Postgraduate research opportunities Evaporation of interacting droplets on inclined surfaces


Key facts

  • Opens: Thursday 3 February 2022
  • Deadline: Friday 30 September 2022
  • Number of places: 1
  • Duration: 3 years
  • Funding: Home fee, Stipend


The proposed project aims to tackle novel aspects of evaporating drops on an inclined plane by using a combination of numerical simulations and asymptotic theory. The outcomes of the mathematical modelling will be complemented by experiments done in the Department of Mechanical and Aerospace Engineering in University of Strathclyde.
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Applicants should have, or be expecting to obtain in the near future, a first class or good 2.1 honours degree (or equivalent) in mathematics or a mathematical science.

THE Awards 2019: UK University of the Year Winner
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Project Details

The proposed project aims to tackle novel aspects of evaporating drops on an inclined plane by using a combination of numerical simulations and asymptotic theory.  The outcomes of the mathematical modelling will be complemented by experiments done in the Department of Mechanical and Aerospace Engineering in University of Strathclyde. The projects are inspired by recent work by Dr.Wray on evaporation of multiple droplets (Wray et al., ``Competitive evaporation of multiple sessile droplets'' J. Fluid Mech. 884 A45 (2020) and ``Contact-line deposits from multiple evaporating droplets'' Phys. Rev. Fluids 6, 073604) and supported by numerical work done by Dr.Das (Das and Saintillan, ``A nonlinear small-deformation theory for transient droplet electrohydrodynamics'' J. Fluid Mech.  829, 127-152 (2017) and ``Electrohydrodynamics of viscous drops in strong electric fields: numerical simulations'' J. Fluid Mech.  810, 225-253).

Evaporation of a sessile droplet of water containing nonvolatile colloidal particles is not only a problem of fundamental interest to scientists but also has real life practical applications. The applications are as ancient as writing on papyrus paper using ink invented by Egyptians and Chinese around 4500 years ago. Modern day applications include inkjet printing, forensic analysis of patterns left behind by dried blood drops, manufacture of DNA microarrays, and enhancing agricultural produce by spraying micronutrient with nanofluids onto plant leaves. The canonical example of this phenomena is the “coffee stain” deposit formed by colloid-containing water droplets. This problem was first tackled from a physical chemistry viewpoint by Deegan et al. (1997). Since this pioneering work, there has been an exponential growth in the number of literature on the topic.  Several experiments exploring different aspects of this problem have been performed in the last decade or so. From a fluid mechanics perspective, evaporating drops with suspended particles is a rich problem.  With some simplifying assumptions, the evaporation of a sessile liquid droplet and the colloidal deposition pattern can be described analytically. However, the problem can be made more complicated by incorporating various phenomena like heat conduction and convection, natural convection, viscous and inertial flows or surface-tension-driven flows. It also includes thermal-hydrodynamic instabilities, buoyancy effects, liquid spreading, contact-line pinning and depinning and adhesion. In order to fully explore the nonlinear effects of these phenomena, we needs to perform numerical simulations.  Hence, the problem of evaporating drops has intrigued scientists from various disciplines, including experimentalists, theorists and numerical simulations experts.

In particular, the problem of evaporating drops on inclined and vertical planes is of great significance in many industrial settings. Inclining the plane allows gravitational forces to take a central role in the final deposit patterns. This problem has been solved numerically by Du and Deegan JFM (2015) and Timm et al. Sci Rep. (2019) recently in two- and three-dimensions, respectively.  However, the natural choice for simulations, boundary element method (BEM), has not been employed to study this problem so far and transient drop dynamics remains unaddressed.  Having reproduced previous results by a novel numerical method based on BEM, we will tackle the practical situation where a pair drops on an inclined plane are interacting with each other through the vapour field. Furthermore, building on the very successful paper of Dr.Wray on evaporation of multiple droplets, we will extend the theoretical results to include the effect of gravity.

Further information

The student will take advanced graduate level lectures provided by the Scottish Mathematical Science Training Centre (SMSTC). This will give the student a strong foundation to complete the research projects. They will also participate in Researcher Development Programme and complete the Postgraduate Certificate for Researcher Professional Development (PGCertRPD).

The student will be mentored and also provided with knowledge and skills required to carry out the proposed research activities that are not covered by SMSTC. Successful completion of the projects will require familiarisation with both analytical and numerical methods.  The supervisors have expertise in both these domains. Additionally, we have a bespoke ``low-order modelling'' module on Myplace designed to bridge the gap between SMSTC Continuum Dynamics and Asymptotic models in the literature, designed and taught by Dr. Wray.  The student will be required to present their research work in national and international conferences. They will also be encouraged to undergo various training programmes meant for post-graduate students in the university and elsewhere in the UK organised by the Engineering and Physical Sciences Research Council. 

Within the department, there is an established cohort of students working on problems to fluid mechanics and mathematical modelling, hence the new student will feel like a natural fit in the group. In particular, they will benefit from being a member of Continuum Mechanics and Industrial Mathematics (CMIM) research group within the Department of Mathematics and Statistics.  The student will have the opportunity to interact with internationally renowned speakers invited to give seminars in weekly group meetings organised by CMIM.  And they will benefit from working in a multidisciplinary team by collaborating with experimentalists in the Department of Mechanical and Aerospace Engineering.

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Funding details

The studentship covers home fees and stipend. All candidates are eligible, but international candidates would need to pay the fees difference between Home and Overseas rates.

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Dr Debasish Das

Strathclyde Chancellor's Fellow
Mathematics and Statistics

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Dr Alexander Wray

Mathematics and Statistics

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Dr David Pritchard

Senior Lecturer
Mathematics and Statistics

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Number of places: 1

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Mathematics and Statistics - Mathematics

Programme: Mathematics and Statistics - Mathematics

Start date: Oct 2022 - Sep 2023