The problem of nonlinear instability of interfacial waves between two immiscible conducting cylindrical fluids of a weak Oldroyd 3-constant kind is studied. The system is assumed to be influenced by an axial magnetic field, where the effect of surface tension is taken into account. The analysis, based on the method of multiple scale in both space and time, includes the linear as well as the nonlinear effects. This scheme leads to imposing of two levels of the solvability conditions, which are used to construct like-nonlinear Schrödinger equations (l-NLS) with complex coefficients. These equations generally describe the competition between nonlinearity and dispersion. The stability criteria are theoretically discussed and thereby stability diagrams are obtained for different sets of physical parameters. Proceeding to the nonlinear step of the problem, the results show the appearance of dual role of some physical parameters. Moreover, these effects depend on the wave kind, short or long, except for the ordinary viscosity parameter. The effect of the field on the system stability depends on the existence of viscosity and differs in the linear case of the problem from the nonlinear one. There is an obvious difference between the effect of the three Oldroyd constants on the system stability. New instability regions in the parameter space, which appear due to nonlinear effects, are shown.
The purpose of this study is to establish the effects of insoluble surfactants on the stability of two layers flow down an inclined wall in the limit of Stokes and long-wavelength approximations. The dynamics of the liquid-liquid interface is described for arbitrary amplitudes by evolution equations derived from the basic hydrodynamic equations, in which the fluids are subjected to a uniform electric field. The principle aim of this work is to investigate the interfacial stability as well as the growth rate in the presence of insoluble surfactants. The parameters governing the flow system, such as Marangoni, Weber, capillary numbers and the inclined substrate strongly affect the waveforms and their amplitudes and hence the stability of the fluid. Approximate solutions of this system of linear evolution equations are performed. epending on the selected parameters, the phenomenon of the dual role is found with respect to the electric Weber number as well as the viscosity ratio. The interfacial waves will be more stable due to the growth of the Marangoni number while, while the opposite effect is found for the increase in capillary number. In the longwave perturbations, the stability process is found to confirm the stabilizing effect of the Marangoni number and the destabilizing influence of both capillary and Reynolds numbers, whereas the dual role is observed for the dielectric ratio.
The issue of hydromagnetic instability for a viscoelastic condensate film streaming down on a normal moving cylinder is discussed. As the visco-elasticity, heat-transfer influence or the magnetic field are included into the governing equations, the results become more suitable with the real situation. The nonlinearity enhances the accuracy of the results. The flow stability is weakened by the reduction of the volume of cylinder. However, the response of the system due to the temperature difference is uncertain. Moreover, the increment of magnetic field can enhance the stability. In contrast to the linear-results, the flow can be stabilized by the rheology.
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