In this study, we analyzed the inertia effect on the axisymmetric squeeze flow of slightly viscoelastic fluid film between two disks. A system of nonlinear partial differential equations (PDEs) in cylindrical coordinates, along with nonhomogenous boundary conditions, illustrates the phenomenon of fluid flow caused by squeezing with the inertia effect. The Langlois recursive approach was applied to obtain the analytical solution of the system having a stream function, axial and radial velocities, pressure distribution, normal and tangential stresses and normal squeeze force. These flow variables are also portrayed graphically to describe the effects of the Reynolds number and slightly viscoelastic parameter. The results show that by increasing the Reynolds number, the velocity profile decreases, and both the pressure distribution and shear stresses increase. Moreover, there is a small increase in normal squeeze force. When the slightly viscoelastic parameter approaches zero, the obtained solution of flow variables matches with the classical results. This study can be applied to understand the mechanism of load-bearing features in thrust bearings and in arthrodial human joint (knee and hip) diseases.

The stability of nonlinear transverse vibrations of an axially moving beam resting on a nonlinear elastic foundation is analysed, considering the effects of viscous damping and a harmonically varying time-dependent velocity around a low constant mean speed. The beam is assumed to have simply supported boundary conditions at both ends. The contribution of this paper is the combined study of damping, nonlinear foundation, and harmonic velocity variation on the stability of axially moving beams, which has not been studied before. The governing equation for the transverse dynamics is a nonlinear partial differential equation with variable coefficients, which is solved using the two-timescale perturbation method in combination with the Fourier series method. The stability of the system is investigated for both non-resonant and resonant cases by examining the influence of key parameters, including nonlinear bending stiffness, nonlinear elastic foundation and damping. The analysis reveals that an increase in nonlinear bending stiffness and nonlinear elastic foundation tends to destabilize the system, leading to growing oscillations and instability. In contrast, an increase in damping enhances stability, causing oscillations to decay over time and leading to an asymptotically stable response. Furthermore, validation was carried out through comparison with an existing model.

The goal of the work is to solve the nonlinear convection-diffusion-reaction problem using the variational iteration method with the combination of the Chebyshev wavelet. This work developed a hybrid iterative technique named as Variational iteration method with the Chebyshev wavelet for the solutions of nonlinear convection-diffusion-reaction problems. The aim of applying the derived algorithm is to achieve fast convergence. During the solution of the given problem, the restricted variations will be mathematically justified. The effects of the scaling and other parameters like diffusion parameter, convection parameter, and reaction parameter on the solution are also focused on by their suitable selection. The approximate results include the error profiles and the simulations. The results of variational iteration with the Chebyshev wavelet are compared with variational iteration method, the Modified variational iteration method, and the Variational iteration method with Legendre wavelet. The error profiles allow us to compare the results with well-known existing schemes.