I want to use lattice Boltzmann method to solve advection-diffusion in three-dimensional space. In fact, my problem is related to drug release in human blood vessels and as a results, I'm interested to use lattice Boltzmann to solve this equation:

$$\frac{\partial \phi}{\partial t} + \vec{u} \cdot \nabla \phi = \nabla \cdot (D \nabla \phi)$$

Where $\phi$ is concentration of drug, $\vec{u}$ is blood velocity derived from other lattice Boltzmann flow solver (for a moment let's forgot about the coupling of advection-diffusion and Navier-Stokes and assume that $\vec{u}$ is already a known vector field), and $D$ is diffusion coefficient of drug in blood.

Due to that $\phi$ is just a scalar variable, I use lattice Boltzmann equation with single-relaxation time collision operator like this:

$$f_{i}(\vec{r}+\vec{c}_{i}\Delta t,t+\Delta t) = f_{i}(\vec{r},t) - \frac{f_{i}-f^{eq}_{i}}{\tau}$$

Where $f_{i}$ is distribution function and $f^{eq}_{i}$ is the equilibrium distribution function derived from Maxwell-Boltzmann distribution and expanded up to first order like this:

$$f^{eq}_{i} = \omega_{i}\phi(1+\frac{\vec{u} \cdot \vec{c}_{i}}{c_{s}^{2}})$$

Where $\omega_{i}$ is the ith weight in the lattice and $c_{s}$ is the speed of sound of the lattice.

For my practical applications, $D$ (diffusion coefficient of drug in blood) is in the order of ~$10^{-11}$ $\frac{\mathrm{m}^{2}}{\mathrm{s}}$. The blood velocity is in the order of 0.1 $\frac{\mathrm{m}}{\mathrm{s}}$. The characteristic length of the blood vessel (L), which is its diameter, is 3 mm. As a result, the Peclet number could be calculated as:

$$Pe = \frac{u L}{D} = \frac{0.1 \times 3 \times 10^{-3}}{10^{-11}} = 3\times 10^{7}$$

Typically in lattice Boltzmann simulations, people use $Mach < 0.02$ to ensure stability but due to my extremely complicated case, I want to push the borders and say Mach number could have some freedom to be as high as 0.1. So, $Mach = 0.1$.

On the other hand, we know that $\tau$ the relxation time of lattice Boltzmann scheme is related to diffusion coefficient as:

$$\tau = \frac{\Delta t}{2} + \frac{1}{c_{s}} \frac{D}{c_{s}} = \frac{\Delta t}{2} + \frac{L}{c_{s}} \frac{Mach}{Pe}$$

Due to calculated Pe number, assumed Mach number, and characteristic length of the blood vessel and the fact that at least for stability of this lattice Boltzmann scheme, we need to have $\tau > 0.5 \Delta t$, we would take $\tau$ as $\tau = 0.501 \Delta t$ (which is really small but still acceptable due to linear equilibrium distribution function used in this study). Finally, we have:

$$\tau = 0.501 \Delta t = 0.5 \Delta t + \frac{L}{c_{s}} \frac{Mach}{Pe}$$

$$0.001 = \frac{3\times 10^{-3}\times 0.1}{\frac{1}{\sqrt{3}}\Delta x \times 3 \times 10^{7}}$$

$$\Delta x = 1.732 \times 10^{-8}\mathrm{m}$$

Where $\Delta x$ is the mesh size.

Now, if we assume that the length of the cylinder which represent the blood vessel is 30 mm, we could estimate the total number of meshes that are needed to fill this cylinder as:

$$N = \frac{V}{\Delta x^{3}} = \frac{\frac{\pi}{4} L^{2} H}{\Delta x^{3}} = 4.081 \times 10^{16}$$

It means, I need $4.081 \times 10^{16}$ meshes to fill this cylinder and make sure my simulation by using this scheme will remain stable (still there is a doubt that my simulation will remain stable or not with even this unrealistic number of meshes). So, finally my question: Am I missing something here or something is wrong with my calculations and theoretical scheme, or really lattice Boltzmann method is unsuitable for this purpose (simulating drug diffusion in blood vessels in the presence of blood perfusion)?

I appreciate any insight or suggestion or useful direction.


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