## Project Details

### Description

Waves are ubiquitous in fluid motion. Mathematics provides a formal framework to study and understand water wave propagation. The first major breakthrough in the mathematical description of water waves is typically attributed to Stokes (a paper published in 1848). Despite this, even the simplest descriptions of water waves continue to surprise researchers with rich mathematical structure.

The equations governing wave motion in fluids are oftentimes too difficult to solve analytically. In such cases, approximate "asymptotic" solutions can be constructed by simplifying the equations, at the cost of imposing additional assumptions about the behaviour of the solution. Another way to approximate solutions is through the use of computer simulation (numerical analysis). The goal of the fellowship is to find both asymptotic and numerical approximations of complex water wave behaviour in three different physical settings.

The first concerns internal waves, which are waves which occur inside stratified fluids such as the ocean. Unlike waves seen on the ocean surface, they can be of ginormous size, reaching heights of 100's of metres and being kilometres long. Internal waves have been studied due to their importance in distributing energy, heat, pollutants and biological matter in the worlds' oceans and atmosphere. However, the majority of research concerns so-called "mode-1" waves, the most observed type of internal wave with the simplest vertical structure. Recent field observations have demonstrated that "mode-2" waves are more common than previously believed. Furthermore, novel experimental and mathematical research of these waves has uncovered a plethora of interesting features of these waves that remain underexplored. They have a complex vertical structure and can have a trapped region of recirculation. During the fellowship, a sophisticated numerical algorithm will be constructed to compute mode-2 solutions, and the code will be utilised to explore their wave properties. This code will be made open-source, to provide a bespoke tool for researchers to use in their studies of these waves.

The second project is on rotational surface waves. When studying waves propagating on the surface of water, most literature assumes the vorticity of the fluid is zero (i.e. particles don't 'spin'). This assumption is not always valid, such as when strong winds at the water surface induce non-constant shear currents. Recently, there has been a spell of interest concerning waves with non-zero vorticity, known as rotational waves. Most existing literature concerns constant vorticity, where it is found the water surface and interior flow have exotic structures such as overhanging waves and internal stagnation points. For non-constant, non-zero vorticity, most formulations used do not allow for these interesting features, severely restricting the form of the wave. A recent formulation of the equations was derived which overcomes and shortcomings, and was used to rigorously prove features of the waves. Waves are yet to be recovered numerically using this formulation, which is the second research objective of the fellowship.

The third topic concerns a phenomenon known as "odd viscosity". It has been observed in fluid or fluid-like systems where particles are rotating, such as a fluid composed of magnetic particles spinning on an axis due to an external magnetic field. I wish to explore the role of odd-viscosity on the famous Plateau-Rayleigh instability. This instability is due to a force known as surface tension, and causes a column of fluid to break into droplets (as can be observed of fluid coming from a tap). The correct form of the equations must be recovered, upon which I will perform asymptotic analysis on the system. The role of odd viscosity on fluid flows in a growing field of research which will be helpful in understanding 'active matter', both occurring naturally (bacteria) and being synthetically produced (magnetic particles).

The equations governing wave motion in fluids are oftentimes too difficult to solve analytically. In such cases, approximate "asymptotic" solutions can be constructed by simplifying the equations, at the cost of imposing additional assumptions about the behaviour of the solution. Another way to approximate solutions is through the use of computer simulation (numerical analysis). The goal of the fellowship is to find both asymptotic and numerical approximations of complex water wave behaviour in three different physical settings.

The first concerns internal waves, which are waves which occur inside stratified fluids such as the ocean. Unlike waves seen on the ocean surface, they can be of ginormous size, reaching heights of 100's of metres and being kilometres long. Internal waves have been studied due to their importance in distributing energy, heat, pollutants and biological matter in the worlds' oceans and atmosphere. However, the majority of research concerns so-called "mode-1" waves, the most observed type of internal wave with the simplest vertical structure. Recent field observations have demonstrated that "mode-2" waves are more common than previously believed. Furthermore, novel experimental and mathematical research of these waves has uncovered a plethora of interesting features of these waves that remain underexplored. They have a complex vertical structure and can have a trapped region of recirculation. During the fellowship, a sophisticated numerical algorithm will be constructed to compute mode-2 solutions, and the code will be utilised to explore their wave properties. This code will be made open-source, to provide a bespoke tool for researchers to use in their studies of these waves.

The second project is on rotational surface waves. When studying waves propagating on the surface of water, most literature assumes the vorticity of the fluid is zero (i.e. particles don't 'spin'). This assumption is not always valid, such as when strong winds at the water surface induce non-constant shear currents. Recently, there has been a spell of interest concerning waves with non-zero vorticity, known as rotational waves. Most existing literature concerns constant vorticity, where it is found the water surface and interior flow have exotic structures such as overhanging waves and internal stagnation points. For non-constant, non-zero vorticity, most formulations used do not allow for these interesting features, severely restricting the form of the wave. A recent formulation of the equations was derived which overcomes and shortcomings, and was used to rigorously prove features of the waves. Waves are yet to be recovered numerically using this formulation, which is the second research objective of the fellowship.

The third topic concerns a phenomenon known as "odd viscosity". It has been observed in fluid or fluid-like systems where particles are rotating, such as a fluid composed of magnetic particles spinning on an axis due to an external magnetic field. I wish to explore the role of odd-viscosity on the famous Plateau-Rayleigh instability. This instability is due to a force known as surface tension, and causes a column of fluid to break into droplets (as can be observed of fluid coming from a tap). The correct form of the equations must be recovered, upon which I will perform asymptotic analysis on the system. The role of odd viscosity on fluid flows in a growing field of research which will be helpful in understanding 'active matter', both occurring naturally (bacteria) and being synthetically produced (magnetic particles).

Status | Active |
---|---|

Effective start/end date | 1/04/23 → 31/03/26 |

### Funding

- Engineering and Physical Sciences Research Council

### RCUK Research Areas

- Mathematical sciences
- Process engineering
- Continuum Mechanics
- Fluid Dynamics

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