# [[Introduction to Fluid Mechanics - Fox, McDonald.pdf#page=110&selection=0,3,2,45|Introduction to Fluid Mechanics, 94 - 105]] Control volume coordinates are inertial -> $xyz$ are either at rest or moving at constant speed with respect to an "absolute" set of coordinates $XYZ$ $ \vec{F}=\vec{F}_{S}+\vec{F}_{B}=\frac{\partial}{\partial t} \int_{CV} \vec{u}\rho dV+\int_{CS}\vec{u}\rho \vec{u}\cdot d\vec{A} $ For cases with uniform flow at each in/outlet, we can use $ \vec{F}=\vec{F}_{S}+\vec{F}_{B}=\frac{\partial}{\partial t} \int_{CV} \vec{u}\rho dV+ \sum_{CS} \vec{u}\rho \vec{u}\cdot \vec{A} $ typically, $ \vec{F}_{B}=\int_{CV}\rho \vec{g}dV=\vec{W}_{CV}=M\vec{g},\qquad \vec{F}_{S}=\int_{A}-pd\vec{A} $ The minus sign in $\vec{F}_{S}$ is because we always compute pressure forces acting *onto* the CV Linear momentum equations are given by $ F_x = F_{S_x} + F_{B_x} = \frac{\partial}{\partial t}\int_{CV} u\,\rho\,d\mathcal{V} + \int_{CS} u\,\rho\,\vec{V}\cdot d\vec{A} $ $ F_y = F_{S_y} + F_{B_y} = \frac{\partial}{\partial t}\int_{CV} v\,\rho\,d\mathcal{V} + \int_{CS} v\,\rho\,\vec{V}\cdot d\vec{A} $ $ F_z = F_{S_z} + F_{B_z} = \frac{\partial}{\partial t}\int_{CV} w\,\rho\,d\mathcal{V} + \int_{CS} w\,\rho\,\vec{V}\cdot d\vec{A} $ # Lecture Consider a system of fluid with small pieces of fluid moving with linear momentum $\vec{u}_{XYZ}\rho dV$ $ \vec{F}_{sys}=\frac{d}{dt}(m \vec{u})_{sys}=\frac{D}{Dt}\left( \int_{V_{sys}} \vec{u}_{XYZ}\rho dV \right) $ Apply [[ME 308 Lec 8 - Conservation of mass#^e52112|Reynold's Transport Theorem]]: $ \frac{D}{Dt}\left( \int_{V_{sys}} \vec{u}_{XYZ} \rho dV \right)=\boxed{ \underbrace{ \frac{d}{dt} \int_{CV}\vec{u}_{XYZ}\rho dV }_{ \text{rate of change of LM in CV} }+\underbrace{ \int_{CS}\vec{u}_{XYZ}(\rho \vec{u}_{rel}\cdot d\vec{A}) }_{ \text{net flux of LM thru CS} }=F_{B}+F_{S} } $ ![[ME 308 Lec 7 - Dynamic and kinematic viscosity; viscous stresses#^11d9d9]] ![[ME 308 Lec 7 - Dynamic and kinematic viscosity; viscous stresses#^195fa6]] 1. LME is a vector eqn. (1 eqn. for each direction, 3 total) 2. Identify the CV 3. Identify coord. system for FoR 4. Draw FBDs 5. Carefully evaluate $\vec{u}_{XYZ,CV},\vec{u}_{XYZ,CS},\vec{u}_{rel}$ 6. State significant assumptions (general advice)