scholarly journals Triple Solutions and Stability Analysis of Micropolar Fluid Flow on an Exponentially Shrinking Surface

Crystals ◽  
2020 ◽  
Vol 10 (4) ◽  
pp. 283 ◽  
Author(s):  
Liaquat Ali Lund ◽  
Zurni Omar ◽  
Ilyas Khan ◽  
Dumitru Baleanu ◽  
Kottakkaran Sooppy Nisar
Keyword(s):  
Stability Analysis ◽  
Shooting Method ◽  
Stable Solution ◽  
Governing Equations ◽  
Initial Value ◽  
Suction Parameter ◽  
Linear Ordinary Differential Equations ◽  
Runge Kutta Method ◽  
Micropolar Fluid Flow ◽  
Shrinking Surface

In this article, we reconsidered the problem of Aurangzaib et al., and reproduced the results for triple solutions. The system of governing equations has been transformed into the system of non-linear ordinary differential equations (ODEs) by using exponential similarity transformation. The system of ODEs is reduced to initial value problems (IVPs) by employing the shooting method before solving IVPs by the Runge Kutta method. The results reveal that there are ranges of multiple solutions, triple solutions, and a single solution. However, Aurangzaib et al., only found dual solutions. The effect of the micropolar parameter, suction parameter, and Prandtl number on velocity, angular velocity, and temperature profiles have been taken into account. Stability analysis of triple solutions is performed and found that a physically possible stable solution is the first one, while all leftover solutions are not stable and cannot be experimentally seen.

scholarly journals Numerical Method in Two Dimensional Boundary Layer Theory for Incompressible Viscous Fluid

2013 ◽  
Vol 38 ◽  
pp. 61-73
Author(s):  
MA Haque
Keyword(s):  
Boundary Layer ◽  
Viscous Fluid ◽  
Initial Value Problem ◽  
Shooting Method ◽  
Boundary Layer Equation ◽  
Incompressible Viscous Fluid ◽  
Initial Value ◽  
Box Scheme ◽  
Runge Kutta Method ◽  
Dimensional Boundary

In this paper laminar flow of incompressible viscous fluid has been considered. Here two numerical methods for solving boundary layer equation have been discussed; (i) Keller Box scheme, (ii) Shooting Method. In Shooting Method, the boundary value problem has been converted into an equivalent initial value problem. Finally the Runge-Kutta method is used to solve the initial value problem. DOI: http://dx.doi.org/10.3329/rujs.v38i0.16549 Rajshahi University J. of Sci. 38, 61-73 (2010)

Post-Buckling Analysis of FGM Beam Subjected to Non-Conservative Forces and In-Plane Thermal Loading

2012 ◽  
Vol 152-154 ◽  
pp. 474-479
Author(s):  
Feng Qun Zhao ◽  
Zhong Min Wang ◽  
Rui Ping Zhang
Keyword(s):  
Shooting Method ◽  
Thermal Loading ◽  
Governing Equations ◽  
Temperature Dependent ◽  
Tangential Load ◽  
Simply Supported ◽  
Runge Kutta Method ◽  
Buckling Behavior ◽  
Post Buckling ◽  
Fgm Beam

Based on the Kirchhoff large deformation theory, the post-buckling behavior of right movable simply supported FGM beam subjected to non-conservative forces and in-plane thermal loading was analyzed in this paper. The temperature-dependent and spatially dependent material properties of the FGM beam were assumed to vary through the thickness. The nonlinear governing equations of FGM beam subjected to a uniform distributed tangential load along the central axis and in-plane thermal loading were derived. Then, a shooting method and Runge-kutta method are employed to numerically solve the resulting equations. The post-buckling equilibrium paths of the FGM beam with different parameters were plotted, and the effects of non-conservative force, temperature, gradient index of FGM on the post-buckling behavior of right movable simply supported FGM beams were analyzed.

A New Efficient Ode Solver For Solving Initial Value Problem

1998 ◽  
pp. 47-56
Author(s):  
Nazeeruddin Yaacob ◽  
Bahrom Sanugi
Keyword(s):  
Stability Analysis ◽  
Explicit Formula ◽  
Fourth Order ◽  
Initial Value Problem ◽  
Harmonic Mean ◽  
Kutta Method ◽  
Initial Value ◽  
Order Accuracy ◽  
Runge Kutta Method ◽  
Runge Kutta

In this paper we develop a new three-stage,fourth order explicit formula of Runge-Kutta type based on Arithmetic and Harmonic means.The error and stability analyses of this method indicate that the method is stable and efficient for nonstiff problems.Two examples are given which illustrate the fcurth order accuracy of the method. Keywords: Runge-Kutta method, Harmonic Mean, three-stage, fourth-order, covergence and stability analysis.

Numerical Solution of the Flow Between Two Disks

1997 ◽  
Author(s):  
Wahid S. Ghaly ◽  
Georgios H. Vatistas
Keyword(s):  
Shooting Method ◽  
Static Pressure ◽  
Numerical Solutions ◽  
Kutta Method ◽  
Initial Value ◽  
Third Order ◽  
Order Ordinary Differential Equation ◽  
First Order ◽  
Pressure Distributions ◽  
Runge Kutta Method

Abstract This paper deals with the numerical solutions of converging and diverging flows, between two disks. The results are obtained by solving a nonlinear third order ordinary differential equation using a modified shooting method. The governing equation is written as a system of three nonlinear first order ODE’s and the resulting system is solved as an initial value problem via the Runge-Kutta method. The results are given in terms of velocity profiles and static pressure distributions. These are compared with previously reported experimental data obtained by others.

Propagation of Continuum Damage in a Viscoelastic Ice Bar

1994 ◽  
Vol 116 (2) ◽  
pp. 109-113
Author(s):  
J. G. Shin ◽  
D. G. Karr
Keyword(s):  
Method Of Characteristics ◽  
Stable Solution ◽  
Continuum Damage ◽  
Governing Equations ◽  
Initial Value ◽  
Nonlinear Viscoelastic ◽  
Characteristic Lines ◽  
Polycrystalline Ice ◽  
Speed Of Propagation ◽  
The Continuum

An initial value problem of a semi-infinite nonlinear viscoelastic bar is solved with continuum damage evolution. The evolution law of the continuum damage for a viscoelastic material is used in order to explore the propagation of two crushing mechanisms: grain boundary cracking and transgranular cracking. Using the method of characteristics, the speed of propagation is found to be dependent on the continuum damage. On the wave front, the delayed elastic strain is zero, and only the continuum damage due to the transgranular cracking evolves. A finite difference method is developed to solve the governing equations on the obtained characteristic lines, and gives a stable solution of the propagation of the stress, strain, and damage. Numerical results are obtained and discussed using the material properties of polycrystalline ice.

scholarly journals Influence of a Darcy-Forchheimer porous medium on the flow of a radiative magnetized rotating hybrid nanofluid over a shrinking surface

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Sumera Dero ◽  
Hisamuddin Shaikh ◽  
Ghulam Hyder Talpur ◽  
Ilyas Khan ◽  
Sayer O. Alharbim ◽  
...  
Keyword(s):  
Heat Transfer ◽  
Stability Analysis ◽  
Differential Equations ◽  
Shooting Method ◽  
Volume Fraction ◽  
Solid Volume Fraction ◽  
Dual Solutions ◽  
Hybrid Nanofluid ◽  
Solution Stability ◽  
Shrinking Surface

AbstractIn this paper, the heat transfer properties in the three-dimensional (3D) magnetized with the Darcy-Forchheimer flow over a shrinking surface of the $$Cu + Al_{2} O_{3} /$$ C u + A l 2 O 3 / water hybrid nanofluid with radiation effect were studied. Valid linear similarity variables convert the partial differential equations (PDEs) into the ordinary differential equations (ODEs). With the help of the shootlib function in the Maple software, the generalized model in the form of ODEs is numerically solved by the shooting method. Shooting method can produce non-unique solutions when correct initial assumptions are suggested. The findings are found to have two solutions, thereby contributing to the introduction of a stability analysis that validates the attainability of first solution. Stability analysis is performed by employing if bvp4c method in MATLAB software. The results show limitless values of dual solutions at many calculated parameters allowing the turning points and essential values to not exist. Results reveal that the presence of dual solutions relies on the values of the porosity, coefficient of inertia, magnetic, and suction parameters for the specific values of the other applied parameters. Moreover, it has been noted that dual solutions exist in the ranges of $$F_{s} \le F_{sc}$$ F s ≤ F sc , $$M \ge M_{C}$$ M ≥ M C ,$$S \ge S_{C} ,$$ S ≥ S C , and $$K_{C} \le K$$ K C ≤ K whereas no solution exists in the ranges of $$F_{s} > F_{sc}$$ F s > F sc , $$M < M_{c}$$ M < M c , $$S < S_{c}$$ S < S c , and $$K_{C} > K$$ K C > K . Further, a reduction in the rate of heat transfer is noticed with a rise in the parameter of the copper solid volume fraction.

scholarly journals The analysis of the suction/injection on the MHD Maxwell fluid past a stretching plate in the presence of nanoparticles by Lie group method

Open Physics ◽  
2015 ◽  
Vol 13 (1) ◽  
Author(s):  
Limei Cao ◽  
Xinhui Si ◽  
Liancun Zheng ◽  
Huihui Pang
Keyword(s):  
Lewis Number ◽  
Maxwell Fluid ◽  
Elastic Parameter ◽  
Group Method ◽  
Physical Parameters ◽  
Governing Equations ◽  
Linear Ordinary Differential Equations ◽  
Boundary Layer Equations ◽  
Runge Kutta Method ◽  
Stretching Plate

AbstractIn this paper, the magnetohydrodynamic (MHD) Maxwell fluid past a stretching plate with suction/ injection in the presence of nanoparticles is investigated. The Lie symmetry group transformations are used to convert the boundary layer equations into non-linear ordinary differential equations. The dimensionless governing equations are solved numerically using Bvp4c with MATLAB, which is a collocation method equivalent to the fourth order mono-implicit Runge-Kutta method. The effects of some physical parameters, such as the elastic parameter K, the Hartmann number M, the Prandtl number Pr, the Brownian motion Nb, the thermophoresis parameter Nt and the Lewis number Le, on the velocity, temperature and nanoparticle fraction are studied numerically especially when suction and injection at the sheet are considered.

scholarly journals Stability Analysis of Darcy-Forchheimer Flow of Casson Type Nanofluid Over an Exponential Sheet: Investigation of Critical Points

Symmetry ◽  
2019 ◽  
Vol 11 (3) ◽  
pp. 412 ◽  
Author(s):  
Liaquat Ali Lund ◽  
Zurni Omar ◽  
Ilyas Khan ◽  
Jawad Raza ◽  
Mohsen Bakouri ◽  
...  
Keyword(s):  
Stability Analysis ◽  
Numerical Solutions ◽  
Similarity Transformation ◽  
Shrinking Sheet ◽  
Boundary Values ◽  
Governing Equations ◽  
Runge Kutta Method ◽  
Exponentially Shrinking Sheet ◽  
The Stability ◽  
High Suction

In this paper, steady two-dimensional laminar incompressible magnetohydrodynamic flow over an exponentially shrinking sheet with the effects of slip conditions and viscous dissipation is examined. An extended Darcy Forchheimer model was considered to observe the porous medium embedded in a non-Newtonian-Casson-type nanofluid. The governing equations were converted into nonlinear ordinary differential equations using an exponential similarity transformation. The resultant equations for the boundary values problem (BVPs) were reduced to initial values problems (IVPs) and then shooting and Fourth Order Runge-Kutta method (RK-4th method) were applied to obtain numerical solutions. The results reveal that multiple solutions occur only for the high suction case. The results of the stability analysis showed that the first (second) solution is physically reliable (unreliable) and stable (unstable).

Onset of convection over a transient base-state in anisotropic and layered porous media

2009 ◽  
Vol 641 ◽  
pp. 227-244 ◽  
Author(s):  
SAIKIRAN RAPAKA ◽  
RAJESH J. PAWAR ◽  
PHILIP H. STAUFFER ◽  
DONGXIAO ZHANG ◽  
SHIYI CHEN
Keyword(s):  
Porous Media ◽  
Stability Analysis ◽  
Initial Value Problem ◽  
Heterogeneous Systems ◽  
Critical Time ◽  
Mode Analysis ◽  
Governing Equations ◽  
Initial Value ◽  
Layered Porous Media ◽  
Permeability Field

The topic of density-driven convection in porous media has been the focus of many recent studies due to its relevance as a long-term trapping mechanism during geological sequestration of carbon dioxide. Most of these studies have addressed the problem in homogeneous and anisotropic permeability fields using linear-stability analysis, and relatively little attention has been paid to the analysis for heterogeneous systems. Previous investigators have reduced the governing equations to an initial-value problem and have analysed it either with a quasi-steady-state approximation model or using numerical integration with arbitrary initial perturbations. Recently, Rapaka et al. (J. Fluid Mech., vol. 609, 2008, pp. 285–303) used the idea of non-modal stability analysis to compute the maximum amplification of perturbations in this system, optimized over the entire space of initial perturbations. This technique is a mathematically rigorous extension of the traditional normal-mode analysis to non-normal and time-dependent problems. In this work, we extend this analysis to the important cases of anisotropic and layered porous media with a permeability variation in the vertical direction. The governing equations are linearized and reduced to a set of coupled ordinary differential equations of the initial-value type using the Galerkin technique. Non-modal stability analysis is used to compute the maximum growth of perturbations along with the optimal wavenumber leading to this growth. We show that unlike the solution of the initial-value problem, results obtained using non-modal analysis are insensitive to the choice of bottom boundary condition. For the anisotropic problem, the dependence of critical time and wavenumber on the anisotropy ratio was found to be in good agreement with theoretical scalings proposed by Ennis-King et al. (Phys. Fluids, vol. 17, 2005, paper no. 084107). For heterogeneous systems, we show that uncertainty in the permeability field at low wavenumbers can influence the growth of perturbations. We use a Monte Carlo approach to compute the mean and standard deviation of the critical time for a sample permeability field. The results from theory are also compared with finite-volume simulations of the governing equations using fully heterogeneous porous media with strong layering. We show that the results from non-modal stability analysis match extremely well with those obtained from the simulations as long as the assumption of strong layering remains valid.

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