2023-01

Mathematics and Statistics (القضية : 1) (الحجم : 11)

The Integral Theory approach is used to explore the stability and dynamics of a free double-sided symmetric thin liquid film. For a Newtonian liquid with non-variable density and moving viscosity, the flowing in a thinning liquid layer is analyzed in two dimensions. To construct an equation that governs such flow, the Navier and Stokes formulas are utilized with proper boundary conditions of zero shear stress conjointly of normal stress on the bounding free surfaces with dimensionless variables. After that, the equations that are a non-linear evolution structure of layer thickness, local stream rate, and the unknown functions can be solved by using straight stability investigation, and the normal mode strategy can moreover be connected to these conditions to reveal the critical condition. The characteristic equation for the growth rate and wave number can be analyzed by using MATLAM programming to show the region of stable and unstable films. As a result of our research, we are able to demonstrate that the effect of a thin, free, double-sided liquid layer is an unstable component

2023-01

General Letters in Mathematics (GLM) (القضية : 4) (الحجم : 12)

The Long-Wave Theory is applied to investigate the dynamic stability of free thin fluid films flowing down an inclined plane. We assume that thin supported films have a thickness of 𝐻̅ and less than or equal to one hundred nm. Equations of Navier and Stokes, continuity-equation, and related boundary conditions are used to represent a two-dimensional stream demonstrated as a continuum. Under long-wave approximation, the governing equations for the film interface have been rescaled and simplified to obtain a highly non-linear condition of development for the film interface. A procedure for evaluating the magnitude of the effects of the high-order effects is also used to formulate simplified governing equations. In the future, we can study this problem by adding heat transfer over the stretching plate. In addition, we can also study the stability analysis to twodimension flow of a viscous liquid within a horizontal thin liquid film with neglecting the inertia terms of NavierStokes equations.

2015-05

IOSR Journal of Mathematics (IOSR-JM) (القضية : 3) (الحجم : 11)

In this paper, we consider the unsteady flow within a double-sided symmetric thin liquid film with negligible inertia. We apply the Navier-Stokes equations in two dimensional flows for incompressible fluid. The similarity method is used in which the explicit time dependence can be isolated and thus the shape of the film will depend on one variable only. The differential equation of the film thickness is obtained analytically and the solution of equation that represents the film thickness is obtained numerically by using Rung-Kutta method.

2015-05

IOSR Journal of Mathematics (IOSR-JM) (القضية : 3) (الحجم : 11)

stability and dynamics of a free double-sided symmetric thin liquid film are investigated by using the long-wave method. The flow in thin liquid film is considered in two dimensions for Newtonian liquid with constant density and dynamic viscosity. The Navier-stokes equations is used with appropriate boundary conditions of zero shear stress and also of normal stress on the bounding free surfaces with non-dimensional variables to obtain an equation that governs such flow.