# Definition Newton's 2^nd law ($\vec{F} = m\vec{a}$) for a fluid Assuming Newtonian fluid: $ \underbrace{\frac{\partial \vec{u}}{\partial t}}_{\text{unsteady}}+\underbrace{(\vec{u}\cdot\nabla)\vec{u}}_{\text{convective acceleration}}=\overbrace{-\nabla p}^{\text{pressure gradient}}+\overbrace{\nu\nabla^2\vec{u}}^{\text{viscous diffusion}} $ Where: $ \begin{align} \vec{u} &:\text{any vector field (usually velocity)} \end{align} $ Euler equation, assumes incompressible ($\rho =$ constant), inviscid flow ($\nu = 0$): $\frac{\partial \vec{u}}{\partial t}+(\vec{u}\cdot\nabla)\vec{u}=-\nabla p $ May be written with material derivative: $ \rho\frac{D\vec{u}}{Dt}=-\nabla p+\mu\nabla^2\vec{u} $ # Differential Form ## Conservation of Mass For a *system*: $ \frac{Dm_{sys}}{Dt}=0 $ To extend this to a *control volume* we use the [[Reynolds Transport Theorem]]: $ \frac{\partial}{\partial t}\int_{CV}\rho dV+\int_{CS}\rho (\vec{u}\cdot\vec{n})dA=0 $ Considering a small element $dxdydz$: $ \frac{\partial}{\partial t}\int_{CV}\rho dV=\frac{\partial\rho}{\partial t}dxdydz $ Net flux in $x$: $ \frac{\partial}{\partial x}(\rho u)dxdydz $ Substituting into RTT: $ \frac{\partial\rho}{\partial t}+\frac{\partial}{\partial x}(\rho u)+\frac{\partial}{\partial y}(\rho v)+\frac{\partial}{\partial z}(\rho w)=0 $ Rewritten as: $ \frac{\partial\rho}{\partial t}+\nabla\cdot\rho\vec{u}=0 $