---
# System prepended metadata

title: '***Computational Hydraulics***'
tags: [HW2.3]

---

---
tags: HW2.3
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# ***Computational Hydraulics***
## **Group 01:** N88071053 洪豪男 N88091011 黃彥鈞
## **HW2.3:** Please compare the above (advection-diffusion) equaiton with the Navier-Stokes equatsions

**Navier-Stokes equatsions (NSE) ( for incompressible fluid in X-direction )**

$\rho(\dfrac{\partial{u}}{\partial{t}} +u\dfrac{\partial u}{\partial x}) =-\dfrac{\partial{p}}{\partial{x}} + {\mu}\dfrac{{\partial}^2{u}}{\partial{x}^2}+\rho g_x$
$\Rightarrow\dfrac{\partial{u}}{\partial{t}}+u\dfrac{\partial u}{\partial x} = -\dfrac{1}{\rho}\dfrac{\partial{p}}{\partial{x}} + \dfrac{\mu}{\rho}\dfrac{{\partial}^2{u}}{\partial{x}^2}+g_x$
**Advection-diffusion equation (ADE)**

$\dfrac{\partial u}{\partial t} + u\dfrac{\partial u}{\partial x} =  D\dfrac{\partial^2 u}{\partial x^2}$

**compare**

$\Rightarrow \left\{
\begin{array}{l}
\dfrac{\partial{u}}{\partial{t}}+u\dfrac{\partial u}{\partial x} =   \dfrac{\mu}{\rho}\dfrac{{\partial}^2{u}}{\partial{x}^2}-\dfrac{1}{\rho}\dfrac{\partial{p}}{\partial{x}}+g_x\\
\dfrac{\partial u}{\partial t} + u\dfrac{\partial u}{\partial x} =  D\dfrac{\partial^2 u}{\partial x^2}
\end{array}\right .$
----
ADE: $\frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x}=  D\frac{\partial^2 u}{\partial x^2}$

. $\frac{\partial u}{\partial t}$ : 局部加速度項

. $u\frac{\partial u}{\partial x}$ : 對流加速度項 

. $D\frac{\partial^2 u}{\partial x^2}$ : 擴散項

NSE: $\frac{\partial{u}}{\partial{t}}+u\frac{\partial u}{\partial x} = \frac{1}{\rho}(-\frac{\partial{p}}{\partial{x}} + {\mu}\frac{{\partial}^2{u}}{\partial{x}^2})$

. $\frac{\partial u}{\partial t}$ : 局部加速度項

. $u\frac{\partial u}{\partial x}$ : 對流加速度項 

. $\frac{\partial{p}}{\partial{x}}$ : 壓力項 (亦可表示成 : 動量來源的內部力量)

. ${\mu}\frac{{\partial}^2{u}}{\partial{x}^2}$ : 黏滯力項 (實際上為"動量的**擴散**")

比較ADE與NSE的方程式，兩者相似的地方為，左式可表示為對流項，右式表視為擴散項，這也表示NSE也是一對流擴散方程
