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• The pipe radius is much smaller than the pipe length (i.e. $R/L\ll1$):

The transformed equation then reads: $$\bar{u}\partial_{\bar{x}}\bar{u}+\bar{v}\partial_{\bar{y}}\bar{u}=-\partial_{\bar{x}}\bar{p}+\frac{1}{\mathrm{Re}}\frac{L}{R}\partial_{\bar{y}}^{2}\bar{u}$$ Here we have a problem because the term $\frac{1}{\mathrm{Re}}\frac{L}{R}$ could be very large and a properly scaled equation only has coefficients $O(1)$ or smaller. So we require a rescaling of the $\bar{x}$ coordinate, $\bar{v}$ velocity and $\bar{p}$ pressure: $$\hat{x}=\bar{x}\left(\frac{R}{L}\right)^{\alpha}\quad\hat{v}=\bar{v}\left(\frac{R}{L}\right)^{-\alpha}\quad\hat{p}=\bar{p}\left(\frac{R}{L}\right)^{\beta}$$ This choice of rescaled quantities ensures that the continuity equation remains of the form: $$\partial_{\hat{x}}\bar{u}+\partial_{\bar{y}}\hat{v}=0$$ The Navier-Stokes equations in terms of the rescaled quantities yields: $$\bar{u}\partial_{\hat{x}}\bar{u}+\hat{v}\partial_{\bar{y}}\bar{u}=-\partial_{\hat{x}}\hat{p}+\frac{1}{\mathrm{Re}}\partial_{\bar{y}}^{2}\bar{u}$$ which is properly scaled with coefficients of $O(1)$ or smaller when we take the values $\alpha=-1,\,\beta=0$. This indicates the pressure scale didn't need any rescaling but the length and velocities scales have been redefined: $$\hat{x}=\bar{x}\frac{L}{R}=\frac{x}{R}\quad\hat{v}=\bar{v}\frac{R}{L}=\bar{v}\frac{V}{U}=\frac{v}{U}\quad\hat{p}=\bar{p}=\frac{p}{\rho U^{2}}$$ and we see that the characteristic length and velocity scale for respectively $x$ and $v$ isn't $L$ and $V$ as assumed at the beginning but $R$ and $U$.

• The pipe radius is much larger than the pipe length (i.e. $R/L\gg1$):

The transformed equation then reads: $$\bar{u}\partial_{\bar{x}}\bar{u}+\bar{v}\partial_{\bar{y}}\bar{u}=-\partial_{\bar{x}}\bar{p}+\frac{1}{\mathrm{Re}}\frac{R}{L}\partial_{\bar{x}}^{2}\bar{u}$$ Likewise to the previous case, $\frac{1}{\mathrm{Re}}\frac{R}{L}$ could be very large and requires a rescaling. Except this time we require a rescaling of the $\bar{y}$ coordinate, $\bar{u}$ velocity and $\bar{p}$ pressure: $$\hat{y}=\bar{y}\left(\frac{R}{L}\right)^{\alpha}=\frac{y}{L}\quad\hat{u}=\bar{u}\left(\frac{R}{L}\right)^{-\alpha}\quad\hat{p}=\bar{p}\left(\frac{R}{L}\right)^{\beta}$$ This choice of rescaled quantities again ensures that the continuity equation remains of the form: $$\partial_{\bar{x}}\hat{u}+\partial_{\hat{y}}\bar{v}=0$$ The Navier-Stokes equations in terms of the rescaled quantities yields: $$\hat{u}\partial_{\bar{x}}\hat{u}+\bar{v}\partial_{\hat{y}}\hat{u}=-\partial_{\bar{x}}\hat{p}+\frac{1}{\mathrm{\hat{\mathrm{Re}}}}\partial_{\bar{x}}^{2}\hat{u}$$ which is properly scaled with coefficients of $O(1)$ or smaller when we take the values $\alpha=1\,\beta=-2$. This indicates the length, velocities and pressure scales have been redefined: $$\hat{y}=\bar{y}\frac{R}{L}=\frac{y}{L}\quad\hat{u}=\bar{u}\frac{L}{R}=\bar{u}\frac{U}{V}=\frac{u}{V}\quad\hat{p}=\bar{p}\left(\frac{L}{R}\right)^{2}=\bar{p}\left(\frac{U}{V}\right)^{2}=\frac{p}{\rho V^{2}}$$ and we see that the characteristic length, velocity and pressure scales for respectively $x$, $v$ and $p$ isn't $R$, $U$, $\rho U^{2}$ as assumed at the beginning but $L$, $V$ and $\rho V^{2}$.

• The pipe radius is much smaller than the pipe length (i.e. $R/L\ll1$):

  The transformed equation then reads: $$\bar{u}\partial_{\bar{x}}\bar{u}+\bar{v}\partial_{\bar{y}}\bar{u}=-\partial_{\bar{x}}\bar{p}+\frac{1}{\mathrm{Re}}\frac{L}{R}\partial_{\bar{y}}^{2}\bar{u}$$ Here we have a problem because the term $\frac{1}{\mathrm{Re}}\frac{L}{R}$ could be very large and a properly scaled equation only has coefficients $O(1)$ or smaller. So we require a rescaling of the $\bar{x}$ coordinate, $\bar{v}$ velocity and $\bar{p}$ pressure: $$\hat{x}=\bar{x}\left(\frac{R}{L}\right)^{\alpha}\quad\hat{v}=\bar{v}\left(\frac{R}{L}\right)^{-\alpha}\quad\hat{p}=\bar{p}\left(\frac{R}{L}\right)^{\beta}$$ This choice of rescaled quantities ensures that the continuity equation remains of the form: $$\partial_{\hat{x}}\bar{u}+\partial_{\bar{y}}\hat{v}=0$$ The Navier-Stokes equations in terms of the rescaled quantities yields: $$\bar{u}\partial_{\hat{x}}\bar{u}+\hat{v}\partial_{\bar{y}}\bar{u}=-\partial_{\hat{x}}\hat{p}+\frac{1}{\mathrm{Re}}\partial_{\bar{y}}^{2}\bar{u}$$ which is properly scaled with coefficients of $O(1)$ or smaller when we take the values $\alpha=-1,\,\beta=0$. This indicates the pressure scale didn't need any rescaling but the length and velocities scales have been redefined: $$\hat{x}=\bar{x}\frac{L}{R}=\frac{x}{R}\quad\hat{v}=\bar{v}\frac{R}{L}=\bar{v}\frac{V}{U}=\frac{v}{U}\quad\hat{p}=\bar{p}=\frac{p}{\rho U^{2}}$$ and we see that the characteristic length and velocity scale for respectively $x$ and $v$ isn't $L$ and $V$ as assumed at the beginning but $R$ and $U$.

• The pipe radius is much larger than the pipe length (i.e. $R/L\gg1$):

  The transformed equation then reads: $$\bar{u}\partial_{\bar{x}}\bar{u}+\bar{v}\partial_{\bar{y}}\bar{u}=-\partial_{\bar{x}}\bar{p}+\frac{1}{\mathrm{Re}}\frac{R}{L}\partial_{\bar{x}}^{2}\bar{u}$$ Likewise to the previous case, $\frac{1}{\mathrm{Re}}\frac{R}{L}$ could be very large and requires a rescaling. Except this time we require a rescaling of the $\bar{y}$ coordinate, $\bar{u}$ velocity and $\bar{p}$ pressure: $$\hat{y}=\bar{y}\left(\frac{R}{L}\right)^{\alpha}=\frac{y}{L}\quad\hat{u}=\bar{u}\left(\frac{R}{L}\right)^{-\alpha}\quad\hat{p}=\bar{p}\left(\frac{R}{L}\right)^{\beta}$$ This choice of rescaled quantities again ensures that the continuity equation remains of the form: $$\partial_{\bar{x}}\hat{u}+\partial_{\hat{y}}\bar{v}=0$$ The Navier-Stokes equations in terms of the rescaled quantities yields: $$\hat{u}\partial_{\bar{x}}\hat{u}+\bar{v}\partial_{\hat{y}}\hat{u}=-\partial_{\bar{x}}\hat{p}+\frac{1}{\mathrm{\hat{\mathrm{Re}}}}\partial_{\bar{x}}^{2}\hat{u}$$ which is properly scaled with coefficients of $O(1)$ or smaller when we take the values $\alpha=1\,\beta=-2$. This indicates the length, velocities and pressure scales have been redefined: $$\hat{y}=\bar{y}\frac{R}{L}=\frac{y}{L}\quad\hat{u}=\bar{u}\frac{L}{R}=\bar{u}\frac{U}{V}=\frac{u}{V}\quad\hat{p}=\bar{p}\left(\frac{L}{R}\right)^{2}=\bar{p}\left(\frac{U}{V}\right)^{2}=\frac{p}{\rho V^{2}}$$ and we see that the characteristic length, velocity and pressure scales for respectively $x$, $v$ and $p$ isn't $R$, $U$, $\rho U^{2}$ as assumed at the beginning but $L$, $V$ and $\rho V^{2}$.

where $D$ is the diffusion coefficient and $t$ is the time. As seen, there is no length scale $L$ involved as it is determined completely by the diffusion dynamics of the system. For an example of such a system see my answer to this question.

where $D$ is the diffusion coefficient and $t$ is the time. As seen, there is no length scale $L$ involved as it is determined completely by the diffusion dynamics of the system. For an example of such a system see my answer to this question.

Similarly, let's transform the Navier-Stokes equations ($x$-component only to keep it short): $$\boldsymbol{u}\cdot\boldsymbol{\nabla u}=-\frac{1}{\rho}\boldsymbol{\nabla}p+\nu\triangle\boldsymbol{u}$$ $$\bar{u}\partial_{\bar{x}}\bar{u}+\bar{v}\partial_{\bar{y}}\bar{u}=-\partial_{\bar{x}}\bar{p}+\frac{1}{\mathrm{Re}}\left[\frac{R}{L}\partial_{\bar{x}}^{2}\bar{u}+\frac{L}{R}\partial_{\bar{y}}^{2}\bar{u}\right]$$ We see here the Reynolds number occuring naturally as part of the scaling process. However, depending on the geometric ratio $R/L$, the may require rescaling. Consider the two cases:

The transformed equation then reads: $$u\partial_{\bar{x}}\bar{u}+\bar{v}\partial_{\bar{y}}\bar{u}=-\partial_{\bar{x}}\bar{p}+\frac{1}{\mathrm{Re}}\frac{L}{R}\partial_{\bar{y}}^{2}\bar{u}$$ Here we have a problem because the term $\frac{1}{\mathrm{Re}}\frac{L}{R}$ could be very large and a properly scaled equation only has coefficients $O(1)$ or smaller. So we require a rescaling of the $\bar{x}$ coordinate, $\bar{v}$ velocity and $\bar{p}$ pressure: $$\hat{x}=\bar{x}\left(\frac{R}{L}\right)^{\alpha}\quad\hat{v}=\bar{v}\left(\frac{R}{L}\right)^{-\alpha}\quad\hat{p}=\bar{p}\left(\frac{R}{L}\right)^{\beta}$$ This choice of rescaled quantities ensures that the continuity equation remains of the form: $$\partial_{\hat{x}}\bar{u}+\partial_{\bar{y}}\hat{v}=0$$ The Navier-Stokes equations in terms of the rescaled quantities yields: $$\bar{u}\partial_{\bar{x}}\bar{u}+\hat{v}\partial_{\bar{y}}\bar{u}=-\partial_{\hat{x}}\hat{p}+\frac{1}{\mathrm{Re}}\partial_{\bar{y}}^{2}\bar{u}$$ which is properly scaled with coefficients of $O(1)$ or smaller when we take the values $\alpha=-1,\,\beta=0$. This indicates the pressure scale didn't need any rescaling but the length and velocities scales have been redefined: $$\hat{x}=\bar{x}\frac{L}{R}=\frac{x}{R}\quad\hat{v}=\bar{v}\frac{R}{L}=\bar{v}\frac{V}{U}=\frac{v}{U}\quad\hat{p}=\bar{p}=\frac{p}{\rho U^{2}}$$ and we see that the characteristic length and velocity scale for respectively $x$ and $v$ isn't $L$ and $V$ as assumed at the beginning but $R$ and $U$.

The transformed equation then reads: $$u\partial_{\bar{x}}\bar{u}+\bar{v}\partial_{\bar{y}}\bar{u}=-\partial_{\bar{x}}\bar{p}+\frac{1}{\mathrm{Re}}\frac{R}{L}\partial_{\bar{x}}^{2}\bar{u}$$ Likewise to the previous case, $\frac{1}{\mathrm{Re}}\frac{R}{L}$ could be very large and requires a rescaling. Except this time we require a rescaling of the $\bar{y}$ coordinate, $\bar{u}$ velocity and $\bar{p}$ pressure: $$\hat{y}=\bar{y}\left(\frac{R}{L}\right)^{\alpha}=\frac{y}{L}\quad\hat{u}=\bar{u}\left(\frac{R}{L}\right)^{-\alpha}\quad\hat{p}=\bar{p}\left(\frac{R}{L}\right)^{\beta}$$ This choice of rescaled quantities again ensures that the continuity equation remains of the form: $$\partial_{\bar{x}}\hat{u}+\partial_{\hat{y}}\bar{v}=0$$ The Navier-Stokes equations in terms of the rescaled quantities yields: $$\hat{u}\partial_{\bar{x}}\hat{u}+\bar{v}\partial_{\hat{y}}\hat{u}=-\partial_{\bar{x}}\hat{p}+\frac{1}{\mathrm{\hat{\mathrm{Re}}}}\partial_{\bar{x}}^{2}\hat{u}$$ which is properly scaled with coefficients of $O(1)$ or smaller when we take the values $\alpha=1\,\beta=-2$. This indicates the length, velocities and pressure scales have been redefined: $$\hat{y}=\bar{y}\frac{R}{L}=\frac{y}{L}\quad\hat{u}=\bar{u}\frac{L}{R}=\bar{u}\frac{U}{V}=\frac{u}{V}\quad\hat{p}=\bar{p}\left(\frac{L}{R}\right)^{2}=\bar{p}\left(\frac{U}{V}\right)^{2}=\frac{p}{\rho V^{2}}$$ and we see that the characteristic length, velocity and pressure scales for respectively $x$, $v$ and $p$ isn't $R$, $U$, $\rho U^{2}$ as assumed at the beginning but $L$, $V$ and $\rho V^{2}$.

Similarly, let's transform the Navier-Stokes equations ($x$-component only to keep it short): $$\boldsymbol{u}\cdot\boldsymbol{\nabla u}=-\frac{1}{\rho}\boldsymbol{\nabla}p+\nu\triangle\boldsymbol{u}$$ $$\bar{u}\partial_{\bar{x}}\bar{u}+\bar{v}\partial_{\bar{y}}\bar{u}=-\partial_{\bar{x}}\bar{p}+\frac{1}{\mathrm{Re}}\left[\frac{R}{L}\partial_{\bar{x}}^{2}\bar{u}+\frac{L}{R}\partial_{\bar{y}}^{2}\bar{u}\right]$$ We see here the Reynolds number occuring naturally as part of the scaling process. However, depending on the geometric ratio $R/L$, the equations may require rescaling. Consider the two cases:

The transformed equation then reads: $$\bar{u}\partial_{\bar{x}}\bar{u}+\bar{v}\partial_{\bar{y}}\bar{u}=-\partial_{\bar{x}}\bar{p}+\frac{1}{\mathrm{Re}}\frac{L}{R}\partial_{\bar{y}}^{2}\bar{u}$$ Here we have a problem because the term $\frac{1}{\mathrm{Re}}\frac{L}{R}$ could be very large and a properly scaled equation only has coefficients $O(1)$ or smaller. So we require a rescaling of the $\bar{x}$ coordinate, $\bar{v}$ velocity and $\bar{p}$ pressure: $$\hat{x}=\bar{x}\left(\frac{R}{L}\right)^{\alpha}\quad\hat{v}=\bar{v}\left(\frac{R}{L}\right)^{-\alpha}\quad\hat{p}=\bar{p}\left(\frac{R}{L}\right)^{\beta}$$ This choice of rescaled quantities ensures that the continuity equation remains of the form: $$\partial_{\hat{x}}\bar{u}+\partial_{\bar{y}}\hat{v}=0$$ The Navier-Stokes equations in terms of the rescaled quantities yields: $$\bar{u}\partial_{\hat{x}}\bar{u}+\hat{v}\partial_{\bar{y}}\bar{u}=-\partial_{\hat{x}}\hat{p}+\frac{1}{\mathrm{Re}}\partial_{\bar{y}}^{2}\bar{u}$$ which is properly scaled with coefficients of $O(1)$ or smaller when we take the values $\alpha=-1,\,\beta=0$. This indicates the pressure scale didn't need any rescaling but the length and velocities scales have been redefined: $$\hat{x}=\bar{x}\frac{L}{R}=\frac{x}{R}\quad\hat{v}=\bar{v}\frac{R}{L}=\bar{v}\frac{V}{U}=\frac{v}{U}\quad\hat{p}=\bar{p}=\frac{p}{\rho U^{2}}$$ and we see that the characteristic length and velocity scale for respectively $x$ and $v$ isn't $L$ and $V$ as assumed at the beginning but $R$ and $U$.

The transformed equation then reads: $$\bar{u}\partial_{\bar{x}}\bar{u}+\bar{v}\partial_{\bar{y}}\bar{u}=-\partial_{\bar{x}}\bar{p}+\frac{1}{\mathrm{Re}}\frac{R}{L}\partial_{\bar{x}}^{2}\bar{u}$$ Likewise to the previous case, $\frac{1}{\mathrm{Re}}\frac{R}{L}$ could be very large and requires a rescaling. Except this time we require a rescaling of the $\bar{y}$ coordinate, $\bar{u}$ velocity and $\bar{p}$ pressure: $$\hat{y}=\bar{y}\left(\frac{R}{L}\right)^{\alpha}=\frac{y}{L}\quad\hat{u}=\bar{u}\left(\frac{R}{L}\right)^{-\alpha}\quad\hat{p}=\bar{p}\left(\frac{R}{L}\right)^{\beta}$$ This choice of rescaled quantities again ensures that the continuity equation remains of the form: $$\partial_{\bar{x}}\hat{u}+\partial_{\hat{y}}\bar{v}=0$$ The Navier-Stokes equations in terms of the rescaled quantities yields: $$\hat{u}\partial_{\bar{x}}\hat{u}+\bar{v}\partial_{\hat{y}}\hat{u}=-\partial_{\bar{x}}\hat{p}+\frac{1}{\mathrm{\hat{\mathrm{Re}}}}\partial_{\bar{x}}^{2}\hat{u}$$ which is properly scaled with coefficients of $O(1)$ or smaller when we take the values $\alpha=1\,\beta=-2$. This indicates the length, velocities and pressure scales have been redefined: $$\hat{y}=\bar{y}\frac{R}{L}=\frac{y}{L}\quad\hat{u}=\bar{u}\frac{L}{R}=\bar{u}\frac{U}{V}=\frac{u}{V}\quad\hat{p}=\bar{p}\left(\frac{L}{R}\right)^{2}=\bar{p}\left(\frac{U}{V}\right)^{2}=\frac{p}{\rho V^{2}}$$ and we see that the characteristic length, velocity and pressure scales for respectively $x$, $v$ and $p$ isn't $R$, $U$, $\rho U^{2}$ as assumed at the beginning but $L$, $V$ and $\rho V^{2}$.