Fluids and Waves Flashcards
Define Lagrangian coordinates
A coordinate system with a reference point given by the position of the particle at t = 0.
Define Eulerian coordinates
A coordinate system with a reference point given by the position of the particle at the current time.
Define a convective derivative
The rate of change with X held constant.
Define the velocity of fluid
u = Dx/DT
Define steady flow
Flow where the velocity of fluid is independent of t.
Define a streamline
A snapshot of the flow at a fixed time t, with curves parallel to the velocity field.
Define a stagnation point
A point where the velocity is 0.
Define particle paths
A description of the flow given by following the path of a particle.
Give Euler’s identity
DJ/Dt = J∇ · u, where J is the Jacobian relating the Lagrangian and Eulerian coordinate systems.
Define incompressible flow
Flow is incompressible if infinitesimal volumes are preserved, that is DJ/Dt = 0.
Give Reynold’s Transport Theorem
d/dt of the triple integral of f over a volume V is given by the triple integral over V of ∂f/∂t + ∇ · (fu).
Give the density equation
Dρ/Dt + ρ∇·u = 0
Give the momentum equation
ρ(Du/Dt) = −∇p + ρg.
Give Bernoulli’s equation for steady flow
p/ρ + (1/2)|u|^2 + χ is constant along streamlines for steady flow.
Define vorticity
Vorticity is the curl of the velocity field.
Give the vorticity equation
∂ω/∂t + (u·∇)ω = (ω·∇)u
Define circulation
The circulation around a closed curve C is given by the path integral along C of u with respect to x.
Give Kelvin’s Circulation Theorem
The derivative of circulation with respect to time is 0.
Define irrotational flow
Flow where the vorticity is 0.
Give Bernoulli’s equation for steady, irrotational flow
p/ρ + (1/2)|u|^2 + χ is constant everywhere if the flow is steady and irrotational.
Define a velocity potential
For irrotational flow, there is a potential ϕ such that u = ∇ϕ, this potential is known as the velocity potential.
Give Bernoulli’s equation for irrotational flow
∂ϕ/∂t + p/ρ + (½)|∇ϕ|^2 + χ = F(t), where F is independent of position.
Give the potential of a line source
At a line source, the potential is given by φ = Qlog(r)/2π.
Define the streamfunction
The function ψ(x, y, t) satisfying u = ∇ × (ψk).
Define flux
The flux between two streamlines is given by the path integral along C of u.nds, where C is a smooth path joining the streamlines and n is the unit normal.
Give the streamfunction of a line vortex
At a line vortex, the streamfunction is given by ψ = −Γlog(r)/2π.
Define the complex potential
Provided that both φ and ψ are continuously differentiable, the complex potential is given by w(z) = φ(x, y) + iψ(x, y).
Give the complex potential of a line source
The complex potential of a line source of strength Q at (a, b) is given by w(z) = Qlog(z − c)/2π, where c = a + bi.
Give the complex potential of a line vortex
The complex potential of a line vortex of strength Γ at (a, b) is given by w(z) = -iΓlog(z − c)/2π, where c = a + bi.
Give Milne-Thomson’s circle Theorem
Suppose a velocity potential w(z) = f(z) is given with the property that any singularities of f(z) occur in |z| > a. Then the potential w(z) = f(z) + [f(a^2/z)] has the same singularities as f(z) in |z| > a and the circle |z| = a as a streamline.
Give Blasius’ Theorem
Suppose fluid flows steadily past an obstacle B with simple closed boundary ∂B. If gravity is neglected, the net force (Fx, Fy) exerted on B by the fluid is given by Fx − iFy = iρ/2 time the path integral along ∂B of (dw/dz)^2.
Give the Kutta-Joukowski Lift Theorem
Consider steady uniform flow at speed U past an obstacle B, where there is a circulation Γ about B but there are no singularities in the flow. Then the obstacle experiences a drag force D parallel to the flow and lift force L perpendicular to the flow given by D = 0, L = −ρUΓ.
Give the Joukowski transformation
The map z = G(ζ) = ζ + a^2/ζ.
Define the Kutta condition
The circulation must be chosen such that the velocity at the trailing edge is finite.
State Helmholtz’s Principle
A vortex moves with the velocity field due to everything except itself.
Give the water wave boundary condition far away from the surface
∇ϕ → 0 as y → −∞.
Give the dynamic boundary condition of water waves
p = Patm at y = η, where Patm denotes the atmospheric pressure above the fluid.
Define the amplitude of a water wave
Given a water wave η(x, t) = Acos(kx − ωt − β), the amplitude is given by A.
Give the kinematic boundary condition of water waves
The velocity of the fluid normal to the boundary must equal the velocity of the boundary normal to itself.
Define the frequency of a water wave
Given a water wave η(x, t) = Acos(kx − ωt − β), the frequency is given by ω.
Define the wave number of a water wave
Given a water wave η(x, t) = Acos(kx − ωt − β), the wavenumber is given by k.
Define the wave length of a water wave
Given a water wave η(x, t) = Acos(kx − ωt − β), the wavelength is given by 2π/k.
Define the wave speed of a water wave
Given a water wave η(x, t) = Acos(kx − ωt − β), the wave speed is given by c = ω/k.
Define a dispersive wave
A wave where the velocity is dependent on the frequency of the wave.
