Wednesday, February 01, 2006

Researcher Spotlight: Professor Grigory. I. Barenblatt

By Xanthippi Markenscoff

If there is anybody alive who embodies what is mechanics, I would say that he is G.I . Barenblatt. A fluid mechanician just told me that Barenblatt is to mechanics, as Mozart is to music! Natural! Indeed, encompassing solid and fluid mechanics, and their interaction, over a period of fifty years, G.I. Barenblatt has unraveled the structure of physical phenomena of mechanics with unparalleled insight, by probing into the effect of scales. Indeed, he understood the physical processes taking place at each scale, and developed the mathematical concepts and tools to quantify them. He saw how the effect of scales manifests itself in physical phenomena almost in a unified way, and he embarked to harness them! I will try to outline this scaling issue briefly in some of the areas that he developed, but this is by no means comprehensive of his contributions. His contributions are enormous in a wide range of fields, and had profound influence.

In solid mechanics, the Barenblatt crack assumes a finite material cohesive force acting in a small zone at the tip of the crack (``cohesive zone'') which leads to a dissipation of energy so that the faces close smoothly. Barenblatt provided a beautiful mathematical solution in addition to the modeling, and introduced one of the basic characteristics of fracture toughness, the cohesion modulus. The concept of scaling led to similarity laws for fatigue cracks and multiple fracturing, and mathematical models of damage accumulation. The model of the flow in a fully saturated heterogeneous medium with several distinct spatial scales is the Barenblatt double –diffusion model. To model water injected in a rock containing oil (-in order to extract oil-), Barenblatt modeled elegantly a phenomenon called non-equilibrium filtration by introducing the concept of a quasi-steady stabilized zone around the water-oil displacement front, which determines the structure of the transition between injected water and oil. The stabilized zone disappears altogether if the water flow velocity is too high, reversing results of previous models.

Barenblatt not only provided elegant solutions to the aforementioned problems, but, in the context of this work he introduced (with Zeldovich) the new mathematical area of intermediate asymptotics. This arose in the problem of nonlinear filtration, which has a self-similar solution of the second kind, (in which the similarity exponent is found by solving a nonlinear eigenvalue problem rather than by dimensional analysis), and showed that indeed the self-similar solution is an asymptotic representation of the solution of the non-self similar problem.. Indeed, self-similar solutions are always ``intermediate asymptotics'' to the solution of more general problems valid for times and distances that are large enough that the fine details of the boundary / initial conditions disappear, but small enough that the system is not in equilibrium. If the fine details do not disappear, then we have similarity of the second kind, in which the exponents of the power-type scaling depend on the fine details of the pre-self similar state. His books

Scaling, Self-Similarity, and Intermediate Asymptotics, Cambridge University Press, 1996, 408 pages.

Scaling, Cambridge University Press, 2003, 171 pages.

are of monumental contribution, in their uniqueness of the topics spanned that exemplify and unify the intermediate asymptotics and renormalization, including flow through porous media, traveling and shock waves, combustion, turbulence and geophysical fluid dynamics.

In his work in fluid mechanics, combustion and geophysical fluid dynamics (interaction of gravity waves, density inhomogeneities and stratified flow), the scaling concepts were further extended to understand the structure of turbulent boundary layers, the scaling of turbulence, and phenomena such as hurricanes. Indeed, the scaling techniques when applied to droplets sandwiched between air and water showed that droplets kicked by the sea waves lubricate the winds and reduce turbulence, hence enhancing the wind speed (up to an order of magnitude). By reducing the size of the droplets (possibly by some chemical additive) the speed of the winds can diminish. This recent work, jointly with Chorin and Prostokishin, unifies theory and application in an Archimedean scope, and carries the contributions of Barenblatt to the 21st century where multiscale analysis presents both theoretical physico-mathematical challenges, as well as challenges of applicability to phenomena of global impact.

Grigory Isaakovich Barenblatt was born in Moscow in 1927 and graduated from the elite Moscow State University, Faculty of Mathematics and Mechanics, in 1950, where he subsequently obtained his Ph.D. and D.Sc. degrees under the guidance of the legendary giant of Soviet science A.N. Kolmogorov. He became Professor in 1962 and remained until 1992 when he was offered the G.I Taylor Chair in Fluid Mechanics at Cambridge University, as its inaugural holder. After his retirement in 1994 as Emeritus G.I. Taylor Professor, he was offered several Visiting Professorships at US Universities, including the Timoshenko Chair Professor at Stanford. Since 1997 he is Professor of Mathematics (in Residence) at UC Berkeley. He is a Foreign Member of the National Academies of Sciences and Engineering of the US, a Foreign Member of the Royal Society of London and has received many prizes, awards, and honorary degrees.

I consider it one of the great fortunes of my life that in 1990 I gave a seminar at the Institute of Problems of Mechanics in Moscow, and that Barenblatt walked in. I did not know who he was but he dominated and electrified the room instantaneously. I happened to be talking on the Sternberg-Koiter wedge paradox (concentrated moment on the apex), a problem that when the wedge becomes a re-entrant corner the solution ``remembers'' the details of the distribution of the moment, an intermediate asymptotics paradigm treated in Barenblatt’s book. In the Russian tradition, the next day he brought me flowers. Knowing Barenblatt gives one the sense of the absolute, in the quest for truth and in the quest for beauty in the world.


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