Dynamics of Ideal Fluids Flashcards
State the simple conservation of mass formula
d/dt INT_Dt ρ dV = 0
State the simple conservation of momentum formula
d/dt INT_Dt ρu dV
ρu = F with no d/dt
How can we derive the continuity equation?
Apply transport theorem to the conservation of mass
How do we describe the force on a fluid
The sum of some body force ρf and some surface force/unit area or stress σ · dS acting on the boundary
Define idea (inviscid) fluid
A fluid which neglects internal friction/viscosity between neighboring fluid elements.
Give the stress tensor for an ideal fluid
σij = −pδij
Any cross terms would correspond to tangential shears which are neglected in inviscid fluid model.
State the momentum equation
∂u/∂t + (u · ∇)u = f − 1/ρ ∇p
How are the incompressible Euler equations characterised?
By constant density ρ = ρ0
Incomp since continuity reduces to ∇ · u = 0
State the continuity equation
∂ρ/∂t + ∇ ·ρu = 0
What boundary conditions do we have for incomp Euler?
u·n = 0
No flow through boundaries
State the conservative KE proposition
In a fixed volume (V) with u · n = 0, the incompressible Euler equations with a conservative body force f = −∇U conserve kinetic energy.
Give the expressions for conservation of KE
dE/dt = 0
E = 1/2 INT_V ρ0|u|^2dV
Do we use transport theorem in conservation of KE?
No, we consider a fixed volume, rather than material curve.
State the Bernoulli function
B(x, t) = p/ρ0 + 1/2 |u|^2 +U
State Bernoulli’s theorem
In an incompressible steady flow of an ideal fluid, with conservative body force f = −∇U, the Bernoulli function is constant along streamlines
