I am trying to nondimensionalize the dual-hyperbolic two temperature model, given as
\begin{align} C_e(T_e) \frac{\partial T_e}{\partial t} &= -\frac{\partial q_e}{\partial x} - G(T_e) (T_e - T_l) + S(x,t) \tag{1} \label{ttm}\\ C_l(T_l) \frac{\partial T_l}{\partial t} &= -\frac{\partial q_l}{\partial x} + G(T_e) (T_e - T_l) \tag{2}\\ q_e + \tau_e\frac{\partial q_e}{\partial t} &= -K_e(T_e, T_l) \frac{\partial T_e}{\partial x} \tag{3}\\ q_l + \tau_l\frac{\partial q_l}{\partial t} &= -K_l(T_l) \frac{\partial T_l}{\partial x} \tag{4} \label{ttm4} \end{align}
Where $T_{e,l}$ is the temperature of the electron and lattice subsystems, $C_{e,l}, K_{e,l}, G > 0$ are some parameters of the system that depend on the temperature nonlinearly, $\tau_{e,l}$ are constant parameters with dimensions of time, $x$ is distance, $t$ is time.
Introducing the following variables \begin{align} \bar{T_e} = \frac{T_e}{T^*}, \bar{T_l} = \frac{T_l}{T^*}, \bar{q_e} = \frac{q_e}{q^*}, \bar{q_l} = \frac{q_l}{q^*}, \bar{x} = \frac{x}{x^*}, \bar{t} = \frac{t}{t^*} \tag{5} \end{align} And the functions \begin{align} \bar{C}_e(\bar{T}_e) = C_e(\bar{T_e}T^*)/C^*, \bar{C}_l(\bar{T}_l) = C_l(\bar{T_l}T^*)/C^* \\ \bar{K}_e(\bar{T}_e, \bar{T_l}) = K_e(\bar{T_e}T^*, \bar{T_l}T^*)/K^*, \bar{K}_l(\bar{T}_l) = K_l(\bar{T_l}T^*)/K^*\\ \bar{G}(\bar{T}_e) = G(\bar{T_e}T^*)/G^*, \bar{S}(\bar{x},\bar{t}) = S(\bar{x}x^*,\bar{t}t^*)/S^* \tag{6} \end{align}
The system \eqref{ttm}-\eqref{ttm4} can be rewritten as \begin{align} C^*\bar{C}_e(\bar{T}_e) \frac{T^*}{t^*}\frac{\partial \bar{T_e}}{\partial \bar{t}} &= -\frac{q^*}{x^*}\frac{\partial \bar{q_e}}{\partial \bar{x}} - T^*G^*\bar{G}(\bar{T}_e) (\bar{T_e} - \bar{T_l}) + S^*\bar{S}(\bar{x},\bar{t})\\ C^*\bar{C}_l(\bar{T}_l) \frac{T^*}{t^*}\frac{\partial \bar{T_l}}{\partial \bar{t}} &= -\frac{q^*}{x^*}\frac{\partial \bar{q_l}}{\partial \bar{x}} + T^*G^*\bar{G}(\bar{T}_e) (\bar{T_e} - \bar{T_l})\\ q^*\bar{q_e} + \tau_e \frac{q^*}{t^*}\frac{\partial \bar{q_e}}{\partial \bar{t}} &= -K^*\bar{K}_e(\bar{T}_e, \bar{T}_l) \frac{T^*}{x^*}\frac{\partial \bar{T_e}}{\partial \bar{x}}\\ q^*\bar{q_l} + \tau_l \frac{q^*}{t^*}\frac{\partial \bar{q_l}}{\partial \bar{t}} &= -K^*\bar{K}_l(\bar{T}_l) \frac{T^*}{x^*}\frac{\partial \bar{T_l}}{\partial \bar{x}} \end{align}
After some manipulation, we reach the form: \begin{align} \frac{\partial \bar{T_e}}{\partial \bar{t}} &= -\frac{q^*t^*}{x^*C^*T^*\bar{C}_e(\bar{T}_e)}\frac{\partial \bar{q_e}}{\partial \bar{x}} - \frac{t^*G^*\bar{G}(\bar{T}_e)}{C^*\bar{C}_e(\bar{T}_e)}(\bar{T_e} - \bar{T_l}) + \frac{t^*S^*\bar{S}(\bar{x},\bar{t})}{T^*C^*\bar{C}_e(\bar{T}_e)}\\ \frac{\partial \bar{T_l}}{\partial \bar{t}} &= -\frac{q^*t^*}{x^*C^*T^*\bar{C}_l(\bar{T}_l)}\frac{\partial \bar{q_l}}{\partial \bar{x}} + \frac{t^*G^*\bar{G}(\bar{T}_e)}{C^*\bar{C}_l(\bar{T}_l)}(\bar{T_e} - \bar{T_l})\\ \bar{q_e} + \frac{\tau_e}{t^*}\frac{\partial \bar{q_e}}{\partial \bar{t}} &= -\frac{K^*}{q^*}\frac{T^*}{x^*}\bar{K}_e(\bar{T}_e, \bar{T}_l)\frac{\partial \bar{T_e}}{\partial \bar{x}}\\ \bar{q_l} + \frac{\tau_l}{t^*}\frac{\partial \bar{q_l}}{\partial \bar{t}} &= -\frac{K^*}{q^*}\frac{T^*}{x^*}\bar{K}_l(\bar{T}_l) \frac{\partial \bar{T_l}}{\partial \bar{x}} \end{align}
The initial conditions are as follows: \begin{align} T_{e,l}(t=0, x) = T_0 \\ q_{e,l}(t=0, x) = 0 \end{align} And boundary conditions \begin{align} q_{e,l}(t, x=0) = 0 \end{align}
However, I am confused and completely stuck after this. My questions are:
- Was it correct to use the same $T^*$ for $T_e$ and $T_l$, same $K^*$ for $K_e$ and $K_l$, etc? Is using $K^*$, $G^*$, etc. even needed?
Does the fact that the coefficients are non-constant (and specifically nonlinear functions, I will provide more details if needed) of $T_{e,l}$ change anything?
How do I proceed to select the starred constants? I think setting $t^* = \tau_e$ or $t^* = \tau_l$ could be beneficial, but sometimes I need to set $\tau_{e,l} = 0$ to simulate the one-step parabolic model. For $x^*$ I could use $x^* = L$ where L is the box size for the simulation, and similarly for $t^*$.
For the barred functions (i.e. $\bar{S}, \bar{G}$, etc.), how do I change their definition now? The definition includes some constants which have the dimension of time, space, etc. Those should be multiplied by the correct scaling factor, I think?
What should I do about $\tau_{e,l}$? They have dimension of time and are constant, I have no idea how to deal with them.
Should I take orders of magnitude into consideration when selecting these values? $t$ and $\tau_{e,l}$ for example are on the order of $10^{-14}$, while $G$ and $S$ on the other hand are on the order of $10^{17}$ and $10^{20}$ respectively in SI units (this huge difference in orders of magnitude is one of the main reasons I am interested in nondimensionalizing the system).
I managed to figure it out myself. I am not sure if it's the best nondimensionalization scheme, but it's giving the expected results numerically.
Additionally defining $\bar{\tau}_{e,l} = \tau_{e,l}/t^*$, and with the following definitions: \begin{align} \begin{split} T^* = T_0, \quad x^* = L, \quad t^* = T_f, \quad S^* = S_0/10^4, \quad q^* = S^*x^*, \\ G^* = S^*/T^*, \quad K^* = (x^*)^2S^*/T^*, \qquad C^* = t^*S^*/T^* \label{stars} \end{split} \end{align}
Where $T_0$ is the initial temperature, $L$ is the box size (film thickness), $T_f$ is the simulation time. After lots of algebra you get: \begin{align} \frac{\partial \bar{T_e}}{\partial \bar{t}} &= -\frac{1}{\bar{C}_e(\bar{T}_e)}\frac{\partial \bar{q_e}}{\partial \bar{x}} - \frac{\bar{G}(\bar{T}_e)}{\bar{C}_e(\bar{T}_e)}(\bar{T_e} - \bar{T_l}) + \frac{\bar{S}(\bar{x},\bar{t})}{\bar{C}_e(\bar{T}_e)} \label{TTM1_nodim}\\ \frac{\partial \bar{T_l}}{\partial \bar{t}} &= -\frac{1}{\bar{C}_l(\bar{T}_l)}\frac{\partial \bar{q_l}}{\partial \bar{x}} + \frac{\bar{G}(\bar{T}_e)}{\bar{C}_l(\bar{T}_l)}(\bar{T_e} - \bar{T_l})\\ \bar{q_e} + \bar{\tau_e}\frac{\partial \bar{q_e}}{\partial \bar{t}} &= -\bar{K}_e(\bar{T}_e, \bar{T}_l)\frac{\partial \bar{T_e}}{\partial \bar{x}}\\ \bar{q_l} + \bar{\tau_l}\frac{\partial \bar{q_l}}{\partial \bar{t}} &= -\bar{K}_l(\bar{T}_l) \frac{\partial \bar{T_l}}{\partial \bar{x}} \label{TTM4_nodim} \end{align}
Which is a nondimensional system of equations. To answer my own questions:
Makes the algebra much easier, and there is no need to define two separate scalings for each temperature, etc.
It does make it a bit harder since you can't set your scaling in terms of these parameters, but defining scaling for the parameters themselves somewhat remedies that.
Since the functions themselves are scaled (e.g. $\bar{G}(\bar{T}_e) = G(\bar{T_e}T^*)/G^*$), nothing really changes. Before plugging in the scaled temperature it has to be multiplied by T^*, and then the result of $G$ is scaled down by $G^*$.
This is precisely what I did. Most scaled variables and parameters on the order of magnitude of 1 or 0.1. $\bar{S}$ is on the order of magnitude of $10^3-10^4$, I didn't realize that it's actually on the order of magnitude of $10^{23}$ unscaled. I set $S^*$ to be the peak intensity divided by $10^4$ to scale $C$, $G$ and $K$ appropriately.