## 1st Law $\frac{dE_{\text{cv}}}{dt}=\dot{Q}-\dot{W}+\sum_{\text{in}}\dot{m}_{\text{in}}\left( h+\frac{v^{2}}{2}+gz \right)_{\text{in}}-\sum_{\text{out}}\dot{m}_{\text{out}}\left( h+\frac{v^{2}}{2}+gz \right)_{\text{out}}$ $h = u + \frac{P}{\rho}$ ## Second Law $\frac{dS_{\text{cv}}}{dt}=\frac{\dot{Q}}{T_{\text{s}}}+\dot{S}_{\text{gen}}+\sum_{\text{in}} \dot{m}_{\text{in}}S_{\text{in}} - \sum_{\text{out}} \dot{m}_{\text{out}} S_{\text{out}}$ ## Mass Conservation $\frac{d}{dt} \int_{CV}\rho \,dV + \int_{CS} \rho \vec{v} \cdot \hat{n} dA = 0$ $\frac{\delta \rho}{\delta t} + \vec{\nabla}(\rho \vec{v}) = 0$ ### Incompressible The divergence of the velocity field of an incompressible fluid is zero: $\nabla \cdot\vec{u} = \frac{ \partial u_{x} }{ \partial x } +\frac{ \partial u_{y} }{ \partial y } +\frac{ \partial u_{z} }{ \partial z } = 0$ ### Reynold’s transport theorem $\frac{d}{dt} B_{sys} = \frac{d}{dt} \int_{CV}\beta \rho\,dV + \int_{CS} \beta \rho \vec{v} \cdot \hat{n} dA$ ## Linear Momentum $\sum \vec{F} = \frac{d}{dt} \int_{CV}\vec{v} \rho dV + \sum_{out}\dot{m}_{i}\vec{v}_{i} - \sum_{in} \dot{m}_{i} \vec{v}_{i}$ ## Navier Stokes $\rho \frac{D\vec{v}}{Dt} = -\vec{\nabla}P + \rho \vec{g} + \mu \nabla^{2} \vec{v}$ Assumes incompressible and constant viscosity (newtonian fluid) Derived from momentum conservation, and terms correspond to inertial, pressure, gravity, and viscous forces, from left to right. ### Expanded form $ \begin{align} \rho\left( \frac{\partial v_{x}}{\partial t} + v_{x} \frac{\partial v_{x}}{\partial x} + v_{y} \frac{\partial v_{x}}{\partial y} + v_{z} \frac{\partial v_{x}}{\partial z} \right) &= -\frac{\partial p}{\partial x} + \rho g_{x} + \mu\left( \frac{\partial^{2}v_{x}}{\partial x^{2}} + \frac{\partial^{2}v_{x}}{\partial y^{2}} + \frac{\partial^{2}v_{x}}{\partial z^{2}} \right) \\ \rho\left( \frac{\partial v_{y}}{\partial t} + v_{x} \frac{\partial v_{y}}{\partial x} + v_{y} \frac{\partial v_{y}}{\partial y} + v_{z} \frac{\partial v_{y}}{\partial z} \right) &= -\frac{\partial p}{\partial y} + \rho g_{y} + \mu\left( \frac{\partial^{2}v_{y}}{\partial x^{2}} + \frac{\partial^{2}v_{y}}{\partial y^{2}} + \frac{\partial^{2}v_{y}}{\partial z^{2}} \right) \\ \rho\left( \frac{\partial v_{z}}{\partial t} + v_{x} \frac{\partial v_{z}}{\partial x} + v_{y} \frac{\partial v_{z}}{\partial y} + v_{z} \frac{\partial v_{z}}{\partial z} \right) &= -\frac{\partial p}{\partial z} + \rho g_{z} + \mu\left( \frac{\partial^{2}v_{z}}{\partial x^{2}} + \frac{\partial^{2}v_{z}}{\partial y^{2}} + \frac{\partial^{2}v_{z}}{\partial z^{2}} \right) \end{align} $ see equation sheet for all expansions in different coordinate systems | Zero Condition | Term to drop | | ------------------------------- | ---------------------------------------- | | Steady state | $\frac{ \partial }{ \partial t } = 0$ | | Symmetry in direction $\hat{i}$ | $\frac{ \partial }{ \partial x_{i} }=0$ | | Negligible flow in $\hat{i}$ | $v_{i} = 0$ | | Inviscid (air) | $\mu \nabla^{2}v=0$ | ### Material Derivative $\frac{D}{Dt} = \left( \frac{\delta}{\delta t} + v_{x} \frac{\delta}{\delta x} + v_{y} \frac{\delta}{\delta y} + v_{z} \frac{\delta}{\delta z} \right) = \frac{\delta}{\delta t} + (\vec{v} \cdot \vec{\nabla})$ ### Bournoulli For inviscid fluids, drop $\mu \nabla^{2}v=0$ and integrate N-$S$ to get $\frac{P}{\rho} + \frac{v^{2}}{2}+gz =$ constant along stream