u {\displaystyle \mathbf {u} } is the flow velocity vector, with components in an N-dimensional space u 1 , u 2 , … , u N {\displaystyle u_{1},u_{2},\dots ,u_{N}} , D v D t = ∂ v ∂ t + u ⋅ ∇ v {\displaystyle {D\mathbf {v} \over Dt}={\partial \mathbf {v} \over \partial t}+\mathbf {u} \cdot \nabla \mathbf {v} } , for a generic function (or field) v {\displaystyle \mathbf {v} } denotes its material derivative in time with respect to the advective field u {\displaystyle \mathbf {u} } and ∇ {\displaystyle \nabla } denotes the gradient with respect to space, ⋅ {\displaystyle \cdot } denotes the scalar product, ∇ {\displaystyle \nabla } is the nabla operator, here used to represent the specific thermodynamic work gradient (first equation), and ∇ ⋅ u {\displaystyle \nabla \cdot \mathbf {u} } is the flow velocity divergence (second equation), w {\displaystyle w} is the specific (with the sense of per unit mass) thermodynamic work, the internal source term. g {\displaystyle \mathbf {g} } represents body accelerations (per unit mass) acting on the continuum, for example gravity, inertial accelerations, electric field acceleration, and so on.