We present four studies that use asymptotic methods to understand physical phenomena in boundarylayer and interfacial flows. First, we present solutions for the shapes of static equilibria in rectangular channels that have been perturbed by isolated ridges and grooves and by scattered bump protrusions and intrusions. We solve the YoungLaplace equation to quantify the sensitivity of the meniscus shape to the perturbations using a combination of numerical computations and asymptotic techniques for a linearised model when the amplitude of the perturbations is small relative to the channel height. For smallamplitude ridge/groove and bump perturbations, we derive an equation for the induced pressure difference over the meniscus that depends solely on the boundary data. Thus the total pressure difference over the meniscus (and therefore the mean curvature) can be found without solving the YoungLaplace equation. Mirror symmetric ridge and bump perturbations which change the volume of the channel cause a change in the mean curvature of the meniscus which leads to longrange curvature of the contact line. For ridge perturbations, we show that this longrange curvature matches onto the contact line of a droplet with the same mean curvature as the meniscus. We use this information to choose specific combinations of perturbations to engineer contact line shapes. We present preliminary results for bump intrusions/protrusions which show that the direction of deformation of a meniscus changes as it passes over a bump. Next, we present an asymptotic description of nonlinear equilibrium and travellingwave solutions of the NavierStokes equations in incompressible unsteady boundarylayer and compressible parallel boundarylayer flows. The solutions take the form of selfsustaining vortexwave interactiontype states, known as freestream coherent structures, with the nonlinear interaction between the vortex and the wave taking place in a layer close to the freestream. The interaction produces streaky disturbances that can grow exponentially due to interaction with the base flow. We first extend the asymptotic theory of Deguchi & Hall (2014a) to show that freestream coherent structures can be embedded in unsteady twodimensional boundary layers. The time evolution of the structure is affected strongly by the unsteady base flow, and ultimately it can only persist for a finite time. Next, we describe freestream coherent structures in compressible parallel boundary layer flows in the subsonic and moderate supersonic regimes. These flows are more industrially relevant to laminar flow control than the previously studied incompressible flows. The key result is that the equations for the nonlinear interaction of the vortex and wave in the layer near the freestream are identical to those obtained in the incompressible problem, but the velocity field now also drives a passive thermal field. The resulting disturbances to both the velocity and temperature fields can then grow exponentially towards the wall; the maximum amplitude of the disturbances depends on the Mach number and the Prandtl number.
Date of Award  31 Dec 2022 

Original language  English 

Awarding Institution   The University of Manchester


Supervisor  Andrew Hazel (Supervisor) & Oliver Jensen (Supervisor) 

 Capillary Flows
 Contact Lines
 Turbulence
 Asymptotic methods
 Fluid dynamics
The Use of Asymptotic Methods in BoundaryLayer and Interfacial Phenomena
Johnstone, E. (Author). 31 Dec 2022
Student thesis: Phd