PINNs-Laminar-Flow-Sudden-Construction-Channel
Paper from what the idea came from: https://arxiv.org/pdf/2002.10558.pdf
Original Implementation (Tensorflow 1): https://github.com/Raocp/PINN-laminar-flow
Paper :
https://github.com/sleshworld/PINNs-Laminar-Flow-Sudden-Construction-Channel/blob/main/VKR_TroshinOf.pdf
Navier-Stokes Equation
$$ \nabla \cdot \textbf{v} = 0 $$
$$ \frac{\partial{\textbf{v}}}{\partial{t}} + (\textbf{v} \cdot \nabla)\textbf{v} = -\frac{1}{\rho}\nabla p + \frac{\mu}{\rho}\nabla^2\textbf{v} + \textbf{g} $$
$$ \textbf{v} = (u, v) $$
This representation of the equations is complicated for PINN because of the presence of second-order derivatives, so we transform:
$$\frac{\partial{\textbf{v}}}{\partial{t}} + (\textbf{v} \cdot \nabla)\textbf{v} = -\frac{1}{\rho}\nabla \sigma + \textbf{g}$$
$$\sigma = p \textbf{I} + \mu(\nabla\textbf{v} + \nabla\textbf{v}^T)$$
, where $\sigma$ - is a stress tensor, and $p = -tr(\sigma)/2$ .
$$\sigma = p \textbf{I} + \mu(\nabla\textbf{v} + \nabla\textbf{v}^T), $$
где $\textbf{I}$ - unit tensor.
$$ \sigma = \begin{bmatrix}
s_{11} & s_{12} \\
s_{21} & s_{22}
\end{bmatrix} $$
$s_{11} = -p + 2\mu \frac{\partial u}{\partial x}$
$s_{22} = -p + 2\mu \frac{\partial v}{\partial y}$
$s_{12} = s_{21} = \mu ( \frac{\partial u}{\partial y} + \frac{\partial v}{\partial x})$
$p = - \frac{s_{11} + s_{22}}{2} = -tr(\sigma)/2$
In this paper, instead of the continuity equation, the velocity is calculated from the current function.
$$[u, v, w]=\nabla \times [0, 0, \psi]$$