Linear stability analysis of thermoviscous instability in immiscible displacement

We study the thermoviscous fingering instability problem that develops when an injected fluid is displacing a fluid with different temperature and viscosity in a Hele-Shaw cell or porous medium. Using linear stability theory, we show how the thermal front caused by the different temperature of the injected fluid impacts the growth rate of instabilities on an immiscible radial displacement front. The immiscible front is stabilized by interfacial tension when the front radius is small but becomes unstable as the front radius grows larger. When the thermal front is unstable, it destabilizes the immiscible fluid front and makes instability possible at smaller front radii. When the thermal front is stable, it stabilizes the fluid front and delays the growth of instabilities. However, the thermal front has only a small impact on the maximum obtainable perturbation growth rates when the front radius is large. In porous medium flow we observe that a thermal front that lags further behind the fluid front is more effective at stabilizing the fluid front.