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| Format: Ms Word | 1-5 Chapters | Table of Content|


Study Level: BTech, BSc, BEng, BA, HND, ND or NCE

Amount: ₦4,000.00

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Magneto-Hydrodynamic Natural Convection Flow in a Concentric Annulus with Ramped Temperature and Ramped Motion of the Boundaries | IntechOpen


This project presents an analytical solution for magnetohydrodynamics natural convection flow in a vertical tube due to time-periodic heating of tube surfaces. The fluid is assumed to be fully developed and the mathematical equations governing flow formation and heat transfer are derived, the governing partial differential equations were transformed to their corresponding ordinary differential equation (ODE), representing the steady and periodic regimes. Closed-form expressions for velocity, temperature, skin-friction, mass flow rate and rate of heat transfer which is expressed as Nusselt number are obtained in terms of modified Bessel’s function by solving the present mathematical model after separating into steady component and periodic regime. The effects of pertinent parameters such as Strouhal number (St), Prandtl number (Pr) and radius ratio (λ) are shown with the aid of line graphs. Results indicate that the role of magnetic field is to decrease fluid velocity and skin-friction. Furthermore, increase in Strouhal number increases the temperature phase-lag.


radius of inner cylinder

radius of outer cylinder



constant magnetic flux density

velocity of the fluid

acceleration due to gravity

magnetic field

dimensionless radial distance

Greek Symbols

kinematic viscosity of liquid


electrical conductivity

coefficient of volume expansion

coefficient of viscosity

radius ratio

α                      thermal diffusivity

β                      coefficient of thermal expansion

κ                      thermal conductivity

ψ                     phase of temperature (dimensionless)

ρ                      fluid density

ρ0                    initial fluid density

χ                      phase of velocity (dimensionless)

ν                      kinematic viscosity

ω                     frequency of time-periodic heating

τ                      skin-friction (dimensionless)











1.1Background of the study. 1

1.2 Statement of Problem.. 2

1.3 Aim and objectives. 2

1.4 Significance of the Study. 3

1.5 Definition of Terms. 5

1.6 Justification. 7



2.1 A Comprehensive review on the study of Magnetohydrodynamics (MHD) 9

2.2 Periodic Heat Input in Natural Convection Flow.. 12



3.1 Introduction. 15

3.2 Tools used. 16

3.3 Mathematical analysis. 17

3.3.1 skin friction. 24

3.3.2 Mass flow rate. 24

3.3.4 Amplitude of velocity and temperature for the problem.. 25

3.3.5 Rate of heat transfer 25



4.1 Result and discussion. 26

4.2 Discussion of the problem.. 26



5.1 Summary. 35

5.2 Recommendations. 35

5.3 Conclusion. 36




1.1       Background of the study

Magnetohydrodynamics, commonly referred to as MHD, represents a fascinating and interdisciplinary field that amalgamates the principles of electromagnetism and fluid dynamics. It provides a profound insight into the behavior of conducting fluids when exposed to magnetic fields. This field finds its roots in both astrophysical phenomena like solar flares and terrestrial applications such as nuclear fusion research and industrial processes involving molten metals and plasmas. MHD’s ubiquity is a testament to its vital role in shaping our understanding of natural processes and its practical applications across a myriad of domains. Natural convection, a captivating fluid dynamics phenomenon, arises from the innate tendency of fluids to move due to density variations driven by temperature gradients. This phenomenon is ubiquitous in nature and engineering, influencing diverse systems ranging from the circulation of Earth’s atmosphere to the cooling of electronic components in advanced computing devices. The intricate interplay of thermal gradients, buoyancy forces, and fluid motion characterizes the essence of natural convection, making it a subject of profound scientific and engineering significance.

Tubes, with their cylindrical symmetry, are emblematic geometric configurations found in numerous engineering applications. These configurations are not merely academic abstractions but integral components of heat exchangers, pipelines, and cooling systems. Understanding the nuances of fluid flow and heat transfer within tubes is paramount for optimizing the performance of these systems. The cylindrical confinement of tubes adds an intriguing dimension to the study of natural convection, introducing complexities that require meticulous investigation. Periodic heat input, characterized by cyclic variations in temperature, stands as a hallmark feature in countless real-world scenarios. Systems such as solar collectors, nuclear reactors, and certain chemical processes inherently exhibit periodic heat input. The dynamic nature of these temperature fluctuations can exert a profound influence on the fluid flow and heat transfer characteristics within tubes. This temporal dimension adds a layer of complexity to the study, necessitating a deeper exploration of the interplay between periodic heat input and natural convection.

The structure of this project reflects a systematic approach to unraveling the complexities of MHD natural convection within tubes with periodic heat input. It comprises several meticulously crafted chapters, each contributing to the overarching goal of advancing our knowledge in this field. The chapters encompass a comprehensive literature review, mathematical modeling of the phenomena, in-depth numerical simulations, and rigorous discussions of results. The sequential flow of the thesis ensures a logical progression, facilitating a profound exploration of the subject matter.

1.2 Statement of Problem

Wang (1998) investigated free convection between vertical plates with periodic heat input. In his work, he separated the solutions into steady and unsteady regime and stated the condition on which the periodic heat input is significant. But in real life situations, heat transfer is common through cylindrical tube. It is therefore significant to study the role of magnetic on free convection flow in a vertical tube inspired by periodic heating at the surfaces of the cylinders. Hence, this current work is aimed to investigate the impact of periodic heating at the surfaces of the cylinders on magnetohydrodynamics natural convection flow formation in a cylindrical tube.

1.3 Aim and objectives

The primary aim of this research is to achieve a comprehensive understanding of the complex interplay between magneto hydro dynamics (MHD), natural convection, and periodic heat input within a tube. This understanding will encompass the dynamic behavior of fluid flow, temperature distribution, and heat transfer rates in this intricate we’ll achieve this by pursuing these objectives:

  1. Derive the equations of motion governing the problems.
  2. Investigate the role of periodic heating at the tube surface on the flow formation and skin friction.
  • Examine the impact of magnetic field on flow formation and drag reduction.

1.4 Significance of the Study.

The significance of this study is multifaceted, encompassing both theoretical advancements and practical applications. It addresses a complex and relatively unexplored problem within the realm of fluid dynamics and magnetohydrodynamics, offering profound contributions to several key areas. This study significantly advances our scientific understanding of the intricate interplay between magnetohydrodynamics (MHD), natural convection, and periodic heat input within a confined tube. It adds depth to the existing body of knowledge by elucidating the underlying physical mechanisms and providing insights into how these phenomena synergistically influence each other.

The findings of this project have direct implications for the optimization of engineering systems that involve tubes, including heat exchangers, pipelines, and cooling systems. By enhancing our understanding of how MHD and natural convection interact with periodic heat input, this study empowers engineers and designers to develop more energy-efficient and reliable systems. In era where sustainability is paramount, this research contributes to the development of sustainable energy solutions. By improving the efficiency of systems such as solar collectors and geothermal heat exchangers, it plays a role in harnessing renewable energy sources more effectively, ultimately reducing our reliance on fossil fuels.

The insights gained from this study are pertinent to nuclear reactor cooling systems, where MHD effects and periodic heat input can impact reactor safety and efficiency. Understanding these dynamics is crucial for ensuring the safe and reliable operation of nuclear power plants. In the aerospace industry and space exploration missions, where extreme conditions prevail, understanding the behavior of fluids in the presence of magnetic fields and varying heat inputs is crucial. This research can aid in the design and operation of spacecraft propulsion systems and thermal management solutions.

This study bridges the gap between various scientific disciplines, bringing together concepts from fluid dynamics, electromagnetism, and heat transfer. This interdisciplinary approach fosters knowledge transfer and collaboration among researchers from different fields. In a world striving for sustainability and energy efficiency, the outcomes of this study have the potential to make a global impact. By optimizing energy systems and engineering solutions, it contributes to the broader goal of mitigating climate change and conserving vital resources.

In summary, the significance of this study extends far beyond the confines of academic research. It offers a pathway to advancing knowledge, improving engineering systems, fostering sustainability, and addressing real-world challenges in various industries. It stands as a testament to the power of interdisciplinary research in shaping a more sustainable and technologically advanced future.

1.5Definition of Terms

Magnetohydrodynamics (MHD): MHD is a multidisciplinary field of study that combines principles from electromagnetism and fluid dynamics. It deals with the behavior of electrically conducting fluids (e.g., plasmas, molten metals, or certain liquids) in the presence of magnetic fields.

 Natural Convection: Natural convection is a mode of heat and mass transfer in fluids driven by density differences due to temperature variations. It occurs without any external mechanical forces and is characterized by fluid motion driven by buoyancy forces.

 Periodic Heat Input: Periodic heat input refers to the cyclic variation of heat or thermal energy added to a system over time. This phenomenon is often encountered in systems with time-varying heat sources, such as alternating heating and cooling cycles.

Cylindrical Tube: A cylindrical tube is a three-dimensional geometric shape characterized by its cylindrical symmetry, typically with a circular cross-section. In the context of this study, it represents the confined space within which fluid flow and heat transfer are analyzed.

Modified Bessel function:Is a special form of Bessel Functions, which were defined by Friedrich Bessel and Daniel Bernoulli as solutions.

Periodic Solution: A solution that periodically depends on the independent variable.


Steady-State: Steady-state refers to a condition in which a system’s properties (e.g., temperature, fluid velocity) do not change with time. In the context of this study, steady-state conditions may be reached after the system has undergone transient changes due to periodic heat input.

 Skin Friction: Skin friction, also known as wall friction, is a type of drag force that occurs when a fluid (such as air or water) flows over a solid surface. It is caused by the viscous or sticky nature of the fluid, and it results from the friction between the fluid particles and the surface. Skin friction is an important factor in the study of fluid dynamics and is a key component in the calculation of drag forces on objects moving through a fluid.

Mass Flow Rate: Mass flow rate is a measurement of the amount of mass that passes through a given point in a fluid system per unit of time. It is typically expressed in units such as kilograms per second (kg/s) or pounds per hour (lb/hr). Mass flow rate is a fundamental concept in fluid dynamics and is used to quantify the rate at which a fluid substance, like a gas or a liquid, moves through a pipe, channel, or any other part of a system.

Strouhal Number: The Strouhal number (St) is a dimensionless number used in fluid mechanics and aerodynamics to describe the behavior of fluid flow around solid objects. It is defined as the ratio of the frequency of vortex shedding (alternating flow patterns) behind an object to the object’s characteristic length and the velocity of the fluid flow. The Strouhal number is often used to predict the occurrence of flow-induced vibrations and oscillations, such as in the study of vortex shedding behind cylinders and the aerodynamics of bodies like airfoils and flags.

Prandtl Number: The Prandtl number (Pr) is a dimensionless parameter used in fluid dynamics to characterize the relative importance of momentum diffusivity (kinematic viscosity) to thermal diffusivity (thermal conductivity) in a fluid. It is defined as the ratio of the kinematic viscosity of the fluid to its thermal diffusivity. The Prandtl number is essential in heat transfer and convection problems, as it helps predict the relative thickness of the thermal and velocity boundary layers in fluid flow.

Nusselt Number:The Nusselt number (Nu) is a dimensionless number used to characterize the convective heat transfer between a solid surface and a fluid. It is defined as the ratio of the convective heat transfer rate to the conductive heat transfer rate for a given system. The Nusselt number is particularly useful in understanding and analyzing heat transfer in situations involving forced convection, natural convection, and boiling or condensation. It depends on the fluid properties, flow conditions, and the geometry of the system.

 MATLAB: MATLAB, which stands for “MATrixLABoratory,” is a high-level programming language and software environment primarily used for numerical computing, data analysis, and visualization.



Account Number: 0709546102

Access Bank: Savings
Account Name: Emmanuel Idorenyin Samuel.