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16. Januar 2024

Navier-Stokes equations

Navier-Stokes equations

\frac{𝜕 u}{𝜕 t} + \frac{1}{V_{F}} \{u A_{x} \frac{𝜕 u}{𝜕 x} + ν A_{y} \frac{𝜕 u}{𝜕 y} + w A_{z} \frac{𝜕 u}{𝜕 z}\} + ξ \frac{ν^{2} A_{y}}{x V_{F}} = - \frac{1}{ρ} \frac{𝜕 p}{𝜕 x} + G_{x} + f_{x} - b_{x} - \frac{R_{S O R}}{ρ V_{F}} (u - u_{w} - δ u_{s}) [11]

\frac{𝜕 ν}{𝜕 t} + \frac{1}{V_{F}} \{u A_{x} \frac{𝜕 v}{𝜕 x} + ν A_{y} \frac{𝜕 v}{𝜕 y} + w A_{z} \frac{𝜕 v}{𝜕 z}\} + ξ \frac{u ν A_{y}}{x V_{F}} = - R \frac{1}{ρ} \frac{𝜕 p}{𝜕 y} + G_{y} + f_{y} - b_{y} - \frac{R_{S O R}}{ρ V_{F}} (v - v_{w} - δ v_{s}) [12]

\frac{𝜕 w}{𝜕 t} + \frac{1}{V_{F}} \{u A_{x} \frac{𝜕 w}{𝜕 x} + ν A_{y} \frac{𝜕 w}{𝜕 y} + w A_{z} \frac{𝜕 w}{𝜕 z}\} = - \frac{1}{ρ} \frac{𝜕 p}{𝜕 z} + G_{z} + f_{z} - b_{z} - \frac{R_{S O R}}{ρ V_{F}} (w - w_{w} - δ w_{s}) [13]

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