The Equation of Motion

In (2.9) the external force, ${\bf F}$, consists of several terms. The importance of these terms depends on the particular situation being modelled. The dominant term for magnetised plasmas is the magnetic force, the Lorentz force, given by

$${\bf j} \times {\bf B}.$$

In addition, the gravitational force $\rho {\bf g}$ is frequently included as well as a viscous force. The actual form of the viscous term is complicated (see Braginskii, 19 ) but it may be approximated by a kinematic viscosity of the form

$$\rho \nu \nabla^{2}{\bf v},$$

for an incompressible flow. Here $\nu$ is the coefficient of kinematic viscosity which Spitzer (1962) gives as

$$\rho \nu = 2.21 \times 10^{-16}{T^{5/2}\over \hbox{ln} \Lambda} \hbox{kg m}^{-1}\hbox{s}^{-1}.$$

Thus, (2.9) becomes

$$\rho {D{\bf v}\over Dt} = - \nabla p + {\bf j} \times {\bf B} + \rho {\bf g} + \rho \nu \nabla^{2}{\bf v}.$$ (2.14)

Note that the Lorentz force couples the fluid equations to the electromagnetic equations.