## Fluid Mechanics

Researchers: John Butler; Chris Hills; Rossen Ivanov; Brendan Redmond

Collaborators: Environmental Sustainability & Health Institute (ESHI)

### Description of area

Slow viscous ﬂows & rotationally-driven ﬂows (Chris Hills)

Slow viscous flows are classical fluid flows associated with low Reynolds numbers when the effects of viscosity dominate inertial effects. Our research in this area has been focused on the flow phenomena and topology associated with such flows, especially when fluid motion is driven through viscosity by the rotation of boundaries. Slow viscous flows are particularly relevant to small-scale systems and systems where secondary flows arise but also have far-reaching implications for many physical phenomena that arise in new technologies, manufacturing and the environment. For example, applications of slow viscous flows and rotationally-driven flows occur in microfluidics, biofluidics and biomonitoring, mixing processes and technologies, and industrial fluid processing where thermal and magnetohydrodynamic effects can also play a role.

A particular focus of research in this area has been the formation of viscous eddies – recirculation regions and cells that form at low Reynolds number due to the physical geometry. These structures inhibit mixing and can explain the interactions at fluid-fluid or fluid-air boundaries. These eddy structures, intrinsic to the geometry and arising prior to the onset of instability, have been widely studied in cylindrical, conical and corner geometries. The structures may evolve in time-dependent flows and the creation and annihilation of viscous eddies is an important topic of study.

More widely, research has also focussed on rotationally-driven flows up to and including the onset of instability. Rotation can provide a stabilising influence on flows and research has considered the evolution of flow regimes throughout parameter space and the influence of different physical processes. By understanding the important parameters and dominant influences for slow viscous flows and rotationally-driven flows, we gain insight into the control of fluid behaviour and the potential for significant benefit, through improved design, to a wide range of fluid and mixing processes.

Nonlinear waves, wave-current interactions, geophysical fluid dynamics & fluid flows with vorticity (Rossen Ivanov, Chris Hills)

The world’s climate depends on the subtle balance between oceanic and atmospheric geophysical flows. The dynamics of extreme climate-related events, like tsunamis and hurricanes, have been studied extensively through simplified linearised models. While these models often give valuable short-term predictions, they are not reliable for predicting longer-term phenomena and complex patterns. For example, important phenomena,such as turbulence, the breaking of internal waves, and flow reversal, cannot be adequately understood by neglecting nonlinear effects.

One aspect of our research has been to consider the nonlinear fluid equations for geophysical waves and currents, exploiting a simplified model of several layers to describe the interaction between both surface and internal waves and currents. One typical example of such a system are the surface and internal waves that propagate along the equator in the Pacific Ocean and the Equatorial Undercurrent (EUC). Our goal is a comprehensive description of wave-current systems and an investigation of how their structure affects wave properties and one of the main objectives is to put the full nonlinear model in a Hamiltonian framework in order to facilitate its systematic treatment.

The analysis will apply to arbitrary wavelengths and finite depths. Waves of larger amplitude require solutions of the exact governing equations and possibly numerical simulations. The long wavelength, or small amplitude, regimes allow for integrable weakly nonlinear models. For example, small amplitude and long-wave regimes allow for smooth solitons arising from the KdV-type approximation regime as well as breaking waves for very large wavelengths, when the equations are asymptotically equivalent to the dispersionless Burgers' equation. Such approximate equations admit exact solutions, derivable by soliton theory methods and the results have the potential to provide new insights into the mechanisms of the Earth’s environment and serve as the theoretical backbone for future attempts at understanding climate change.

Boundary layer theory with applications to melt spinning (Brendan Redmond, Chris Hills, John Butler)

When fluid flows past a solid object, a boundary layer is often formed in the immediate neighbourhood of that object. The fluid sticks to the surface of the object because of its viscosity; this is a manifestation of the no slip condition whereby, on the surface of the solid, there is no relative motion between it and the fluid. Moving radially away from the surface of the solid, the velocity of the fluid is seen to rapidly approach the velocity of the main body of the fluid, until it reaches the same velocity. This region is known as the momentum boundary layer. A similar layer exists if there is heat transfer from the object to the environment and is called the thermal boundary layer.

Momentum and thermal boundary layers play a very significant role in many physical processes such as in aerodynamics and the melt spinning of synthetic fibres, both of which represent huge global industries. Fibres created by melt spinning are used on a daily basis in industries as diverse as clothing, textiles and telecommunications. In the melt spinning process, thick viscous liquids are extruded from a spinneret; they then pass through an air chamber and are then wound onto a drum. The air resistance on the surface of the fibre and the rate of heat loss from the fibre to the surrounding environment, characterised by the Bingham Number and the Nusselt Number respectively, play an important role on the final properties of the fibre. The application of boundary layer theory to analyse quantities such as these is a valuable tool in understanding the melt spinning process and can be used to improve fibre quality and reduce costs in the production of melt spun fibres.

Prandtl boundary layers (John Butler)

In 1905 Prandtl introduced boundary layer theory to describe flow near a surface. The flow is described by a system of non-linear second order differential equations which takes into account the boundary region of rapid change of flow around a surface. Variations of these equations can be solved analytically or numerically approximated. The system equations can be converted into a higher order single variable differential equations using a transformation of variables. Well known examples of this are Blasius Equation which describes flow along a plate with heat and mass transfer, and the Faulkner Skan Equation which describes flow along a wedge.

Numerically the solution of Prandtl boundary layer equations can be approximated by finite difference methods. To account for the boundary region of rapid change along the surface an adaptive mesh such as the Shishkin mesh is required to ensure a stable numerical solution.

For more information about any of these areas and to discuss opportunities for research please contact the individuals above.