5. Euler's Equation Flashcards
Forces Acting on a Fluid
- the forces acting on a fluid can be divided into two types:
i) body forces
ii) surface forces
Body Forces
-act on all the particles throughout V, e.g. gravity:
Fv = ∫ ρ |g dV
Surface Forces
- caused by interactions on the surface S
- we will only consider fluid pressure
Pressure
-collisions between fluid molecules on either side of the surface S produce a flux of momentum across the boundary in the direction of the normal n
-the force exerted on the fluid into V by the fluid on the other side of S is, by convention, written as:
Fs = ∫ -p |n dS
-where p(|x)>0 is the fluid pressure
Newton’s Second Law of Motion
-newton’s second law of motion tells us that the sum of the forces acting on the volume of fluid V is equal to the rate of change of its momentum
-since D|u/Dt is the acceleration of the fluid particles, or fluid elements, within V we have:
∫ρ D|u/Dt dV = ∫ -p|ndS + ∫ρ|gdV
Deriving Euler’s Equation
-starting with newton’s second law of motion:
∫ρ D|u/Dt dV = ∫ -p|ndS + ∫ρ|gdV
-apply divergence theorem
∫ρ D|u/Dt dV = ∫(-∇p+ρ|g)dV
-since V is arbitrary, both integrands must be equal:
ρ D|u/Dt = -∇p + ρ|g
Euler’s Equation
ρ D|u/Dt = -∇p + ρ|g
Equation of Hydrostatic Balance
-in the case of a fluid at rest, |u=0 so Euler's equation is reduced to: 0 = -∇p + ρ|g -so the equation of hydrostatic balance: ∇p = ρ|g => p(|x) = -ρ|g.|x + C -where C is a constant
State Archimedes Theorem
-the force on a body in a fluid is an upthrust equal to the weight of fluid displaced
The Vorticity Equation
D|ω/Dt = ∂|ω/∂t + (|u.∇)|ω
= (|ω.∇)|u
-for incompressible flows
What does the vorticity equation show?
-it shows that the vorticity of a fluid particle changes because of gradients of |u in the direction of |ω
Properties of the Vorticity Equation
1) if |ω=0 everywhere initially, then |ω remains zero, thus flows that start off irrotational remain so
2) in a two-dimensional planar flow, |u=(u(x,y),v(x,y),0), the vector vorticity has only one non-zero component:
|ω = (∂v/∂x - ∂u/∂y) ^ez so that:
(|ω.∇)|u = ω d|u(x,y)/dz = 0
-hence the vorticity equation reduced to:
D|ω/Dt = ∂|ω/∂t + (|u.∇)|ω = 0
-which shows that the vorticity of a particle remains constant, if in addition the flow is steady then ∂|ω/∂t=0 and vorticity is constant along streamlines
Kelvin’s Circulation Theorem
Words
-the circulation around a closed material curve remains constant in an inviscid fluid of uniform density and subject to conservative forces
Kelvin’s Circulation Theorem
Formula
dΓ/dt = d/dt ∮|u . d|l = 0
-where the integral is over C(t), a closed curve formed of fluid particles following the flow
Flux of Vorticity Through a Surface That Spans a Material Curve
-the circulation around a closed curve C is equal to the flux of vorticity through an arbitrary surface S that spans C
-so from Kelvin’s Circulation Theorem:
dΓ/dt = d/dt ∮|u . d|l
= d/dt ∫|ω . |n dS = 0
-this demonstrates that the flux of vorticity through a surface that spans a material curve is constant
