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2023-12-29

Navier-Stokes Equations

Navier-Stokes Equations

\frac{𝜕 u}{𝜕 t} + \frac{1}{V_{F}} \{{uA}_{x} \frac{𝜕 u}{𝜕 x} + {νA}_{y} \frac{𝜕 u}{𝜕 y} + {wA}_{z} \frac{𝜕 u}{𝜕 z}\} + ξ \frac{ν^{2} A_{y}}{{xV}_{F}} = - \frac{1}{ρ} \frac{𝜕 p}{𝜕 x} + G_{x} + f_{x} - b_{x} - \frac{R_{SOR}}{{ρV}_{F}} (u - u_{w} - {δu}_{s}) [11]

\frac{𝜕 ν}{𝜕 t} + \frac{1}{V_{F}} \{{uA}_{x} \frac{𝜕 v}{𝜕 x} + {νA}_{y} \frac{𝜕 v}{𝜕 y} + {wA}_{z} \frac{𝜕 v}{𝜕 z}\} + ξ \frac{{uνA}_{y}}{{xV}_{F}} = - R \frac{1}{ρ} \frac{𝜕 p}{𝜕 y} + G_{y} + f_{y} - b_{y} - \frac{R_{SOR}}{{ρV}_{F}} (v - v_{w} - {δv}_{s}) [12]

\frac{𝜕 w}{𝜕 t} + \frac{1}{V_{F}} \{{uA}_{x} \frac{𝜕 w}{𝜕 x} + {νA}_{y} \frac{𝜕 w}{𝜕 y} + {wA}_{z} \frac{𝜕 w}{𝜕 z}\} = - \frac{1}{ρ} \frac{𝜕 p}{𝜕 z} + G_{z} + f_{z} - b_{z} - \frac{R_{SOR}}{{ρV}_{F}} (w - w_{w} - {δw}_{s}) [13]

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