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In particular, how are we to interpret the time derivative 3 Mass Conservation and the Equation of Continuity 23 d/dt that appears in Eq. 13)? Since these equations each express the law of conservation of mass, they must be consistent. Note that we can write Eq. 14) which will be consistent with Eq. 13) if we identify ∂ρ dρ = + V · ∇ρ. 15) Generally, the operator d/dt = ∂/∂t + V · ∇ is called the total time derivative or the Lagrangian derivative. It measures the total change in a quantity associated with a fluid element as it moves about in space.

21) We can use the resistive form of Ohm’s law to eliminate V × B in favor of J and E. Then writing ρV 2 ∇ · V = ∇ · (ρV 2 V) − V · ∇ ρV 2 and using the definition of P, d dt 1 ρV 2 V − V · ∇ 2 1 ρV 2 + ∇ · 2 1 ρV 2 2 = −∇ · (P · V) + p∇ · V − Π : ∇V + J · E − η J 2 . 22) If we now transform to the Eulerian frame, the last term on the left-hand side cancels, and we obtain the final expression for the rate of change of kinetic energy: ∂ ∂t 1 ρV 2 2 =−∇ · 1 ρV 2 I + P · V 2 flux through surface + p∇ · V + J · E − PV work EM work Π : ∇V − viscous dissipation ηJ2 .

A. Newcomb (unpublished), 1975. The author was privileged to attend these lectures. : The Equation of Motion. Lect. Notes Phys. 1007/978-3-642-00688-3 4 26 Lectures in Magnetohydrodynamics We consider three orientations for S, along each of the three coordinate directions. 4) which is a vector. 5) F = P31 eˆ 1 + P32 eˆ 2 + P33 eˆ 3 . 6) and if S = eˆ 3 , then It therefore takes nine numbers to define the force on the surface S. These are the components of the stress tensor, Pi j . The total surface force acting on a fluid element is the sum of the forces on its faces.

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