Capillary Waves and Young-Laplace Equation in Miscible Fluids

We probably all have observed tiny waves forming on the surface of a body of water on a windy day. The onset and propagation of these ripples are driven by interfacial tension, a force traditionally understood as the result of the different mutual interactions between the molecules of two immiscible fluids, such as air and water, which effectively makes their shared interface "elastic." But what happens if the surface in question separates two miscible fluids that gradually mix over time, like lemon juice and tea, eventually forming a homogeneous fluid in which no trace of the interface remains?
We have recently discovered [1] that capillary waves can also be observed at the boundary between two miscible fluids when they are pumped at different flow rates in a microfluidic channel (Fig. 1), long before they completely mix. Most importantly, the dispersion relation of these waves, which links their frequency and wavelength, allowed us to measure an effective tension between the fluids. We have therefore revealed that a tension exists even between miscible molecular fluids, which governs wave propagation shortly after the first fluid-fluid contact; moreover, this tension depends strongly on time since bringing the two fluids in contact.
This ambitious PhD project aims to study the onset of capillary waves in miscible fluids under different physico-chemical conditions by measuring their propagation speed, frequency, and amplitude. The final goal is to establish, for the first time, a relationship between effective tension and pressure field in miscible fluids.

i)Short/Medium-Term: The first step will be to explore in greater detail the effect of confinement, which determines the lower bound for the accessible wave number of capillary waves: the larger the channel, the smaller the wave number accessible in experiments. Testing our current results across different microchannel geometries will be the natural continuation of our most recent work in this field [1].

ii) Medium-Term: The second step will be to investigate the role of liquid chemistry. We have already observed that spinning droplets of triethylene glycol (TEG) in a reservoir of glycerol do not deform in the same way as droplets in water-glycerol mixtures, and this phenomenon cannot be explained solely by fluid dynamics arguments related to density or viscosity differences [2], pointing to the presence of different interfacial stresses. In our co-flow experiments, we will primarily test for the presence of capillary waves and measure the effective tension at alcohol-glycerol boundaries—specifically in TEG-Glycerol, Ethylene Glycol-Glycerol, Ethanol-Glycerol, Methanol-Glycerol, and Octanol-Glycerol systems. This will allow us to unveil the effect of liquid structure on wave propagation and on the effective tension.

iii) Medium/Long-Term: The third and most challenging step will be to test the validity of the Young-Laplace equation for miscible fluids, which links the pressure in the fluids and the effective tension. 
To obtain the pressure, we will measure the velocity field of the fluids by adding tracer particles and using methods such as Particle Imaging Velocimetry (PIV) and/or Ghost Particle Velocimetry (GPV). Our focus will be on two kinds of systems: a) molecular co-flowing liquids (as in point (ii)), and b) colloidal fluids in contact with their own solvent. For Newtonian fluids, the local pressure tensor can be obtained by integrating the Poisson equation [3] using finite difference schemes [4], once the velocity field is known. For non-Newtonian fluids, integration schemes of the full Navier-Stokes equation [5] require knowledge of the shear rate-dependent viscosity, which can be measured using standard shear rheometry.

References:
[1] A. Carbonaro, G. Savorana, L. Cipelletti, R. Govindarajan, and D. Truzzolillo, Phys. Rev. Lett., 134, 054001 (2025).
[2] A. Carbonaro, L. Cipelletti, D. Truzzolillo, Phys. Rev. Fluids, 5, 074001 (2020) - Editor's suggestion.
[3] B. W. Van Oudheusden. Meas. Sci. Technol., 24(3):032001, (2013).
[4] J. D. Hoffman. Numerical Methods for Engineers and Scientists. Marcel Dekker, New York, 2nd ed., rev. and expanded edition, (2001)
[5] N. Tiwari, Y. Tasaka, and Y. Murai. Flow Measurement and Instrumentation, 77:101852, (2021)

The PhD project will be carried on within the Soft Matter team of the Laboratoire Charles Coulomb (L2C) (https://www.softmatter-l2c.fr/), in Montpellier (France), under the supervision of Dr. Domenico Truzzolillo and Prof. L. Cipelletti. The final awarding of the PhD scholarship involves a selection process by a dedicated committee.

The student must hold a master’s degree in physics, chemical Engineering, or materials science. Proficiency in programming languages (Python, C), optical microscopy, and basic fluid dynamics is a plus for the application.