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+---
+title: "Reynolds number"
+date: 2021-05-04
+categories:
+- Physics
+- Fluid mechanics
+- Fluid dynamics
+layout: "concept"
+---
+
+The [Navier-Stokes equations](/know/concept/navier-stokes-equations/)
+are infamously tricky to solve,
+so we would like a way to qualitatively predict
+the behaviour of a fluid without needing the flow $\va{v}$.
+Consider the main equation:
+
+$$\begin{aligned}
+    \pdv{\va{v}}{t} + (\va{v} \cdot \nabla) \va{v}
+    = - \frac{\nabla p}{\rho} + \nu \nabla^2 \va{v}
+\end{aligned}$$
+
+In this case, the gravity term $\va{g}$
+has been absorbed into the pressure term:
+$p \to p\!+\!\rho \Phi$,
+where $\Phi$ is the gravitational scalar potential,
+i.e. $\va{g} = - \nabla \Phi$.
+
+Let us introduce the dimensionless variables $\va{v}'$, $\va{r}'$, $t'$ and $p'$,
+where $U$ and $L$ are respectively a characteristic velocity and length
+of the system at hand:
+
+$$\begin{aligned}
+    \va{v} = U \va{v}'
+    \qquad
+    \va{r} = L \va{r}'
+    \qquad
+    t = \frac{L}{U} t'
+    \qquad
+    p = \rho U^2 p'
+\end{aligned}$$
+
+In this non-dimenionsalization, the differential operators are scaled as follows:
+
+$$\begin{aligned}
+    \pdv{}{t}
+    = \frac{U}{L} \pdv{}{t'}
+    \qquad \quad
+    \nabla
+    = \frac{1}{L} \nabla'
+\end{aligned}$$
+
+Putting everything into the main Navier-Stokes equation then yields:
+
+$$\begin{aligned}
+    \frac{U^2}{L} \pdv{\va{v}}{t'} + \frac{U^2}{L} (\va{v}' \cdot \nabla') \va{v}'
+    = - \frac{U^2}{L} \nabla' p' + \frac{U \nu}{L^2} \nabla'^2 \va{v}'
+\end{aligned}$$
+
+After dividing out $U^2/L$,
+we arrive at the form of the original equation again:
+
+$$\begin{aligned}
+    \pdv{\va{v}}{t'} + (\va{v}' \cdot \nabla') \va{v}'
+    = - \nabla' p' + \frac{\nu}{U L} \nabla'^2 \va{v}'
+\end{aligned}$$
+
+The constant factor of the last term
+leads to the definition of the **Reynolds number** $\mathrm{Re}$:
+
+$$\begin{aligned}
+    \boxed{
+        \mathrm{Re}
+        \equiv \frac{U L}{\nu}
+    }
+\end{aligned}$$
+
+If we choose $U$ and $L$ appropriately for a given system,
+the Reynolds number allows us to predict the general trends.
+It can be regarded as the inverse of an "effective viscosity":
+when $\mathrm{Re}$ is large, viscosity only has a minor role,
+but when $\mathrm{Re}$ is small, it dominates the dynamics.
+
+Another way is thus to see the Reynolds number
+as the characteristic ratio between the advective term
+(see [material derivative](/know/concept/material-derivative/))
+to the [viscosity](/know/concept/viscosity/) term,
+since $\va{v} \sim U$:
+
+$$\begin{aligned}
+     \mathrm{Re}
+     \approx \frac{\big| (\va{v} \cdot \nabla) \va{v} \big|}{\big| \nu \nabla^2 \va{v} \big|}
+     \approx \frac{U^2 / L}{\nu U / L^2}
+     = \frac{U L}{\nu}
+\end{aligned}$$
+
+In other words, $\mathrm{Re}$
+describes the relative strength of intertial and viscous forces.
+Returning to the dimensionless Navier-Stokes equation:
+
+$$\begin{aligned}
+    \pdv{\va{v}}{t'} + (\va{v}' \cdot \nabla') \va{v}'
+    = - \nabla' p' + \frac{1}{\mathrm{Re}} \nabla'^2 \va{v}'
+\end{aligned}$$
+
+For large $\mathrm{Re} \gg 1$,
+we can neglect the latter term,
+such that redimensionalizing yields:
+
+$$\begin{aligned}
+    \pdv{\va{v}}{t} + (\va{v} \cdot \nabla) \va{v}
+    = - \frac{\nabla p}{\rho}
+\end{aligned}$$
+
+Which is simply the main [Euler equation](/know/concept/euler-equations/)
+for an ideal fluid, i.e. a fluid without viscosity.
+
+
+
+## Stokes flow
+
+A notable case is so-called **Stokes flow** or **creeping flow**,
+meaning flow at $\mathrm{Re} \ll 1$.
+In this limit, the Navier-Stokes equations can be linearized:
+since $\mathrm{Re}$ is the advective-to-viscous ratio,
+$\mathrm{Re} \ll 1$ implies that we can ignore the advective term, leaving:
+
+$$\begin{aligned}
+    \boxed{
+        \pdv{\va{v}}{t}
+        = - \frac{\nabla p}{\rho} + \nu \nabla^2 \va{v}
+    }
+\end{aligned}$$
+
+This equation is called the **unsteady Stokes equation**.
+Usually, however, such flows are assumed to be steady
+(i.e. time-invariant), leading to the **steady Stokes equation**,
+with $\eta = \rho \nu$:
+
+$$\begin{aligned}
+    \boxed{
+        \nabla p
+        = \eta \nabla^2 \va{v}
+    }
+\end{aligned}$$
+
+This equation is much easier to solve than the full Navier-Stokes equation
+thanks to being linear,
+and has some interesting properties, such as time-reversibility.
+
+
+
+## References
+1.  B. Lautrup,
+    *Physics of continuous matter: exotic and everyday phenomena in the macroscopic world*, 2nd edition,
+    CRC Press.
+2.  R. Fitzpatrick,
+    [Dimensionless numbers in incompressible flow](https://farside.ph.utexas.edu/teaching/336L/Fluid/node17.html),
+    University of Texas.