The Navier-Stokes equation in a magnetic field

The following equations are for a magnetorheological fluid undergoing adiabatic demagnetization and remagnetization. The symbols retain their usual meaning, and ∇ and ρ are the curl vector and charge density, respectively.

∇.B = ∇ . (∇ x A) = 0

∇ x E = ∇ x (-∇φ - ∂A / ∂t) = - ∂ / ∂t(∇ x A)

∇ . A + (1/c²)(∂φ / ∂t) = 0

∇²φ – (1/c²)( ∂²φ / ∂t²) = ρ/Є_o

∇²A – (1/c²)(∂²A / ∂t²) = - µ_oJ

∇ x (∇/µ x A) = J

ρ(∂v / ∂t + v . ∇v) = -∇p + µ∇²v + f

ρ{∂v / ∂t + v(∇v)} = -∇p + η∇v + ρg + M∇B

{ρ = ρ_o(1 – βT)

M = M_o(1 – KΘ)}