19revey / pinn_granular_segregation Goto Github PK

• [1] General Model For Segregation Forces in Flowing Granular Mixtures
• [2] Diffusion, mixing, and segregation in confined granular flows
• [3] On Mixing and Segregation: From Fluids and Maps to Granular Solids and Advection–Diffusion Systems

• https://github.com/nanditadoloi/PINN
• https://github.com/omniscientoctopus/Physics-Informed-Neural-Networks
• https://github.com/maziarraissi/PINNs

An advection-diffusion transport equation has been successfully used to model the segregation. Within this continuum framework, the concentration of species $i$ can be expressed as

$$\frac{\partial c_i}{\partial t} + {\nabla \cdot (\pmb u c_i)}={\nabla \cdot (D\nabla c_i)}.$$

With assumption of incompressible flow and negligible vertical acceleration, the above equation in the $z$ direction can be written as

$$\frac{\partial c_i}{\partial t} +\frac{\partial (w+w_{i})c_i}{\partial z}=\frac{\partial}{\partial z} \Big( D\frac{\partial c_i}{\partial z} \Big),$$

or, rearranging, as

$$ w_{i}c_i-D\frac{\partial c_i}{\partial z}=0, $$

where $w_{i}$ is the segregation velocity relative to the bulk velocity $w$.

The full model prediction is compared to the previous model by Schlick et al. 2015 with different values of $\lambda$.

$$\frac{d_c}{d_z}=\frac{1}{\lambda}c_l(1-c_l)$$