The existence of a pair of oncoming quasi-stationary longitudinal thermonuclear combustion waves is demonstrated in a plasma column with cold, dense ends while undergoing uniform radial expansion. It is shown that if a suitable set of parameters is selected, an exponential expansion of a cylindrical column is possible, during which the temperature, density and longitudinal velocity distributions remain unchanged with time and in which a continuous inflow of thermonuclear fuel is produced by convection driven by the combustion waves near the ends of the plasma column. A numerical procedure is used to find a self-consistent distribution of fusion power in the plasma with non-local thermalization of α-particles. — The usefulness of these thermonuclear combustion waves for increasing the effectiveness of an imploding liner system is discussed.