scholarly journals Electrohydrodynamic Instability of Two Thin Viscous Leaky Dielectric Fluid Films in a Porous Medium

2011 ◽  
Vol 2011 ◽  
pp. 1-35 ◽  
Author(s):  
M. F. El-Sayed ◽  
M. H. M. Moussa ◽  
A. A. A. Hassan ◽  
N. M. Hafez
Keyword(s):  
Porous Medium ◽  
Evolution Equations ◽  
Nonlinear Evolution Equations ◽  
Dielectric Fluid ◽  
Physical Parameters ◽  
Fluid Films ◽  
Linearized Stability ◽  
Leaky Dielectric ◽  
The Stability ◽  
Wave Limit

The effect of an applied electric field on the stability of the interface between two thin viscous leaky dielectric fluid films in porous medium is analyzed in the long-wave limit. A systematic asymptotic expansion is employed to derive coupled nonlinear evolution equations of the interface and interfacial free charge distribution. The linearized stability of these equations is determined and the effects of various parameters are examined in detail. For perfect-perfect dielectrics, the various parameters affect only for small wavenumber values. For dielectrics, the various parameters affect only for small wavenumber values. For effect for small wavenumbers, and a stabilizing effect afterwards, and for high wavenumber values for the other physical parameters, new regions of stability or instability appear. For leaky-leaky dielectrics, the conductivity of upper fluid has a destabilizing effect for small or high wavenumbers, while it has a dual role on the stability of the system in a wavenumber range between them. The effects of all other physical parameters behave in the same manner as in the case of perfect-leaky dielectrics, except that in the later case, the stability or instability regions occur more faster than the corresponding case of leaky-leaky dielectrics.

scholarly journals Thermosolutal Instability in Compressible Viscoelastic Dusty Fluid through Porous Medium

2013 ◽  
Vol 18 (1) ◽  
pp. 99-112 ◽  
Author(s):  
P. Kumar ◽  
H. Mohan
Keyword(s):  
Porous Medium ◽  
Viscoelastic Fluid ◽  
Normal Mode Analysis ◽  
Suspended Particles ◽  
Mode Analysis ◽  
Stationary Convection ◽  
Linearized Stability ◽  
Medium Permeability ◽  
The Stability ◽  
Solute Gradient

Thermosolutal instability in a compressible Walters B’ viscoelastic fluid with suspended particles through a porous medium is considered. Following the linearized stability theory and normal mode analysis, the dispersion relation is obtained. For stationary convection, the Walters B’ viscoelastic fluid behaves like a Newtonian fluid and it is found that suspended particles and medium permeability have a destabilizing effect whereas the stable solute gradient and compressibility have a stabilizing effect on the system. Graphs have been plotted by giving numerical values to the parameters to depict the stability characteristics. The stable solute gradient and viscoelasticity are found to introduce oscillatory modes in the system which are non-existent in their absence.

Surfactant Effects on Fluid-Elastic Instabilities of Liquid-Lined Flexible Tubes: A Model of Airway Closure

1993 ◽  
Vol 115 (3) ◽  
pp. 271-277 ◽  
Author(s):  
D. Halpern ◽  
J. B. Grotberg
Keyword(s):  
Film Thickness ◽  
Surfactant Concentration ◽  
Evolution Equations ◽  
Liquid Films ◽  
Nonlinear Evolution Equations ◽  
Small Airways ◽  
Airway Closure ◽  
Elastic Tubes ◽  
Critical Film Thickness ◽  
The Stability

A theoretical analysis is presented predicting the closure of small airways in the region of the terminal and respiratory bronchioles. The airways are modelled as thin elastic tubes, coated on the inside with a thin viscous liquid lining. This model produces closure by a coupled capillary-elastic instability leading to liquid bridge formation, wall collapse or a combination of both. Nonlinear evolution equations for the film thickness, wall position and surfactant concentration are derived using an extended version of lubrication theory for thin liquid films. The positions of the air-liquid and wall-liquid interfaces and the surfactant concentration are perturbed about uniform states and the stability of these perturbations is examined by solving the governing equations numerically. Solutions show that there is a critical film thickness, dependent on fluid, wall and surfactant properties above which liquid bridges form. The critical film thickness, εc, decreases with increasing mean surface-tension or wall compliance. Surfactant increases εc by as much as 60 percent for physiological conditions, consistent with physiological observations. Airway closure occurs more rapidly with increasing film thickness and wall flexibility. The closure time for a surfactant rich interface can be approximately five times greater than an interface free of surfactant.

scholarly journals Fourth order nonlinear evolution equations for gravity-capillary waves in the presence of a thin thermocline in deep water

2002 ◽  
Vol 43 (4) ◽  
pp. 513-524 ◽  
Author(s):  
Suma Debsarma ◽  
K.P. Das
Keyword(s):  
Fourth Order ◽  
Deep Water ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Capillary Wave ◽  
Capillary Waves ◽  
Nonlinear Evolution Equations ◽  
Marginal Stability ◽  
Stability And Instability ◽  
The Stability

AbstractFor a three-dimensional gravity capillary wave packet in the presence of a thin thermocline in deep water two coupled nonlinear evolution equations correct to fourth order in wave steepness are obtained. Reducing these two equations to a single equation for oblique plane wave perturbation, the stability of a uniform gravity-capillary wave train is investigated. The stability and instability regions are identified. Expressions for the maximum growth rate of instability and the wavenumber at marginal stability are obtained. The results are shown graphically.

Soliton-like solutions to the generalized Burgers-Huxley equation with variable coefficients

2013 ◽  
Vol 3 (4) ◽  
Author(s):  
Houria Triki ◽  
Abdul-Majid Wazwaz
Keyword(s):  
Evolution Equations ◽  
Soliton Solutions ◽  
Dark Soliton ◽  
Travelling Wave ◽  
Variable Coefficients ◽  
Nonlinear Evolution Equations ◽  
Time Dependent ◽  
Physical Parameters ◽  
Ansatz Method ◽  
Huxley Equation

AbstractIn this paper, we consider the generalized Burgers-Huxley equation with arbitrary power of nonlinearity and timedependent coefficients. We analyze the traveling wave problem and explicitly find new soliton-like solutions for this extended equation by using the ansatz of Zhao et al. [X. Zhao, D. Tang, L. Wang, Phys. Lett. A 346 (2005) 288–291]. We also employ the solitary wave ansatz method to derive the exact bright and dark soliton solutions for the considered evolution equation. The physical parameters in the soliton solutions are obtained as function of the time-dependent model coefficients. The conditions of existence of solitons are presented. As a result, rich exact travelling wave solutions, which contain new soliton-like solutions, bell-shaped solitons and kink-shaped solitons for the generalized Burgers-Huxley equation with time-dependent coefficients, are obtained. The methods employed here can also be used to solve a large class of nonlinear evolution equations with variable coefficients.

Construction of the numerical and analytical wave solutions of the Joseph–Egri dynamical equation for the long waves in nonlinear dispersive systems

2020 ◽  
Vol 34 (30) ◽  
pp. 2050289
Author(s):  
Abdulghani R. Alharbi ◽  
M. B. Almatrafi ◽  
Aly R. Seadawy
Keyword(s):  
Solitary Wave ◽  
Numerical Solution ◽  
Evolution Equations ◽  
Dynamical Equation ◽  
Nonlinear Evolution ◽  
Three Dimensional ◽  
Nonlinear Evolution Equations ◽  
Solitary Wave Solutions ◽  
Wave Solutions ◽  
The Stability

The Kudryashov technique is employed to extract several classes of solitary wave solutions for the Joseph–Egri equation. The stability of the achieved solutions is tested. The numerical solution of this equation is also investigated. We also present the accuracy and the stability of the numerical schemes. Some two- and three-dimensional figures are shown to present the solutions on some specific domains. The used methods are found useful to be applied on other nonlinear evolution equations.

Nonlinear dynamics of vorticity waves in the coastal zone

1996 ◽  
Vol 326 ◽  
pp. 181-203 ◽  
Author(s):  
Victor I. Shrira ◽  
Vyacheslav V. Voronovich
Keyword(s):  
Evolution Equation ◽  
Potential Vorticity ◽  
Large Scale ◽  
Evolution Equations ◽  
Short Wave ◽  
Nonlinear Evolution Equations ◽  
Principal Mechanism ◽  
Weakly Nonlinear ◽  
The Mean ◽  
Wave Limit

Vorticity waves are wave-like motions occurring in various types of shear flows. We study the dynamics of these motions in alongshore shear currents in situations where it can be described within weakly nonlinear asymptotic theory. The principal mechanism of vorticity waves can be interpreted as potential vorticity conservation with the background vorticity gradient provided both by the mean current shear and the variation of depth. Under the assumption that the mean potential vorticity distibution is monotonic in the cross-shore direction, the nonlinear stage of the dynamics of weakly nonlinear vorticity waves, long in comparison with the current cross-shore scale, is found to be governed by an evolution equation of the generalized Benjamin–Ono type. The dispersive terms are given by an integro-differential operator with the kernel determined by the large-scale cross-shore depth and current dependence. The derived equations form a wide new class of nonlinear evolution equations. They all tend to the Benjamin–Ono equation in the short-wave limit, while in the long-wave limit their asymptotics depend on the specific form of the depth and current profiles. For a particular family of model bottom profiles the equations are ‘intermediate’ between Benjamin–Ono and Korteweg–de Vries equations, but are distinct from the Joseph intermediate equation. Solitary-wave solutions to the equations for these depth profiles are found to decay exponentially. Taking into account coastline inhomogeneity or/and alongshore depth variations adds a linear forcing term to the evolution equation, thus providing an effective generation mechanism for vorticity waves.

The shock peakon wave solutions of the general Degasperis–Procesi equation

2019 ◽  
Vol 33 (29) ◽  
pp. 1950351 ◽  
Author(s):  
Lijuan Qian ◽  
Raghda A. M. Attia ◽  
Yuyang Qiu ◽  
Dianchen Lu ◽  
Mostafa M. A. Khater
Keyword(s):  
Stability Property ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Dynamical Behavior ◽  
Nonlinear Evolution Equations ◽  
Wave Solutions ◽  
Generalized Kudryashov Method ◽  
Kudryashov Method ◽  
Different Types ◽  
The Stability

This research paper applies the modified Khater method and the generalized Kudryashov method to the general Degasperis–Procesi (DP) equation, which is used to describe the dynamical behavior of the shallow water outflows. Some shock peakon wave solutions are obtained by using these methods. Moreover, some figures are sketched for these solutions to explain more physical properties of the general DP equation and to figure out the coincidence between different types of obtained solutions. The stability property by using the features of the Hamiltonian system is tested to some obtained solutions to show their ability for applying in the model’s applications. The obtained solutions were verified with Maple 16 & Mathematica 12 by placing them back into the original equations. The performance of these methods shows their power and effectiveness for applying to many different forms of the nonlinear evolution equations with an integer and fractional order.

scholarly journals Hybrid Nanofluid Flow over a Permeable Shrinking Sheet Embedded in a Porous Medium with Radiation and Slip Impacts

Mathematics ◽  
2021 ◽  
Vol 9 (8) ◽  
pp. 878
Author(s):  
Shahirah Abu Bakar ◽  
Norihan Md Arifin ◽  
Najiyah Safwa Khashi’ie ◽  
Norfifah Bachok
Keyword(s):  
Porous Medium ◽  
Differential Equations ◽  
Similarity Transformation ◽  
Lower Solution ◽  
Shrinking Sheet ◽  
Physical Parameters ◽  
Hybrid Nanofluid ◽  
Uniqueness Of Solutions ◽  
Nanofluid Flow ◽  
The Stability

The study of hybrid nanofluid and its thermophysical properties is emerging since the early of 2000s and the purpose of this paper is to investigate the flow of hybrid nanofluid over a permeable Darcy porous medium with slip, radiation and shrinking sheet. Here, the hybrid nanofluid consists of Cu/water as the base nanofluid and Al2O3–Cu/water works as the two distinct fluids. The governing ordinary differential equations (ODEs) obtained in this study are converted from a series of partial differential equations (PDEs) by the appropriate use of similarity transformation. Two methods of shooting and bvp4c function are applied to solve the involving physical parameters over the hybrid nanofluid flow. From this study, we conclude that the non-uniqueness of solutions exists through a range of the shrinking parameter, which produces the problem of finding a bigger solution than any other between the upper and lower branches. From the analysis, one can observe the increment of heat transfer rate in hybrid nanofluid versus the traditional nanofluid. The results obtained by the stability of solutions prove that the upper solution (first branch) is stable and the lower solution (second branch) is not stable.

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