# Dynamics of the capillary rise: a peculiar occurrence of oscillations

My name is Mateusz Świtała and I am at my fifth year on Faculty of Pure and Applied Mathematics, Wrocław University of Science and Technology, Poland. My subject of study is Applied Mathematics and I am also giving tutorials in mathematical software programming. This work with younger students is very rewarding and inspiring for further scientific activity which I would very like to pursue.

My scientific interests are closely related to fields of mathematics such as differential equations (both PDE and ODE), integro-differential equations, numerical methods, fluid mechanics and optimization theory. The applications of these areas of mathematics in mathematical modelling of real-world phenomena are equally important. I participated in ESGI 121, which took place in DTU, Kongens Lyngby, Denmark. In my opinion, the participation in study groups is the best way to learn advanced topics in science and its applications. It is also a great opportunity to learn how does the scientific work looks like. Together with students from other countries and under a supervision of the instructor, we tried to solve or optimize real problems using the toolbox of applied mathematics. These problems were posed by industrial companies and were meaningful so that they could introduce some improvements in the existing structure of industrial equipment.

Over the past year, together with Łukasz Płociniczak, PhD, we were investigating time- fractional nonlinear diffusion equation which describes the anomalous diffusion in porous media. Analysis of this equation is especially important because of its wide applications in natural science and in industry. The next blog post will give some more details about this case.

Another problem which I studied together with Łukasz Płociniczak over the past few months, was the dynamics of capillary rise. To be more precisely, we analysed the governing equation in the fluid column rising due to capillary presure. It is a very common phenomenon in nature and its properties are also used in the industry (ex. inkjet printing and some capillary pumps). The governing equation for the fluid column height $h=h(t)$ is associated with Washburn’s equation and is given by

$\displaystyle\frac{8 \mu }{r^2} h h'+\rho g h+\rho \displaystyle\frac{\,d}{\,d t}\left(h h'\right)=\displaystyle\frac{2 \gamma \cos{\theta}}{r},$

where $\mu$ is the viscosity, $\rho$ is the density, $\gamma$ is the surface tension, $\theta$ being the contact angle while $r$ is the radius of the capillary tube. The initial conditions are $h(0)=0$ and $h'(0)=\frac{2 \gamma \cos{\theta}}{\rho r}$. The question about the value of initial velocity has previously been discussed in the literature and some approaches lead to a contradiction. Our choice is a natural one which comes from the structure of the ODE.

The most important results which we obtained were: the proof of the existence and uniqueness of the solution of governing equation and the determination of the critical value of the nondimensional parameter for which the solution changes its behaviour from being monotone to oscillatory. If we define the Jurin’s height $h_e=\frac{P_c}{\rho g}$ to be the equilibrium height, the essential nondimensional parameter takes the form

$\omega = \displaystyle\frac{\rho ^2 g r^4}{64 \mu ^2 h_e}.$

We have rigorously shown that for $\omega$ increasing through $0.25$ the solution changes its behaviour from being monotone to oscillatory. This phenomenon has previously been spotted in the experiments.

Moreover, during our progress we have found the asymptotic behaviour of the solution and analysed the stability of the critical point (Lyapunov’s Theorem + LaSalle’s Invariance Principle). Some of the results from our work were part of my undergraduate thesis.

In my opinion, both anomalous diffusion and capillary rise are very interesting topics  which are worth to study. There is still a number of unknowns in each of them and the deeper analysis of each topic may result in new findings in different fields of mathematics and fluid mechanics.