ADVERTISEMENT

What Can Mathematical Analysis Reveal About The Behavior Of A Fluid?

The study of flows or fluid mechanics began since antiquity for everyday applications such as irrigation in agriculture. Several innovative and complex irrigation systems have been discovered in ancient cities. The inclusion of mathematics has revolutionized the fluid mechanics owing to the works of Daniel Bernoulli, Jean Rond d’Alembert, and Leonhard Euler. The introduction by Henry Navier in 1820 of the concept of viscosity, allowed George Gabriel Stokes to introduce the Navier-Stokes equations which will mark the whole history of fluid mechanics.

The Navier-Stokes equations are nonlinear partial differential equations that describe the motion of gases and most liquids. Joseph Boussinesq proposes an alternative model, which also takes into account the temperature variations. Indeed, a fluid particle is driven by the flow, but also exchanges heat with the surrounding environment and undergoes thermal expansion which has an influence on its motion. For instance, a fluid particle heated at the base expands and rises under the effect of buoyancy. Arriving at the top and in contact with a cold source, the fluid particle cools, contracts and creates a return heat transfer. These phenomena create currents in the fluid called convection movements.

ADVERTISEMENT

The Boussinesq system is a system of partial differential equations that describes the coupling between fluid flow and temperature variations. For a given initial configuration of speed and temperature, the resolution of the system provides information on the flow, and also on the evolution of temperature at any point of the fluid.

Our work provides a mathematical analysis of this system of equations and in particular the behavior of its solutions in the long run. More precisely, we try to answer the following questions: Can we construct solutions to this system that are defined for all times? How does the flow behave when time goes to infinity? What are the conserved quantities? Is there any quantity that could become indefinitely large over time?

In our article, we expose several interesting conclusions. We were able to build suitable solutions for small initial data. But we also discovered a very interesting property of the Boussinesq system. In particular, we have shown that in an unbounded domain, if the initial temperature has non zero mean, the energy of the system can become indefinitely large (blow up) over time. In addition, we provide a quantified measure of the speed at which this phenomenon occurs. This counter-intuitive result raises many questions about the Boussinesq model and its limits and may help to understand it more deeply.

These findings are described in the article entitled A Short Proof of the Large Time Energy Growth for the Boussinesq System, published in the Journal of Nonlinear Science. This work was led by Charafeddine Mouzouni & Lorenzo Brandolese from Institut Camille Jordan.

ADVERTISEMENT

Comments

READ THIS NEXT

Catalysts In The Chemical Industry

The chemical industry influences all our lives daily, everything from the components in the screen you are reading this article […]

The Surface Roughness Of 433 Eros From The NEAR Laser Rangefinder

The Near-Earth (not hazardous to Earth) asteroid 433 Eros was explored by the NEAR-Shoemaker spacecraft in 2000-2001. This mission, built […]

Snake Eggs: How To Identify With Pictures

The eggs of snakes can be identified by a number of different characteristics including the appearance, hardness, and shape of […]

Concentric Crater Fill: How Martian Craters Filled With Ice And Why It’s Still There

Many mid-latitude impact craters on Mars are filled to variable degrees with a combination of ice, dust, and rocky debris. […]

How Do Coral Reef Fish Behave After Being Caught And Released From Fishing Gear?

What happens to coral reef fish after they’re caught by fishers and released back into the water? In a new […]

Intelligent Image-Based Cell Sorting And Beyond

Cells have been the building block of all organisms on Earth ever since they presumably emerged more than 3.5 billion […]

Corrosion Inhibition By Essential Oil Extracts: A Sustainable Method Of Controlling Carbon Steel Degradation

Published by Roland Loto Department of Mechanical Engineering, Covenant University, Ota, Ogun State, Nigeria These findings are described in the […]

Science Trends is a popular source of science news and education around the world. We cover everything from solar power cell technology to climate change to cancer research. We help hundreds of thousands of people every month learn about the world we live in and the latest scientific breakthroughs. Want to know more?