2. Mass Conservation & Streamfunction Flashcards
The Continuity Equation
Equation
∂ρ/∂t + ∇ . (ρ|u) = 0
-can be applied to quantities other than mass density e.g. charge density, energy density, concentration etc.
The Continuity Equation
Derivation
-consider a volume V fixed in space and bounded by a surface S
-the only way that the mass in V can change is if mass flows in or out of the surface S
-one expression for the total mass in V is to integrate density ρ(x,t) over V, rate of change of mass is then:
dM/dt = ∫ ∂ρ/∂t dV
-also the mass crossing S in time dt is:
dM/dt = - ∫ρ|u . ^n dS
-apply the divergence theorem and equate both sides:
dM/dt = 0 = ∫ ∂ρ/∂t dV
= - ∫ ∇ . (ρ|u) dV
-rearrange, and note that if the intergral of a function is zero then the function must also be zero:
∂ρ/∂t + ∇ . (ρ|u) = 0
The Continuity Equation
Lagrange Form
∂ρ/∂t + ∇ . (ρ|u) = 0
= Dρ/Dt + ρ |∇.|u = 0
Incompressible Fluids
-if |∇.|u = 0 then the density of the fluid ‘blob’ doesn’t change with time, i.e. Dρ/Dt = 0
-the fluid is incompressible if and only if |∇.|u = 0
-note that although the shape of an incompressible fluid can be changed, its volume will always remain constant
-an incompressible flow can always be written as:
|u = |∇ x |S
Incompressible 2D Flows
-for a 2D flow there is no movement in the z direction:
|u = (u(x,y,z) , v(x,y,z) , 0)
-|u can be written in the form: |u = |∇x|S where:
|S = (0, 0, ψ(x,y) )
-where ψ(x,y) is called the stream function
-in terms of the stream function:
|u = (u,v,0) = (∂ψ∂y,-∂ψ/∂x,0)
Stream Function
-the stream function is such that a 2D flow:
|u = (u,v,0) = |∇x|S
-where |S = (0,0,ψ)
-so |u = (∂ψ/∂y,-∂ψ/∂x,0)
-also |u.|∇ψ = 0 so the greatest change of ψ is always perpendicular to |u and ψ does not change along direction |u
-ψ is constant along streamlines
-conversely consider a curve of constant ψ:
∂ψ/∂s = 0 = ∂ψ/∂s dx/ds + ∂ψ/∂s dy/ds
=|u . |∇ψ
Incompressibility, Streamlines and Stream Function
-we can write a flow |u in terms of stream function ψ, if |u is incompressible then streamlines are lines of constant ψ
2D Flow Polar Coordinates
|u(r,θ) = ur(r,θ)^er + uθ(r,θ)^eθ
2D Incompressible Flow Polar Coordinates
-substitute |S = ψ(r,θ,t)*^ez in to |u=|∇x|S gives: |u = |∇ x (ψ*^ez) => ur = 1/r * ∂ψ/∂θ uθ = - ∂ψ/∂r
Physical Significance of the Stream Function
-the stream function is constant on streamlines
-consider two stream lines defined by ψ1 = ψ(x,y) and ψ2 = ψ(x,y)
-the flow rate or volume flux through an arbitrary curve C connecting the two streamlines is given by:
ψ1-ψ2
Axisymmetric Flows
-in cylindrical polar coordinates have only two non-zero components and two effective coordinates
-no ^eθ component
i.e. ;
|u(r,z) = ur(r,z)^er + uz(r,z)^ez
Stokes Stream Functions
-for axisymmetric, incompressible flows we define:
S = 1/r * Ψ(r,z,t)*^eθ
-where Ψ is a (scalar) Stokes stream function such that:
|u = (1/r Ψ ^eθ)
=>
uz = 1/r * ∂Ψ/∂r
ur = -1/r * ∂Ψ/∂z
Stokes Streamfunction and Stream Tubes
-Ψ is constant on stream tubes
Volume Flux and Streamtubes
-the volume flux between two streamtubes with Ψ=Ψi and Ψ=Ψo is:
Q = ∫ |u.^n dS
Q = 2π (Ψo-Ψi)
Streamfunction in Plane Polar Coordinates
ur = 1/r * ∂Ψ/∂θ uθ = - ∂Ψ/∂r
