Exact solutions for thermal problems: buoyancy, marangoni, vibrational and magnetic-field-controlled flows

In general, the thermal‐convection (Navier‐Stokes and energy) equations are nonlinear partial differential equations that in most cases require the use of complex algorithms in combination with opportune discretization techniques for obtaining reliable numerical solutions. There are some cases, however, in which such equations admit analytical solutions. Such exact solutions have enjoyed a widespread use in the literature as basic states for determining the linear stability limits in some idealized situations. This review article provides a synthetic review of such analytical expressions for a variety of situations of interest in materials science (especially crystal growth and related disciplines), including thermogravitational (buoyancy), thermocapillary (Marangoni), thermovibrational convection as well as “mixed” cases and flow controlled via static and uniform magnetic fields.