How are the Navier-Stokes equations derived?

They arise from the application of Newton’s second law in combination with a fluid stress (due to viscosity) and a pressure term. The Navier-Stokes equations can be derived from the basic conservation and continuity equations applied to properties of fluids.

Why is Navier-Stokes unsolvable?

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 assumption is used to derive the Navier-Stokes equations from the general equations of motion?

The Navier–Stokes equations are based on the assumption that the fluid, at the scale of interest, is a continuum, in other words is not made up of discrete particles but rather a continuous substance.

Does Navier-Stokes assume incompressible?

Incompressible flow The incompressible 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.

Is Navier-Stokes Eulerian or Lagrangian?

The latter perspective is indeed the Lagrangian perspective which is widely used in solid mechanics like elasticity. Thus, the Navier-Stokes equations should be and indeed they are always used in the Eulerian perspective.

What is convective term in Navier-Stokes equation?

The terms on the left hand side of the momentum equations are called the convection terms of the equations. The terms on the right hand side of the momentum equations that are multiplied by the inverse Reynolds number are called the diffusion terms.

Will Navier-Stokes ever be solved?

Even more basic properties of the solutions to Navier–Stokes have never been proven. For the three-dimensional system of equations, and given some initial conditions, mathematicians have not yet proved that smooth solutions always exist. This is called the Navier–Stokes existence and smoothness problem.

Who is Diane Adler mathematician?

Diane Adler was a brilliant mathematician, a prodigy whose talent had put her on the fast track to scholarly fame and immortality. She is on the verge of solving one of the most difficult and significant of all mathematical problems.

Is Navier-Stokes equation Lagrangian?

Most researches on fluid dynamics are mostly dedicated to obtain the solutions of Navier-Stokes equation which governs fluid flow with particular boundary conditions and approximations. We propose an alternative approach to deal with fluid dynamics using the lagrangian.

When was the derivation of the Navier Stokes equations?

Derivation of the Navier-Stokes Equations and Solutions. In this chapter, we will derive the equations governing 2-D, unsteady, compressible viscous flows. These equations (and their 3-D form) are called the Navier-Stokes equations. They were developed by Navier in 1831, and more rigorously be Stokes in 1845.

How is shear stress directed in the Navier Stokes equation?

The fluid element is slowed down on the right side, so to speak, and on the left side it is carried along by the flow, i.e. accelerated. The shear stress at the point y is thus directed in negative x direction and at the point y+dy in positive direction.

Which is the integral continuity equation in Navier Stokes equation?

This is expressed by the following integral continuity equation: where u is the flow velocity of the fluid, n is the outward-pointing unit normal vector, and s represents the sources and sinks in the flow, taking the sinks as positive.

How are the Stokes equations related to Newton’s second law?

This appears to simply be an expression of Newton’s second law ( F = ma) in terms of body forces instead of point forces. Each term in any case of the Navier–Stokes equations is a body force. A shorter though less rigorous way to arrive at this result would be the application of the chain rule to acceleration: