# New techniques for studying turbulent fluid mixtures

|One of the great achievements of mathematics is to offer models that predict the evolution of physical systems based on basic principles. Since **Isaac Newton's time** , the so-called differential equations have been one of the most efficient models. Advanced methods of partial derivative equation theory, well understood by researchers, are commonly used to study these equations. But when unstable situations appear, they fail. It is therefore necessary to invent radically different approaches, the development of which is a challenge for current research.

Instabilities appear constantly both in mathematical theory and in nature. An example is the evolution of two different fluids that mix, such as water and wine, or water and oil. This last situation – especially when it occurs in a porous medium – is of great practical importance. In 1930 the engineer Morris Muskat was looking for a way to extract oil by injecting water under pressure into the earth. To model the evolution of fluids, he devised a system of three coupled partial derivative equations, which correspond to the law of conservation of mass, the conservation of the volume occupied by fluids (incompressible) and Darcy's law (constitutive). The latter postulates that, in a porous medium, the flow velocity is proportional to the sum of the forces acting on it; in this case, pressure and gravity.

What we know as the Muskat problem involves a special difficulty in the case in which the densest fluid lies on top of the other, because the force of gravity introduces an instability in the system that causes that the classical methods cannot be used. Apply

Solving these equations, which is what we now know as the **Muskat problem** , involves a special difficulty in the case in which the densest fluid lies on top of the other, due to the force of gravity introducing a instability (that of Rayleigh-Taylor) in the system, which means that the classical methods cannot be applied.

**Solutions to the problem** have recently been found. These solutions predict the formation of a zone where the fluids mix developing a strange, chaotic and irregular pattern called "viscous fingers", experimentally observed . Somehow, the solutions find a certain order in the turbulence, predicting the size and shape of this mixing zone.

To arrive at these solutions, the answer to a famous problem in differential geometry, about deformations of the sphere, has been key. It is easy to reduce a sphere of radius one to another of radius 0.25 simply by shrinking it. It also seems easy to fit a sphere of radius one into one of 0.25 without shrinking it if they allow us to wrinkle it a lot. But what if they don't allow us to shrink or wrinkle it a lot? Can we still fit one into the other? In the 50s of the last century, the mathematician **John Nash** demonstrated that yes, you can deform the sphere of radius one to put it within a sphere of radius as small as you want without appearing very pronounced corners or wrinkles and without shrinking it, through infinite carefully selected folds.

In 2008, two young mathematicians, Camillo De Lellis and László Székelyhidi Jr., discovered that the theory of convex integration could be adapted to fluid mechanics.

This impossible result constituted the embryo of a new mathematical concept, convex integration, crucial since then in differential geometry. In 2008, two young mathematicians, **Camillo De Lellis and László Székelyhidi Jr.,** **discovered** that the theory of convex integration could be adapted to fluid mechanics. With it, they obtained new solutions for Euler's equations (the mother equation of all fluid equations) with surprising behavior and which were dubbed "wild solutions", due to their irregularity. Its construction is a true work of mathematical goldsmithing and has given rise to tools that allow tackling other problems, such as that of Muskat's unstable regime.

From a physical point of view, the key is the duality between the macroscopic and microscopic description of the fluid. Macroscopically, the evolution is that of a classic movement: the heavier fluid is gradually being exchanged with the lighter. Microscopically the evolution is highly irregular and describes a pattern similar to that present in the elaboration of modern metamaterials. Returning to Nash's ideas, the macroscopic interpretation of the fluid corresponds to the natural shrinkage of the sphere that does not preserve distances, and the microscopic version to Nash folds.

In turbulent regimes the macroscopic behavior of solutions is described by the *relaxation* –that is, less demand– of the non-linear relations of the problem. In the macroscopic version of the Muskat problem, the relation of flow equal to mass for velocity is replaced by flow "not too different" of mass for velocity. Rigorously this relationship is expressed geometrically by imposing that the flow, velocity and density belong to a set of five dimensions known as the relaxation of the Muskat problem.

After a purely geometric analysis, the difficulties continue, because there is no theory to solve the new relaxed equations

Therefore, the first objective is to understand the geometry of this space. After this purely geometric analysis the difficulties continue, because there is no theory to solve the new *relaxed equations* . However, in **a work** that will be published in the journal *Inventiones Mathematicae* , a mathematically innovative strategy has been postulated to address and solve these relaxed equations.

In this way, the program we are developing proposes a systematic way of addressing physically unstable problems. , which until now were inaccessible to theory. Of course, these works generate new fascinating questions: Should we modify the traditional constitutive relationships in turbulent regimes? Can we arrive at new concepts in theoretical physics from these wild solutions? … As always happens in science, an answer is the door for a thousand new questions

**Ángel Castro***and* **Diego Córdoba***are* **ICMAT***researchers,* **Daniel Faraco***is a professor at the* **Autonomous University of Madrid***and a member of the* **ICMAT**

*Editing and coordination:* **Ágata A. Timón G-Longoria***(ICMAT)*

**Café y Teoremas***es a section dedicated to mathematics and the environment in which it is created, coordinated by the Institute of Mathematical Sciences (ICMAT), in which researchers and members of the center describe the latest advances in this discipline, share meeting points between mathematics and other social and cultural expressions and remember those who marked its development and knew how to transform coffee into theorems. The name evokes the definition of the Hungarian mathematician Alfred Rényi: "A mathematician is a machine that transforms coffee into theorems."*

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