Fluid Dynamics

Navier-Stokes Equation

\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

  1. For a system:

        \[ \frac{d}{dt}M_{SYS}=0 \]

  2. 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

  1. For a system:

        \[\vec{F}=\frac{d}{dt}\int_{SYS}\vec{V} dm\]

  2. 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

  1. Normal stresses:

        \[\sigma_{aa}=-P+2\mu\frac{\partial v_a}{\partial a}\]


  2. 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

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