De Ris formulae for the flame spread rate over solid fuels that propagate against an external fluid flow for both the thermally thin and thick regimes have become well-known. In some cases, it is expected that predictions from these formulae will fail since they are independent of parameters that may modify the burning velocity. Based on the technique developed by Benguria and Depassier, we derive in this article new analytical expressions that correspond to upper bounds for the speed of the flame front propagating downwards. These upper bounds are functions of several parameters that are not taken into consideration in the de Ris expressions, such as the activation energy for solid- and gas-phase reactions. Burning velocity values obtained for these upper bounds are compared with new data and with other experimental results found in the literature, showing a better agreement than the de Ris formulae for a wide variety of cases.