On estimates for short wave stability and long wave instability in three-layer Hele-Shaw flows

► An exact formula for an upper bound of the growth rates of all unstable waves is given. ► This formula can be used for the selection of parameter values in order to combat the effects of instabilities without resorting to any kind of computation. ► An exact formula for a lower bound of the growth rates of all stable waves is given. ► This formula can be used for the selection of parameter values to enhance the stability of these stable waves. ► The upper and lower bound results can used to control the bandwidth of unstable waves.