# Analytic Calculation of Capillary Bridge Properties Deduced as an Inverse Problem from Experimental Data

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## Abstract

In this work, we propose an original resolution of Young–Laplace equation for capillary doublets from an inverse problem. We establish a simple explicit criterion based on the observation of the contact point, the wetting angle and the gorge radius, to classify in an exhaustive way the nature of the surface of revolution. The true shape of the admissible static bridges surface is described by parametric equations; this way of expressing the profile is practical and well efficient for calculating the binding forces, areas and volumes. Moreover, we prove that the inter-particle force may be evaluated on any section of the capillary bridge and constitutes a specific invariant.

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## Notes

1. The boundary conditions are not concerned for the moment.

2. When he studied the complicated integral form given by Monge as a solution of minimal surface problem.

3. When the gravity is taken into account.

4. The terminology of conic roulette comes from this property.

5. Erle et al. (1971) have limited their study to nil contact angle.

6. A negative suction device, as will be shown later.

7. Redundant information when suction is known in axisymmetric case.

8. Directly as the surface is additionally known to be axisymmetric, Eq. (10) can be written $$\dfrac{1}{y}\dfrac{d}{dy}\dfrac{y}{\sqrt{1+y^{\prime 2}}}=\dfrac{\Delta p}{\gamma }.$$

9. Note that we have $$\displaystyle { \frac{d}{dy} (y^{'2}) = \frac{1}{y^{'}} \frac{d }{d x} (y^{'2}) =2y^{''} }$$.

10. The wetting angle is a characteristic of the liquid and of the surfaces in contact.

11. At the contact points, the tangent to the meridian is denoted $$t$$ (Fig. 3).

12. The degenerate case $$y$$ constant ($$y = r\sin \delta$$) corresponds to the right circular cylinder.

13. With a camera with macrozoom for instance.

14. A typical key observation to discriminate the various cases.

15. C. Delaunay in 1841 is the first author to propose in Delaunay (1841), p. 313 another parameterization using the angle $$\varphi$$ between the tangent to the meridian curve and $$x$$-axis:

$$y\left( \varphi \right) =-a\cos \varphi +a\sqrt{1+\alpha -\sin ^{2}\varphi }$$

$$x\left( \varphi \right) =\beta +a\sin \varphi -a\tan \varphi \sqrt{1+\alpha -\sin ^{2}\varphi }+ {\displaystyle \int _{0}^{\varphi }} \dfrac{a\alpha ~d\theta }{\cos ^{2}\theta \sqrt{1+\alpha -\sin ^{2}\theta }}$$

with $$\dfrac{1}{a}=\dfrac{\Delta p}{\gamma }$$, $$\ \alpha$$ and $$\beta$$ any constants. Note that Lindelöf (1869) has also studied in 1861 these curves using the calculus of variations.

16. Equivalently, we have a positive suction $$s=-\Delta p$$ and therefore $$p_{int}<p_{ext}$$ inside the capillary bridge.

17. More generally, when the radii $$r_1$$ and $$r_2$$ are different, the same principle demonstrates that the filling angles $$\delta _1$$ and $$\delta _2$$ are linked. More specifically, the following relationship applies:

\begin{aligned} r_1 \sin \delta _1 (2a \sin (\delta _1+\theta ) + r_1 \sin \delta _1 ) = r_2 \sin \delta _2 (2a \sin (\delta _2+\theta ) + r_2 \sin \delta _2 ). \end{aligned}
18. In particular, $$a$$ given by (22) is strictly positive.

19. Note that, as for the previous case, the condition $$b^{2} = -y^{*2} + 2ay^{*}>0$$ would lead to $$\displaystyle { y^{*}<r\frac{\sin \delta }{\sin \left( \delta +\theta \right) } }$$ which is already satisfied.

20. It cannot be encountered in the Proof of Result 2, as it would lead to infinite negative values of $$H$$ and $$\lambda$$, that is in contradiction with the beginning assumption $$\lambda >0$$.

21. From Young–Laplace equation (10), as $$y^{''}\ge 0$$ and $$y\ge rsin\delta$$, we have necessarily $$\Delta p>0$$.

22. Compatible with the capillary bridges with concave meridian addressed in this section.

23. Providing exact results for any polynomial function of degree three or less.

24. Accounting to the sign of the main curvatures.

25. Corresponding to $$\lambda >0$$.

26. Corresponding to $$H>0$$ and $$\lambda >0$$.

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Correspondence to Olivier Millet.

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Gagneux, G., Millet, O. Analytic Calculation of Capillary Bridge Properties Deduced as an Inverse Problem from Experimental Data. Transp Porous Med 105, 117–139 (2014). https://doi.org/10.1007/s11242-014-0363-y

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• DOI: https://doi.org/10.1007/s11242-014-0363-y