ0v0d0/μ = v0*d0/v

Question 5

When viscous dominates the flow, what happens to Re?

Answer 5

The Reynolds number is small when the flow is viscous dominant.

Question 6

What does the Reynold’s number equal for the Poiseuille flow shown before?

Answer 6

Re = Q*d0^3 * ρ0/32μ

Question 7

How could you use the Reynolds number to find the flow speed of something?

Answer 7

Rearrange the Reynolds number equation for the velocity.

Question 8

What 4 things do we need to find the flow speed if the viscosity isn’t dominant and the flow is not laminar? What is this called?

Answer 8

• Mass conservation
• F = ma
• Viscous stress in full
• Boundary conditions

These equations are called the Navier-States equations.

Question 9

Which quantities do we average in the continuum hypothesis?

Answer 9

v(r,t), ρ, P, T, μ etc

Question 10

What do we consider if ρ is not constant?

Answer 10

A closed volume V with bounding surface S.

Question 11

What is the mass flux out of the closed volume V with bounding surface S? What must this equal?

Answer 11

Integral over S of ρ*v.ds

This must equal the rate of decrease of mass inside the volume: d/dt of the integral over V of ρ dv

Question 12

What is the divergence theorem?

Answer 12

integral over S of A.ds = integral over V of (Δ.A) dV

Question 13

How can we incorporate the divergence theorem into the closed volume problem?

Answer 13

Since the mass flux out is equal to the rate of decrease of mass: d/dt of the integral over V of ρ dv = -Integral over S of ρ*v.ds

We can sub in the divergence theorem for the second part.

Question 14

What happens to the equation if there is a fixed Eulerian control volume?

Answer 14

d/dt of the integral over V of ρ dv = integral of dρ/dt dV, and can then rearrange and set this integral equal to zero.

Question 15

What is the continuity equation?

Answer 15

dρ/dt + ∇.(ρv) = 0

Question 16

What does it mean if a fluid is incompressible in terms of the density and velocity?

Answer 16

Density is constant and ∇.v = 0

Question 17

When can we assume incompressibility?

Answer 17

When the flow speed is less than the sound speed.

Question 18

What is Dv/Dt equal to for a fluid moving with velocity v(r(t), t)?

Answer 18

Dv/Dt = lim as Δt goes to 0 of (v(r(t+Δt), t + Δt) - v(r(t), t))/Δt

Question 19

How can we expand the equation for Dv/Dt?

Answer 19

Take the 1/Δt out and then have v(r,t) + dv/dx * vx + dv/dy * vy + dv/dz * vz + dv/dt = (v.∇ + d/dt)v

Question 20

How does D/Dt differ from d/dt?

Answer 20

D/Dt is the rate of change moving with fluid (Lagrangian), whereas d/dt is at a fixed r (Eulerian)

Question 21

What are the 3 possible forces in a fluid element?

Answer 21

Gravity, pressure and viscous force.

Question 22

What is the equation for the gravity force on a fluid element?

Answer 22

Fg = -mg k = -ρg ΔxΔyΔz k

Question 23

What is the equation for the pressure force on a fluid element in the x-direction and in 3D?

Answer 23

• x-direction: Fp(x) = (P(x)-P(x+Δx)) ΔyΔz = -dP/dx ΔxΔyΔz

- 3D: Fp = -∇P ΔxΔyΔz

Question 24

What is the equation for the viscous force from stress on a fluid element?

Answer 24

Fv = μ ∇^2v ΔxΔyΔz

Question 25

What is the equation for the mass of a fluid element?

Answer 25

m = ρΔxΔyΔz

Question 26

How can we combine all of these forces on a fluid element?

Answer 26

Using F=ma and set them all equal to eachother

Question 27

What is the Navier-States equation?

Answer 27

ρdv/dt + ρ(v.∇)v = -∇P + μ ∇^2v - ρg k

Question 28

What 3 things do we need to solve th Navier states equation?

Answer 28

Need ρ(r,t) aswell from dρ/dt + ∇.(ρv) = 0, need an equation for the pressure P and need boundary conditions to solve PDEs