Colloquium 2024: Book of Abstracts
The Optimal Control of Geomagnetic Disturbances in Power Network
Montaz Ali
University of the Witwatersrand, Johannesburg
Geomagnetic Induced Current (GIC) is known for causing so much trouble when it comes to power networks and transmission lines. This has been a challenge to so many countries as they lose billions of dollars when it comes to fixing the damages. It is almost impossible to eliminate GIC in a network, however one can mitigate the impact it has to the network as well as man-made equipment. We, mathematically, model the GIC using optimal control in a power transformer. Since the flow of current in a transformer depends on the voltage coming from the voltage source, one can control the GIC in a power network through the incoming voltage in the primary winding of a transformer. One of the key findings was that the control managed to keep the GIC as low as possible hence leading to minimum energy in the GIC signal. We also compared the results obtained to the actual GIC data to see the validity of our method and it shows that our method performed good as the approximated GIC is not far from the measured GIC.
Turbines - Transparency and Turbulence
Kendall Born
University of the Witwatersrand, Johannesburg
As the world adopts sustainable energy solutions, wind turbine-generated energy has gained traction as our largest renewable source of power, further reducing our dependence on fossil fuels. However, it is important to approach wind turbine implementation with transparency, acknowledging both the benefits and potential drawbacks. This talk covers the fluid mechanics of wind turbines, with a focus on turbulence and efficiency. Environmental implications, such as the threat to bird and bat populations, as well as recycling and decommissioning of old turbines, will be covered briefly. By discussing these and other challenges openly, strategies to mitigate negative impacts while harnessing the full potential of renewable energy can be developed. A balance between advancing renewable energy technologies and preserving the environment and wildlife, ensuring a holistic approach to combating global warming, can be achieved.
On the Versatility of Lie Symmetry Analysis over Power Series in Solving Second-Order Variable Coefficient ODEs
Mosa Chaisi
Walter Sisulu University
Many second-order ordinary differential equations (ODEs) with variable coefficients resist solutions in terms of standard functions. However, a significant subset of these equations can be solved using power series methods. In this talk, we compare the power series method for solving ODEs with variable coefficients with the Lie symmetry analysis approach. We illustrate both methods through a series of examples. The Lie symmetry approach appears to be more versatile and efficient, as it allows for the application of a single routine to tackle various types of ODEs that would typically require distinct cases when using the power series method
Fluid driven hydraulic fracture: Mathematical models and solutions
Adewumni Fareo
University of the Witwatersrand, Johannesburg
Hydraulic fracturing is a well stimulation technique used to improve the productivity of crude oil and natural gas from underground reservoirs. The technique involves injecting fluid at ultra-high pressure into the reservoir rock to propagate pre-existing fracture networks or create new fractures. A crucial goal in hydraulic fracturing is the use of high-pressure fluid to continually create and propagate new fractures in the rock formation, to continually liberate the crude oil and natural gas that may have been trapped. However, because of the permeable nature of the rock formations, fluid leak-off into the formations do occur, the rate of which increases as more and more fractures are created. The effect of fluid leak-off on fracture extension or propagation is significant and several research in this area has been done. In this talk, a detailed account of mathematical models for the propagation of a pre-existing hydraulic fracture with fluid leak-off in a permeable rock will be presented. Two models describing the elastic response of rocks due to the injected fluid will be discussed. These are the PKN model and the Cauchy singular integral model.
A relativistic mixture of formulation of two-phase flow
Sheldon Herbst
University of Johannesburg, Johannesburg
In the following work, we present a relativistic formulation of compressible two-phase flow. Using the variable found in the classical formulation of two-phase flow, it is seen that an additional variable is required to provide sub-luminal velocities in the presence of high pressures. We present some numerical solutions to the hyperbolic conservation laws that arise from the formulation and compare the results to the single-phase relativistic problem as well as the classical (non-relativistic) two-phase problem.
Fractures and fingers in shear-thinning fluids
Ashleigh Hutchinson
University of Manchester, United Kingdom
When a less viscous fluid is injected into a more viscous one within a narrow gap between two solid boundaries, instabilities resembling finger-like structures can emerge. The origin of these instabilities is as a result of the more viscous fluid having a higher pressure gradient, and so any protrusion of the less viscous fluid - which has greater mobility - into it results in the formation of an extended finger.
These so-called Saffman-Taylor instabilities typically occur in shear-dominated flows.
Another type of instability, whose origin is of an entirely different nature, arises when a shear-thinning fluid is injected into a lubricated narrow gap, where shear stress is negligible in comparison to extensional stress.
In this talk, I will explore the origins of this second type of instability, which is closely linked to the fracturing behaviour of shear-thinning fluids and contrast it with Saffman-Taylor instabilities.
I will conclude with an example that illustrates the presence of both types of instabilities in one single laboratory experiment.
On the Application of the Double Reduction Theory to (n+1)-Dimensional Scalar PDEs and Systems of PDEs
Molahlehi Kakuli
Walter Sisulu University
The Double Reduction method, as proposed by Sjoberg, is a powerful
algorithm in the analysis of partial differential equations (PDEs), which offers a systematic approach for the reduction of higher-dimensional scalar PDEs and systems of PDEs. The algorithm extends the classical method for finding invariant solutions by using symmetries associated with conservation laws. In this presentation, we explore the double reduction method and reflect on some interesting aspects on the application of the algorithmic method, drawing examples from (n + 1)-dimensional scalar PDEs and systems of PDEs.
Linear hydraulic fracture with tortuosity: Conservation laws and fluid extraction
Rahab Kgatle-Maseko
University of the Witwatersrand, Johannesburg
Fluid extraction from a pre-existing two-dimensional hydraulic fracture with tortuosity is investigated. The tortuous fracture is replaced by a symmetric open fracture without asperities (deformations) on opposite crack walls but with a modified Reynolds flow law and a modified crack law (the linear crack law). The Perkins–Kern–Nordgren approximation is made in which the normal stress at the fracture walls is proportional to the half-width of the symmetric model fracture. By using the multiplier method two conservation laws for the non-linear diffusion equation for the half-width are derived. Two analytical solutions generated by the Lie point symmetries associated with the conserved vectors are obtained. One is the known solution for a fracture with constant volume. The other is new and is the limiting solution for fluid extraction. A jet of fluid escapes from the fracture entry and the volume of the fracture decreases. There is a dividing cross-section between fluid flowing towards the fracture entry and fluid flowing towards the fracture tip which explains why the length of the fracture continues to grow as fluid is extracted. As tortuosity increases the position of the dividing cross-section moves closer to the entry. A numerical solution is presented for the other cases of fluid extraction. Comparison of the fluid flux for different operating conditions within the fluid extraction region shows that the limiting solution yields the maximum rate of fluid extraction from the fracture. As the fracture becomes more tortuous its length becomes less dependent on the operating conditions at the fracture entry. For fluid extraction working conditions close to the constant volume operating condition the width averaged fluid velocity increases approximately linearly along the whole length of the fracture. For these working conditions, an approximate analytical solution for the half-width for fluid extraction, which agrees well with the numerical solution, is derived by assuming that the width averaged fluid velocity increases exactly linearly along the fracture.
The roles of symmetry in the study of differential equations.
Peter Leach
University of Kwazulu-Natal/ Durban University of Technology
A characteristic of differential equations is that many are invariant under some sort of transformation. In increasingly more difficult to examine are point, contact and finally nonlocal transformations. These can be applied equally to ordinary, partial and integral equations, naturally with usually increasing degree of difficulty. In theory it is immaterial whether the equation is ordinary or partial, single or a system. It is more convenient to use infinitesimal operators to generate the transformations. Under the operation of taking the Lie Bracket of the infinitesimal generators, the set of generators for a given equation (aeq. system) constitutes a Lie algebra. We examine the principal ideas and illustrate them with simple examples.
Singularities of some models in Physics and Mechanics
Jean M-S Lubuma
University of the Witwatersrand, Johannesburg
One of the famous results of S. Agmon, A. Douglis and L. Nirenberg (1959-1964) in the study of linear partial differential equations can roughly be stated as follows: for a boundary value problem (BVP) on a bounded domain Ω of (n = 2 or 3), the solution u belongs to the Sobolev (energy) space of optimal order corresponding to the orders of both the partial differential equation and the Sobolev space in which the right-hand side f of the BVP lives. This result has far-reaching implications on the convergence and accuracy of numerical methods such as the Finite Element Method (FEM). However, the smoothness condition imposed on Ω excludes domains with angular, conical, edge, etc., corners, which naturally arise in applications. The natural question of interest is to investigate how bad the solution u is. In order to have explicit results, we restrict ourselves in this talk to three partial differential equations of mathematical physics, namely the Dirichlet problem for the Laplace operator, the Stokes system and the elasticity (Lame) system. We start by studying the singular functions of these problems on two-dimensional domains with angular points. Next, we use these results to describe the edge behaviour of the solutions of the three models. We conclude by showing how the knowledge of the singularities can be used to restore the order of convergence in the construction of finite element method.
Exact Solutions of the generalized Kuramoto–Sivashinsky (GKS) Equation from Classical and Nonclassical Symmetries
Vernon Lucwaba
Walter Sisulu University
Lie symmetry analysis is a powerful tool for finding exact solutions of partial differential equations (PDEs). Both classical and nonclassical symmetries can be used to construct invariant solutions of a given PDE. Moreover, point transformations arising from classical symmetries can be used to transform nonclassical solutions into new solutions. In this work, we determine the classical and nonclassical symmetries of the GKS equation. We construct an optimal system of the classical symmetries and use it to derive invariant solutions and to map nonclassical solutions into additional exact solutions of the GKS Equation. The nature of the obtained additional exact solutions
is investigated with a view to exploring the effectiveness of combining classical and nonclassical symmetry methods for finding exact solutions of nonlinear PDEs.
Interesting features of relativistic fluids in astrophysics
Sunil Maharaj
Astrophysics Research Centre
University of KwaZulu-Natal, Durban
We consider the role of relativistic fluids in general relativity and modified gravity theories. In general, the gravitational behaviour is governed by Abelian differential equations. Some interesting geometrical and physical features are highlighted. We find that the dynamical behaviour of the fluid is determined by spacetime dimension and the specific theory of gravity. We show how this affects astrophysical applications.
Continuum Mechanics in the School of CSAM: Past, present, future
David Mason
University of the Witwatersrand, Johannesburg
Research in Continuum Mechanics has been undertaken in the School for over fifty years. In solid mechanics (elasticity) areas of investigation have included rock mechanics and support for excavations in mines, finite elasticity, nonlinear oscillations, applications to medicine. In fluid mechanics areas of research have included slow viscous flow, boundary layers, wakes and jets, thin fluid film theory, turbulent flow. Research has also been undertaken on the interaction of fluid with solids in porous elastic media and hydraulic fracturing. The MISG from 2004 onwards supplied problems from the mining, glass and sugar industries. Highlights from past research will be presented, present work will be reviewed and future plans discussed.
Double diffusive convection in rotating fluids under gravity modulation
Alfred Mathunyane
University of the Witwatersrand, Johannesburg
This study employs the method of normal modes and linear stability analysis to investigate double-diffusive convection in a horizontally layered, rotating fluid, specifically focusing on its application to oceanic dynamics. Double diffusive convection arises when opposing gradients of salinity and temperature interact within a fluid, a phenomenon known as thermohaline convection, and it is crucial for the understanding of ocean circulation and its role in climate change. With the increasing mass of water due to glaciers melting, fluid pressure variations occur, leading to slight fluctuations in gravity. We conduct both stationary and oscillatory stability analyses to determine the onset of double-diffusive convection under gravity modulation. Our analysis reveals that time-dependent periodic modulation of gravitational fields can stabilize or destabilize thermohaline convection for both stationary and oscillatory convection, with amplitude stabilizing and frequency destabilizing. The wave number in the y- direction also affects convection in the equatorial regions. This wavenumber exhibits destabilizing effects for large values and stabilizing effects for small values for both stationary and oscillatory convection. Rotation along with gravity modulation tends to destabilize the system for both stationary and oscillatory convection. The key difference between stationary and oscillatory convection is that oscillatory convection exhibits large values of the Rayleigh number, thus susceptible to over-stability while stationary convection tends to have relatively smaller Rayleigh numbers and thus more stable. This research provides insights into the complex interplay between gravity modulation and thermohaline convection, contributing to our understanding of ocean dynamics and their implications for climate change.
Nonclassical Potential Symmetries are Non-existent for the Inhomogeneous Nonlinear Differential Equations
Joel Moitsheki
University of the Witwatersrand, Johannesburg
We show that nonclassical potential symmetries for the inhomogeneous nonlinear differential equation (INDE) are just classical Lie point symmetries.
An evaluation of the finite difference method for solving Cauchy singular integral equations of the first kind
Mathibele Nchabeleng
University of Pretoria, Pretoria
The numerical solution of Cauchy singular integral equations (CSIEs) has continued to gain increasing interest among researchers. For linear Cauchy singular integral equations, a great deal of theory which provides closed form solutions exists. Literature based on numerical techniques for this class of integral equations have mainly been based on methods other than finite difference. Simply very little research has been done on the use of numerical techniques such as finite difference method. Using a collection of examples of linear Cauchy singular integral equations arising from mechanics, this paper presents a discussion exploring the use of the finite difference method for solving linear Cauchy-type singular integral equations of the first kind over a finite interval. Discretization often results in an overdetermined system of linear algebraic equations. It is shown that the finite difference method can easily produce
solutions that do not converge as the grid is refined. However, convergence can be obtained by solving the relevant set of linear equations. The crucial importance of asymptotic analysis in the determination of solution behaviour near end points of concern in the solution domain is presented.
Recent models for double-diffusive convective flow through porous media with rotational modulation
Precious Sibanda
School of Mathematics, Statistics & Computer Science
University of KwaZulu-Natal
We give an overview of some recent results on two-dimensional double-diffusive Rayleigh-Benard-type convection within a rotating anisotropic porous layer. In this talk we will discuss the impact of time-varying rotation and anisotropy on heat and mass transfer in the porous layer. The effect of rotation modulation is seen through changes in the amplitude and frequency that influence the system behaviour. In addition to the normal mode approach to determining conditions for the onset of convective motions, we demonstrate the use of a recent numerical technique, the local quasilinearization block hybrid method (LQBHM) for initial value problems through solving the Lorentz-type equations obtained for the weakly nonlinear regime. This method is chosen for its accuracy, efficiency, and reliability.
Determining Lie Point Symmetries of First-Order ODEs by Mapping to Second-Order ODEs via Total Differentiation
Winter Sinkala
Walter Sisulu University
Finding Lie point symmetries of first-order ODEs presents challenges,
primarily because the determining equation derived from the invariance condition does not contain derivatives. This absence complicates the decomposition into simpler linear partial differential equations necessary for solving for the infinitesimals. A common strategy for finding Lie point symmetries of first-order ODEs involves hypothesising the form of the infinitesimals. [Equations]
Understanding the dynamics of HIV with Mother-to-Child transmission
Arsene Tasse
University of the Witwatersrand, Johannesburg
In this project, we propose an HIV model involving both vertical and sexual transmissions. The new feature of this model is the account of two groups of women, the unpregnant and the pregnant, to capture the mother-to-child (MC) of HIV. The model is quantitatively and qualitatively analysed. We investigate the conditions for the local and the global asymptotic stability of the disease-free equilibrium and prove the possible existence of several interior equilibria. Due to the complexity of the model, we develop a Nonstandard finite difference (NSFD) scheme to preserve the qualitative properties of the model. Numerical simulations that support the theory are provided. Moreover, we explore the relevance of antiretroviral therapy to prevent MC HIV transmission. The sensitivity analysis suggests that one should educate HIV-infected women to reduce the number of their infants and educate people to use condoms during sexual intercourse to lessen the number of infected infants. This motivates to set an optimal control problem that combines safe sex practices (SSP) education through the use of condoms (C) and ART for pregnant infected women. The solution to this problem shows that SSP-C mitigates both vertical and horizontal transmissions, while the use of ART reduces MC transmission only, but its impact is more sensitive than SSP-C on the dynamics of infected infants.
The KdV-Burgers equation in the small dispersion/diffusion limits
Andre Weideman
University of Stellenbosch, Stellenbosch
The viscous Burgers equation models the interaction of a quadratic nonlinearity and diffusion. In the limit of small diffusion, the Burgers equation approaches the entropy solution of the inviscid Burgers equation. On the other hand, the Korteweg-de Vries (KdV) equation models the interaction of the same nonlinearity but with dispersion instead of diffusion. In the limit of small dispersion, the KdV equation does not approach the entropy solution. Instead, the limiting solution has a highly oscillatory profile. In this talk the focus is on the situation when both diffusive and dispersive terms are present. Because the entropy solution has a jump discontinuity (shock), computing nearby solutions is highly challenging. Preliminary results will be presented.
The Aerodynamics of Wind Turbine Blades
Nicholas Whittaker
University of the Witwatersrand, Johannesburg
With concerns over climate change and nuclear energy, global powers began to invest in greener sources of energy as early as the 1960s, with research into wind turbines as a source of electrical power starting in earnest in the second half of the 20th century. Since then, much progress has been made and now wind energy is the third largest source of renewable energy in the world and is growing rapidly. In 2023 wind turbines generated over 7.8% of the world’s power. To further increase the power output of wind turbines it is critical to maximise their efficiency. A. Betz showed that the theoretical maximum efficiency of a wind turbine is 59.26%. However, the efficiency of a wind turbine is reduced by factors such as tip losses, wake effects, drive train/mechanical inefficiencies, and sub-optimal blade shape. Most modern-day wind turbines have 3 aerofoil blades that rotate about an axis parallel to the ground as this is the most efficient, structurally sound, controllable, and aesthetically pleasing configuration. These turbines generate torque through the aerodynamic lift generated by the aerofoils used as blades. For this reason, a detailed analysis of aerofoils is performed to derive an equation for the aerodynamic force generated by the blades. The rotation of the blades causes the relative air velocity over them to be dependent on the radial position. For this reason, wind turbine blades are twisted, and their width (chord) changes with the distance from the axis of rotation. We use blade element momentum theory (BEM) also known as strip theory, to determine the optimal angle of twist and chord distribution for a blade given certain design conditions. We then investigate this blade shape by evaluating its efficiency in design conditions and behaviour in off-design conditions.