Navier-Stokes equation supplementary (2)

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$$\dfrac{1}{r}\dfrac{\partial}{\partial r}\left(r\dfrac{\partial u_{r}}{\partial r}\right) - \dfrac{u_{r}}{r^{2}} = \dfrac{\partial}{\partial r}\left(\dfrac{1}{r}\dfrac{\partial}{\partial r}(r\,u_{r})\right)$$

 

Left term

$$\dfrac{1}{r} \dfrac{\partial}{\partial r}\left(r\dfrac{\partial u_{r}}{\partial r}\right) = \dfrac{1}{r} \dfrac{\partial u_{r}}{\partial r} + \dfrac{1}{r}\cdot r \dfrac{\partial^{2} u_{r}}{\partial r^{2}} \cdots (1) $$

 

Right term

$$\dfrac{\partial}{\partial r}(r u_{r}) = u_{r}\left(\dfrac{\partial}{\partial r}r\right) + r\left(\dfrac{\partial}{\partial r}u_{r}\right)$$

 

$$\dfrac{1}{r}\dfrac{\partial}{\partial r}(r u_{r}) = \dfrac{u_{r}}{r} + \dfrac{\partial}{\partial r}u_{r}$$

 

$$\dfrac{\partial}{\partial}\left(\dfrac{1}{r}\dfrac{\partial}{\partial r}(r u_{r})\right) = \dfrac{\partial}{\partial r}\left(\dfrac{u_{r}}{r}\right) + \dfrac{\partial^{2} u_{r}}{\partial r^{2}} \cdots (2) $$

 

이때 위 식에서 우항의 첫 번째 항을 풀어보면 다음과 같아진다.

$$\dfrac{\partial}{\partial r} \dfrac{u_{r}}{r} = \dfrac{1}{r}\dfrac{\partial u_{r}}{\partial r} - \dfrac{u_{r}}{r^{2}} \cdots (3)$$

 

따라서 식 \((1),\, (2),\, (3)\)을 잘 정리해주면 둘이 같음을 알 수 있다.

 

Reference

Fluid mechanics 3rd edition in SI units p.469