Stability analysis for multiple solutions of boundary layer flow towards a shrinking sheet: Analytical solution by using least square method (original) (raw)

Linear Stability Analysis of Two Dimensional MHD Unsteady Flow of Viscous Fluid on a Shrinking Sheet

2020

In this article, the work of Lund et al. (2019) has been extended for stability analysis, which was not considered in their study. In this study, the stability analysis of dual solutions for Caputo fractional-order-two dimensional MHD generalized viscous fluid over a shrinking sheet has been considered. The system of governing partial differential equations is reduced to the linearized system of eigenvalue problems. The resultant equations have been solved by using three stages Lobatto IIIa formula. The results revealed that the first solution is more stable as compared to the second solution, as expected. Further, it has been observed that the behavior of the initial growth of the disturbance is noticed for the unstable solution. Keywords— Stability Analysis; Dual solutions; three stages Lobatto III a formula; Stable solution.

Nonlinear instability analysis of a vertical cylindrical magnetic sheet

2021

This paper concerns with the nonlinear instability analysis of double interfaces separated three perfect, incompressible cylindrical magnetic fluids. The cylindrical sheet is acted upon by an axial uniform magnetic field. The current nonlinear approach depends mainly on solving the linear governing equations of motion and subjected to the appropriate nonlinear boundary conditions. This procedure resulted in two nonlinear characteristic equations governed the behavior of the interfaces deflection. By means of the Taylor expansion, together with the multiple time scales, technique, the stability analysis of linear as well as the nonlinear is achieved. The linear stability analysis reveals a quadratic dispersion equation in the square of growth rate frequency of the surface wave. On the other hand, the nonlinear analysis is accomplished by a coupled nonlinear Schrodinger equation of the evolution amplitudes of the surface waves. The stability criteria resulted in a polynomial of the e...

Analytical and numerical solutions to an electrohydrodynamic stability problem

Applied Mathematics and Computation, 2010

A linear hydrodynamic stability problem corresponding to an electrohydrodynamic convection between two parallel walls is considered. The problem is an eighth order eigenvalue one supplied with hinged boundary conditions for the even derivatives up to sixth order. It is first solved by a direct analytical method. By variational arguments it is shown that its smallest eigenvalue is real and positive. The problem is cast into a second order differential system supplied only with Dirichlet boundary conditions. Then, two classes of methods are used to solve this formulation of the problem, namely, analytical methods (based on series of Chandrasekar-Galerkin type and of Budiansky-DiPrima type) and spectral methods (tau, Galerkin and collocation) based on Chebyshev and Legendre polynomials. For certain values of the physical parameters the numerically computed eigenvalues from the low part of the spectrum are displayed in a table. The Galerkin and collocation results are fairly closed and confirm the analytical results.

On absolute linear instability analysis of plane Poiseuille flow by a semi-analytical treatment

Journal of the Brazilian Society of Mechanical Sciences and Engineering, 37 (2): 495–505 (2015), DOI: 10.1007/s40430-014-0187-2, 2015

"The absolute linear hydrodynamic instability of the plane Poiseuille flow is investigated by solving the Orr– Sommerfeld equation using the semi-analytical treatment of the Adomian decomposition method (ADM). In order to use the ADM, a new zero-order ADM approximation is defined. The results for the spectrum of eigenvalues are obtained using various orders of the ADM approximations and discussed. A comparative study of the results for the first, second and third eigenvalues with the ones from a previously published work is also presented. A monotonic trend of approach of decreasing relative error with the increase of the orders of ADM approximation is indicated. The results for the first, second and third eigenvalues show that they are in good agreement within 1.5 % error with the ones obtained by a previously published work using the Chebyshev spectral method. The results also show that the first eigenvalue is positioned in the unstable zone of the spectrum, while the second and third eigenvalues are located in the stable zone."

The effect of a periodic tangential magnetic field on the stability of a horizontal magnetic fluid sheet

Heat Transfer-Asian Research, 2019

The current article aims at investigating the effect of a periodic tangential magnetic field on the stability of a horizontal flat sheet. The media were considered porous, the three viscous-fluid layers were initially streaming with uniform velocities, and the magnetic field admitted the presence of free-surface currents. Furthermore, the transfer of mass and heat phenomenon was taken into account. The analysis, in this paper, was followed by the viscous potential theory. Moreover, the stability of the boundary-value problem resulted in coupled secondorder linear differential equations with damping and complex coefficients. In regard to the uniform and periodic magnetic field, the standard normal mode approach was applied to deduce a general dispersion relation and judge the stability criteria. In addition, several unfamiliar cases were reported, according to appropriate data choices. The stability conditions were theoretically analyzed, and the influences of the various parameters in the stability profile were identified through a set of diagrams. In accordance wth the oscillating field, the coupled dispersion equations were combined to give the established Mathieu equation. Therefore, the governed transition curves were, theoretically, obtained. Finally, the results were numerically confirmed.

THE STABILITY OF SOME PHYSICALLY REALISTIC MODELS USING CHANDRASEKHER'S TECHNIQUES

isara solutions, 2016

The paper examines the stability of an in compressible, non-viscous fluid with horizontal magnetic field and vertical rotation confined between two parallel plates. Here an attempt has been made to investigate the stability of some physically realistic models using Chandrashekhar's technique. A number of results have been obtained which help is a better understanding of a physical situation.

Comparative Study on Sixth Order Boundary Value Problems with Application to Linear Hydrodynamic Stability Problem and Benard Layer Eigenvalue Problem

Differential Equations and Dynamical Systems, 2019

Numerical estimation for higher order eigenvalue problems are promising and has accomplished significant importance, mainly due to existence of higher order derivatives and boundary conditions relating to higher order derivatives of the unknown functions. In this article, we perform a numerical study of linear hydrodynamic stability of a fluid motion caused by an erratic gravity field. We employ two methods, collocation and spectral collocation based on Bernstein and Legendre polynomials to solve the linear hydrodynamic stability problems and Benard type convection problems. In order to handle boundary conditions, our techniques state all the unknown coefficients of boundary conditions derivatives in terms of known coefficient. The schemes have been carried out to several test problems to establish the efficiency of the two methods.

The stability of eigenvalues and eigenvectors and their impact on differential systems

Journal of Pure & Applied Sciences

In this article, we apply the stability of eigenvalues and eigenvectors and their impact on differential systems. To achieve this goal, the eigenvalues and eigenvectors are studied and their differential systems, nature in terms of being different real values, compound eigenvalues, or equal eigenvalues.And to identify how to solve linear differential systems with fixed coefficients with the initial condition a complete solution, which depends on the eigenvalues and the corresponding eigenvectors, finding the general solution and the geometry of teigenvectors graphically and the effect of theigenvalues for the three eigenvalues cases, by drawing the paths and the phase plane and clarifying the state of equilibrium contract and stability

Closed-form linear stability conditions for magneto-convection

Journal of Fluid Mechanics, 2003

extensively investigated the linear dynamics of Rayleigh-Bénard convection in an electrically conducting fluid exposed to a uniform vertical magnetic field and enclosed by rigid, stress-free, upper and lower boundaries. He determined the marginal stability boundary and critical horizontal wavenumbers for the onset of convection as a function of the Chandrasekhar number Q or Hartmann number squared. No closed-form formulae appeared to exist and the results were tabulated numerically. We have discovered simple expressions that concisely describe the stability properties of the system. When the Prandtl number Pr is greater than or equal to the magnetic Prandtl number Pm the marginal stability boundary is described by the curve Q = π −2 [R − R 1/3 c R 2/3 ] where R is the Rayleigh number and R c = (27/4)π 4 is Rayleigh's famous critical value for the onset of stationary convection in the absence of a magnetic field (Q = 0). When Pm > Pr the marginal stability boundary is determined by this curve until intersected by the curve Q = 1 π 2

An alternative approach to linear and nonlinear stability calculations at finite Reynolds numbers

Journal of Fluid Mechanics, 1984

An extended version of the interactive boundary-layer approach which has been used widely in steady-flow calculations is applied here to the linear and nonlinear stability properties of channel flows and boundary layers in the moderate-to-large Reynolds-number regime. This is the regime of most practical concern. First, for linear stability the agreement found between the interactive approach and Orr-Sommerfeld results remains fairly close even at Reynolds numbers as low as about frac110\frac{1}{10}frac110 of the critical value for plane Poiseuille flow, or frac15\frac{1}{5}frac15 for Blasius flow. Secondly, nonlinear unsteady calculations and comparisons with full solutions obtained by enlarging the same method are also presented. Overall the work suggests that, at the finite Reynolds numbers where real interest lies, the dominant physical processes of instability in channel flow and boundary layers are of boundary-layer form, with interaction, and it suggests also an alternative numerical technique for...