Using the continuity equations, the rate of entropy production may now be written: ∂ ∂ s c ∂ ∂ t = J u ⋅ ⋅ ∇ ∇ 1 T + J ρ ρ ⋅ ⋅ ∇ ∇ − − μ μ T = ∑ ∑ α α J α α ⋅ ⋅ ∇ ∇ f α α and, incorporating the phenomenological equations: ∂ ∂ s c ∂ ∂ t = ∑ ∑ α α ∑ ∑ β β L α α β β ( ∇ ∇ f α α ) ⋅ ⋅ ( ∇ ∇ f β β )