## Appendix A. Application of lubrication theory to an air film under a levitating droplet

Applying lubrication theory to a levitating droplet in the same way as in previous studies (Lhuissier *et al.*
2013; Saito & Tagawa 2015) requires a Reynolds number
$Re$
inside the air film sufficiently smaller than 1 and a film thickness that is sufficiently thinner than the characteristic length along the mainstream. In this paper,
$Re$
including a geometrical aspect ratio (
$=\unicode[STIX]{x1D70C}Uh^{2}/\unicode[STIX]{x1D707}d$
) is used, and
$Re$
is sufficiently smaller than 1 for the airflow under a levitating droplet in our experiments. Thus, it is appropriate to apply lubrication theory to the air film under a levitating droplet. However, if general
$Re=\unicode[STIX]{x1D70C}Uh/\unicode[STIX]{x1D707}$
is used,
$Re$
is not sufficiently smaller than 1 in our experiments. Hence, it is questionable whether lubrication theory can be applied since the inertia term in the Navier–Stokes equation does not seem to be negligible. Therefore, we discuss here whether lubrication theory can be applied to the air film under a levitating droplet by performing numerical simulations. For this discussion, we numerically solve the Navier–Stokes equation including or neglecting the inertia term with a continuity equation, and then compare the pressure distributions in the air film.

We conduct the numerical simulation using COMSOL Multiphysics, which performs computations by the finite element method. We investigate whether the inertia term affects the pressure distribution generated inside the air film based on the results of numerical simulations obtained by two kinds of governing equations: (i) the general Navier–Stokes equation and continuity equation and (ii) the Navier–Stokes equation neglecting the inertia term and continuity equation. For each approach, we use about 700 000 triangle-shape meshes. We simulate an air film between a moving wall and the three-dimensional shape of the air film as measured in our experiment with droplet diameter
$d=3.16\pm 0.02~\text{mm}$
, viscosity
$\unicode[STIX]{x1D708}=100~\text{cSt}$
and wall velocity
$U=1.57~\text{m}~\text{s}^{-1}$
. We assume that the gas–liquid interface is a solid wall with a no-slip condition since the air film has a steady shape and the surface velocity of the droplet is negligible in comparison with the wall velocity (see § 4). We use
$U=1.57~\text{m}~\text{s}^{-1}$
for the moving wall along the
$x$
-axis positive direction and atmospheric pressure on the rim of the thin-film area as boundary conditions, as in our experiment.

Figure 14. Pressure distribution computed by (*a*) general Navier–Stokes equation and continuity equation and (*b*) Navier–Stokes equation neglecting the inertia term and continuity equation.

Figures 14(*a*) and 14(*b*) show the pressure distributions numerically simulated by the governing equations (i) and (ii), respectively, where the horizontal and vertical axes are the
$x$
- and
$y$
-axes and the arrow and black line represent the direction of wall velocity and the rim of the thin-film area, respectively. Both pressure distributions have positive pressure except in the vicinity of the rim of the thin-film area. The peak values of the positive and negative pressures each agree between the two distributions within an error of 0.3 %. In addition,
$L$
calculated by (2.2) agrees for both distributions within an error of 0.3 %. Therefore, we obtain quantitative agreement between the two pressure distributions shown in figure 14(*a*,*b*). This agreement indicates that the inertia term in the Navier–Stokes equation does not largely affect the lubrication pressure.

## Appendix B. The propagation of errors from the experimental parameters

We have the following two equations, namely for pressure gradients in the direction of
$x$
-axis and
$y$
-axis to derive the equation of motion in the air film (2.1):

where
$Q_{x}$
and
$Q_{y}$
indicate flow rate per unit thickness in the direction of
$x$
-axis and
$y$
-axis, respectively. By the law of propagation of errors, the errors of pressure gradients in the direction of
$x$
-axis
$\unicode[STIX]{x1D70E}_{x}$
and
$y$
-axis
$\unicode[STIX]{x1D70E}_{y}$
from (B 1) and (B 2) are

where
$\unicode[STIX]{x1D70E}_{Q_{x}}$
,
$\unicode[STIX]{x1D70E}_{Q_{y}}$
,
$\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D707}}$
,
$\unicode[STIX]{x1D70E}_{h}$
and
$\unicode[STIX]{x1D70E}_{U}$
indicate the errors of each parameter, respectively. We estimate that the air film thickness
$h$
, the dynamic viscosity of the air
$\unicode[STIX]{x1D707}$
and the wall velocity
$U$
are
$O(10^{-6})~\text{m}$
,
$O(10^{-5})~\text{Pa}~\text{s}$
and
$O(10^{0})~\text{m}$
, respectively. The flow rate
$Q_{x}$
is
$O(10^{-6})~\text{m}^{2}~\text{s}^{-1}$
which comes from the wall velocity
$U$
and the air film thickness
$h$
. We also estimate that
$\unicode[STIX]{x1D70E}_{h}$
,
$\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D707}}$
and
$\unicode[STIX]{x1D70E}_{U}$
are
$O(10^{-6})~\text{m}$
,
$O(10^{-7})~\text{Pa}~\text{s}$
and
$O(10^{-2})~\text{m}$
, respectively. Error
$\unicode[STIX]{x1D70E}_{Q_{x}}$
is
$O(10^{-2})~\text{m}^{2}~\text{s}^{-1}$
which is same as
$\unicode[STIX]{x1D70E}_{U}$
, because
$\unicode[STIX]{x1D70E}_{h}$
is sufficiently small in comparison with
$\unicode[STIX]{x1D70E}_{U}$
. The flow rate
$Q_{y}$
is assumed to be negligible. As a result, we find that the error of the air film thickness
$h$
is dominant in comparison with those of the other parameters.

## Appendix C. Decision procedure of the thin-film area

We consider the area inside the air film where the lubrication pressure is generated as the thin-film area. The rim of the thin-film area has thickness
$H$
which is equal to
$z_{0}$
(see figure 2) and experiences atmospheric pressure as the boundary condition. When
$H$
changes with the thin-film area diameter
$d_{t}$
, the thin-film area where the lubrication theory is applied also changes. Therefore, we need to select the proper thickness
$H$
to accurately calculate the lubrication pressure
$p$
generated inside the air film. In this appendix, we determine the thin-film area and the thickness at its rim
$H$
focusing on the lift
$L$
. We calculate
$L$
for a range of thicknesses
$H$
, in steps of
$1~\unicode[STIX]{x03BC}\text{m}$
. When the variation of
$L$
with
$H$
is smaller than the measurement error, that value of
$H$
is defined as the thickness of the thin-film area. Next, we numerically compute
$d_{t}$
corresponding to each value of
$H$
.

We show an example of our results in figure 15, where the horizontal axis is the thickness at the rim of the thin-film area,
$H$
, and the vertical axis is the calculated
$L$
. The result is obtained for a droplet diameter of
$d=3.16\pm 0.02~\text{mm}$
, viscosity
$\unicode[STIX]{x1D708}=100~\text{cSt}$
and wall velocity
$U=1.57~\text{m}~\text{s}^{-1}$
. The variation in
$L$
is smaller than 1 % when
$H$
is in the range
$20$
–
$24~\unicode[STIX]{x03BC}\text{m}$
. However, the measurement error of the calculated
$L$
is 5 % for
$H=22~\unicode[STIX]{x03BC}\text{m}$
, which is larger than the variation in
$L$
with
$H$
. Based on the above, we choose
$22~\unicode[STIX]{x03BC}\text{m}$
for
$H$
at the rim of the thin-film area. Note that the lift
$L$
decreases with
$H>30~\unicode[STIX]{x03BC}\text{m}$
and becomes negative with
$H>300~\unicode[STIX]{x03BC}\text{m}$
, since the lubrication approximation becomes inappropriate. Similarly, we investigate the relation between
$H$
and
$L$
for the whole range of values and find that
$H$
and the thin-film area can be determined for every experimental condition, where the two kinds of balance are verified as mentioned in §§ 5.1 and 5.2. Consequently, we conclude that the thin-film area is appropriately determined in our method.

Figure 15. Calculated lift
$L$
versus thickness at the rim of the thin-film area
$H$
.

Figure 16. A part of a sphere formed by closed black lines which are substituted for interference fringes (*a*) before reconstruction and (*b*) after reconstruction.

Figure 17. A part of a sphere formed by opened black lines which are substituted for interference fringes (*a*) before reconstruction and (*b*) after reconstruction.

## Appendix D. Calculation and error estimation for mean curvature

We calculate the mean curvature
$\unicode[STIX]{x1D705}$
by using the patch curvature function written in MATLAB. In the function, the least-squares curved surface including the local neighbourhood is calculated at a certain point. By using the eigenvectors and eigenvalues of the Hessian to the curved surface, the mean curvature
$\unicode[STIX]{x1D705}$
is obtained.

We determine the error of the mean curvature
$\unicode[STIX]{x1D705}$
as follows. We conduct a smoothing process between fringes to obtain the mean curvature
$\unicode[STIX]{x1D705}$
. This smoothing process causes the error of the mean curvature
$\unicode[STIX]{x1D705}$
. To investigate the error, we reconstruct a known three-dimensional shape from certain interference fringes, and calculate the mean curvature
$\unicode[STIX]{x1D705}$
. A part of a sphere of 1 mm in radius is used, where the mean curvature
$\unicode[STIX]{x1D705}$
is
$10^{3}~\text{m}^{-1}$
. This mean curvature
$\unicode[STIX]{x1D705}$
is of the same order as the mean curvature around the minimum thickness of the measured air film. It is inferred that the absolute value of the mean curvature
$\unicode[STIX]{x1D705}$
is large around the minimum thickness, and the error of the mean curvature
$\unicode[STIX]{x1D705}$
is the largest. We set the minimum height of the interference fringes at
$5.01~\unicode[STIX]{x03BC}\text{m}$
from the wall, and 19 fringes between
$5.42$
and
$13~\unicode[STIX]{x03BC}\text{m}$
at regular intervals. Additionally, the maximum height of the air film (i.e. the rim of the air film) is set at
$20~\unicode[STIX]{x03BC}\text{m}$
. We calculate the mean curvature
$\unicode[STIX]{x1D705}$
, when the interference fringes consist of (i) only closed lines which are circles and (ii) closed lines and opened lines which are semicircles. Figures 16(*a*) and 16(*b*) show the interference fringes and the reconstructed shape for condition (i), respectively. Figures 17(*a*) and 17(*b*) show the interference fringes and the reconstructed shape for condition (ii), respectively. In condition (ii), the fringes at the minimum and maximum height are closed lines, and the others are opened lines. Figures 18(*a*) and 18(*b*) show the relative error
$\unicode[STIX]{x1D705}_{err}$
of the mean curvature
$\unicode[STIX]{x1D705}$
for conditions (i) and (ii), respectively. Black lines and colour bars indicate the interference fringes and relative error for theoretical mean curvature of the sphere
$\unicode[STIX]{x1D705}_{theory}=10^{3}~\text{m}^{-1}$
, respectively. The relative error
$\unicode[STIX]{x1D705}_{err}$
can be calculated by the equation

Figure 18. Relative error
$\unicode[STIX]{x1D705}_{err}$
of calculated curvature using (*a*) closed and (*b*) opened black lines which are substitutes for interference fringes for theoretical mean curvature
$\unicode[STIX]{x1D705}_{theory}=10^{3}~\text{m}^{-1}$
.

Figure 19. (*a*) The pressure
$p_{s}$
overlaid with obtained interference fringes and (*b*) enlarged view of (*a*) around the smallest thickness.

where
$\unicode[STIX]{x1D705}$
is mean curvature calculated by our method. First, we focus on condition (i) in which the interference fringes consist of only closed lines. As shown in figure 18(*a*), the error of the mean curvature
$\unicode[STIX]{x1D705}$
is large around the maximum height of the air film, i.e. the rim of the air film. This is because the area is near to the rim of the air film, in other words, the boundary of calculation. However, when the mean curvature
$\unicode[STIX]{x1D705}$
is calculated in our experiments, interpolation is conducted between the interference fringes and the rim of the air film. Therefore, the significant error of the mean curvature
$\unicode[STIX]{x1D705}$
occurs inside the area of the interference fringes in experiments. In this area, the maximum and average of the error of the mean curvature
$\unicode[STIX]{x1D705}$
are about 50 % and 7 %, respectively. Now, we focus on condition (ii). As shown in figure 18(*b*), the error of the mean curvature
$\unicode[STIX]{x1D705}$
is large at the ends of the opened lines. The absolute pressure of pressure
$p_{s}$
also increases at the ends of the opened lines in figure 19(*b*). Thus, the error of the pressure
$p_{s}$
is due to the increase of the error of the mean curvature
$\unicode[STIX]{x1D705}$
at the ends of the interference fringes. In summary, we find that the smoothing process affects on the error of the mean curvature
$\unicode[STIX]{x1D705}$
, and the error increases at the open ends of the interference fringes.

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