SA Graduate Modelling Camp Problems

Problem 1: Multiscale Modelling of Mucociliary Clearance Using Thin-Film Theory: Incorporating Viscoelastic Mucus, Periciliary Layer Mechanics, and Airflow Coupling

Presenters:

Dr Avnish Bhowan Magan, School of Computer Science and Applied Mathematics, University of the Witwatersrand. (Computational Modelling in Medicine and Physiology Research Programme)
Mr Rohail Hershil Bhana, School of Computer Science and Applied Mathematics, University of the Witwatersrand. (Computational Modelling in Medicine and Physiology Research Programme).

Problem Statement

The airway surface liquid (ASL) forms a critical protective barrier lining the human respiratory tract. It comprises two distinct sublayers: a viscoelastic mucus layer that traps inhaled pathogens and particulates, and a low-viscosity periciliary layer that allows coordinated ciliary beating. The airway epithelium is covered with dense mats of motile cilia that beat in an asymmetric pattern to propel the mucus layer distally— a process known as mucociliary clearance. This mechanism is the primary innate defence of the respiratory system. When clearance is impaired, mucus transport slows or fails, leading to stagnation or plug formation. Such dysfunction is central to a range of pathologies, including cystic fibrosis, chronic bronchitis, and acute infections such as Respiratory Syncytial Virus (RSV) bronchiolitis.

The goal of this project is to develop a physiologically informed, two-layer thin-film model of the ASL that captures:

1. the viscoelastic and shear-thinning behaviour of the mucus layer,
2. the fluid mechanics of the periciliary layer, including explicit ciliary forcing, and
3. coupling to airflow shear at the mucus–air interface.

This multiscale framework will enable systematic investigation of the conditions under which mucociliary transport is enhanced or impaired, the mechanisms leading to mucus stagnation, and the onset of airway-blocking plug formation. Through parameter sensitivity analysis, the model can further elucidate how disease-associated changes—such as altered rheology, increased secretion, or reduced ciliary activity—affect transport efficiency.

Research Question

How do mucus rheology, periciliary layer thickness, ciliary driving forces, and airflow-induced shear interact to determine whether mucociliary clearance succeeds or fails, and under what parameter regimes does this failure manifest as mucus stagnation or airway plug formation?

Presenter:

Dr Lombuso Precious Shabalala, Department of Applied Management, UNISA

Problem Statement

The Injaka Dam is situated in Mpumalanga Province, within the Bushbuckridge Local Municipality (Figure 1).  It was initiated in 2001 mainly for relegation and domestic use. The Mgwaritjie River and the Mgwaritjana River form this dam.  The buffer zone surrounding the dam is 100% claimed by Injaka Watervaal Communal Property Association (CPA). 

The local community members were compensated for the Dam, but they hold title deeds for the buffer zones. The dam has the potential to be one of the province's top attractions, offering a range of water activities. A decision to develop this area into a resort has been supported by management.

A Memorandum of Agreement is in place between the Department of Water and Sanitation (DWS) and Bushbuckridge Local Municipality (BLM) for the dam management. The Resource Management Plan is in place.

Figure 1:  Injaka Dam. Source: Bushbuckridge Local Municipality, 2023.

Facts about Injaka Dam

• 4000 Hectares, which includes the dam and buffer zone.
• It is regarded as a national key point.
• Conducive for water sport facilities and activities such as fishing and boating.
• There are alien trees underneath the dam (some of which surface above the water), which could cause serious damage to boats.
• Level-height: 35 meters.
• Capacity: 132 843 m3.

Tourist attractions such as Injaka Dam are expected to implement user-pay strategies such as to entry which will require an appropriate regulatory framework and the ability to physically restrict access only to those willing to pay for the experience (Morgan & Lok, 1999:1-2). There are tourist attractions in Mpumalanga Province that are deserted.  These tourist attractions have the potential to realise their sustainable development while practicing responsible tourism in their daily operational activities and generate healthy revenue.  Tourism development in South Africa is guided by the key principles of Responsible Tourism stipulated in the 1996 White Paper.

• South Africa is the first country to include Responsible Tourism in its national tourism policy, the 1996 White Paper on the Development and Promotion of Tourism in South Africa.  
• Goodwin (2007) notes that Responsible Tourism is about “making better places for people to live in and better places for people to visit.” Responsible Tourism requires that operators, hoteliers, governments, local people and tourists take responsibility and take action to make tourism more sustainable.

Visitors’ expectations

• Safe environment when participating or enjoying recreational activities (Customer satisfaction & value for money).
• Not an overpopulated attraction.
• Enough safe car parking.
• Camping site and picnic spots/braai areas.
• Adherence to environmental guidelines.
• Boat parking (fishing competition).

Fishing Tournament

The Injaka Dam is a popular venue for bass fishing competitions hosted by the South African Bass Angling Association (SABAA).  The Team Tournament Trail Nationals was scheduled for late 2024 and early 2025.   These events showcased the potential of the dam.  It is worth noting that between late 2024 and 2025, several smaller-scale tournaments, such as Joey's Towing Trail, have regularly used Injaka Dam as their stop within event.

The problem to investigate

• Currently, there is a challenge with alien plants growing inside the dam rooted in the heel of the dam. These alien plants are a threat to the water activities currently taking place and potentially cause physical harm to competition participants or tourists using boats.  In addition, it could jeopardise development and growth opportunities.
• Secondly, there is a possibility of overtourism, which is regarded as congestion from too many tourists in an area or attraction, resulting in conflict with locals and impacting customer satisfaction. Solving this problem could translate to a green economy reaching local communities and contribute to genuine sustainable development.

Building on these two points, the problem seeks to develop:

1. A mathematical model to identify the alien plants and the best way to remove them at an early stage and to optimise the conservation of life forms (fish) and tourism recreational activities.
2. A mathematical model to determine the maximum number of people Injaka Dam tourist attraction can take at one time (carrying capacity) to prevent overtourism.

References

• Bushbuckridge Local Municipality,2023. Injaka Dam Map.
• Goodwin, H., 2007.Taking responsibility for Tourism. Accessed date: 21 November 2022. Available from: https://haroldgoodwin.info/responsible-tourism/.
• Morgan, D. and Lok, L., 1999. Social Comfort Within Natural Tourist Attractions: A Case Study of Visitors to Hanging Rock, Victoria.

Additional Material

• Bushbuckridge Tourism Organisation Facebook:
  https://www.facebook.com/bushbuckridgeLTO/videos/sabaa-ttt-2024-nationals07-09-november-2024inyaka-dam/1512301189441872/. Access date: 10 Dec 2025.
• Goodwin, H., 2017. The challenge of overtourism. Responsible tourism partnership, 4, pp.1-19.
• Linder.L. 2024. Lowveld Bass Trail triumphs on Injaka Dam.  Available from: https://www.citizen.co.za/lowvelder/sports-news/2024/11/24/lowveld-bass-trail-team-triumphs-on-injaka-dam/ . Access Date: 10 Dec 2025.
• SAABAA. Custodian of the Competition. Available from:  https://sabaa.co.za/ . Access date: 10 Dec 2025.

Acknowledgements to Mr Mkhonto and Mr Thwala (Injaka Dam employees) and Mr Modipane (BLM Tourism Officer).

Presenter:

Mr Kendall Born, School of Computer Science and Applied Mathematics, University of the Witwatersrand

Problem Statement

As the global demand for low-carbon energy increases, ways are needed to compare different renewable and non-renewable energy sources using both consistent and meaningful metrics. Two commonly used measures are:

• Efficiency, defined as the ratio of useful energy extracted to energy available, and
• Capacity factor, defined as the ratio of actual energy produced to the maximum possible energy that could be produced over a given time period.

Different technologies exhibit vastly different efficiencies and capacity factors. For example, solar energy is globally available but intermittent; hydroelectric power is highly efficient but geographically constrained; nuclear energy provides near-continuous output but has relatively low thermal efficiency, and wind energy depends strongly on atmospheric conditions.

Wind energy is harvested using wind turbines, which are usually classified into horizontal axis wind turbines (HAWTs) and vertical axis wind turbines (VAWTs). HAWTs extract energy primarily through aerodynamic lift and are subject to a theoretical upper bound on efficiency. Albert Betz derived the limit which states that no HAWT can extract more than 16/27 ≈ 59.3% of the kinetic energy from the wind.

While Betz’s limit is widely accepted and can be derived using relatively simple control volume and momentum arguments, no universally agreed theoretical maximum efficiency exists for VAWTs. These turbines may operate using lift, drag, or a combination of both, and their flow interactions are more complex. Additionally, VAWTs exhibit increased efficiency output when placed near other VAWTs. This is not true for HAWTs.

The problem to investigate

During the modelling camp, students will explore a range of mathematical modelling techniques that lead to the derivation of Betz’s limit for horizontal-axis wind turbines. Building on this foundation, the central challenge of this problem is to investigate whether analogous theoretical limits can be derived for vertical-axis wind turbines.

In particular, students are encouraged to:

• Examine the assumptions underlying the derivation of Betz’s limit
• Identify which assumptions fail or must be modified for VAWTs
• Propose simplified physical or mathematical models for VAWTs
• Explore whether one or more theoretical efficiency bounds can be derived, and under what assumptions.

Is a single universal efficiency limit for VAWTs plausible, or do different operating regimes imply different bounds?

Additional Component

In addition to the mathematical modelling content, students in this group will receive guidance on how to effectively communicate ideas through presentations. This includes strategies for clarity, audience engagement, and visual explanations.

Presenter:

Professor Graeme Hocking, Mathematics and Statistics, Murdoch University, Perth, Western Australia.

Problem Statement

Water is used for many purposes in the mining industry.  It is often obtained from groundwater aquifers and is used in leaching and transport and in stabilization of tailings dams. One problem that frequently arises is that the water is often mixed with clay and minerals creating a slurry. The mixture is often very thick to save water. However, such fluids are often non-Newtonian, which is to say there is a nonlinear relationship between stress and strain. As a result, they may not flow very well, causing clogging of pipes and blockages. For example, Bingham fluids have a “yield stress” which means they will not flow at all until a certain amount of force is applied.

In addition to Bingham fluids, there are fluids such as Herschel-Buckley, and power-law fluids.

In this problem, examine the properties of these fluids and then use various simple one-dimensional models to gain a better understanding of how and when these fluids flow. In a real situation, we would like to understand what happens when such a fluid is in a two-dimensional channel.  We will attempt to model flow of a non-Newtonian fluid in a two-dimensional channel to assist in improving the transport of slurry flows on a mine site.

Presenter:

Professor David Mason, School of Computer Science and Applied Mathematics, University of the Witwatersrand

Problem Statement

An axisymmetric thermal plume consists of rising fluid due to a point source of heat. The fluid is heated and less dense than its surroundings. Plumes can occur in the atmosphere, in the ocean and in the mantle of the Earth. An important example is the urban thermal plume which consists of rising warm air caused by urban areas being warmer than surrounding areas  due to heat absorbed by buildings and roads. Urban heat islands form. Other examples include industrial power plants with exhaust gases from turbines and cooling towers, geysers which consist of hot water and steam rising from underground heat sources and mantle plumes which consist of upwelling of molten rock from heat sources deep in the Earth’s mantle.

Thermal plumes are modelled mathematically by the Navier-Stokes equation with a buoyancy body force term, the conservation of mass equation, and the energy equation. There are three characteristic numbers, the Reynolds number, Grashof number and the Prandtl number. The boundary layer approximation is made. Conserved quantities are investigated. The Navier-Stokes equation does not generate a conserved quantity due to the buoyancy term. Analytical solutions are derived for special values of the Prandtl number.  The buoyancy term plays an important part in the derivation of these special solutions.

The Study Group will look at some examples of thermal plumes and investigate if the analytical solutions yield new results.

Presenter:

Professor Montaz Ali, School of Computer Science and Applied Mathematics, University of the Witwatersrand

Problem Statement

A single-commodity inventory problem is considered. Customers arrive according to Poisson process at a single queue, single service channel. The arrival rate¹ is modelled as 3 customers per hour. In the problem discussed here the customer demand and replenishment lead time are also random, as they follow distributions, making it a stochastic problem. Decision variables are reorder point r and reorder quantity Q, both discrete. These variables must be changed over predefined ranges (see Table 1) to optimize the two objective functions, which are service level (maximized) and (average) inventory cost (minimized). The inventory level is monitored, and when it falls below the recorder

point r, a reorder is initiated. After the lead time has elapsed, the inventory level is increased by the reorder quantity Q. The average inventory level is calculated over time to determine the average inventory cost, which is minimized. The unit time or the time interval can be considered as each day for 6 months, and so the total number of time intervals is 180. The cost for keeping one inventory per unit time is 1. One can minimize the total inventory cost but, in this case, it is better to minimize the average inventory cost i.e. cost per interval. The service level is defined as

Service level=Number of Customer serviced/Number of Customers requiring Service

The customers have a finite stochastic demand for the single commodity being sold following a Weibull distribution, namely ⌊Weibull(λ, k)⌋ = ⌊Weibull(1, 8)⌋. For the lead time the following distributions can be considered:

• Probability(Lead time=1)=0.25; Probability(Lead time=2)=0.5; Probability(Lead time=3)=0.25.
• Discrete Uniform Lead Time in {1, 2, 3}.

¹In a Poisson distribution, the arrival rate (λ) is the average number of customers that arrive per unit of time or an interval, assuming these events happen randomly and independently.

Figure 1: Inventory level I(t) vs Time

Simulation to be Carried Out:

The above problem is a stochastic programming problem involving two objective functions, which will not be so easy to solve during the modelling Camp period. However, students can simulate the process by writing a computer program where iid sample can

be drawn from Poisson as well as Weibull distribution. In particular, the participants of the modelling Camp can use r ∈ [100, 400] and Q ∈ [100, 400]. Using the pair (r, Q) value they can simulate the process, see Figure 1, for say 10 times and calculate the average (of average inventory cost) and the average service level over 10 simulations. This process can be repeated for a set of pairs of (r, Q) values and then try to locate pareto set using the corresponding average (cost, service level).

References

• Braglia, M., Castellano, D., Marrazzini, L. & Song, D. (2019). Continuous review, (q, r) inventory model for a deteriorating item with random demand and positive lead time. Computers Operations Research, 109, 102-121.
• Jackson, I., Tolujevs, J. & Kegenbekov, Z. (2020). Review of inventory control models: A classification based on methods of obtaining optimal control parameters. Transport and Telecommunication Journal, 21, 191-202.