FDM Solutions for the flow of Micropolar Nanofluid inside a partially heated Square Cavity

Abstract

Heat transfer in cavities with non-Newtonian fluid is the hot research topic over the decade and great interest is shown by the scientists and thermal engineers by investigation of the heat transfer in non-Newtonian fluid and various shapes of geometry. Non-Newtonian fluid is the more complex and complicated fluids which exhibits different nature than that of Newtonian fluid. Complex fluid has applications in the thermal system devices, lubricating procedures, device cooling the ventilation system of buildings Engineers are interested in the high heat transfer rate. They created numerous types of models in order to decrease drag force and skin fraction coefficient. The computation of both simple fluids and complex fluid are performed Additionally. heat transfer analysis in cavities, ducts, cylindrical pipes, are common scenarios for the validation of the numerical techniques or outcomes from mathematical models and simulations. Utilizing computer programming or programs through software such as MATLAB[1]. Nowadays a lot of CFD commercial software is available based on different Numerical Techniques like COMSOL [2], ANSYS [3], And Open FOAM[4], etc. Convective transport and heat transfer in a square enclosure have been studied broadly and simulated for a long time [5]. Heat transfer in cavities by convective mode has got huge consideration in last two decades considering wide available application in modern and designing fields, for example, in heat exchanger, geothermal, and geophysical frameworks [6], repositories, building protections, [7]. Various kinds of ducts and cavities are the flow problem geometries which have been discussed by many researchers and tested physically by the engineers. The rectangular shape cavity in which one of its sides moves in its own reference plane is called Lid driven cavity. And the two side of the cavity is free to move in opposite direction in its reference frame of plane which is called cavity driven by two opposite lid. The cavity lid driven problem is considered the benchmark problem in the history of numerical simulation of the cavity problems proving grounds for concentrating on specific actual effects of a numerical simulation consideration. On searching web of sciences for subject top driven yields more than 2000 hits. Considerable efforts have been done on the research topic of transfer of heat in different fluid in enclosure due to importance its use in real life in exchanger of heat, room ventilation, and the designing of nuclear reactor and various other convective movement phenomena. The study due to difference in temperature i.e. buoyancy driven flow in rectangular cavity is the fundamental importance in the natural convection phenomena. Some of the basic studies associated with the natural convection phenomena are Drummond and Korpela [8] and Ostrach [9]. Investigate the natural convection that occurs naturally in shallow cavity, and finds out that results essential Grashofs numbers doesn’t significantly varies with isothermal effects along with the condition of the side walls of respective cavity boundaries because the stability mostly varies due to applied force not by the buoyancy effects. Hossain and Rees [10] examined natural convection of viscous incompressible fluid in a porous rectangular cavity that is heated from lower wall and side walls is at constant temperature. For convective terms they use the 2nd iii upwind order accuracy scheme. The study also examined the effect of zero Darcy effect. During this study it revealed that the cell formation property of cavity aspects ratio and Grashofs numbers it determines how the cell can be forms in the cavity and the partial heating process is only occurring in earth and atmosphere. In the early, 1990s the scientists developed the interest in the use of the nanomaterial for the heat transfer and get some useful productive results on heat transfer phenomena. That was Choi [11] who gave the idea of Nano particle mixture in the fluid which is named as “Nanofluid” in 1995. Nanofluid are the mixture of one or more nanotechnology-based liquid set up by spreading and suspending nanometer sized molecules with ordinary size scaled on request of one tenth thousand human hair for example (1-100 nm) in standard heat transfer liquid. Nanofluid has exceptionally valuable application in the heat transfer and energy components crossover power motions, atomic reactor coolant in space innovation, protection of ships and so on. Nanoparticles likewise exists in unadulterated metals like 𝐶𝑢 ,𝐴𝑔,𝐹𝑒, as some metal oxides 𝐹𝑒𝑂,𝐶𝑢𝑂,𝑇𝑖𝑂 𝑎𝑛𝑑 𝑆𝑖𝑂, carbides and nitrides,𝑆𝑖𝐶,𝑇𝑖𝐶,𝐴𝑙𝑁,𝑆𝑖𝑁 and some carbon containing iotas like jewels , graphite and other carbon nanotubes, Nanofluid is additionally exists in fluid structures also. For example, ethylene glycol, motor oil and water. Lately scientists and research analysts mostly focus on the warm conductivity and steady nature of nano fluids and its expanding application in enterprises. As analyst are attempting to sum up the new essential studies of the nanofluids. Keblinski [12] tested the nanofluids thermal as well as physical properties and its difficulties in future. Wang and Mujumdar [13] studied the qualities of the nano fluid in free and force convection, in such types of flows. In force convection flow heat transfer of the nanofluids, its tested and numerical mathematical word was studied by Weerapun and Somachai [14]. Heat transfer is critical in the warm designing field. For a long time, the work is done for the coolants to upgrades the heat transfer and to further develop heat transfer of the fouling. The impact of fluid particle macrostructure in classical Navier-Stokes (NS) equations cannot taken into account for study. Characteristic of fluid particles or simply the fluid expands contracts circulate around its own line of action and it changes physical shape because of applied force on it. Such type properties are found in the animal blood, liquid crystals, suspensions and polymer type fluids. This fluid type are named as Micropolar fluid. The idea of Micropolar fluid was presented by Eringen [15]. A micropolar fluid is the fluid which includes the inflexible and randomly coincide suspended particles traveling in a viscous flow, and the fluid particle deformation is ignored. The microrotation vector, is represented by the path of rotation and is associated with the usual components of velocity. Therefore, angular momentum equation included in the flow model. The only coupling parameter in Micropolar fluid is the material parameter Κ. The earlies study of Micropolar fluid in enclosure is was considered by Bhattachariya and Jena [16]. They use the Galerkin residual method to find the vital Rayleigh number for Newtonian as well as micropolar fluid in a cavity through thermally isotherm from below and the other sides are assumed at constant temperature. Hsu and Wang [17] examined the buoyancy effect on micropolar fluid iv in vertical drawn cavity and get the results that heat transfer rate is effected due to the value of which Nusselt number is decreases by the increment in value of coupling parameter for an angle of inclination. Chen and Hsu [18] studies the buoyancy effects in Micropolar fluid in a rectangular enclosure in which upper wall of geometry is isothermal and bottom is kept at constant temperature, and vertical walls are kept at the zero-temperature gradient. There are so many works in which Micropolar fluid is taken into account. This thesis consists of three chapters in which the first chapter deals with the basics of fluid mechanics associated with the fluid mechanics fluid classification types of fluids, non dimensional numbers, and General law of conservation of continuity, momentum and energy, Introduction to CFD and some properties of the fluid are also discussed. Second chapter deals with the steady, laminar, and viscous incompressible flow of “Natural convection of Water based Cupper nanofluid in square cavity differentially heated form below”. The heat source of length 𝜖 is localized at the bottom surface while the top wall is at zero temperature gradient, and the side walls are at constant temperature. The nano particles here taken are 𝐶𝑢 with base fluid as water. We model the equation for nanofluid by using the Nanofluid Model of Brinkman [19], and solve the governing equation of the flow by FDM using 2nd order accuracy difference scheme for diffusion part and apply upwind scheme for convective part. The discretize equation is solved by Tridiagonal Matrix algorithm line by line Method. The simulation is contrasted with the results of Calcagni [20]. The result is computed for different Rayleigh numbers as and on different heat source length. And the effect of different parameters is presented. Third chapter deals with the Extension study of the “FDM solutions for the flow of Micropolar nanofluid inside a partially heated square cavity.” The heat source of length 𝜖 is placed at the middle position on lower wall while the upper wall of the cavity is adiabatic in nature, and the side walls are at some temperature. The nano particle here is taken to be cupper(𝐶𝑢) and the base fluid is taken as water and the fluid is taken as Micropolar fluid. The flow is assumed as steady, laminar, and incompressible. The Governing partial differential equation of mathematical flow model is discretized by finite difference method. The 2nd order central accuracy Difference scheme is used for the diffusion part and Upwind scheme is employed to convective terms. The discretized equation is solved by Tridiagonal Matrix Algorithm through line-by-line Method which is called Thomas Algorithm. The results simulated for different heat source length is visualized. For different Material parameter,Κ solid Volume fraction 𝜙, aspect ratio of heat source 𝜖, Rayleigh Number 𝑅𝑎, Material parameter Κ on Local Nusselt numbers 𝑁𝑢 and mean Nusselt numbers 𝑁𝑢