Stability Analysis of Cross Diffusion for the Walters B Fluid Model Saturated With Permeable Nanofluid

2019 ◽  
Vol 11 (4) ◽  
Author(s):  
J. C. Umavathi ◽  
Ali J. Chamkha
Keyword(s):  
Stability Analysis ◽  
Rayleigh Number ◽  
Fluid Model ◽  
Lewis Number ◽  
Double Fourier Series ◽  
Elastic Parameter ◽  
Cross Diffusion ◽  
Nusselt Numbers ◽  
Approach To Steady State ◽  
The Impact

Stability analysis for the Walters-B model saturated with permeable nanofluid is taken under study including cross diffusion effects. The porous medium is defined using modified Darcy model, and the nanofluid is considered to have the impact of thermophoresis and Brownian motion. The thermal energy equation includes the effects of diffusion and also cross diffusion. For the study of linear theory, normal mode procedure is applied and to understand the nonlinear theory, the method of minimal representation of double Fourier series is utilized. The effects of nondimensional parameters such as concentration Rayleigh number, Lewis number, Soret and Dufour parameters, Solutal Rayleigh number, elastic parameter, Prandtl number, viscosity ratio, and conductivity ratio on the stationary and oscillatory convections are represented graphically. The effect of time on transient Nusselt numbers is also taken under investigation. It is concluded that when time is small, the three Nusselt numbers oscillate for all the governing parameters and approach to steady-state as time increases.

Natural convection effects on thermal ignition in a porous medium. II. Lumped thermal model-I

1990 ◽  
Vol 429 (1877) ◽  
pp. 555-567 ◽  
Keyword(s):  
Thermal Conductivity ◽  
Rayleigh Number ◽  
Thermal Model ◽  
Lewis Number ◽  
Reaction Model ◽  
Heat Of Reaction ◽  
Bifurcation Diagrams ◽  
Diffusion Term ◽  
The Impact ◽  
Species Diffusion

In this paper, we examine the disappearance of criticality, ignition locus and bifurcation diagrams of temperature against Rayleigh number of a one-dimensional diffusion-convection-reaction model with the assumption of infinite thermal conductivity and zero species diffusivity. The predictions of this model are compared with those of the Semenov model to determine the impact of the species diffusion term. It is shown that for large values of the Rayleigh number (R* ≫ 1), the ignition locus may be expressed in a parametric form B Ls = t /ln t + t /( t - 1) (1 < t ≼ 3.4955), ψ / R * = ( B Ls ) 2 (( t - 1)/ t ) exp{ - B Ls + B Ls / t } ln t , where B is the heat of reaction parameter, ψ is the Semenov number and Ls is a (modified) Lewis number. Criticality is found to disappear at B Ls = 4.194. When these results are compared with those of the Semenov model, it is found that neglecting the species diffusion term gives conservative approximations to the ignition locus, and criticality boundary. It is found that the lumped thermal model-I has five different types of bifurcation diagrams of temperature against Rayleigh number (single­-valued, isola, inverse S , mushroom, inverse S + isola). These diagrams are qualitatively identical to the bifurcation diagrams of temperature against flow rate for the forced convection problem under the assumption of infinite thermal conductivity and zero species diffusivity.

Cross-Diffusion Induced Turing Instability and Amplitude Equation for a Toxic-Phytoplankton–Zooplankton Model with Nonmonotonic Functional Response

2017 ◽  
Vol 27 (06) ◽  
pp. 1750088 ◽  
Author(s):  
Renji Han ◽  
Binxiang Dai
Keyword(s):  
Stability Analysis ◽  
Functional Response ◽  
Spatiotemporal Distribution ◽  
Spatiotemporal Pattern ◽  
Amplitude Equations ◽  
Toxic Phytoplankton ◽  
Cross Diffusion ◽  
The Cross ◽  
Nonmonotonic Functional Response ◽  
The Impact

The spatiotemporal pattern induced by cross-diffusion of a toxic-phytoplankton–zooplankton model with nonmonotonic functional response is investigated in this paper. The linear stability analysis shows that cross-diffusion is the key mechanism for the formation of spatial patterns. By taking cross-diffusion rate as bifurcation parameter, we derive amplitude equations near the Turing bifurcation point for the excited modes in the framework of a weakly nonlinear theory, and the stability analysis of the amplitude equations interprets the structural transitions and stability of various forms of Turing patterns. Furthermore, we illustrate the theoretical results via numerical simulations. It is shown that the spatiotemporal distribution of the plankton is homogeneous in the absence of cross-diffusion. However, when the cross-diffusivity is greater than the critical value, the spatiotemporal distribution of all the plankton species becomes inhomogeneous in spaces and results in different kinds of patterns: spot, stripe, and the mixture of spot and stripe patterns depending on the cross-diffusivity. Simultaneously, the impact of toxin-producing rate of toxic-phytoplankton (TPP) species and natural death rate of zooplankton species on pattern selection is also explored.

scholarly journals The Coriolis Effect on Thermal Convection in a Rotating Sparsely Packed Porous Layer in Presence of Cross-Diffusion

Coatings ◽  
2021 ◽  
Vol 12 (1) ◽  
pp. 23
Author(s):  
Suman Shekhar ◽  
Ravi Ragoju ◽  
Gudala Janardhana Reddy ◽  
Mikhail A. Sheremet
Keyword(s):  
Rayleigh Number ◽  
Eigenvalue Problem ◽  
Porous Layer ◽  
Lewis Number ◽  
Darcy Number ◽  
Taylor Number ◽  
Wave Number ◽  
Physical Parameters ◽  
Critical Wave Number ◽  
Cross Diffusion

The effect of rotation and cross-diffusion on convection in a horizontal sparsely packed porous layer in a thermally conducting fluid is studied using linear stability theory. The normal mode method is employed to formulate the eigenvalue problem for the given model. One-term Galerkin weighted residual method solves the eigenvalue problem for free-free boundaries. The eigenvalue problem is solved for rigid-free and rigid-rigid boundaries using the BVP4c routine in MATLAB R2020b. The critical values of the Rayleigh number and corresponding wave number for different prescribed values of other physical parameters are analyzed. It is observed that the Taylor number and Solutal Rayleigh number significantly influence the stability characteristics of the system. In contrast, the Soret parameter, Darcy number, Dufour parameter, and Lewis number destabilize the system. The critical values of wave number for different prescribed values of other physical parameters are also analyzed. It is found that critical wave number does not depend on the Soret parameter, Lewis number, Dufour parameter, and solutal Rayleigh number; hence critical wave number has no impact on the size of convection cells. Further critical wave number acts as an increasing function of Taylor number, so the size of convection cells decreases, and the size of convection cells increases because of Darcy number.

scholarly journals On the flow of MHD generalized maxwell fluid via porous rectangular duct

Open Physics ◽  
2020 ◽  
Vol 18 (1) ◽  
pp. 989-1002
Author(s):  
Aamir Farooq ◽  
Muhammad Kamran ◽  
Yasir Bashir ◽  
Hijaz Ahmad ◽  
Azeem Shahzad ◽  
...  
Keyword(s):  
Fluid Model ◽  
Maxwell Fluid ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Rectangular Duct ◽  
Modified Darcy’S Law ◽  
Initial Boundary Value ◽  
Generalized Maxwell Fluid ◽  
Limiting Case ◽  
The Impact

Abstract The purpose of this proposed investigation is to study unsteady magneto hydrodynamic (MHD) mixed initial-boundary value problem for incompressible fractional Maxwell fluid model via oscillatory porous rectangular duct. Considering the modified Darcy’s law, the problem is simplified by using the method of the double finite Fourier sine and Laplace transforms. As a limiting case of the general solutions, the same results can be obtained for the classical Maxwell fluid. Also, the impact of magnetic parameter, porosity of medium, and the impact of various material parameters on the velocity profile and the corresponding tangential tensions are illuminated graphically. At the end, we will give the conclusion of the whole paper.

Two-Phase Flows in Mini-/Micro-Gas Flow Channels of PEM Fuel Cells

2013 ◽  
Author(s):  
Sung Chan Cho ◽  
Yun Wang
Keyword(s):  
Pressure Drop ◽  
Gas Flow ◽  
Fluid Model ◽  
Pem Fuel Cells ◽  
Two Phase ◽  
Micro Gas Flow ◽  
Two Fluid Model ◽  
The Impact ◽  
Two Fluid ◽  
Phase Pressure

In this paper, two-phase flow dynamics in a micro channel with various wall conditions are both experimentally and theoretically investigated. Annulus, wavy and slug flow patterns are observed and location of liquid phase on different wall condition is visualized. The impact of flow structure on two-phase pressure drop is explained. Two-phase pressure drop is compared to a two-fluid model with relative permeability correlation. Optimization of correlation is conducted for each experimental case and theoretical solution for the flows in a circular channel is developed for annulus flow pattern showing a good match with experimental data in homogeneous channel case.

Effect of the Condition Number of the Inverse Fourier Transform Matrix on the Solution Behavior of the Time Spectral Equation System

2020 ◽  
Author(s):  
Sen Zhang ◽  
Dingxi Wang ◽  
Yi Li ◽  
Hangkong Wu ◽  
Xiuquan Huang
Keyword(s):  
Fourier Transform ◽  
Stability Analysis ◽  
Condition Number ◽  
Inverse Fourier Transform ◽  
Real Life ◽  
Equation System ◽  
Time Instant ◽  
Transform Matrix ◽  
Spectral Equation ◽  
The Impact

Abstract The time spectral method is a very popular reduced order frequency method for analyzing unsteady flow due to its advantage of being easily extended from an existing steady flow solver. Condition number of the inverse Fourier transform matrix used in the method can affect the solution convergence and stability of the time spectral equation system. This paper aims at evaluating the effect of the condition number of the inverse Fourier transform matrix on the solution stability and convergence of the time spectral method from two aspects. The first aspect is to assess the impact of condition number using a matrix stability analysis based upon the time spectral form of the scalar advection equation. The relationship between the maximum allowable Courant number and the condition number will be derived. Different time instant groups which lead to the same condition number are also considered. Three numerical discretization schemes are provided for the stability analysis. The second aspect is to assess the impact of condition number for real life applications. Two case studies will be provided: one is a flutter case, NASA rotor 67, and the other is a blade row interaction case, NASA stage 35. A series of numerical analyses will be performed for each case using different time instant groups corresponding to different condition numbers. The conclusion drawn from the two real life case studies will corroborate the relationship derived from the matrix stability analysis.

Nonlinear mixed convective Williamson nanofluid flow with the suspension of gyrotactic microorganisms

2021 ◽  
pp. 2150145
Author(s):  
Shuang-Shuang Zhou ◽  
M. Ijaz Khan ◽  
Sumaira Qayyum ◽  
B. C. Prasannakumara ◽  
R. Naveen Kumar ◽  
...  
Keyword(s):  
Heat Source ◽  
Lewis Number ◽  
Solution Procedure ◽  
Energy Concentration ◽  
Gyrotactic Microorganisms ◽  
Dimensionless Parameters ◽  
Flow Equations ◽  
Fluid Parameter ◽  
Source Sink ◽  
The Impact

This investigation aims to present the thermally developed bioconvection flow of Williamson nanoliquid over an inclined stretching cylinder in presence of linear mixed convection and nonuniform heat source/sink. The activation energy and suspension of gyrotactic microorganisms are accounted with applications of bioconvection phenomenon. Appropriate nondimensional variables are opted to attain the dimensionless form of flow equations. The resulting momentum, energy, concentration and motile density equations are abridged to highly coupled and nonlinear in nature. The numerical treatment is followed for the solution procedure by employing the shooting method. The influence of some relevant dimensionless parameters is discoursed graphically along with physical justifications. Moreover, the impact of several dimensionless parameters on skin friction and Nusselt number is obtained and listed in tables. It is observed that the velocity of fluid shows a decreasing variation for Williamson fluid parameter. The change in unsteadiness parameter and heat source parameter enhanced the nanofluid temperature. The motile microorganisms profile declines with bioconvection constant and bio-convection Lewis number.

The Graetz Problem for the Ellis Fluid Model

2018 ◽  
Vol 74 (1) ◽  
pp. 15-24 ◽  
Author(s):  
N. Ali ◽  
M.W.S. Khan
Keyword(s):  
Energy Equation ◽  
Fluid Model ◽  
Separation Of Variables ◽  
Surface Heat ◽  
Thermal Boundary Conditions ◽  
Nusselt Numbers ◽  
Graetz Problem ◽  
Developing Flow ◽  
Sturm Liouville

AbstractThe determination of temperature and auxiliary quantities such as local and average Nusselt numbers for thermally developing flow is referred as the Graetz problem. In the classical Graetz problem, the fluid entering the tube or channel is Newtonian in nature. Here, an extension of the classical Graetz problem is presented by assuming that the fluid entering the tube or channel obeys the Ellis constitutive equation. The energy equation for the considered problem is solved using the separation of variables technique supplemented with the MATLAB routine bvp4c for computation of the eigenvalues and numerical solution of the associated Sturm-Liouville boundary value problem. The problem is solved for two types of thermal boundary conditions, namely, uniform surface temperature and uniform surface heat flux for both flat and circular geometries. Expressions for bulk mean temperature and local and average Nusselt numbers are presented and discussed through tables and graphs.

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