Conservation of Mass (Continuity Equation)

For an incompressible fluid

• $\nabla \cdot \mathbf{u}=0$

For a compressible fluid

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

where:

• $\rho =\text{velocity field}$
• $\mathbf{u}=(u,v,w)=\text{velocity field}$

Conservation of Momentum (Navier-Stokes Equation)

• $\rho \left(\frac{\partial \mathbf{u}}{\partial t}+(\mathbf{u}\cdot \nabla )\mathbf{u}\right)=-\nabla p+\mu {\nabla }^2\mathbf{u}+\left(\frac{\mu }{3}+\lambda \right)\nabla (\nabla \cdot \mathbf{u})+\rho \mathbf{f}$

where:

• $p=\text{pressure}$
• $\mu =\text{dynamic viscosity}$
• $\lambda =\text{second viscosity (often combined into bulk viscosity}$
• $\mathbf{f}=\text{external body force per unit mass (e.g. gravity)}$

Incompressible Navier-Stokes Simplification

For. constant density and incompressibility (𝛻·𝐮 = 0)

• $\frac{\partial \mathbf{u}}{\partial t}+(\mathbf{u}\cdot \nabla )\mathbf{u}=-\frac{1}{\rho }\nabla p+\nu {\nabla }^2\mathbf{u}+\mathbf{f}$

where:

• $\nu =\mu /\rho \text{ is the kinematic viscosity}$