# Mathematical models for heat and mass transfer in nanofluid flows.

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## Date

2018

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## Abstract

The behaviour and evolution of most physical phenomena is often best described using
mathematical models in the form of systems of ordinary and partial differential equations.
A typical example of such phenomena is the flow of a viscous impressible fluid which
is described by the Navier-Stokes equations, first derived in the nineteenth century using
physical approximations and the principles of mass and momentum conservation. The flow
of fluids, and the growth of flow instabilities has been the subject of many investigations because
fluids have wide uses in engineering and science, including as carriers of heat, solutes
and aggregates. Conventional heat transfer fluids used in engineering applications include
air, water and oil. However, each of these fluids has an inherently low thermal conductivity
that severely limit heat exchange efficiency. Suspension of nanosized solid particles in
traditional heat transfer fluids significantly increases the thermophysical properties of such
fluids leading to better heat transfer performance.
In this study we present theoretical models to investigate the flow of unsteady nanofluids,
heat and mass transport in porous media. Different flow configurations are assumed including
an inclined cylinder, a moving surface, a stretching cone and the flow of a polymer
nanocomposite modeled as an Oldroyd-B fluid. The nanoparticles assumed include copper,
silver and titanium dioxide with water as the base fluid. Most recent boundary-layer
nanofluid flow studies assume that the nanoparticle volume fraction can be actively controlled
at a bounding solid surface, similar to temperature controls. However, in practice,
such controls present significant challenges, and may, in practice, not be possible. In this
study the nanoparticle flux at the boundary surface is assumed to be zero.
Unsteadiness in fluid flows leads to complex system of partial differential equations. These
transport equations are often highly nonlinear and cannot be solved to find exact solutions
that describe the evolution of the physical phenomena modeled. A large number of numerical
or semi-numerical techniques exist in the literature for finding solutions of nonlinear
systems of equations. Some of these methods may, however be subject to certain limitations
including slow convergence rates and a small radius of convergence. In recent years, innovative
linearization techniques used together with spectral methods have been suggested as
suitable tools for solving systems of ordinary and partial differential equations. The techniques
which include the spectral local linearization method, spectral relaxation method
and the spectral quasiliearization method are used in this study to solve the transport equations,
and to determine how the flow characteristics are impacted by changes in certain
important physical and fluid parameters. The findings show that these methods give accurate
solutions and that the speed of convergence of solutions is comparable with methods
such as the Keller-box, Galerkin, and other finite difference or finite element methods.
The study gives new insights, and result on the influence of certain events, such as internal
heat generation, velocity slip, nanoparticle thermophoresis and random motion on the flow
structure, heat and mass transfer rates and the fluid properties in the case of a nanofluid.

## Description

Doctoral Degree. University of KwaZulu-Natal, Pietermaritzburg.