The linear and weakly nonlinear stability of viscoelastic film flowing down a slippery inclined plane is investigated analytically. Under the assumption of the long wave approximation, the first-order Benny equation of Oldroyd-B fluid thin film with slip condition is obtained. Through the normal mode analysis, the neutral stability curve and the temporal growth rates are calculated to explore the linear stability of the film. Linear results show that the critical Reynolds number decreases with the increase in slip length and viscoelastic parameter and that the liquid film may exhibit pure elastic instability. For the nonlinear stability analysis, both hydrodynamic instability and elastic instability are discussed. The primary bifurcations in the phase plane are identified by calculating the Landau coefficient, i.e., the unconditional stable region, the supercritical region, the subcritical region, and the explosive region. The dependence of primary bifurcation regions upon the slip length and Deborah number are studied, and the results indicate that the slip boundary and viscoelasticity destabilizes the flow. According to the Ginzburg–Landau equation, the threshold amplitude of the nonlinear equilibrium solution is analyzed as well.

We establish both the uniqueness and the existence of the solutions to a hidden-memory variable-order fractional stochastic partial differential equation, which models, e.g., the stochastic motion of a Brownian particle within a viscous liquid medium varied with fractal dimensions. We also investigate the inverse problem concerning the observations of the solutions, which eliminates the analytic assumptions on the variable orders in the literature of this topic and theoretically guarantees the reliability of the determination and experimental inference.

The finite-amplitude instability of the natural convection in a vertical porous slab filled with variable permeability porous medium is investigated analytically. The side walls of the slab are kept at different temperatures, and the permeability in the horizontal direction is assumed to be exponential heterogeneous models. Two-dimensional, finite-amplitude solutions for the thermal buoyant flow are obtained for Darcy–Rayleigh numbers close to the critical values by using the amplitude expansion method. The dependence of the fundamental mode, the distortion of the mean flow, and the second harmonic upon the variable permeability constant are discussed. By calculating the first Landau coefficient, the primary bifurcations in the vicinity of the neutral stability curves are identified. The results show that only supercritical bifurcations are found to occur, rather than subcritical instabilities. In terms of the well-known Landau equation, the threshold amplitude of the nonlinear equilibrium solution is analyzed as well.