# What are the limitations of the Navier-Stokes equation?

Table of Contents

- 1 What are the limitations of the Navier-Stokes equation?
- 2 What is the physical significance of Navier-Stokes equation?
- 3 Are the Navier Stokes equations valid?
- 4 What is Navier-Stokes equation derived from?
- 5 Does Navier-Stokes apply to turbulent flow?
- 6 How can I derive the NSE of a given volume?
- 7 Is Euler’s equation for fluid flow too simplistic?

The Navier-Stokes equations can technically apply to problems involving all of those variables, both compressible and incompressible. The two most important limitations on the Navier-Stokes equations is that they only apply to (a) fluids that can adequately be modeled by a continuum and (b) Newtonian fluids.

**Why cant the Navier-Stokes equations be solved?**

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.

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.

**What are the assumptions of Navier-Stokes equation?**

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.

The NS equations are valid for Kn<0.01. For 0.010.1, they are not valid. At the ambient pressure of 1 atm – for instance, the mean free path of air molecules – is 68 nanometers.

**Is it possible to solve the Navier Stokes equation?**

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 equations are derived from the basic principles of continuity of mass, momentum, and energy. Sometimes it is necessary to consider a finite arbitrary volume, called a control volume, over which these principles can be applied. This finite volume is denoted by Ω and its bounding surface ∂Ω.

**What is Navier-Stokes derived from?**

The Navier-Stokes equations can be derived from the basic conservation and continuity equations applied to properties of fluids. The basic continuity equation is an equation which describes the change of an intensive property L. An intensive property is something which is independent of the amount of material you have.

Even though the basic equations of motion of fluid turbulence, the Navier-Stokes equations, are known for nearly two centuries, the problem of predicting the behaviour of turbulent flows, even only in a statistical sense, is still open to this day.

**What are the Navier-Stokes equations?**

The Navier-Stokes equations consists of a time-dependent continuity equation for conservation of mass , three time-dependent conservation of momentum equations and a time-dependent conservation of energy equation. There are four independent variables in the problem, the x, y, and z spatial coordinates of some domain, and the time t.

### How can I derive the NSE of a given volume?

The traditional approach is to derive teh NSE by applying Newton’s law to a \\fnite volume of uid.

**How do scientists and mathematicians solve fluid dynamics equations?**

Until the advent of scientific computing engineers, scientists and mathematicians could really only rely on very approximate solutions. In modern computational fluid dynamics (CFD) codes the equations are solved numerically, which would be prohibitively time-consuming if done by hand.

## Is Euler’s equation for fluid flow too simplistic?

While, this approach allowed Euler to find solutions for some idealised fluids, the equation is rather too simplistic to be of any use for most practical problems. A more realistic equation for fluid flow was derived by the French scientist Claude-Louis Navier and the Irish mathematician George Gabriel Stokes.