Proof: Assume that Ω {\displaystyle \Omega } is an arbitrary fixed control volume. Then u n = 0 {\displaystyle u_{n}=0} and the derivative can be taken inside the integral to give ∫ Ω ∂ ∂ t ( ρ   η )   dV ≥ − ∫ ∂ Ω ρ   η   ( v ⋅ n )   dA − ∫ ∂ Ω q ⋅ n T   dA + ∫ Ω ρ   s T   dV   . {\displaystyle \int _{\Omega }{\frac {\partial }{\partial t}}(\rho ~\eta )~{\text{dV}}\geq -\int _{\partial \Omega }\rho ~\eta ~(\mathbf {v} \cdot \mathbf {n} )~{\text{dA}}-\int _{\partial \Omega }{\cfrac {\mathbf {q} \cdot \mathbf {n} }{T}}~{\text{dA}}+\int _{\Omega }{\cfrac {\rho ~s}{T}}~{\text{dV}}~.} Using the divergence theorem, we get ∫ Ω ∂ ∂ t ( ρ   η )   dV ≥ − ∫ Ω ∇ ⋅ ( ρ   η   v )   dV − ∫ Ω ∇ ⋅ ( q T )   dV + ∫ Ω ρ   s T   dV   . {\displaystyle \int _{\Omega }{\frac {\partial }{\partial t}}(\rho ~\eta )~{\text{dV}}\geq -\int _{\Omega }{\boldsymbol {\nabla }}\cdot (\rho ~\eta ~\mathbf {v} )~{\text{dV}}-\int _{\Omega }{\boldsymbol {\nabla }}\cdot \left({\cfrac {\mathbf {q} }{T}}\right)~{\text{dV}}+\int _{\Omega }{\cfrac {\rho ~s}{T}}~{\text{dV}}~.} Since Ω {\displaystyle \Omega } is arbitrary, we must have ∂ ∂ t ( ρ   η ) ≥ − ∇ ⋅ ( ρ   η   v ) − ∇ ⋅ ( q T ) + ρ   s T   . {\displaystyle {\frac {\partial }{\partial t}}(\rho ~\eta )\geq -{\boldsymbol {\nabla }}\cdot (\rho ~\eta ~\mathbf {v} )-{\boldsymbol {\nabla }}\cdot \left({\cfrac {\mathbf {q} }{T}}\right)+{\cfrac {\rho ~s}{T}}~.} Expanding out ∂ ρ ∂ t   η + ρ   ∂ η ∂ t ≥ − ∇ ( ρ η ) ⋅ v − ρ   η   ( ∇ ⋅ v ) − ∇ ⋅ ( q T ) + ρ   s T {\displaystyle {\frac {\partial \rho }{\partial t}}~\eta +\rho ~{\frac {\partial \eta }{\partial t}}\geq -{\boldsymbol {\nabla }}(\rho _{\eta })\cdot \mathbf {v} -\rho ~\eta ~({\boldsymbol {\nabla }}\cdot \mathbf {v} )-{\boldsymbol {\nabla }}\cdot \left({\cfrac {\mathbf {q} }{T}}\right)+{\cfrac {\rho ~s}{T}}} or, ∂ ρ ∂ t   η + ρ   ∂ η ∂ t ≥ − η   ∇ ρ ⋅ v − ρ   ∇ η ⋅ v − ρ   η   ( ∇ ⋅ v ) − ∇ ⋅ ( q T ) + ρ   s T {\displaystyle {\frac {\partial \rho }{\partial t}}~\eta +\rho ~{\frac {\partial \eta }{\partial t}}\geq -\eta ~{\boldsymbol {\nabla }}\rho \cdot \mathbf {v} -\rho ~{\boldsymbol {\nabla }}\eta \cdot \mathbf {v} -\rho ~\eta ~({\boldsymbol {\nabla }}\cdot \mathbf {v} )-{\boldsymbol {\nabla }}\cdot \left({\cfrac {\mathbf {q} }{T}}\right)+{\cfrac {\rho ~s}{T}}} or, ( ∂ ρ ∂ t + ∇ ρ ⋅ v + ρ   ∇ ⋅ v )   η + ρ   ( ∂ η ∂ t + ∇ η ⋅ v ) ≥ − ∇ ⋅ ( q T ) + ρ   s T   . {\displaystyle \left({\frac {\partial \rho }{\partial t}}+{\boldsymbol {\nabla }}\rho \cdot \mathbf {v} +\rho ~{\boldsymbol {\nabla }}\cdot \mathbf {v} \right)~\eta +\rho ~\left({\frac {\partial \eta }{\partial t}}+{\boldsymbol {\nabla }}\eta \cdot \mathbf {v} \right)\geq -{\boldsymbol {\nabla }}\cdot \left({\cfrac {\mathbf {q} }{T}}\right)+{\cfrac {\rho ~s}{T}}~.} Now, the material time derivatives of ρ {\displaystyle \rho } and η {\displaystyle \eta } are given by ρ  ̇ = ∂ ρ ∂ t + ∇ ρ ⋅ v   ;     η  ̇ = ∂ η ∂ t + ∇ η ⋅ v   . {\displaystyle {\dot {\rho }}={\frac {\partial \rho }{\partial t}}+{\boldsymbol {\nabla }}\rho \cdot \mathbf {v} ~;~~{\dot {\eta }}={\frac {\partial \eta }{\partial t}}+{\boldsymbol {\nabla }}\eta \cdot \mathbf {v} ~.} Therefore, ( ρ  ̇ + ρ   ∇ ⋅ v )   η + ρ   η  ̇ ≥ − ∇ ⋅ ( q T ) + ρ   s T   . {\displaystyle \left({\dot {\rho }}+\rho ~{\boldsymbol {\nabla }}\cdot \mathbf {v} \right)~\eta +\rho ~{\dot {\eta }}\geq -{\boldsymbol {\nabla }}\cdot \left({\cfrac {\mathbf {q} }{T}}\right)+{\cfrac {\rho ~s}{T}}~.} From the conservation of mass ρ  ̇ + ρ   ∇ ⋅ v = 0 {\displaystyle {\dot {\rho }}+\rho ~{\boldsymbol {\nabla }}\cdot \mathbf {v} =0} . Hence, ρ   η  ̇ ≥ − ∇ ⋅ ( q T ) + ρ   s T   . {\displaystyle {\rho ~{\dot {\eta }}\geq -{\boldsymbol {\nabla }}\cdot \left({\cfrac {\mathbf {q} }{T}}\right)+{\cfrac {\rho ~s}{T}}~.}} Proof: Using the identity ∇ ⋅ ( φ   v ) = φ   ∇ ⋅ v + v ⋅ ∇ φ {\displaystyle {\boldsymbol {\nabla }}\cdot (\varphi ~\mathbf {v} )=\varphi ~{\boldsymbol {\nabla }}\cdot \mathbf {v} +\mathbf {v} \cdot {\boldsymbol {\nabla }}\varphi } in the Clausius-Duhem inequality, we get ρ   η  ̇ ≥ − ∇ ⋅ ( q T ) + ρ   s T or ρ   η  ̇ ≥ − 1 T   ∇ ⋅ q − q ⋅ ∇ ( 1 T ) + ρ   s T   . {\displaystyle \rho ~{\dot {\eta }}\geq -{\boldsymbol {\nabla }}\cdot \left({\cfrac {\mathbf {q} }{T}}\right)+{\cfrac {\rho ~s}{T}}\qquad {\text{or}}\qquad \rho ~{\dot {\eta }}\geq -{\cfrac {1}{T}}~{\boldsymbol {\nabla }}\cdot \mathbf {q} -\mathbf {q} \cdot {\boldsymbol {\nabla }}\left({\cfrac {1}{T}}\right)+{\cfrac {\rho ~s}{T}}~.} Now, using index notation with respect to a Cartesian basis e j {\displaystyle \mathbf {e} _{j}} , ∇ ( 1 T ) = ∂ ∂ x j ( T − 1 )   e j = − ( T − 2 )   ∂ T ∂ x j   e j = − 1 T 2   ∇ T   . {\displaystyle {\boldsymbol {\nabla }}\left({\cfrac {1}{T}}\right)={\frac {\partial }{\partial x_{j}}}\left(T^{-1}\right)~\mathbf {e} _{j}=-\left(T^{-2}\right)~{\frac {\partial T}{\partial x_{j}}}~\mathbf {e} _{j}=-{\cfrac {1}{T^{2}}}~{\boldsymbol {\nabla }}T~.} Hence, ρ   η  ̇ ≥ − 1 T   ∇ ⋅ q + 1 T 2   q ⋅ ∇ T + ρ   s T or ρ   η  ̇ ≥ − 1 T ( ∇ ⋅ q − ρ   s ) + 1 T 2   q ⋅ ∇ T   . {\displaystyle \rho ~{\dot {\eta }}\geq -{\cfrac {1}{T}}~{\boldsymbol {\nabla }}\cdot \mathbf {q} +{\cfrac {1}{T^{2}}}~\mathbf {q} \cdot {\boldsymbol {\nabla }}T+{\cfrac {\rho ~s}{T}}\qquad {\text{or}}\qquad \rho ~{\dot {\eta }}\geq -{\cfrac {1}{T}}\left({\boldsymbol {\nabla }}\cdot \mathbf {q} -\rho ~s\right)+{\cfrac {1}{T^{2}}}~\mathbf {q} \cdot {\boldsymbol {\nabla }}T~.} Recall the balance of energy ρ   e  ̇ − σ : ∇ v + ∇ ⋅ q − ρ   s = 0 ⟹ ρ   e  ̇ − σ : ∇ v = − ( ∇ ⋅ q − ρ   s )   . {\displaystyle \rho ~{\dot {e}}-{\boldsymbol {\sigma }}:{\boldsymbol {\nabla }}\mathbf {v} +{\boldsymbol {\nabla }}\cdot \mathbf {q} -\rho ~s=0\qquad \implies \qquad \rho ~{\dot {e}}-{\boldsymbol {\sigma }}:{\boldsymbol {\nabla }}\mathbf {v} =-({\boldsymbol {\nabla }}\cdot \mathbf {q} -\rho ~s)~.} Therefore, ρ   η  ̇ ≥ 1 T ( ρ   e  ̇ − σ : ∇ v ) + 1 T 2   q ⋅ ∇ T ⟹ ρ   η  ̇   T ≥ ρ   e  ̇ − σ : ∇ v + q ⋅ ∇ T T   . {\displaystyle \rho ~{\dot {\eta }}\geq {\cfrac {1}{T}}\left(\rho ~{\dot {e}}-{\boldsymbol {\sigma }}:{\boldsymbol {\nabla }}\mathbf {v} \right)+{\cfrac {1}{T^{2}}}~\mathbf {q} \cdot {\boldsymbol {\nabla }}T\qquad \implies \qquad \rho ~{\dot {\eta }}~T\geq \rho ~{\dot {e}}-{\boldsymbol {\sigma }}:{\boldsymbol {\nabla }}\mathbf {v} +{\cfrac {\mathbf {q} \cdot {\boldsymbol {\nabla }}T}{T}}~.} Rearranging, ρ   ( e  ̇ − T   η  ̇ ) − σ : ∇ v ≤ − q ⋅ ∇ T T   . {\displaystyle {\rho ~({\dot {e}}-T~{\dot {\eta }})-{\boldsymbol {\sigma }}:{\boldsymbol {\nabla }}\mathbf {v} \leq -{\cfrac {\mathbf {q} \cdot {\boldsymbol {\nabla }}T}{T}}~.}} ← Balance of energy Nonlinear finite elements Objectivity of constitutive relations → ← Balance of energy Nonlinear finite elements Objectivity of constitutive relations →
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