There are three references in this model.

\rho_w=\frac{\dot{m}_p}{4\pi\|\mathbf{r'}\|^2\mathbf{v}_w}

Outside the computation box:

\mathbf{a_r}=\mathbf{a_i}-2\mathbf{\Omega}\times\mathbf{v}_w-\mathbf{\Omega}\times(\mathbf{\Omega}\times\mathbf{r'})-\frac{d\mathbf{\Omega}}{dt}\times\mathbf{r'}

\mathbf{r'}-\mathbf{r}=\mathbf{L}

We can get \mathbf{u} field in the reference of co-rotating primary after integration. So that we can get the consistent flux on the boundary of the computation box.

Inside the computation box (include the boundaries): First compute the \mathbf{v} field in the reference of co-rotatng companion.

\mathbf{v}=\mathbf{u}+\mathbf{\Omega}\times\mathbf{r}

Special attention should be paid to the boundaries since they may swell or shrink along the orbital motion. Which introduce additional flux.

\frac{\partial \rho}{\partial t}+\nabla\cdot(\rho\mathbf{v})=0

\frac{(\partial \rho \mathbf{v})}{\partial t}+\nabla\cdot(\rho\mathbf{v}\mathbf{v})=-\nabla p-\rho\nabla\phi-2\rho\mathbf{\Omega}\times\mathbf{v}-\rho\mathbf{\Omega}\times(\mathbf{\Omega}\times\mathbf{r})-\rho\frac{d\mathbf{\Omega}}{dt}\times\mathbf{r}

\frac{\partial E}{\partial t}+\nabla\cdot[(E+p)\mathbf{v}]=-\rho \dot{Q}_{cool}+\rho\frac{\partial\phi}{\partial t}

E=\rho(\frac{1}{2}\mathbf{v}^2+e)

e=\frac{p}{(\gamma-1)\rho}

T=\frac{p}{\rho R}

To include orbital change:

\frac{1}{2} m(\dot{r}^2+r^2\dot{\theta}^2)+Q(r)=C+\int_{\mathbf{s}(t)}^{\mathbf{s}(t+\triangle t)}\mathbf{f}(\mathbf{s})\cdot \mathbf{ds}

\frac{d(mr^2\dot{\theta})}{dt}=L(\mathbf{s})

m may be changing. \mathbf{s}(t) is the relative location of companion to the primary, and is basically in the inertial reference. It's initial condition is exactly \mathbf{L}.

Therefore, f(\mathbf{s}) and L(\mathbf{s}) is important to incorporate drag force and drag moment. These will give coupled orbital equation.

Question: In low-mass companion binary systems, what is the order in all these phenomenons below :

thermal non-uniformity, orbital change, mass change, frictional force, rotation of the companion and even EM field

Remark: Resolution is important as computational matter, order is the "resolution" in physics and modelling.