Is the Navier Stokes equation proven?
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In particular, solutions of the Navier–Stokes equations often include turbulence, which remains one of the greatest unsolved problems in physics, despite its immense importance in science and engineering. Even more basic (and seemingly intuitive) properties of the solutions to Navier–Stokes have never been proven.
The Navier-Stokes equation is difficult to solve because it is nonlinear. This word is thrown around quite a bit, but here it means something specific. You can build up a complicated solution to a linear equation by adding up many simple solutions.
What is Navier-Stokes equation used for?
The Navier–Stokes equations are useful because they describe the physics of many phenomena of scientific and engineering interest. They may be used to model the weather, ocean currents, water flow in a pipe and air flow around a wing.
Is Navier-Stokes equation valid for compressible flow?
Compressible flow The compressible momentum Navier–Stokes equation results from the following assumptions on the Cauchy stress tensor: the stress is Galilean invariant: it does not depend directly on the flow velocity, but only on spatial derivatives of the flow velocity.
Detailed Solution The conservation of mass also expressed as Continuity Equation. N-S equation also represents the momentum conservation equation.
Starts here14:01Applying the Navier-Stokes Equations, part 1 – Lecture 4.6YouTube
What is the Navier-Stokes equation?
The Navier-Stokes equation is named after Claude-Louis Navier and George Gabriel Stokes. This equation provides a mathematical model of the motion of a fluid. It is an important equation in the study of fluid dynamics, and it uses many core aspects to vector calculus.
What is the Navier-Stokes existence and smoothness problem?
This is called the Navier–Stokes existence and smoothness problem. The Clay Mathematics Institute has called this one of the seven most important open problems in mathematics and has offered a US$ 1 million prize for a solution or a counterexample.
Is dynamic viscosity constant in incompressible flow?
Dynamic viscosity μ need not be constant – in incompressible flows it can depend on density and on pressure. Any equation that makes explicit one of these transport coefficient in the conservative variables is called an equation of state. The divergence of the deviatoric stress is given by:
What are the basic assumptions of viscous flow?
They arise from applying Isaac Newton’s second law to fluid motion, together with the assumption that the stress in the fluid is the sum of a diffusing viscous term (proportional to the gradient of velocity) and a pressure term—hence describing viscous flow.