Fluid Equations

Abstract

first 4 are scalars the 4 following are vectors which makes 12 more for a total of: 16 unknowns Equations: a)Maxwell: ∇ • B = 0 , ∇ × B = µ 0 ε 0 J + ε 0 ∂E ∂t , ∇ • D = ρ , ∇ × E = − ∂B ∂t as pointed out these four vector equations are linearly independent so there is only two of them each. Thus for a) we have 2 vector equations X3 components = 6 equations b) Continuity of mass ∂p m ∂t + ∇ • ρ m u () = 0 2 species X 1 scalar equation = 2 equations c) Force: ρ m (∂u ∂t       + (u •∇)u) = −∇p + qn(E + u × B) 3 equations/species X2 species = 6 equations The Grand total is 14 equations We need 2 more equations to close the fluid set. These must be the equations of state that relate the Thermodynamic variables. Which equation we choose is dictated by the problem and is an assumption. (The use of fluid theory is also an assumption) Some equations of state are: 1) P = Cρ m γ adiabatic, γ = c p /c v ratio of specific heats at constant volume and at constant pressure. (May be written ∂ ∂t Pρ m − γ () = 0) 2) ∇ • v = 0 incompressible fluid 3) Isothermal plasma ∂ ∂t P ρ m       = 0 Each choice adds two more equations which closes the loop