 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,where $\varphi$ 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 $h\dot{\varphi }=-W_s\varphi$, 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 $\gamma \varphi -\theta$, 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.