Numerical Solutions of Nanofluid Flow Problems in Crisp and Uncertain Environments

Abstract

Fluids are often used as heat carriers in heat transfer equipments. Examples of important uses of heat transfer fluids include cooling systems in the transportation industry, hydronic heating, and cooling systems in buildings, industrial process heating and cooling systems in the petrochemical, textile, etc. In all these applications, the thermal conductivity of fluid plays an important role in the development of energy-efficient heat transfer equipment. However, the existing heat transfer fluids, viz. water, oil, ethylene glycol mixture, etc., have poor thermal conductivity, which is a limitation in improving the performance of many heat transfer equipments. But nowadays, due to various recent applications, industries have a strong need to develop advanced heat transfer fluids with a higher thermal conductivity compared to the thermal conductivity of existing heat transfer fluids. As such, a new class of fluid has been introduced by researchers, called nanofluid, which is a fluid with suspended nanoparticles. The introduction of nanofluid made researchers to think beyond the traditional fluid flow problems for handling the challenging nanofluid dynamics problems. As such, this dissertation addresses the nanofluid flow between two plates, viz. vertical parallel plates, horizontal parallel plates, and inclined plates. These problems are usually governed by nonlinear partial differential equations, which may be converted into nonlinear ordinary differential equations with the aid of appropriate similarity variables. It may not always be possible to get analytical solutions to these nonlinear differential equations. Accordingly, different semi-analytical and numerical methods such as homotopy perturbation method, optimal homotopy analysis method, Adomian decomposition method, Galerkin’s method, and least square method are applied here to handle the corresponding governing differential equations. Convergence for each method is discussed numerically, and residual errors have been computed to see the efficacy of the mentioned methods. Effects of the involved parameters on velocity and/or temperature profiles have been illustrated graphically for all the considered models, and the results are discussed in detail. Generally, the physical parameters used in different nanofluid dynamics models are assigned crisp values. But in practical scenarios, these parameters may be uncertain in nature due to errors in the measurements, observations, or experiments. The involvement of uncertain parameters mimics the actual practical problems and leads to uncertain differential equations. For example, nanoparticle volume fraction plays an important role in many nanofluid flow problems. A small change in the value of nanoparticle volume fraction or any other parameter may affect the velocity and/or temperature profiles of nanofluid. As such, it will be important and challenging to study nanofluid flow problems by considering nanoparticle volume fraction as an uncertain parameter. In this regard, this thesis also aims to investigate the above-mentioned nanofluid problems in uncertain environment. The uncertain parameter(s) may be considered in terms of interval or fuzzy number. Accordingly, fuzzy number has been used here to express the impreciseness in the value of parameters such as the volume fraction. Further, homotopy perturbation and Adomian decomposition methods have been extended with the aid of the double parametric concept of fuzzy numbers to handle uncertain differential equations. Further, one may have recorded the velocity and/or temperature profile from experiments, but they may demand the value of one or more parameters involved in the models. In such cases, it will be helpful to compute the values of the unknown parameters from the recorded velocity and/or temperature profiles, which is known as the inverse problem. In this regard, this dissertation introduces the inverse problem related to nanofluid flow model also. Here, the desired inclined angle between two plates has been computed using the given velocity profile of nanofluid flow between the channel. It is worth mentioning that the inverse problem in uncertain environment may be more challenging. Accordingly, desired fuzzy volume fraction has been computed here for a given fuzzy velocity profile of nanofluid flow in an inclined channel. The homotopy perturbation method has been used here to study the inverse case for both crisp and uncertain environments.