Approximate Analytical Solution of Natural Convection Flow With Viscous Dissipation and Variable Viscosity Effects
Chapter One
Aim and Objectives of the study
The aim of this work is to investigate the effects of variable viscosity and viscous dissipation on steady, fully developed natural convection flow of incompressible fluids between infinite vertical parallel plates. This aim will be achieved through the following objectives; which are to:
- study the effect of varying viscosity on both the velocity and temperature profiles for the fluids in a vertical
- examine the contribution of suction/injection to both the energy and momentum of the
- investigate the general effects of viscous dissipation on the fluid flow pattern and
CHAPTER TWO
LITERATUREREVIEW
Introduction
Natural convection flow under suction and injection between vertical infinite parallel plates has enormous applications in scientific and engineering processes such as in the operations of air conditioners, nuclear reactors, the cooling of engines and electrical and electronic components as well as in heat exchangers to mention a few. A lot of works exist in literature that investigated the different physical situations considered in this dissertation. Some of these works are presented in this chapter under different sub-headings.
Natural convection flow
The study of natural convection flow due to steady-periodic heating has received attention due to its applicability in different scientific and engineering systems. Natural convection is still a means of cooling various engineering and control systems as well as electrical and electronic components which generate heat and are subjected to heating periodically. Viscosity is a property of fluid which changes with changing temperatures. Gebhart (1962) showed that the heating of fluids by viscous dissipation cannot be neglected for fluids with high Prandtl number when steady free convective flow is considered because it can greatly affect the flow characteristics. Again, when fluids are sheared, viscous dissipation is induced by the friction between the different layers of particles of the fluid as they move against each other, generating heat in the process. It therefore follows that the rate of shear- induced heat generation within a fluid is responsive to variations in viscosity. Inspired by
the separate works of Sahin (1999) and Tasnim and Mahmud (2009), Jha and Ajibade (2011) stated that viscosity is the most sensitive property of fluid to rising temperatures. For instance, 10% rise in the temperature of water results to as much as 140% reduction in viscosity, at least, Schlitchting and Gersten (2000). Bar-Cohen and Roshenow (1984) studied the fully developed fluid flow between two periodically heated parallel plates to capture miniaturizations of electrical and electronic components. Nanda and Sharma (1963) while studying the effects of sinusoidal variations in surface temperatures on free convective laminar flow were able to separate the temperature and velocity into steady and periodic parts. Wang (1988) investigated the effect of Strouhal number on the development of boundary layers between two periodically heated parallel plates by also separating the steady and unsteady parts and concluded that Strouhal number exerts an inverse effect on unsteady flow, but the effect tends to zero as Strouhal number tends to infinity. Barletta and Rossi di Schio (2004) studied a mixed convection flow in a circular duct whose wall is subjected to periodic sinusoidal heating and were able to show that there exists a resonance frequency where velocity oscillations reach a maximum. Several other authors who have taken in hand to investigate natural convection flow under different physical phenomena include Muhuri and Gupta (1979), Zacchini (1998), Watanabe and Pop (1994), Barletta (1998), Jha and Ajibade (2009), Soundalgekar et al (2004) among others.Jha and Ajibade (2011) investigated the effects of viscous dissipation on natural convection flow in a vertical channel formed by two infinite parallel plates and concluded that the viscous dissipation heating becomes stronger than those of the plates when the thermal diffusivity of the fluid is relatively small causing transfer of heat from fluid to plates instead, but their results are limited to very narrow range of values of E. Again the viscous dissipation parameter, E, is limited in its role in the work of Jha and Ajibade (2011) as it was used as a perturbation parameter only, being an off-shoot of the bottle necks of the general perturbation method.
Suction and Injection
The significance of suction and injection is well recognized in space science and aerodynamics in the area of boundary layer control. In the area of mass transfer cooling, suction/injection can significantly alter the flow field and by extension affect the rate of heat transfer on the porous boundaries. The flow with heat transfer of viscous incompressible fluid between parallel plates has important applications in many systems and devices such as electrostatic precipitation, aerodynamics heating, polymer technology and the petroleum industry. Shojaefard et al (2005) investigated flow control on a subsonic airfoil by suction and injection and concluded that suction significantly increases the lift coefficient while injection decreases the surface skin friction resulting in significant reduction in energy consumption of such aircrafts. This by extension will amount to reduced level of air pollutant emission as well as cost for such air crafts. Attia (2005) also investigated the effect of suction and injection on unsteady flow between parallel plates. By and large, physical phenomena in Medicine, Engineering and several other fields of research naturally involves fluids that are Newtonian in nature. Jha and Ajibade (2009) studied the influence of suction/injection on natural convection flow of viscous fluid when the convection current is induced by the heating of the porous boundaries, but considers fluid viscosity to be a constant. Recently, Rundora and Makinde (2013) studied the effects of suction and injection on unsteady reactive variable viscosity fluid flow with porous medium but considers the fluid to be non-Newtonian in nature.
CHAPTER THREE
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MATHEMATICAL ANALYSES
Introduction
This chapter is preoccupied with the mathematical formulation, derivation and solution of both the energy and momentum equations of the two problems under investigation in this dissertation. The dimensional energy and momentum equations are first put in dimensionless forms by the introduction of appropriate non-dimensional quantities. The Homotopy Perturbation method is then applied to linearize and uncouple the non-linear and coupled differential equations before solving by the undetermined coefficient method to obtain close form expressions for both the velocity and temperature fields in the two problems (see Kryszig E., 1990). The skin friction and Nusselt number are also calculated as well as the phases and amplitudes of the unsteady velocity and temperature.
Mathematical formulation and Geometrical Description
In this subsection, we formulate a mathematical modelof natural convection flowbetween infinite vertical parallel plates with viscous dissipation and variable viscosity effects.
Considering the flow of viscous dissipating incompressible fluid with temperature dependent viscosity between two vertical and parallel plates of infinite length, the channel formed by the boundary plates are at fixed distances ± on the horizontal axis from the center line, = 0. The direction of flow is taken as the vertically while the direction of heat penetration from the ambient (plates) is taken as perpendicular to the direction of flow. The initial temperature of the fluid within the channel is taken as 0.
CHAPTER FOUR
SOLUTION OF THE PROBLEMS
Introduction
This chapter provides details of how the solutions of the non-dimensional momentum and energy equations describing the two problems in this theses were solved as well as the obtained solutions. It also presents the determination of other quantities of interest such as the phases and amplitudes of temperature and velocity fields, skin friction at both bounding plates as wellas the Nusselt number.
CHAPTER FIVE
RESULTS AND DISCUSSION
Introduction
This chapter presents the discussion of the results obtained from the previous chapter in the form of values and line graphs that clearly reveal the influences of the governing parameters on the flow behaviours. A computer program was written and used in the simulation of the periodic temperature and velocity, Skin friction and Nusselt number for each of the studied problems. The Phase and amplitude of fluid temperature and velocity were also simulated for both cases.
CHAPTER SIX
SUMMARY, CONCLUSION AND RECOMMENDATION
Summary
In this dissertation, a study of natural convection flow between infinite vertical parallel plates with viscous dissipation and variable viscosity effects have been studied. The problem is further extended to investigate natural convection heat transfer flow in a vertical channel when the boundary permits movement of fluid in and out (suction/injection). The momentum and energy equations of these problems were first separated into steady and periodic parts before putting in dimensionless form. The strongly non-linear and coupled ode‟s obtained were first subjected to the Homotopy Perturbation technique to break the non-linearity and coupling, resulting in linear ode‟s which were then solved by means of the undetermined coefficient method to obtain expressions for both temperature and velocity fields in the two problems. Simulation of the results was carried out using MATLAB (R2012b) program and generated numerical values and line graphs were used in the discussion, inference and drawing of conclusions.
Conclusion
In this dissertation, we have been able to overcome the limitations of coupling and non- linearity posed by the governing differential equations to obtain, by the Homotopy Perturbation method, model solutions for both the energy and momentum of fluidsin a vertical channel under the effects of viscous dissipation, variable viscosity and suction/injection.
Based on the analysis of the obtained results, the following conclusions are drawn from this dissertation:
Firstly, for fluids with > 1, the effect of variable viscosity is dominant over that of viscous dissipation.
Secondly, viscous dissipation heating becomes stronger and exerts greater influence on both the energy and momentum of the working fluid when viscosity has been increased.
Thirdly, fluid temperature decreases with growing viscous dissipation if the viscosity of the fluid decreases with increasing temperature.
Fourthly, when viscosity is raised, the rate of heat transfer on the plates decreases with increasing viscous dissipation for all values of .
In addition, suction on a plate increases the velocity and decreases the temperature near it while the other plate witnesses decrease in velocity and increased temperature under injection.
Finally, the skin friction on the plates increases with increased suction while it decreases with increasing injection.
Recommendation
It is hereby recommended that the contribution of variations in conductivity of the fluidcan be studied on each of the two problems as a further investigation.
Mixed convection or pressure driven flow situation can be studied in either of the cases to investigate the effect of pressure gradient on the flow pattern.
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