How to measure the thickness of a lubrication film in a pancake bubble with a single snapshot?

(1)

OATAO is an open access repository that collects the work of Toulouse

researchers and makes it freely available over the web where possible

Any correspondence concerning this service should be sent

to the repository administrator:

tech-oatao@listes-diff.inp-toulouse.fr

This is an author’s version published in:

http://oatao.univ-toulouse.fr/23461

To cite this version:

Atasi, Omer and Haut, Benoit and Dehaeck, Sam and

Dewandre, Adrien and Legendre, Dominique and Scheid,

Benoit How to measure the thickness of a lubrication film in a

pancake bubble with a single snapshot? (2018) Applied

Physics Letters, 113 (17). 173701. ISSN 0003-6951

Official URL:

https://doi.org/10.1063/1.5051057

(2)

(3)

(4)

(5)

lower values of this variable due to slight blurring of the images at higher translation velocities. For this reason, the precision on the calculation of r/l is better for small values of h1ðdrÞ, and here, only the data points corresponding to a

translation velocity lower than 0.007 m/s have been kept for validation purpose. The results are again in excellent agreement with independent techniques as shown in Table

II.

Note that to increase the accuracy on dr, we have used the value averaged over the range of angles # 615_{, as}

represented on the right of Fig.2. Despite the fact that the shape functionF(x, y) used in this work is strictly valid for # ¼ 0, the corresponding variation made on the shape for small variations along the perimeter of the bubble remains small. Indeed, using the model of Burgess and Foster,17 we have estimated the maximum of the variation Dh1/a at 60.004, i.e., well below the uncertainty of our experimental measure-ments, as it can be deduced from the amplitude of the error bars in Fig.5.

Now, our method can easily be adapted to other situa-tions such as a pancake drop instead of a pancake bubble, at the sole condition thatm > 1. To this purpose, the new shape function should be constructed depending additionally on the viscosity ratio, denoted k¼ ld/l, between the viscosity of the drop, ld, and the one of the surrounding liquid.16 Updating the ray tracing algorithm with these new shapes should allow one to extend the correlating function in Eq.(5)

for non-zero viscosity ratio, i.e., f(m, k). Nevertheless, as mentioned earlier, the correlating function f(m) is already valid for inviscid pancake droplets. Practically, Balestra et al.16 have demonstrated that the shape of droplets for kⱗ 10 2_{is almost identical to the shape obtained for k}_{¼ 0,}

as it has been considered in this work. This said, it is worth mentioning that the ratio of refractive indicesm being lower for droplets than for bubbles, the sensitivity of the method

would also be lower, as inferred from Fig. 4by comparing the slopes of the solid and dashed lines.

In conclusion, we have shown how one can measure the lubrication film thickness around a pancake bubble or an inviscid drop, from a single bright-field image, as can be typ-ically obtained with a transmission microscope. We also demonstrated how one can combine this technique with the measurement of the bubble velocity to extract a measure of the visco-capillary velocity. It is believed that the method presented in this work can be of wide use in the future as it requires a standard microscope and a visual inspection of a single image, hence a smaller amount of less sophisticated instruments, as summarized in TableI.

Seesupplementary materialfor a detailed description of the ray tracing procedure.

O.A. and B.S. thank the F.R.S-FNRS for financial support and Innoviris (RBC) through the project Microeco. O.A. also thanks the “Fonds Alice et David van Buuren” and “Jaumotte-Demoulin” Foundation.

1_{H. Chen, Q. Meng, and J. Li,}_{Appl. Phys. Lett.}_{107, 141608 (2015).} 2_{A. Huerre, O. Theodoly, A. M. Leshansky, M. P. Valignat, I. Cantat, and}

M. C. Jullien,Phys. Rev. Lett.115, 064501 (2015).

3_{O. Atasi, S. Khodaparast, B. Scheid, and H. A. Stone,}_{Phys. Rev. Fluids} 2, 094304 (2017).

4_{M. J. F. Warnier, E. V. Rebrov, M. H. J. M. de Croon, V. Hessel, and J. C.} Schouten,Chem. Eng. J.135, S153 (2008).

5_{J. A. Howard, P. A. Walsh, and E. J. Walsh,}_{Int. J. Heat Mass Transfer} 54, 4752 (2011).

6_{S. Khodaparast, O. Atasi, A. Deblais, B. Scheid, and H. A. Stone,}

Langmuir34, 1363 (2018).

7_{A. van der Net, L. Blondel, A. Saugey, and W. Drenckhan,}_{Colloids Surf.,}

A309, 159 (2007).

8_{H. C. Hulst and R. T. Wang,}_{Appl. Opt.}_{30, 4755 (1991).}

9_{S. Dehaeck, J. P. A. J. v. Beeck, and M. L. Riethmuller,}_{Exp. Fluids} 39, 407 (2005).

10_{G. Kœnig, K. Anders, and A. Frohn,}_{J. Aerosol Sci.}

17, 157 (1986). 11

A. R. Glover, S. M. Skippon, and R. D. Boyle,Appl. Opt.34, 8409 (1995). 12

S. Dehaeck and J. van Beeck,Appl. Opt.46, 5957 (2007). 13_{M. R. Gavrilovic,}_{Eur. Phys. J. D}_{71, 316 (2017).} 14_{F. P. Bretherton,}_{J. Fluid Mech.}

10, 166 (1961). 15

M. Van Dyke, Perturbation Methods in Fluid Mechanics (Academic Press, 1964).

16

G. Balestra, L. Zhu, and F. Gallaire,Microfluid. Nanofluid.22, 67 (2018). 17

D. Burgess and M. R. Foster,Phys. Fluids A2, 1105 (1990). TABLE II. Measured values of the visco capillary velocity using direct

techniques as given in TableIand (in bold) the indirect method that com bines the classical and present methods.

Mineral oil Silicon oil r

l ½m=s 1.35 6 0.03 1.08 6 0.03 see TableI 1:2660:26 1:0860:09 Eqs.(3) (5)