Introduction of Critical Filling Parameter on Faraday Instability and Its Dependencies on Fluid Densities

In the present work, we have studied two immiscible Newtonian fluid systems having different pairs of comparable densities on a rectangular container undergoing vertical oscillation. Inspired by the Floquet analysis of Kumar and Tuckerman (J Fluid Mech 279:49–68, 1994 [1]), stability analysis is carried out to calculate the threshold. A two-fluid system filled with different volumes of fluids is defined by a filling parameter \(\left( {\psi = h_{1} /\left( {h_{1} + h_{2} } \right)} \right)\) , where \(h_{1}\) and \(h_{2}\) are the lower and upper-layer heights, respectively. It has been found that the threshold curve for Faraday instability shows bidirectional behavior with filling parameter \(\left( \psi \right)\) for a two-fluid system with comparable densities. The point about which the threshold curve shows bidirectional nature has been first time introduced here as a critical filling parameter \(\left( {\psi_{{\text{c}}} } \right)\) . An investigation on effect of density difference \(\left( {\Delta \rho } \right)\) and density ratio \(\left( {\rho_{{\text{r}}} } \right)\) on threshold curve ( \(f {-} A_{{\text{c}}}\) curve) and \(\psi_{{\text{c}}}\) has been performed, in which an increment in \(\psi_{{\text{c}}}\) with \({\Delta }\rho\) and \(\rho_{{\text{r}}}\) has been observed.