Fluid Dynamics

$\rho$ [ $kg/m^3$ ] : fluid density

$\mu$ [ $Pa\cdot s$ ] : dynamic viscosity

$\nu$ [ $N/\rho=m^2/s$ ] : kinematic viscosity

$\vec{V}=(u,v,w)$ [ $m/s$ ] : velocity

$\forall$ : volume

Conservation of mass

For a system:
$\frac{d}{dt}M_{SYS}=0$
For a C.V. (Reynolds transport theorem):\]

$\frac{\partial}{\partial t}\int_{CV}{\rho d\forall}+\int_{CS}{\rho \vec{V}\cdot \hat{n} dA}=0$

Differential form (\delta x\delta y\delta z):

$\frac{\partial}{\partial t}\int_{CV}{\rho d\forall}=\frac{\rho}{\partial t} \delta x\delta y\delta z$

$\frac{\partial}{\partial x}(\rho u)+ \frac{\partial}{\partial y}(\rho v)+ \frac{\partial}{\partial z}(\rho w) =\frac{\rho}{\partial t}$

$\frac{\partial \rho}{\partial t}+\nabla\cdot\rho\vec{V}$

Conservation of momentum

For a system:
$\vec{F}=\frac{d}{dt}\int_{SYS}\vec{V} dm$
For a C.V.:

$\sum{\vec{F}_{CV}}=\frac{\partial}{\partial t}\int_{CV}{\vec{V}\rho d\forall}+ \int_{CS}{\rho\vec{V}\cdot\hat{n}dA=0$

Differential form ( $\delta\vec{F}=\delta m \frac{d}{dt}\vec{V}$ , $\delta m=\rho\delta x\delta y\delta z$ )

Normal stress ( $\sigma$ ):

$\delta F\bot \delta A$

$\sigma_n=\lim_{\delta A=0}\frac{\delta F_n}{\delta A}$

Shearing stress ( $\tau$ ):

$\delta F\parallel\delta A$

$\tau_s=\lim_{\delta A=0}{\frac{\delta F_s}{\delta A}}$

General Equations (momentum & mass)

$\frac{\partial}{\partial x}(\rho u)+ \frac{\partial}{\partial y}(\rho v)+ \frac{\partial}{\partial z}(\rho w) =\frac{\rho}{\partial t}$

$\rho g_x + \frac{\partial \sigma_{xx}}{\partial x} +\frac{\partial \tau_{yx}}{\partial y} +\frac{\partial \tau_{zx}}{\partial z}= \rho(\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}+w\frac{\partial u}{\partial z})$

$\rho g_y + \frac{\partial \tau_{xy}}{\partial x} +\frac{\partial \sigma_{yy}}{\partial y} +\frac{\partial \tau_{zy}}{\partial z}= \rho(\frac{\partial v}{\partial t}+u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}+w\frac{\partial v}{\partial z})$

$\rho g_z + \frac{\partial \tau_{xz}}{\partial x} +\frac{\partial \tau_{yz}}{\partial y} +\frac{\partial \sigma_{zz}}{\partial z}= \rho(\frac{\partial w}{\partial t}+u\frac{\partial w}{\partial x}+v\frac{\partial w}{\partial y}+w\frac{\partial w}{\partial z})$

Navier-Stokes Equation

Newtonian fluids: linear relationship between stresses & rates of deformation.

Incompressible fluid: $\nabla \cdot \vector{V}=0$

Normal stresses:
$\sigma_{aa}=-P+2\mu\frac{\partial v_a}{\partial a}$
$\frac{1}{3}(\sigma_{xx}+\sigma_{yy}+\sigma_{zz})$
Shearing stresses (simmetry):

$\tau_{xy}=\tau_{yx}=+\mu(\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x})$

$\tau_{yz}=\tau_{zy}=+\mu(\frac{\partial v}{\partial z}+\frac{\partial w}{\partial y})$

$\tau_{xz}=\tau_{zx}=+\mu(\frac{\partial u}{\partial z}+\frac{\partial w}{\partial x})$

General equation

$\rho(\frac{\partial u}{\partial t}+ u\frac{\partial u}{\partial x}+ v\frac{\partial u}{\partial y}+ w\frac{\partial u}{\partial z})= \rho g_x- \frac{\partial P}{\partial x}+ \mu(\frac{\partial^2u}{\partial x^2}+ \frac{\partial^2 u}{\partial y^2}+ \frac{\partial^2 u}{\partial z^2})$

$\rho(\frac{\partial v}{\partial t}+ u\frac{\partial v}{\partial x}+ v\frac{\partial v}{\partial y}+ w\frac{\partial v}{\partial z})= \rho g_y- \frac{\partial P}{\partial y}+ \mu(\frac{\partial^2v}{\partial x^2}+ \frac{\partial^2 v}{\partial y^2}+ \frac{\partial^2 v}{\partial z^2})$

$\rho(\frac{\partial w}{\partial t}+ u\frac{\partial w}{\partial x}+ v\frac{\partial w}{\partial y}+ w\frac{\partial w}{\partial z})= \rho g_z- \frac{\partial P}{\partial z}+ \mu(\frac{\partial^2w}{\partial x^2}+ \frac{\partial^2 w}{\partial y^2}+ \frac{\partial^2 w}{\partial z^2})$

Vector notation

$\rho(\frac{\partial \Vec{V}}{\partial t} + (\Vec{V}\cdot \nabla)\Vec{V})=\rho \vec{g} - \nabla P + \mu \nabla^2 \vec{V}^2$

3 equations (momentum) + continuity $\propto$ $u$ , $v$ , $w$ , $P$ , $\rho$