# 1. Fluid Equations and Linear Wave Theory Flashcards

Density Definition

-let Q denote the amount of a quantity in a volume V

-the density of Q, q, is defined as:

Q = ∫ q(x_,t) dV

-this also applies to vectors

Mass Density

Q = mass q = ρ

Kinetic Energy Density

Q = KE q = 1/2 ρu²

Volume Density

Q = V q = 1

Linear Momentum Density

Q_ = linear momentum q = ρu_

Flux Definition

-flux of a quantity Q through a surface S is the rate at which Q passes through S

flux = ∫ q (u_ . n_) dS

Mass Flux

mass flux = ∫ ρ (u_ . n_) dS

Momentum Flux

flux = ∫ (ρu_) (u_ . n_) dS

Conservation Laws

-the evolution of any quantity Q in a volume V is governed by a conservation law of universal form

dQ/dt = d/dt ∫qdV

= ∫qu_.(-n_)dS + ∫K.(-n_)dS + ∫adV

-first term is flux through S, second term includes any other surface sources of Q and the third term is for body sources

Mass Conservation

-substitute q=ρ into the universal conservation law

-physics tells us that the second and third terms are 0

=>

d/dt ∫ρ dV = ∫ρu_.(-n_) dS

-put under single integarl using divergence theorem

∫∂ρ/∂t + ∇.(ρu_) dV = 0

-this is true for any V, thus:

∂ρ/∂t + ∇.(ρu_) = 0

Momentum Conservation

q_ = ρu_

-sub into universal conservation law:

d/dt ∫ρ dV = ∫ρu_.(-n_) dS + ∫ρu_(u_.(-n_)) dS + ∫F_ dV

-put into suffix notation to put all terms under one volume integral, set equal to 0

=>

ρ Du_/Dt = -∇p + F_

-this is essentially Newton’s 2nd Law

Equations of State

- physics => p = f(ρ,T)

- where f is material dependent

Ideal Gas Law

p = RρT

- doubling the density will double the frequency of particle collisions so pressure would be twice as great
- doubling the temperature doubles the energy so you would get twice the momentum transfer per collision and thus double the pressure

Linear Approximation to the Ideal Gas Law

-when we make a small perturbation from the base state we can use a linear approximation, a Taylor expansion of (3) about the base state (ρ,T,p) = (ρo,To,po): p = po + Aρ (ρ-ρo) + At (T-To) -where: Aρ = ∂f/∂ρ|ρo,To At = ∂f/∂T|ρo,To -this is commonly written as: ρ = ρo [1 + αp(p-po) + αt(T-To)] -where αp is the coefficient of compressibility and αt is the thermal expansion coefficient

Fluid Mechanical Equations for a Compressible or Incompressible Fluid

∂ρ/∂t + ∇.(ρu_) = 0

ρ Du_/Dt = -∇p + F_

p = F(ρ), assuming T is constant

Boundary Conditions

Moving Rigid Boundary

u_.n_ = U_.n_

-where u_ is the fluid velocity and U_ is the boundary velocity

Boundary Conditions

Fluid-Fluid Interface

u1_.n_ = u2.n_ -also need stress to be continuous -for an invicid fluid this implies: p1=p2 on S -generally: σ1_.n_ = σ2_.n_