Inertial effects on flow dynamics near a moving contact line

Abstract

This study investigates the role of inertia in moving contact lines using experiments, theoretical analysis, and numerical simulations. Experiments are conducted using a plate immersion configuration over a wide range of Reynolds numbers from $O(10^{-3})$ to $O(10)$. Flow configurations and quantitative measurements are obtained using high-speed imaging and particle image velocimetry. The streamfunction contours reconstructed from the experimental velocity fields are compared with the viscous modulated wedge solution (viscous-MWS) and inertial-MWS theory. Experimental observations show that the streamfunction contours agree well with viscous predictions at low Reynolds numbers; however, systematic deviations emerge as the Reynolds number increases. The inertial-MWS theory, an inertial extension of the Huh and Scriven framework, accounts for these deviations, but only within a narrow range of Reynolds numbers $10^{-1} < Re < 1$. At higher Reynolds numbers, inertial theory fails to accurately capture the deviations in the streamfunction contours observed in the experiments. Moreover, simulations conducted using the volume of fluid method support our findings, exhibiting deviations in streamfunction contours consistent with experimental observations. We demonstrate that inertia does not fundamentally alter the underlying flow configuration but instead induces a systematic deviation in the streamfunction contours. At finite $Re$, the interfacial speed transitions from a nearly constant value in the viscous regime to a monotonic decay along the interface. These findings expose the need for more sophisticated models of moving contact lines.

Figures (10)

• Figure 1: An illustration of the experimental setup. (i) A transparent acrylic tank filled with liquid (ii) A glass plate traversing into the liquid bath (iii) A linear traversing system (iv) A combination of laser and optical lens for forming a thin laser sheet illuminating the region of interest (v) High-speed camera with a macro lens. The inset provides a magnified view of the field of interest, showing the fluid phase B forming an obtuse contact angle with the solid surface.
• Figure 2: (a) Schematic of geometry showing plate immersion resulting in an advancing contact line. Cylindrical coordinate system ($r$,$\theta$) used for flow near a moving contact line, a flat interface with a constant angle $\phi$, caused by a plate moving at constant speed $U$. (b) Coordinate system for flow near a moving contact line with a curved interface $\beta(r)$.
• Figure 3: Comparison of streamfunction contours between the flat interface (regression fit) and the curved interface. (a) Comparison between HS71 theory and inertia-corrected theory for the flat interface; linear fit: $y=0.25x$; The flat interface is obtained by performing a linear regression fit ($R^2$ =0.99) to the outlined curved interface. (b) comparison between HS71 theory and inertia-corrected theory for the curved interface; fitting parameters for interface: $c_1$ = 0.40, $c_2$ = 0.09, $c_3$ = -0.41 and $c_4$ = -0.77. The blue dashed curve denotes the streamfunction contour from HS71 theory, and the red solid curve denotes the streamfunction contour from inertia-corrected theory. The blue solid curve denotes the interface (both flat and curved) in the respective plots.
• Figure 4: [Experiments (black solid curve) + viscous theory (blue dashed curve) + inertial correction (red dashed curve)] Contours of the streamfunction obtained from experiments, viscous MWS theory, and inertial MWS theory are superimposed. The grey rectangle represents the solid plate moving downwards, and the blue solid curve represents the interface between air and the liquid phase. Contours are shown for the following Reynolds and capillary numbers: (a) 500 cSt silicone oil at $Re$ = $1.3\times 10^{-3}$ and $Ca$ = $1.4\times 10^{-2}$, fitting parameters for interface: $c_1$ = -0.70, $c_2$ = 0.01, $c_3$ = 0.70 and $c_4$ = -0.81; (b) 50% sucrose-water at $Re = 3.55 \times 10^{-2}$ and $Ca = 2.6 \times 10^{−5}$, fitting parameters for interface: $c_1$ = 2.66, $c_2$ = -0.19, $c_3$ = -2.39 and $c_4$ = -0.25; (c) 48% sugar-water at $Re = 1.25 \times 10^{-1}$ and $Ca = 8.16 \times 10^{−5}$, fitting parameters for interface: $c_1$ = 1.64, $c_2$ = -0.22, $c_3$ = -1.64 and $c_4$ = -0.28; (d) 40% sucrose-water at $Re = 3.02$ and $Ca = 3.3 \times 10^{−4}$, fitting parameters for interface: $c_1$ = 0.40, $c_2$ = 0.09, $c_3$ = -0.41 and $c_4$ = -0.77.
• Figure 5: [High $Re$ experiments (black solid curve) + viscous theory (blue dashed curve) + inertial correction (red dashed curve)] Contours of the streamfunction obtained from experiments, viscous MWS theory, and inertial MWS theory are superimposed. The grey rectangle represents the solid plate moving downwards, and the blue solid curve represents the interface between air and 40% sucrose-water mixture. Contours are shown for the following Reynolds and capillary numbers: (a) $Re = 3.02$ and $Ca = 3.3 \times 10^{−4}$, fitting parameters for interface: $c_1$ = 0.40, $c_2$ = 0.09, $c_3$ = -0.41 and $c_4$ = -0.77; (b) $Re = 6.04$ and $Ca = 6.6 \times 10^{−4}$, fitting parameters for interface: $c_1$ = 1.03, $c_2$ = -0.02, $c_3$ = -1.02 and $c_4$ = -0.38; (c) $Re = 9.05$ and $Ca = 9.9 \times 10^{−4}$, fitting parameters for interface: $c_1$ = 0.82, $c_2$ = -0.008, $c_3$ = -0.31 and $c_4$ = -0.56; (d) $Re = 12.1$ and $Ca = 13.3 \times 10^{−3}$, fitting parameters for interface: $c_1$ = 1.15, $c_2$ = -0.05, $c_3$ = -0.54 and $c_4$ = -0.49.