Reynold's Number & Continuity Equation Flashcards
What are the dimensions of μ?
M*L^-1 * T^-1
What are the dimensions of ρ0v0d0, where ρ is the mass density, v is the speed and d is the scale-length?
M*L^-1 * T^-1
How do we make a dimensionless number using these dimensions?
ρ0v0d0/μ = v0*d0/v
What is the Reynolds number?
Dimensionless number Re = ρ0v0d0/μ = v0*d0/v
When viscous dominates the flow, what happens to Re?
The Reynolds number is small when the flow is viscous dominant.
What does the Reynold’s number equal for the Poiseuille flow shown before?
Re = Q*d0^3 * ρ0/32μ
How could you use the Reynolds number to find the flow speed of something?
Rearrange the Reynolds number equation for the velocity.
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?
- Mass conservation
- F = ma
- Viscous stress in full
- Boundary conditions
These equations are called the Navier-States equations.
Which quantities do we average in the continuum hypothesis?
v(r,t), ρ, P, T, μ etc
What do we consider if ρ is not constant?
A closed volume V with bounding surface S.
What is the mass flux out of the closed volume V with bounding surface S? What must this equal?
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
What is the divergence theorem?
integral over S of A.ds = integral over V of (Δ.A) dV
How can we incorporate the divergence theorem into the closed volume problem?
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.
What happens to the equation if there is a fixed Eulerian control volume?
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.
What is the continuity equation?
dρ/dt + ∇.(ρv) = 0
What does it mean if a fluid is incompressible in terms of the density and velocity?
Density is constant and ∇.v = 0
When can we assume incompressibility?
When the flow speed is less than the sound speed.
What is Dv/Dt equal to for a fluid moving with velocity v(r(t), t)?
Dv/Dt = lim as Δt goes to 0 of (v(r(t+Δt), t + Δt) - v(r(t), t))/Δt
How can we expand the equation for Dv/Dt?
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
How does D/Dt differ from d/dt?
D/Dt is the rate of change moving with fluid (Lagrangian), whereas d/dt is at a fixed r (Eulerian)
What are the 3 possible forces in a fluid element?
Gravity, pressure and viscous force.
What is the equation for the gravity force on a fluid element?
Fg = -mg k = -ρg ΔxΔyΔz k
What is the equation for the pressure force on a fluid element in the x-direction and in 3D?
- x-direction: Fp(x) = (P(x)-P(x+Δx)) ΔyΔz = -dP/dx ΔxΔyΔz
- 3D: Fp = -∇P ΔxΔyΔz
What is the equation for the viscous force from stress on a fluid element?
Fv = μ ∇^2v ΔxΔyΔz
What is the equation for the mass of a fluid element?
m = ρΔxΔyΔz
How can we combine all of these forces on a fluid element?
Using F=ma and set them all equal to eachother
What is the Navier-States equation?
ρdv/dt + ρ(v.∇)v = -∇P + μ ∇^2v - ρg k
What 3 things do we need to solve th Navier states equation?
Need ρ(r,t) aswell from dρ/dt + ∇.(ρv) = 0, need an equation for the pressure P and need boundary conditions to solve PDEs
