On the travelling wave instability caused by moving planar heat sources: the rigid problem

This analysis deals with the convective travelling wave instability appearing in a fluid medium at rest and contained between two horizontal rigid plates, subjected to the same sinusoidal temperature distribution, moving at a uniform speed and in the same direction. The temperature distribution is caused by travelling planar heat sources with a time harmonic output. A three-dimensional coordinate system is used and the small parameter ε in this problem represents the ratio between the buoyancy and inertial forces. For a finite, yet small ε, asymptotic expansions are assumed for the velocity, pressure, temperature and the Reynolds number. The mean motion generated by the Reynolds stresses is calculated separately. By keeping the Prandtl number fixed and by using long length and time scales, successive linearized perturbation equations are considered. Two successive amplitude equations are analyzed and their solution yields the mathematical form of these travelling waves, their group velocity and the elevation above the critical Reynolds number.