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$$\int \int {\int }_V\frac{D\rho }{Dt}dv=\int \int {\int }_V\frac{\partial \rho }{\partial t}dv+\int {\int }_{S=\partial V}\rho u\cdot nds=0$$

$$\int \int {\int }_V\frac{Du}{Dt}dv=\int \int {\int }_V\frac{\partial \left(\rho u\right)}{\partial t}dv+\int {\int }_{S=\partial V}[\rho \overline{\overline{u\otimes u}}]nds=\int {\int }_{S=\partial V}[\overline{\overline{\tau }}-p\overline{\overline{I_3}}]nds+\int \int {\int }_V\rho fdv$$

$$\int \int {\int }_V\frac{D\left(\rho e\right)}{Dt}dv=\int {\int }_{S=\partial V}[\overline{\overline{\tau }}-p\overline{\overline{I_3}}]u\cdot nds+\int \int {\int }_V\rho f\cdot udv+\int {\int }_{S=\partial V}\stackrel{.}{q}\cdot nds+\int \int {\int }_{V}rdv$$

$$\frac{\partial \rho }{\partial t}+\nabla \cdot \left(\rho u\right)=0$$

$$\frac{\partial \left(\rho u\right)}{\partial t}+\nabla \cdot [\rho \overline{\overline{u\otimes u}}]=-\nabla p+\nabla \cdot \overline{\overline{\tau }}+\rho f$$

$$\frac{\partial \left(\rho e\right)}{\partial t}+\nabla \cdot \left(\right(\rho e+p\left)u\right)=\nabla \cdot \left(\overline{\overline{\tau }}u\right)+\rho f\cdot u+\nabla \cdot \stackrel{.}{q}+r$$

$$\frac{\partial \rho }{\partial t}+\frac{\partial \left(\rho u_i\right)}{\partial x_i}=0$$

$$\frac{\partial \left(\rho u_i\right)}{\partial t}+\frac{\partial [\rho u_i{u}_j]}{\partial x_j}=-\frac{\partial p}{\partial x_i}+\frac{\partial {\tau }_{ij}}{\partial x_j}+\rho f_i$$

$$\frac{\partial \left(\rho e\right)}{\partial t}+(\rho e+p)\frac{\partial u_i}{\partial x_i}=\frac{\partial \left({\tau }_{ij}{u}_j\right)}{\partial x_i}+\rho f_i{u}_i+\frac{\partial \left(\stackrel{.}{q_i}\right)}{\partial x_i}+r$$