## Singularities in Mathematical Models of Physical Phenomena

``How wonderful that  we have met with a paradox.
Now we have some hope of making progress.''
N. Bohr

The section explains the essence of this direction of research and briefly describes some results obtained by now.

The term `singularity' is used in applied mathematics to indicate that a conventional way of modelling a certain physical process mathematically leads to consequences which for some reasons cannot be accepted. This definition implies that the assessment is made on the basis of some physical criteria associated with what features of the process should be modelled, i.e. the criteria which lie outside the mathematics of the problem, though in the `most singular' cases non-existence of a solution makes the formulation of the problem unacceptable also from the purely mathematical point of view.

A necessary remark. If a solution exists but some physical arguments indicate a `singularity', one has to find out whether it appeared due to simplifications and/or additional assumptions made in the process of obtaining the solution (see [1] for a review of such cases) or whether it is inherent in the very formulation of the problem. In the former case, the singularity can be removed mathematically by relaxing `extra' constraints, i.e. without altering the underlying model, while in the latter, one has to look for those physical processes which are not accounted for in the conventional formulation and generalize the model itself. It is necessary to emphasize that the singularities in question are the problems of modelling but not of mathematical techniques. In particular, they are the fundamental difficulty of principle on the way of computational fluid dynamics. Any attempt to bypass the difficulty using some ad hoc criteria would reduce the modelling to a semiempirical level thus making the whole numerical exercise entirely pointless.

The goal of research is (i) to find the physical factors whose absence in the conventional model has lead to a singularity and (ii) to remove the singularity by incorporating them into the mathematical model in the simplest self-consistent way.

## Some results

The main result obtained in the last decade is that a number of seemingly different fluid motions where the conventional approach of fluid mechanics gives rise to `singularities' or `paradoxes' were found to be particular cases of a more general physical phenomenon which is not accounted for in the classical model. As a consequence, the simplest theory incorporating this phenomenon removes paradoxes in all these problems in a physically plausible way without any ad hoc assumptions. The phenomenon in question is the interface formation/disappearance process. Some examples are given below.

### Moving contact-line problem

Figure 1. A liquid drop sliding down an inclined plane is an example of flow with moving contact lines.

The essence of this famous (or, according to some researchers, infamous) problem is that the displacement of a liquid or gas by another liquid from a solid surface is not allowed in classical fluid mechanics: the corresponding problem has no solution. Numerous attempts to improve the situation lead to a number of models whose features are in qualitative conflict with experimental observations, in particular on the flow kinematics. The final blow was delivered by recent experiments [2] describing the effect of the fluid motion on the dynamic contact angle.

Figure 2. A sketch of the flow kinematics known from experiments showing that dynamic wetting is an interface disappearance-formation process..

At the same time, a look at the flow kinematics observed in experiments makes it clear that dynamic wetting is indeed an interface disappearance-formation process, namely the process by which the liquid-fluid interface transforms into the liquid-solid interface after passing through the three-phase-interaction region (the `contact line'). This idea has been developed in a series of papers [3-9]. The theory removes inconsistencies of early models and is in agreement with experiments reported by different authors over the last three decades. An example is given below.

Figure 3. Comparison of the theory with experiments by Fermigier & Jenffer [10].

### Free-surface cusps

Experiments [11] show that convergent flows near a gas-liquid interface can lead to the formation of two-dimensional free-surface cusps where (i) the free-surface curvature becomes singular, and - more importantly - (ii) the flow kinematics undergoes a qualitative change: the fluid particles initially belonging too the free surface start to travel through the cusp into the interior [12]. An important feature is that this flow corresponds to a range of capillary numbers between the state with a smooth free surface and the situation where ythe system looses stability and bubbles start to penetrate into the liquid.

Figure 4.  As a smooth gas-liquid interface (a) turns into a cusped one (b), a stagnation line initially present on the free surface disappears. Fluid particles belonging to the free surface start to travel through the cusp line into the interior..

Since the standard kinematic boundary condition prescribes the fluid particles to stay on the free surface at all times, the kinematics specific to flows with cusps cannot be adequately described in the framework of the conventional model in principle. Furthermore, the necessity to balance the capillary force acting on the cusp ensures that the above shortcoming of the standard formulation is not the only one.

Figure 5. The capillary force acting on the cusp (a) cannot be balanced by the bulk stress. The balancing force comes from the surface-tension-relaxation `tail' (b) where the interfaces, after passing through the cusp, gradually loose their surface tension.

Indeed, exact solutions obtained in the framework of the classical formulation show that there is either no cusp at finite capillary numbers [12] or, if a cusp is imposed, the flow has to be singular with the infinite rate of dissipation of energy [13]. However, the very kinematics of the flow described above suggests the way out. Indeed, it shows that one is dealing with an interface disappearance process by which the free surface, after being driven through the cusp into the interior, looses its surface properties (the surface tension), and the fluid particles initially belonging to it become `ordinary' particles of the bulk. This idea allows one to apply the theory of interface disappearance/formation developed in [2-9] without any ad hoc changes. The results are reported in [14].

### Topological transitions in the flow domains

Typical examples of fluid motion where the flow domain undergoes a topological transition are coalescence of drops and capillary breakup of liquid jets. These processes treated in the framework of conventional formulation lead to solutions which become unphysically singular as the topology change is approached (the issue first raised in [15]; see also [16, 17, 20] for reviews).

Fig 6. A sketch of coalescence/breakup process.

A standard remedy for the singularity is to use `microscopic cutoffs' to stop the solution on its way to the singularity which is inherent in the very formulation of the problem. Clearly, this `remedy' reduces the modelling to a semi-empirical level.

The problem has been addressed recently in the framework of the theory of capillary flows with forming interfaces [18-20]. Even on a qualitative level one can see that coalescence is a process of the interface disappearance where a part of the free surface becomes trapped between the coalescing bodies. The cusp formed when the two bodies are brought in contact will propagate outwards leaving behind a portion of the former free surface which will gradually (though very quickly in physical terms) loose its surface properties (such as the surface tension) and become part of the bulk. The cusp will disappear at a finite distance from the point of the initial contact where the standard model takes over.

As one would expect, the breakup turns out to be the process of interface formation. The standard model gives that, as the moment of the breakup is approached, the rate at which the fresh free surface area is created tends to infinity. As this rate becomes comparable with the inverse surface-tension-relaxation time, the free surface will no longer be in equilibrium and the full dynamics of the interface formation will unfold [20] to take the flow through the topological transition without singularities.

It should be noted here that the study of singularities, no matter how unphysical they might be, has become a popular activity (see, e.g. [16] for a review) and the work that is aimed at removing singularities by means of the proper modelling naturally meets opposition from those who invested their careers in this activity. As an instance of this, one can mention a recent “comment” by Eggers & Evans [21] on one of the works applying the theory of flows with forming interfaces. As an argument against the theory Eggers & Evans state their, in their words, ‘belief’ that the relaxation of the surface tension to its equilibrium value happens… at the speed of light! A rebuttal  of this argument [22] points out that relaxation of the surface tension (as indeed any relaxation) is a dissipative process whereas the speed of light has nothing to do with dissipation. Obviously, an attempt to estimate the rate of a dissipative process by considering a nondissipative process is a mistake of principle, betraying the unfamiliarity of its authors with basic thermodynamics. Regardless of this ‘commenting activity’, the theory has now become the subject of investigation and application in a number of institutions worldwide (e.g. Germany, France to mention the most recent examples). .

References

1. Goldshtik, M. A. 1990 Viscous-flow paradoxes. Annu. Rev. Fluid Mech.22, 441-472.
2. Blake, T. D., Bracke, M. and Shikhmurzaev, Y. D. 1999 Experimental evidence of nonlocal hydrodynamic influence on the dynamic contact angle. Phys. Fluids 11, 1995-2007.
3. Shikhmurzaev, Y. D. 1991a Hydrodynamics of wetting. Displacement of a gas by a liquid from a solid surface. Invited lecture at The 7th All-Union Congr. Theor. & Appl. Mech., Moscow, 15-21 August, 1991.
4. Shikhmurzaev, Y. D. 1991b Spreading flow of a viscous liquid over a solid surface. Sov. Phys. Dokl. 36, 749--752.
5. Shikhmurzaev, Y. D. 1993 The moving contact line on a smooth solid surface. Intl J. Multiphase Flow 19, 589-610.
6. Shikhmurzaev, Y. D. 1994 Mathematical modeling of wetting hydrodynamics. Fluid Dyn. Res. 13, 45-64.
7. Shikhmurzaev, Y. D. 1996 Dynamic contact angles in gas/liquid/solid systems and flow in vicinity of moving contact line. AIChE J. 42, 601-612.
8. Shikhmurzaev, Y. D. 1997 Spreading of drops on solid surfaces in a quasi-static regime. Phys Fluids 9, 266-275.
9. Shikhmurzaev, Y. D. 1997 Moving contact lines in liquid/liquid/solid systems. J. Fluid Mech. 334, 211-249.
10. Fermigier, M. & Jenffer, P. 1991 Experimental investigation of the dynamic contact angle in liquid-liquid systems. J. Colloid Interface Sci. 146, 226-241.
11. Joseph, D. D., Nelson, J., Renardy, M. and Renardy, Y. 1991 Two-dimensional cusped interfaces. J. Fluid Mech. 223, 383-409.
12. Jeong, J.-T. and Moffatt, H. K. 1992 Two-dimensional cusped interfaces. J. Fluid Mech. 241, 1-22.
13. Richardson, S. 1968 Two-dimensional bubbles in slow viscous flow. J. Fluid Mech. 33, 476-493.
14. Shikhmurzaev, Y. D. 1998 On cusped interfaces. J. Fluid Mech. 359, 313-328.
15. Hopper, R. W. 1992 Stokes flow of a cylinder and half-space driven by capillarity. J. Fluid Mech. 243, 171-181.
16. Eggers, J. 1997 Nonlinear dynamics and breakup of free-surface flows. Rev. Mod. Phys. 69, 865-929.
17. Lister, J. R., Brenner, M.P., Day, R. F., Hinch. E. J. and Stone, H. A. 1997 Singularities and similarity solutions in capillary breakup. In IUTAM Symp. on Non-linear Singularities in Deformation and Flow, (D. Durban and J. R. A. Pearson, eds), Kluwer Acad. Publ., 257-269.
18. Shikhmurzaev, Y. D. 2000 Coalescence and capillary breakup of liquid volumes, Phys. Fluids, 12, 2386-2396.
19. Shikhmurzaev, Y. D. 2005 Capillary breakup of liquid threads: A singularity-free solution. IMA Journal of Appl. Maths, 70, 880-907.
20. Shikhmurzaev, Y. D. 2005 Fluid dynamics with transitions in the topology of the flow domain: breakup of jets and rupture of films. Doklady Phys. 400, No. 1, 49-52.
21. Eggers, J. & Evans, R. 2004 Comment on “Dynamic wetting by liquids of different viscosity,” by T.D. Blake and Y.D. Shikhmurzaev. J. Colloid Interface Sci. 280, 537-538.
22. Shikhmurzaev, Y. D. & Blake, T. D. 2004 Response to the comment on [J. Colloid Interface Sci. 253 (2002) 196] by J. Eggers and R. Evans. J. Colloid & Interface Sci. 280, 539-541.