Suppose we have a liquid with a thermal conductivity that is $$ \lambda =\dfrac{\lambda_0}{1-b(\rho-\rho_0)}, \qquad \lambda_0, b, \rho_0 >0. $$ (I don't know what liquids verify this experimental equation, if anyone knows...)

I do some approximations:

• It's incompressible, so the state equation is $$\text{d}\rho = -\alpha \rho \text{d}T$$ and of course, $\kappa_T \sim 0$. A not too deep layer of this liquid is heated on its free surface ($z=0$) so $$ \lambda \left( \dfrac{\text{d}T}{\text{d}z}\right)_{z=0} = \phi_0 >0.$$

I know that the differential equations of $\rho(z)$ and $T(z)$ (are stationary) are $$ \dfrac{\text{d}\rho(z)}{\text{d}z} = \bigl( \rho(z) -\rho_F \bigr) g\tag{1}$$ and $$ \lambda (z) \dfrac{\text{d}T(z)}{\text{d}z} = C = \phi_0. \tag{2}$$ The solution of equation (1) is $$ \rho(z) = (\rho_0 -\rho_F) \text{e}^{gz} +\rho_F$$ where $\rho_0, T_0$ are the density and the temperature of the free surface ($z=0$) and $\rho_F, T_F$ are the density and the temperature of the bottom (for example $z \longrightarrow -\infty$).

• Density at the free surface: $$ \rho (0) = \rho_0$$ and at the bottom $$ \underset{z \longrightarrow -\infty}{\lim} \rho(z) = \rho_F < \infty.$$

But the solution of (2) at the bottom diverges and it's not physically acceptable, where's my mistake? $$ \underset{z\longrightarrow -\infty}{\lim} T(z) = -\infty$$ where $T(z) = A + Bz + C\text{e}^{gz}.$