Nonlinear Models with Darcy Forchheimer Relation

Abstract

Transport of fluid through porous space is quite important topic. Such importance is quite prevalent in various engineering process. Mathematical and analytical techniques to model flows in porous media vary from algebraic expressions to fluid models. Extensive studies have been undertaken for porous medium employing classical Darcy’s expression. Darcy assumes a continuum approximation of both medium and fluid to study the fluid transport in porous media. Additional features are incorporated to increase its accuracy and validity for a wider range of porous media. Darcy-Forchheimer, Darcy-Brinkman and Darcy-Brinkman-Forchheimer are few examples which incorporate inertia and boundary features additionally. Modified Darcy’s law is another continuum approach which is based on rheological properties of fluid. Rotating flows has significance in the field of meteorology and oceanography. It is because of the effects of Coriolis and centrifugal forces. Thus it is imperative to study the effect of rotating flows by continuously moving surfaces. In applications related to transport of blood, foam, emulsion or suspension the no slip boundary condition is not appropriate. In these situations, it is essential to use slip conditions which defines a relation between particles adjacent to surface to normal component of velocity at surface. Prescribed heat flux condition at the boundary is also utilized in this thesis. Present thesis focuses on the characteristics of nonlinear models through Darcy Forchheimer porous space. For more general view of engineering applications, the flows by different surfaces are also studied. This thesis is structured as follows: Chapter one provides a detailed literature review and fundamental expressions. Chapter two discussed the rotating flow of two-phase nanofluid through porous space. Velocity and thermal slip conditions are employed at the boundary. Darcy-Brinkman expression is utilized to capture the effect of porous space. Inclined magnetic field is applied. Heat transfer aspects are studied in presence of viscous dissipation. Numerical solutions are computed through NDSolve technique. The contents of this chapter are submitted in Numerical Methods for Partial Differential Equations. Darcy-Forchheimer flow of nanofluid subject to rotating frame is analyzed in Chapter three. Nanofluid consisting of carbon nanotubes is utilized. Exponential stretching sheet creates disturbance in flow. Prescribed heat flux condition is employed at the boundary. Behaviors of emerging variables on flow and physical quantities are physically interpreted. The relevant observations are published in Physica Scripta 96 (2021) 025217. Chapter four aims to compute optimal series solutions for chemically reactive flow of carbon nanotubes through Darcy-Forchheimer porous space. Carbon nanotubes consisting of single and multiple layers of graphene are used in analysis. Heat generation/absorption and viscous dissipation are also accounted. Entropy generation in a system is modelled through second law of thermodynamics. Comparative results are obtained for single wall and multi wall carbon nanotubes. Optimal solutions are approximated through OHAM. The data of this chapter is published in Physica Scripta 96 (2021) 095209. Chapter five presents the numerical investigation of carbon nanotubes through porous space. Carbon nanotubes namely single and multi walls are utilized in the analysis. Disturbance in flow is generated by the stretching sheet whose curvature is altered in a controllable manner. Flow in porous space is characterized by Darcy-Forchheimer relation. Graphical illustration for behavior of emerging variables on flow fields is provided. Materials of this chapter are published in Journal of Central South University 26 (2019) 865-872. Chapter six elaborates the impact of prescribed heat flux condition in flow of water-based carbon nanotubes. Exponential curved stretching sheet creates disturbance in flow. Heat transfer aspect is analyzed in presence of heat generation/absorption. Porous space effect is characterized by Darcy Forchheimer relation. NDSolve technique is employed for computation of numerical solutions. The contents of this chapter are published in Physica A: Statistical Mechanics and its Applications 554 (2020) 124002. Features of hybrid nanofluid through Darcy-Forchheimer porous space is illustrated in Chapter seven. Molybdenum disulfide and Silicon dioxide are utilized in flow analysis. Comparative results are obtained for hybrid nanofluid and nanofluid. Porous space with variable characteristics is analyzed. Additional effects of nonlinear thermal radiation, heat generation/absorption and viscous and porous dissipation are considered. Entropy generated in a system is modelled by second law of thermodynamics. Observations of this chapter are published in Entropy 23 (2021) 89. Chapter eight provides the comparative analysis for flow of carbon nanotubes due to a rotating disk. Boundary conditions for velocity and temperature are set so that slip effects are not ignored. Flow in porous space is described by Darcy Forchheimer relation. Viscous dissipation is also considered. Optimum series solutions are computed by optimal homotopy analysis technique. Data of this chapter is published in International Communications in Heat and Mass Transfer 116 (2020) 104641. Chapter nine develops the numerical solution for nanofluid flow filling porous space with variable characteristics. Mass transfer aspect is studied in presence of activation energy. Buongiorno model is utilized for nanoliquid transport phenomenon. Permeability and porosity of porous space are linear functions of space variable. Disturbance in flow is created by rotating disk. Variations of flow fields against emerging variables are interpreted through graphs. Numerical data of physical quantities is obtained and analyzed. Material of this chapter is published in International Communications in Heat and Mass Transfer 119 (2020) 104904. Simultaneous features of thermal stratification and nonlinear thermal radiation in flow of hybrid nanofluid are interpreted in Chapter ten. Nanoparticles of two types namely Titanium dioxide and Aluminum oxide are accounted. Velocity slip conditions are employed at the boundary. Variable aspects of porosity and permeability are utilized through porous space effect. Contents of this chapter are published in Alexandria Engineering Journal 60 (2021) 3047-3056. Chapter eleven aims to analyze the features of Carreau fluid through porous space with variable characteristics. Flow is created by a rotating disk. Flow properties are discussed subject to viscous dissipation. Rate of entropy generation is also calculated. Keeping in view the rheological characteristics of Carreau fluid, modified Darcy’s law is utilized to capture the effect of porous space. Numerical solutions are computed. Observations of this chapter are published in International Communications in Heat and Mass Transfer 120 (2021) 105073. Chapter twelve consists of the concluding remarks of present thesis.