What is Navier-stokes: Definition and 84 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. casparov

    A Navier-Stokes solutions: Beltrami flow

    There are some known solutions for 3D Navier-Stokes such as Beltrami flow. In the literature these Beltrami flow solutions are said to not take into account viscosity, however when I read the information on Beltrami flow, they do seem to involve (kinematic) viscosity: From incompressible...
  2. JungleKing

    Resources for learning more about Navier-Stokes Equations

    Summary: I'm looking to learn more about the Navier Stokes Equations and Laplace's Tidal Equations. Do you know of any books/resources/ problems I can go over to learn how they work. I've studied physics and math in school and I'm looking to learn more about fluid mechanics and Laplace's Tidal...
  3. J

    A Why do the Navier-Stokes equations give us non-existent results for 3D flow?

    Navier-Stokes equations for 3D flow gives us wrong/non existent results, results that don't exist in nature. Does that mean equations that describe flow of fluids in a wrong way or how we can explain this situation? Because math is allways 100% correct, 2+2 is always 4, math is apsolute TRUTH...
  4. Hari Seldon

    A Deriving Navier-Stokes: Lagrangian & Hamiltonian Methods

    Is that possible to derive the Navier-Stokes equations with Lagrangian and Hamiltonian methods? If yes, how? and if it is not possible, why?
  5. casualguitar

    Modelling of two phase flow in packed bed using conservation equations

    Previously, I have seen the derivation of the energy conservation equations for simulation of single phase flow in a porous media (a packed bed). These are the energy equations for the solid and fluid respectively: I understand the derivation, however, these equations will only work when the...
  6. M

    I Connection between Set Theory and Navier-Stokes equations?

    Hi, I saw this video by numberphile, and near the end they mention how at the point of a right angle the equation shows infinite velocity for fluids. I'm wondering if this isn't perhaps related to Cantor's solution to Zeno's Paradox of distance (there's always a midpoint). Because I feel like at...
  7. taktoa

    A Navier-Stokes with spatially varying viscosity

    Does anyone know of a differential equation, similar to Navier-Stokes, that encompasses incompressible fluid flow with spatially varying viscosity? Viscosity is treated as a global constant in NS; I've found some papers online that address NS with viscosity as a function of velocity, but I can't...
  8. person123

    I Viscosity Term in Navier-Stokes Equation

    I'm a bit confused about the viscosity term in the Navier-Stokes equation; my intuitive understanding of what it would is different from what it actually is. I took the z component of the stress on an infinitesimal cube, but the same approach should apply in the x and y direction. I think my...
  9. S

    A Stressing Over Stress Tensor Symmetry in Navier-Stokes

    How do we know that the stress tensor must be symmetric in the Navier-Stokes equation? Here are some papers that discuss this issue beyond the usual derivations: Behavior of a Vorticity Influenced Asymmetric Stress Tensor In Fluid Flow http://www.dtic.mil/dtic/tr/fulltext/u2/a181244.pdf...
  10. snoopies622

    I Outer product of flow velocities in Navier-Stokes equation

    Reading the Wikipedia entry about the Navier–Stokes equation, and I don't understand this second term, the one with the outer product of the flow velocities. I mean, I understand the literal mathematical meaning, but I don't have an intuitive idea of what it physically represents. When I make...
  11. P

    Exploring the Fascinating Properties of Non-Newtonian Liquids

    Hello to the community, I'm 12th grade student in an IB school. I love physics and I am doing a research project on non-Newtonian liquids, and there are concepts which are so complex to understand so I was hoping that some of you could help :)
  12. 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...
  13. 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...
  14. tom.stoer

    Hydrostatic equilibrium and Navier-Stokes equations

    Is it possible to derive the condition for hydrostatic equilibrium or the Navier-Stokes equation for a self-gravitating fluid - e.g. for water on a planet with non-homogeneous density - based on a variational principle? (the planet itself is assumed to be a fixed hard core not subject to the...
  15. W

    Why do we assume rho*gz=0 in the Navier-Stokes equations?

    Homework Statement Homework Equations Navier-Stokes equations of motion The Attempt at a Solution I did everything well but, my question is, why we assume last term rho*gz=0? in the N-S equation? Also why do we use Navier Stokes equations in terms of velocity gradients for Newtonian...
  16. M

    A How to Solve Isothermal Incompressible Navier-Stokes for 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...
  17. 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:http://citeseerx.ist.psu.edu/viewdoc/download?doi=] and he mentions after time discretization he ends...
  18. M

    Bernoulli Equation and Navier-Stokes

    Hi PF! I was reading about Bernoulli's equation for steady, inviscid, incompressible flow. Now it's my understanding this equation is derived from the Navier-Stokes (momentum balance); then these two equations are identical regarding information offered. However, while thinking about...
  19. F

    I Integral form of Navier-Stokes Equation

    The Navier-Stokes equation may be written as: If we have a fixed volume (a so-called control volume) then the integral of throughout V yields, with the help of Gauss' theorem: (from 'Turbulence' by Davidson). The definition of Gauss' theorem: Could someone show me how to go from the...
  20. 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...
  21. G

    Weak form of Navier-Stokes and heat transfer coupling(COMSOL)

    I am asked to simulate a 2-D coupled problem in COMSOL(Navier stokes with Heat transfer) of a simple room. I'm not sure if COMSOL already has preexisting physics for navier stokes and heat tranfer that I could use directly but I am provided with two differential equations and boundary...
  22. M

    Navier-Stokes Problem: Solving for Pressure Gradient in a 2D Rectangular Cavity

    Hi PF! Assume we have a rectangular cavity (2D) filled with a liquid of dimensions ##L \times H## and that the top plate of the cavity moves with some velocity ##V_0##. Also assume ##L \gg H##. I'll also assume ##L \gg H## implies flow is roughly 1-dimensional, and thus a pressure gradient...
  23. C

    [Fluid mechanics] Navier-Stokes and Hagen-Poiseuille

    Can the Hagen-Poiseuille equation be used for a vertical flow in the water tab or any flow that has circular cross-sections with varying diameters? If not, how can the Navier-Stokes equations or any other equations be used to describe a viscous incompressible free-falling vertical jet?
  24. Domenico94

    Exploring the Role of PDEs in Cancer Modeling: A Comprehensive Overview

    Which are the most frequently used PDEs in cancer modelling? Are navier-stokes' equations and fluidodynamics equations used there?
  25. K

    Validity of Navier-Stokes at Extreme Mach Nos. (M = 100,000)

    So, speaking theoretically, if you could accelerate a fluid to extreme Mach number at sea level, then at what point does the Navier Stokes number lose its validity? What equations would you then use to model this potentially extreme momentum? I presume based on the fact that Newton's Law's are...
  26. 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...
  27. A

    Chorin Artificial Compressibility Equations

    Hi! I have the following problem: pt + (c2u)x + (c2v)y = 0 ut + (u2+p)x + (uv)y = α(uxx+uyy) vt + (uv)x + (v2+p)y = α(vxx+vyy) It is a formulation of the incompressible Navier-Stokes equations. I would like to know an exact solution. Can anyone help me? Thanks
  28. P

    Navier-Stokes Equations for a Compressible Fluid

    Hello, I don't know if this question belonged here or in General Physics, so I apologize if I made a mistake. My question is simple, what are the Navier-Stokes Equations for a Compressible Fluid? I don't mean from a conceptual point of view, what I mean are the mathematical equations themselves...
  29. U

    How Do Velocity and Pressure Relate in Linear Sound Wave Equations?

    Taken from my lecturer's notes, how did they make the jump from 8.5 to 8.6 and 8.7? Even after differentiating (8.5) with time I get \rho_0 \frac{\partial^2 \vec u'}{\partial t^2} + \nabla \frac{\partial p '}{\partial t} = 0 \frac{\partial^2 p'}{\partial t^2} + \rho_0 c^2 \nabla \cdot...
  30. U

    Pressure gradient term in Navier-Stokes

    Hi, I've been thinking about the Navier-Stokes equations and trying to build skill in implementing it in various situations. In a particular situation, if I have a fluid flowing down an inclined surface such that it forms a film of finite height which is smaller than the length of flow, there...
  31. C

    Divergence Operator on the Incompressible N-S Equation

    Hello All, If I apply the Divergence Operator on the incompressible Navier-Stokes equation, I get this equation: $$\nabla ^2P = -\rho \nabla \cdot \left [ V \cdot \nabla V \right ]$$ In 2D cartesian coordinates (x and y), I am supposed to get: $$\nabla ^2P = -\rho \left[ \left( \frac...
  32. Feodalherren

    Newtonian fluid mechanics: Navier-Stokes equation

    Homework Statement Homework Equations Navier-Stokes The Attempt at a Solution Not really trying to solve a problem, trying to understand what is going on in my textbook. So look at the stuff in red first. I see where all that is coming from, it's clear to me. However, the stuff in green...
  33. Y

    Navier-Stokes and why DP/Dz is constant

    So in trying to solve for Poiseuille's Law using the continuty eq and Navier-Stokes, the differential EQ becomes DP/dz = some function of R only. The professor says that because DP/dz is ONLY a function of Z, and the left side is ONLY a function of R, the only way they can be equal is if both...
  34. U

    Navier Stokes Equation - Flow of waves

    Homework Statement [/B] (a) Show that for an incompressible flow the velocity potential satisfies ##\nabla^2 \phi = 0##. Show further the relation for the potential to be ## \frac{\partial \phi}{\partial t} + \frac{\nabla \phi \dot \nabla \phi}{2} + \frac{p}{\rho} + gz = const.## (b)Write out...
  35. U

    How Does Gravity Affect Convection Between Two Plates?

    Homework Statement From my lecture notes, here are the equations for convection between two plates. I have derived equations 9.6, 9.7 and 9.8. But for 9.4 there's a problem when gravity becomes involved. Homework Equations Navier stokes: ## \rho \frac{D \vec u}{D t} = -\nabla p + \mu...
  36. H

    Laplacian term in Navier-Stokes equation

    I am trying to derive part of the navier-stokes equations. Consider the following link: http://www.gps.caltech.edu/~cdp/Desktop/Navier-Stokes%20Eqn.pdf Equation 1, without the lambda term, is given in vector form in Equation 3 as \eta\nabla^2\mathbf{u}. However, when I try to get this from...
  37. A

    Is stress tensor symmetric in Navier-Stokes Equation?

    Hello, In CFD computation of the Navier-Stokes Equation, is stress tensor assumed to be symmetric? We know that in NS equation only linear momentum is considered, and the general form of NS equation does not assume that stress tensor is symmetric. Physically, if the tensor is asymmetric then...
  38. Z

    Steady-state incompressible Navier-Stokes discretization

    Hi, I would like to solve the steady-state incompressible Navier-Stokes equations by a spectral method. When I saw the classic primitive-variable finite element discretization of the time-dependent incompressible N-S, it turned out that the coefficient matrix of the derivatives of the unknowns...
  39. M

    Navier-stokes flow around a sphere

    hi pf! basically, i am wondering how to find the velocity profile of slow flow around a sphere in terms of a stream function ##\psi = f(r,\theta)## where we are in spherical coordinates and ##\theta## is the angle with the ##z##-axis. (i think this is a classical problem). i understand the...
  40. M

    Navier-stokes derivation question

    hey pf! so i have a small question when deriving the navier-stokes equations from Newton's 2nd law. specifically, Newton states that $$\Sigma \vec{F} = m \vec{a} = m \frac{d \vec{v}}{dt}$$ when setting a control volume of fluid and dealing with the time rate-of-change of momentum we write...
  41. G

    An exact solution for the Navier-Stokes?

    source: http://ru-facts.com/news/view/30934.html I understand the source means to say Mujtarbay Otelbayev has found a solution to Navier-Stokes equations. The only reference I've found is the article itself (in Russian), so I don't understand a word...
  42. M

    How Does the Boundary Layer Simplify the Navier-Stokes Equations?

    hey pf! so i have a question concerning navier-stokes equations in a boundary layer, which, as a refresher, is \frac {D \vec{V}}{Dt} = - \nabla P + \nu \nabla^2 \vec{V} where we know the x-component of \nabla^2 \vec{V} may be re-wrote as \frac{\partial^2 u}{\partial x^2}+\frac{\partial^2...
  43. D

    Why this solution to the 3D periodic Navier-Stokes?

    Who can provide a physical understanding to this solution to the 3D periodic Navier-Stokes equation: http://purvanced.wordpress.com?
  44. Q_Goest

    Navier-Stokes from Quantum Mechanics?

    In Victor Stenger's book, "Quantum Gods" he states: Do you agree with the second part of this (part that is in bold)? Why or why not? If yes, how would you suggest deriving those equations? If not, what makes the NS equations underivable. Edit: If you have references that back up your...
  45. W

    Pressure variation in Navier-Stokes Equation

    Hello everyone, I have a concern regarding the conservation of momentum for an incompressible Newtonian fluid with constant viscosity. Say you have a volume of fluid sliding down an inclined plane with a velocity Vx with the perpendicular axis facing upward in the y-direction. When you try...
  46. L

    Landau-Lifshitz Navier-Stokes equation on which page of the textbook?

    I see some papers mentioning the fluctuating hydrodynamics, which originates from the Landau-Lifshitz Navier-Stokes (LLNS) equation. I know that the LLNS equation has added a thermal fluctuation term to the N-S equation, but I still couldn't find it on the Landau-Lifshitz Fluid Mechanics book...
  47. C

    About Turbulence and Navier-Stokes

    I was discussing this with my friend the other night. I mentioned that proving existence of smooth solutions for the Navier-Stokes equations would win a $1,000,000 prize from the Clay Mathematics Institute, and also that turbulence is considered one of the unsolved problems of physics; as a...
  48. G

    Pompeiu Problem: Solved & Related to Navier-Stokes?

    Hi, http://en.wikipedia.org/wiki/Pompeiu_problem Can someone rephrase the problem so I better understand its meaning? And has it been solved? Solution to Pompeiu Problem http://arxiv.org/abs/1304.2297 Thanks PS. does this problem have anything to do with Navier-Stokes?
  49. H

    Understanding the Navier-Stokes Equations: Conservation of Momentum and More

    Hi, Stupid question: the Navier-Stokes equations, do they only consist of the equations for a statement of the conservation of momentum or do they also include the equations for conservation of mass (continuity equation), conservation of energy and an equation of state? Thanks.
  50. D

    Solution of Navier-Stokes eq for a single particle?

    Solution of Navier-Stokes eq for a single particle? Hi! I'm reading this paper on fluid dynamics: http://jcp.aip.org/resource/1/jcpsa6/v50/i11/p4831_s1 Its equation (13) is the velocity distribution around a single bead of radius a subjecting to force fi in solution. (the subscript i is...