By Batyrkhan Alimzhanov
Overview:
Initially derived by physicists Claude-Louis Navier and George Gabriel Stokes, the Navier-Stokes equations have been established to model the motion of Newtonian, incompressible, isothermal fluids. For a newtonian fluid, it can be seen that shear stress has no effect on viscosity. In contrast, a non-newtonian fluid’s viscosity decreases exponentially in response to an increase or shear stress. For a fluid to be incompressible and isothermal would mean that it would not loose or gain heat, and that with added pressure, the amount of the liquid wouldn’t change. The Navier-Stokes equations belong to a discipline of physics called fluid dynamics. Consisting of aero and hydrodynamics, the field of fluid dynamics allows us to understand the movement of gases and liquids. The principles of fluid dynamics are the First Law of Thermodynamics, the law of conservation of energy, conservation of linear momentum, and conservation of mass.
What are the Navier-Stokes Equations? Will they help me gain immortality?
There are a grand total of 2 equations:
∇⋅v⃗= 0
&
ρ(Dv⃗/Dt) = −∇p + ∇⋅T + f⃗
No, these equations wouldn’t give you immortality but will give you the power to understand the behavior of many liquids. So you ask: what do all of these symbols mean? The first equation establishes the idea that in the continuity equation, mass is preserved. The continuity equation simply shows the change in the total amount of property as the amount that flows through, and is lost or gained through sources, or sinks inside the boundary. When applying the continuity equation to the one of conservation of mass, you arrive at the first equation. The second equation is modeled after Newton’s Second Law, which states that F = ma, where F is force, m is mass, and a is acceleration. In the second equation, the mass is replaced by density, acceleration (convection term) is represented by the derivative of the velocity vector field, and F (diffusion term) is the sum of the forces of pressure, friction, and other external forces. When the convection term is greater than the diffusion term, there exists turbulent flow, but when the diffusion term is greater than the convection term, the flow would be smooth.
What were the attempts to solve them?
The first attempt at solving the Navier Stokes equations was made by Hiroshi Fujita & Tosio Kato, in their 1964 paper. In their paper, they assumed that if velocity is sufficiently slow, there exists smooth solutions. Thus, they established that if the convection term is of the lowest possible value, it can be simplified in the the calculations, making the calculations much easier. Later, the second major attempt was made by M.Cannone, Y.Meyer, and F.Phlanchon in their 1994 paper. In this case, the mathematicians showed that if velocity oscillates rapidly by a large magnitude, there would exist smooth solutions. By doing this, the mathematicians proved that turbulence truly exists only at a very small scale. Finally, Olga Ladyzhenskaya has used inequalities in order to solve the Navier Stokes equation. However, she was only able to do so in 2 dimensions. Due to this research, there is now grater understanding of the Navier Stokes equations among the scientific community and better methods for solving them. However, these equations are yet to be solved for 3 dimensional spaces.
Why try to solve these equations?
The Navier Stokes equations are fundamental to the functioning of many aspects of our life. Think of a flowing river, a hurricane, and even the plane. All of them involve the moving of or moving through liquids. The equations can help to model how exactly air would move around an airplane’s wings, how a hurricane would develop, more specifically, where it would go. If we would be able to solve the Navier Stokes equations, it would mean that we would be able to truly quantify the world, to define nearly all interactions mathematically. This equation would in turn show us how everything would develop, how each fluid in the universe would interact with each other. Although the Navier Stokes equations are already being used for 2D spaces, the scientific community has not fundamentally ‘solved’ them yet. It is in a sense like predicting the future. If all fluids can be simplified to a set of equations, we would understand what will happen, how we can act, how we could feel. Thus, the scientific community has offered a $1 million dollar prize for anyone who can elegantly prove and solve these equations. Good luck!!
Works Cited:
Gibiansky, Andrew. Fluid Dynamics: The Navier-Stokes Equations – Andrew Gibiansky, 7 May
2011, https://andrew.gibiansky.com/blog/physics/fluid-dynamics-the-navier-stokes-equations/.
Li, Jun. “Fluid Mechanics Based on Continuum Assumption.” SpringerLink, Springer
International Publishing, 28 Aug. 2019,
https://link.springer.com/chapter/10.1007/978-3-030-26466-6_1.
Cannone, Marco, et al. Solutions Auto-Similaires Des Équations De Navier-Stokes, 18 Jan.
1994, http://www.numdam.org/item/SEDP_1993-1994____A8_0.pdf.
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