A comprehensive study on dynamical analysis and numerical simulation of foam drainage equation using time-fractional derivative

This study is concerned with the dynamical behaviour of foams that arises due to the liquid flow in the nodes and channels between the bubbles. The time-fractional foam drainage equation that specifies the enhancement of vertical density in foams due to gravitational acceleration has been examined using an efficient method called the approximate analytical method. Three examples with different initial conditions are illustrated, along with a numerical simulation to explore the efficacy of the proposed technique. Existence and error analysis for the solution using the proposed technique have been presented. The suggested approach is well-organised for studying the physical behaviour of the equation, and yields fast-converging series solutions. The equation’s behaviour has been presented through surface plots and graphical representations for diverse fractional orders. It highlights that the considered equation is time-dependent and emphasises the role of memory effects inherent in fractional systems. Moreover, this method helps in analysing the nonlinear partial differential equations of fractional order arising in various scientific and technological sectors.