For fluid modelling of plasma inside the ion thruster RIT-2.5 we use a following set of equations for mass flow, momentum and energy. $$\begin{array}{}& \frac{\mathrm{\partial <!--\; \partial \; -->}{\mathrm{\rho <!--\; \rho \; -->}}_{\mathrm{\alpha <!--\; \alpha \; -->},m}(\mathbf{x},t)}{\mathrm{\partial <!--\; \partial \; -->}t}+\mathrm{\nabla <!--\; \nabla \; -->}\cdot <!--\; \cdot \; -->\left({\mathrm{\rho <!--\; \rho \; -->}}_{\mathrm{\alpha <!--\; \alpha \; -->},m}{\mathbf{u}}_{\mathrm{\alpha <!--\; \alpha \; -->}}(\mathbf{x},t)\right)& =S_{\mathrm{\alpha <!--\; \alpha \; -->}}\end{array}$$ $$\begin{array}{}& \frac{\mathrm{\partial <!--\; \partial \; -->}{\mathrm{\rho <!--\; \rho \; -->}}_{\mathrm{\alpha <!--\; \alpha \; -->},m}{\mathbf{u}}_{\mathrm{\alpha <!--\; \alpha \; -->}}(\mathbf{x},t)}{\mathrm{\partial <!--\; \partial \; -->}t}+\mathrm{\nabla <!--\; \nabla \; -->}\cdot <!--\; \cdot \; -->({\mathrm{\rho <!--\; \rho \; -->}}_{\mathrm{\alpha <!--\; \alpha \; -->},m}{\mathbf{u}}_{\mathrm{\alpha <!--\; \alpha \; -->}}{\mathbf{u}}_{\mathrm{\alpha <!--\; \alpha \; -->}})+\mathrm{\nabla <!--\; \nabla \; -->}\cdot <!--\; \cdot \; -->p& =\frac{{\mathrm{\rho <!--\; \rho \; -->}}_{\mathrm{\alpha <!--\; \alpha \; -->},m}{q}_{\mathrm{\alpha <!--\; \alpha \; -->}}}{m_{\mathrm{\alpha <!--\; \alpha \; -->}}}(\mathbf{E}+{\mathbf{u}}_{\mathrm{\alpha <!--\; \alpha \; -->}}\times <!--\; \times \; -->\mathbf{B})\end{array}$$ $$\begin{array}{}& \frac{\mathrm{\partial <!--\; \partial \; -->}{\mathcal{E}}^{tot}}{\mathrm{\partial <!--\; \partial \; -->}t}+\mathrm{\nabla <!--\; \nabla \; -->}\cdot <!--\; \cdot \; -->({\mathcal{E}}^{tot}+p)\mathbf{u}& =\mathbf{j}\cdot <!--\; \cdot \; -->\mathbf{E}\end{array}$$ The equation of the state (EOS) is given as $$\begin{array}{}& {\mathcal{E}}^{tot}=\frac{p}{(\mathrm{\gamma <!--\; \gamma \; -->}-<!--\; -\; -->1)}+\frac{1}{2}{\mathrm{\rho <!--\; \rho \; -->}}_m\Vert <!--\; \Vert \; -->\mathbf{u}{\Vert <!--\; \Vert \; -->}^2\end{array}$$

The Rayleigh-Bénard convection involves a variable-temperature incompressible fluid. The temperature variation within the fluid temperature affects the momentum balance influenced by a buoyancy term. This leads non-linear coupling of Navier-Stokes equation giving rise to convection-diffusion equation for temperature.

The Gierer-Meinhardt model is a reaction-diffusion system of the activator-inhibitor type that appears to account for many important types of pattern formation and morphogenesis observed in development.