= Heated particle-laden layer
A heated <particle suspension> can have competing compositional and thermal contributions to its <mass density>. Let $\rho_0$ be the carrier-fluid <mass density> at reference <temperature> $T_0$, $\rho_p$ the particle <mass density> and $\alpha$ the carrier-fluid <thermal expansion coefficient>. In a dilute mixture, neglecting products of small changes,
$$
\rho\simeq(1-\phi)\rho_0[1-\alpha(T-T_0)]+\phi\rho_p,
\qquad \frac{\rho-\rho_0}{\rho_0}=\gamma\phi-\theta,
$$
where $\phi$ is <particle volume fraction>, $\gamma=(\rho_p-\rho_0)/\rho_0$ and $\theta=\alpha(T-T_0)$. Heavy particles increase <mass density>, whereas warming an ordinarily expanding carrier fluid decreases it.
For a well-mixed layer of fixed depth $h$ with an absorbing horizontal bed, <particle deposition flux> gives $h\dot\phi=-W_s\phi$, if $W_s$ is a constant downward <settling velocity> magnitude and other particle fluxes vanish. A separate <heat> balance determines $\theta$. Their different time dependences can change the sign of $\gamma\phi-\theta$, reversing the layer's <reduced gravity> relative to fluid at the reference state. A resulting <unstable density stratification> can invalidate the well-mixed or one-dimensional settling assumptions, so the density balance is also a consistency check on the model.
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