Navier stokes equation Definition and 16 Discussions

In physics, the Navier–Stokes equations () are a set of partial differential equations which describe the motion of viscous fluid substances, named after French engineer and physicist Claude-Louis Navier and Anglo-Irish physicist and mathematician George Gabriel Stokes.
The Navier–Stokes equations mathematically express conservation of momentum and conservation of mass for Newtonian fluids. They are sometimes accompanied by an equation of state relating pressure, temperature and density. 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. The difference between them and the closely related Euler equations is that Navier–Stokes equations take viscosity into account while the Euler equations model only inviscid flow. As a result, the Navier–Stokes are a parabolic equation and therefore have better analytic properties, at the expense of having less mathematical structure (e.g. they are never completely integrable).
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. The Navier–Stokes equations, in their full and simplified forms, help with the design of aircraft and cars, the study of blood flow, the design of power stations, the analysis of pollution, and many other things. Coupled with Maxwell's equations, they can be used to model and study magnetohydrodynamics.
The Navier–Stokes equations are also of great interest in a purely mathematical sense. Despite their wide range of practical uses, it has not yet been proven whether smooth solutions always exist in three dimensions – i.e. they are infinitely differentiable (or even just bounded) at all points in the domain. 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.

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  1. steve1763

    A Green's function for Stokes equation

    So I've just started learning about Greens functions and I think there is some confusion. We start with the Stokes equations in Cartesian coords for a point force. $$-\nabla \textbf{P} + \nu \nabla^2 \textbf{u} + \textbf{F}\delta(\textbf{x})=0$$ $$\nabla \cdot \textbf{u}=0$$ We can apply the...
  2. person123

    Inertial Force in Fluid Mechanics

    According to one explanation, the left hand acceleration terms of Navier Stokes equations are the called the inertial terms. If you were to balance forces on the fluid particle, they would have to be equal and opposite to the forces on the right hand side (pressure gradient, viscous, and body)...
  3. Where Does River Water Go? - Numberphile

    Where Does River Water Go? - Numberphile

    Tom Crawford on the mathematics of where river water goes when it hits the sea.
  4. nomadreid

    Requesting recommendations for specialized journal for....

    A physicist I know is looking to publish a paper on a special case of the Navier-Stokes equation; he submitted it to Physics Review Letters, who rejected it for two reasons: (1) it was too specialized, so not suitable for the broad readership that PRL targets, and (2) the physicist had made used...
  5. A

    Should the non-relativistic Navier Stokes Equations be modified?

    Choking mass flow seems to reflect the fact that fluid momentum density has a maximum value (in stationary conditions) equal to ##\rho_* c_*## where ##\rho_*## is the critical mass density and ##c_*## is the critical velocity, which is closely related to the speed of sound (see...
  6. K

    Solving the Navier-Stokes equation

    Hi all, My first post. I am not sure how does Chorin's Projection method for coupling pressure-velocity differ from the Issa's method of of Pressure Implicit with Splitting of Operators (PISO)? Franckly speaking both the methods look to solve the poisson equation for pressure and update a...
  7. W

    I Vector-wavelet Galerkin projection of Navier-Stokes equation

    Hi, I am having a little trouble understanding a minor step in a paper by [V. Zimin and F. Hussain][1]. They define a collection of divergence-free vector wavelets as $$\mathbf{v}_{N\nu n}(\mathbf{x}) = -\frac{9}{14}\rho^{1/2}_N...
  8. U

    I Derivation of non-dimensional Navier Stoke equation

    Take the first three terms of Navier Stoke equation: $$\rho \cdot \left ( v_{x}\cdot \frac{\partial \vec{v}}{\partial x} + v_{y}\cdot \frac{\partial \vec{v}}{\partial y} + v_{z}\cdot \frac{\partial \vec{v}}{\partial z}\right )$$ Define the length ##v## of the velocity vector field...
  9. M

    A How to solve (or close) the incompressible Navier-Stokes flow eqs for a compressible fluid?

    I did a lot of googling but could not find a satisfying answer to my question, hence a post here. Question: How to solve (or close) the isothermal incompressible Navier-Stokes equations for an isothermal compressible fluid? Situation: We have a compressible fluid, for example a gas. The flow...
  10. K

    What does the Navier-Stokes equation look like after time discretization?

    Hi, I know the general form of the Navier Stokes Equation as follows. I am following a software paper of "Gerris flow solver written by Prof. S.Popinet" [Link:] and he mentions after time discretization he ends...
  11. W

    I Spectral representation of an incompressible flow

    Hi PH. Let ##u_i(\mathbf{x},t)## be the velocity field in a periodic box of linear size ##2\pi##. The spectral representation of ##u_i(\mathbf{x},t)## is then $$u_i(\mathbf{x},t) = \sum_{\mathbf{k}\in\mathbb{Z}^3}\hat{u}_i(\mathbf{k},t)e^{\iota k_jx_j}$$ where ι denotes the usual imaginary...
  12. H

    Is this equation conservative or non-conservative?

    Homework Statement This is the Navier-Stokes equation for compressible flow. nj is the unit normal vector to the surface 'j', and ni is the unit normal vector in the 'i' direction. Is this equation written for a control volume or a material volume? Homework Equations The Attempt at a...
  13. F

    I Divergence of the Navier-Stokes Equation

    The Navier-Stokes equation is: (DUj/Dt) = v [(∂2Ui/∂xj∂xi) + (∂2Uj/∂xi∂xi)] – 1/ρ (∇p) where D/Dt is the material (substantial) derivative, v is the kinematic viscosity and ∇p is the modified pressure gradient (taking into account gravity and pressure). Note that the velocity field is...
  14. C

    Newton's law of viscosity in 3D, used to derive Navier-Stoke

    I'm trying to understand how the Navier-Stokes equations are derived and having trouble understanding how the strain rates are related to shear stresses in three dimensions, what a lot of texts refer to as the 'Stokes relations'...
  15. K

    Navier-Stokes equation for parallel flow

    Homework Statement [/B] Find an equation for the flow velocity of a river that is parallel to the bottom as a function of the perpendicular distance from the surface. Apply the boundary conditions given and solve, and find the velocity at the surface. Note that the coordinates are: x is the...
  16. G

    Navier Stokes Eqn for const. density and viscosity

    I was watching a lecture in which the professor derived the Navier Stokes Equations for const density and viscosity. He however skipped a step and directly went from one equation to another without giving any explanation. I have attached an image file in which the 2nd equation is derived from...