It is known that an increased flow rate can be achieved in channel flows when smooth walls
are replaced by superhydrophobic surfaces. Applications include the thermal management of microelectronics, where a competition between convective and conductive resistances must be accounted for to evaluate any advantages of these surfaces. Additionally, it is important to address the stability of the Cassie-Baxter state for future work. Fluid properties are central in such problems, and in applications; these can range from those of water to liquid metals.
Of particular interest is the hydrodynamic stability, thermal resistance and boundary layers of said flows, something that has been widely overlooked within the literature. This is
of relevance to applications in microelectronics and marine transport, that typically base
designs on steady-states or apparent-slip models that approximate them.
The global stability problem where longitudinal grooves are periodic in the spanwise direction
is studied. The flow is driven by either the motion of a smooth upper lid or a constant
pressure gradient. In the case of smooth walls, the former problem (plane Couette flow) is
linearly stable at all Reynolds numbers, whereas the latter (plane Poiseuille flow) becomes unstable above a relatively large Reynolds number. When ridges are present, we show that additional instabilities arise in both cases. Generally, for lid-driven flows, an unstable mode is found that becomes neutral as the Reynolds number increases. The critical Reynolds numbers increase as the channel height increases or the slip fraction decreases. For pressure driven flows, two modes can coexist and exchange stability depending on the channel height and slip fraction. The first mode remains unstable as the Reynolds number increases and corresponds to an unstable mode of the Rayleigh equation, while the second mode becomes neutrally stable at infinite Reynolds numbers. Comparisons of critical Reynolds numbers with experimental observations are encouraging.
The stability of a pressure-driven flow over a longitudinally ridged superhydrophobic surface
is studied; including isoflux heating and curvature at the liquid-gas contact line. Three modes are identified that exchange stability characteristics depending on the channel geometry and protrusion angle. A mode arises due to meniscus curvature which exists predominantly for small channel heights. The viscous and inviscidly unstable mode from the flat meniscus case are perturbed by these new flow features, where the latter has the lowest critical Reynolds number when the channel height is small and the liquid-gas interface protrudes into the channel. When the channel height is small, positive protrusions into the gas cavity induce the largest growth rates and the lowest critical Reynolds numbers (for all slip fractions). When the channel height is increased, the most unstable protrusion angle depends on the slip fraction. The critical Reynolds number attains a global minimum for large gas fractions and small channel heights; where these Re = O(10) values are substantially less then those imposed for laminar flows in experiments.
The optimisation of a lid- and pressure-driven flow, in a heated channel textured with a
superhydrophobic surface, is studied. The dependence of mean flow quantities on the:
protrusion angle, slip fraction, channel height, thermocapillary stress, thermal Péclet number
and channel length is established. For large channel heights, the lid-driven problem favours
internally protruding menisci and large slip fractions. However, a liquid-gas interface that
enters the gas cavity is optimal when the height of the channel is small. The same is true
for the pressure-driven case. In both configurations, increasing the thermocapillary stress reduces the flow rate. Considering the diabatic problem where one would like to maximise heat transfer within the cross-plane, ridged channels are detrimental. It is known that a design compromise exists between the convective and caloric resistance, therefore, one must evaluate the total thermal resistance. It has been established that in the liquid-metal
regime, where Pé << 1, a ridged channel is generally favoured. If one was instead to
consider other liquid-species, where Pé = O(1), this may not be the case. The benefits of
superhydrophobic surfaces are shown to increase as the length of the channel is increased and the thermocapillary stress approaches zero.
The laminar boundary layer over a longitudinally grooved superhydrophobic surface is studied. The displacement and momentum thickness increase down the surface for the: solid, gas and periodic ridge paradigms considered. Far downstream, these develop similarly but become less than those characteristic of the boundary layer over a semi-infinite flat plate. In the periodic paradigm, their magnitude decreases with decreasing solid ridge width; such that at its minimum, the downstream displacement thickness is 78% of the two-dimensional
boundary layer. The average wall shear stress decreases and the skin friction increases with
the streamwise coordinate for all configurations; developing similarly but surpassing those characteristic of the boundary layer over a semi-infinite flat plate. In the periodic case, their magnitude increases with decreasing solid ridge width; such that at its maximum, the skin friction is 105% of the two-dimensional boundary layer. The slip length has a non-trivial dependence on the longitudinal coordinate; but for a fixed downstream location in the periodic problem, it grows as the solid ridge width decreases. The isothermal boundary layer over a periodically ridged superhydrophobic surface is
studied; including transverse variation and constant curvature at the interfacial contact line. The displacement and momentum thickness increase along the ridged surface. They are maximised for small ridge widths and large negative contact angles (that protrude into the bulk flow). The average wall shear stress decreases and the skin friction increase along the superhydrophobic surface. They are maximised when the solid-to-gas ridge ratio is large and the meniscus deforms into the bulk flow. The slip length is non-monotonic down the surface for all slip fractions, but is maximised for large gas ridges and negative contact angles. The average Nusselt number decreases along the superhydrophobic surface. To attain its global maximum, one wishes for the geometry to consist of small solid ridges with a liquid-gas interface that protrudes into the bulk flow. Its magnitude scales with the Prandtl number, such that one may choose a fluid to maximise this quantity.