Methods to evaluate allelopathic effects on nodulating bacteria and nodulation in legumes

A linear analysis of oscillatory thermocapillary instability associated with deformation of the free surface in a finite-depth horizontal liquid layer heated from above is carried out. First, we numerically solve the problem and determine the conditions of marginal stability. Then, assuming that Galileo and Marangoni numbers are large enough while the capillary number is small enough, which corresponds to the time scales of transverse and longitudinal oscillations related to the free surface being much smaller than the characteristic viscous and thermal time scales, we construct an asymptotic solution of the problem. Two high-frequency overstable modes - transverse (capillarygravity waves) and longitudinal - are identified and analyzed. Mode-mixing occurring at their possible resonance is also studied.