updated dims

lattice-boltzmann.com/content/1-dimensionless-lbm.md

@@ -12,17 +12,17 @@ toc = true
 +++
 
 
-## The dimensionless Navier-Stokes
+## The non-dimensional Navier-Stokes equations
 
 In order to write the dimensionless lattice Boltzmann equation, we start by reminding the basis of dimensional analysis for fluid flows by writing the dimensionless incompressible Navier--Stokes which are more common in the literature.
 
 The incompressible Navier--Stokes equations reads
 
 $$
-\begin{aligned}
+\begin{align}
 &\bm{\nabla}\cdot\bm{u}=0,\\\\
 &\frac{\partial}{\partial t}\bm{u}+\bm{u}\cdot\bm{\nabla}\bm{u}=-\frac{1}{\rho}\bm{\nabla}p+\nu\bm{\nabla}^2\bm{u},
-\end{aligned}
+\end{align}
 $$
 where $p$, $\rho$, $\nu$, and $\bm{u}$ are respectively the pressure, density, kinematic viscosity, and velocity of the flow.
 
@@ -30,41 +30,45 @@ In order to transform this equation into its dimensionless form a certain amount
 
 We can the write all the above quantities dimensionless form
 $$
+\begin{equation}
 \begin{aligned}
 &\bm{u}^\ast=\frac{\bm{u}}{U}, &p^\ast=\frac{p}{\rho U^2},\\\\
 &t^\ast=\frac{U}{L}t, &\bm{x}^\ast=\frac{\bm{x}}{L},\\\\
 &\frac{\partial}{\partial t^\ast}=\frac{L}{U}\frac{\partial}{\partial t}, &\bm{\nabla}^\ast=L\bm{\nabla}
 \end{aligned}
+\end{equation}
 $$
 Replacing these equations into the Navier--Stokes equations one gets
 $$
-\begin{aligned}
+\begin{align}
 &\bm{\nabla}^\ast\cdot\bm{u}^\ast=0,\\\\
 &\frac{\partial}{\partial t^\ast}\bm{u}^\ast+\bm{u}^\ast\cdot\bm{\nabla}^\ast\bm{u}^\ast=-\bm{\nabla}^\ast p^\ast+\frac{1}{\mathrm{Re}}\bm{\nabla}^{\ast 2}\bm{u}^\ast,
-\end{aligned}
+\end{align}
 $$
 where $\mathrm{Re}=U\cdot L/\nu$ is the _famous_ Reynolds number which represents the ratio of the inertial over the viscous forces.
 
-It must me noted that this non-dimensionnalization procedure heavily relies on the choice of the characteristic timescale of the flow 
-which is this case can be constructed through the characteristic velocity of the flow, $T=L/U$. By chosing the kinematic viscosity instead of the
+It must me noted that this non dimensionalization procedure heavily relies on the choice of the characteristic timescale of the flow 
+which is this case can be constructed through the characteristic velocity of the flow, $T=L/U$. By choosing the kinematic viscosity instead of the
 velocity, the characteristic time of the flow becomes, $T=L^2/\nu$. With this new characteristic time the dimensionless quantities become
 $$
+\begin{equation}
 \begin{aligned}
 &\bm{u}^\ast=\frac{L}{\nu}\bm{u}, &p^\ast=\frac{L^2}{\nu^2\rho}p,\\\\
 &t^\ast=\frac{\nu}{L^2}t, &\bm{x}^\ast=\frac{\bm{x}}{L},\\\\
 &\frac{\partial}{\partial t^\ast}=\frac{L^2}{\nu}\frac{\partial}{\partial t}, &\bm{\nabla}^\ast=L\bm{\nabla}
 \end{aligned}
+\end{equation}
 $$
 which in turn give the following dimensionless Navier--Stokes equations
 $$
-\begin{aligned}
+\begin{align}
 &\bm{\nabla}^\ast\cdot\bm{u}^\ast=0,\\\\
 &\frac{\partial}{\partial t^\ast}\bm{u}^\ast+\bm{u}^\ast\cdot\bm{\nabla}^\ast\bm{u}^\ast=-\bm{\nabla}^\ast p^\ast+\bm{\nabla}^{\ast 2}\bm{u}^\ast,
-\end{aligned}
+\end{align}
 $$
 where we see that no dimensionless number is present anymore. This case represents creeping flows that move only very slowly and will not interest us in this series of tutorials.
 
-## The dimensionless Boltzmann equation
+## The non-dimensional Boltzmann equation
 
 The quantity of interest in the Boltzmann equation is the density probability distribution function
 $$
@@ -74,17 +78,46 @@ which represents the probability of finding a particle at position $\bm{x}$ with
 
 In this tutorial we will only be interested in the Boltzmann-BGK equation, which reads
 $$
+\begin{equation}
 \partial_t f(\bm{x}, \bm{\xi}, t)+\bm{\xi}\cdot \bm{\nabla}f(\bm{x},\bm{\xi}, t)=-\frac{1}{\tau}\left(f(\bm{x}, \bm{\xi}, t)-f^{eq}(\bm{x}, \bm{\xi}, t)\right),
+\end{equation}
 $$
 where $\tau$ is the relaxation time of the fluid, and $f^{eq}$ the equilibrium distribution function which is typically the Maxwell-Boltzmann density distribution
 function. Considering our fluid is made of particles of mass $m$, the Boltzmann distribution reads
 $$
+\begin{equation}
 f^{eq}(\bm{x}, \bm{\xi}, t) = \rho(\bm{x}, t)\left(\frac{m}{2\pi k_B T(\bm{x}, t)}\right)^{D/2}\exp\left(-\frac{m(\bm{\xi}-\bm{u}(\bm{x}, t))^2}{2k_B T(\bm{x}, t)}\right),
+\end{equation}
 $$
 where $\rho$, $\bm{u}$, $T$ are respectively the density, velocity and temperature of the fluid, $k_B$ the Boltzmann constant, and $D$ the dimension of the velocity.
 
-The dimensionless density distribution function is then given by
-
+The dimensions of the density distribution function are therefore
+$$
+\left[M \cdot L^{-2D}\cdot T^D\right]
+$$
+By defining the characteristic velocity of our particles by $\xi_0$ (due to thermal agitation)
+the non-dimensional quantities of interest become
+$$
+\begin{equation}
+\begin{aligned}
+&\bm{\xi}^\ast=\frac{1}{\xi_0}\bm{\xi},&\quad\rho^\ast=\frac{1}{\rho_0}\rho,\\\\
+&t^\ast=\frac{\xi_0}{L}t,&\quad\bm{x}^\ast=\frac{\bm{x}}{L},\\\\
+&\frac{\partial}{\partial t^\ast}=\frac{L}{\xi}\frac{\partial}{\partial t},&\quad\bm{\nabla}^\ast=L\bm{\nabla},\\\\
+&f^\ast=\frac{\xi_0^D}{\rho_0}f,&\quad{f^{eq}}^\ast=\frac{\xi_0^D}{\rho_0}f^{eq},
+\end{aligned}
+\end{equation}
+$$
+With these relations, we obtains the non-dimensional BGK equation
+$$
+\begin{equation}
+\partial_t^\ast f^\ast+\bm{\xi}^\ast\cdot \bm{\nabla}^\ast f^\ast=-\frac{1}{\mathrm{Kn}}\left(f^\ast-{f^{eq}}^\ast\right),
+\end{equation}
+$$
+where the space, time, and microscopic velocity is omitted and where
+$$
+\mathrm{Kn}=\frac{\tau\xi_0}{L},
+$$
+is the Knudsen number.
 
 [^1]: See reference TODO