Spreading viscous fluids on a horizontal surface: project-based learning in fluid mechanics

Abstract

The spreading of a thin viscous fluid film on a horizontal surface is an interesting problem in fluid mechanics with many practical applications ranging from coating processes to biological systems and environmental flows. It can even be observed in everyday situations, such as syrup spreading on a pancake. We present a project-based learning approach to this problem, in which engineering or physics undergraduates apply classroom knowledge to understand and solve it, using dimensional analysis, experiments, and theoretical modeling. First, a dimensional analysis is conducted to guide the design of the experiment suitable for an undergraduate laboratory or even at home. The problem is then simplified to obtain a mathematical model that accounts for the experimental results. Through this process, students are able to obtain a solution compatible with those published in fluid mechanics journals with minimal supervision from the instructor. This project not only develops important skills but also motivates students by showing that they have the ability to solve complex problems.

Figures (6)

• Figure 1: Sketch of the growing process of a fluid layer (density $\rho$, dynamic viscosity $\mu$ and surface tension coefficient $\sigma$) that falls at a constant volume rate $Q$ on a flat surface. The fluid spreads on the surface isotropically, creating a cylinder of radius $R(t)$ and thickness $h$.
• Figure 2: Experimental setup in our laboratory using easily accessible materials such as plastic bottles, mobile phone and tape.
• Figure 3: Frames taken from the videos showing the expansion of the circular fluid plate in time. The top row corresponds to the sugar-water solution ($H_0$ = 1 cm), the middle to olive oil ($H_0$ =1.5 cm), and the bottom to the soap ($H_0$ =1.5 cm). Two circles are visible in each frame. The one with a fixed radius is the bottle, while the circle that gets larger in each panel, highlighted with a red circumference, is the liquid sheet of radius $R$($t$). The black line in left figures is a scale bar of 3 cm.
• Figure 4: Radius of the liquid sheet as a function of time for a sugar--water mixture at different initial liquid heights, $H_0$. The error cross indicates representative uncertainties of 1 mm in $R$ and 0.033 s in $t$.
• Figure 5: Experimental values of the dimensionless parameters for experiments on all three fluids. The equation obtained from dimensional analysis (Eq. \ref{['eq:solexp']}) is represented by dashed lines, with $A=$0.35, 0.21, and 0.12 for sugar--water solutions, soap, and oil, respectively. Our theoretical model (Eq. (\ref{['def']})) is also shown (solid line). Error bars are omitted for clarity; estimated uncertainties are discussed in the text.