What is Navier stokes: Definition and 63 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. MasterOgon

    I Navier-Stokes equation in a triangular coordinate system

    The Navier-Stokes equation is solved in a vector grid in a Cartesian coordinate system. That is, rectangular. But does a rectangular mesh relate to what happens in a gas or liquid, and is it better to use a triangular mesh? Undoubtedly, it is incredibly difficult to take into account all the...
  2. 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...
  3. F

    MHB Divergence of the Navier Stokes equation

    If not, can someone walk me through the steps to get to the results that my professor got? Thank you.
  4. M

    Deriving the 1-D Linear Convection Equation

    With the assumptions of Inviscid flow, no pressure gradient and no body force terms in 1-D Navier Stokes becomes 1-D nonlinear convection equation; And if we assume velocity of wave propagation is constant value c, equation becomes 1-D linear convection equation; This is online derivation and...
  5. M

    Non-dimensional Navier Stokes

    Hi PF! I'm running a CFD software that non-dimensionalizes the NS equations. The problem I'm simulating is a 2D channel flow: relaxation oscillations of an interface between two viscous fluids, shown here. I'm trying to see what they are non-dimensionalizing time with, which is evidently just...
  6. 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...
  7. O

    How to estimate the performance of a pump using Navier Stokes?

    Hello everyone, Attached is the homework problem (FluidHmk.PNG) as well as the attempt (Attempted 1 and 2). Just wanted to know if this is method to approach the problem Thanks in advance.
  8. A

    I Deriving Navier-Stokes Equation

    Just trying to derive the Navier-Stokes equation. (1)The velocity at any point in space of an infinitesimal fluid element is v(x,y,z,t) (2) acceleration ##\frac{dv}{dt}=\frac{\partial v}{\partial t}+\sum_i\frac{\partial v_i}{\partial x_i}{\dot x_i}## ##a=\frac{dv}{dt}=\frac{\partial v}{\partial...
  9. 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...
  10. 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...
  11. Domenico94

    Speed of fluid in a turbulent flow

    Suppose a fluid passes from having laminar flow, to having a turbulent flow (like when passing after an object). How do fluid speed and fluid density change after that?
  12. yecko

    Navier Stokes Thm Homework: Equations & Solutions

    Homework Statement [/B] Homework Equations Navier strokes theorem The Attempt at a Solution May I ask why would there suddenly a "h" in the highlighted part? "h" wasnt existed in the previous steps, which C2=0 shouldn't add height of the liquid as a constant in the formula... thanks
  13. mertcan

    Jacobian matrix and Navier Stokes equation

    Hi, in first attachment/picture you can see the generalized navier stokes equation in general form. In order to linearize these equation we use Beam Warming method and for the linearization process we deploy JACOBİAN MATRİX as in the second attachment/picture. But on my own I can ONLY obtain the...
  14. jedishrfu

    B What makes the Navier Stokes equation so difficult?

    An article in Quanta Magazine discusses the math behind the Navier Stokes equations, why they are so difficult to solve and whether they truly represent fluid flow: https://www.quantamagazine.org/what-makes-the-hardest-equations-in-physics-so-difficult-20180116/
  15. J

    Inviscid flows and the turbulent (eddy) viscosity

    Hello, After Favre averaging the momentum equation for an inviscid flow, the following can be obtained: $$\frac{\partial}{\partial t} \left(\overline{\rho}\tilde{u}_i \right) + \frac{\partial}{\partial x_j}\left( \overline{\rho}\tilde{u}_i \tilde{u}_j \right) + \frac{\partial...
  16. ramzerimar

    When are the Navier Stokes equations invalid?

    I'm studying the Navier Stokes equations right now, and I've heard that those set of equations are invalid in some situations (like almost any mathematical formulation for a physics problem). I would like to know in which situations I cannot apply the NS equations, and what is the common...
  17. A

    Conservation law form of Navier Stokes Equation

    I am pretty confused about how to write Navier-Stokes Equation into conservation form, it seems that from my notes, first, the density term with the pressure gradient dropped out. and second, du^2/dx seems to be equal to udu/dx. Why is it so? I attached my notes here for your reference.
  18. P

    Heat equation from Navier Stokes eqns?

    Can you derive the heat conduction equation from the navier stokes equations (particularly the energy eqn)?
  19. S

    I Solving Navier Stokes & energy equations with different coordinates

    Hi all I am conducting a fluid analysis on water flowing through a subsea pipe. Having used navier stokes equation, i derived the equation for velocity in the r-direction (using cylindrical coordinates. But when initially solving the energy equation to determine temperature distribution I...
  20. 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...
  21. 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...
  22. gfd43tg

    Navier Stokes two infinite parallel plates

    Homework Statement Compare the COMSOL results to the analytical solution for laminar flow between flat plates. Assume no effect of gravity on the flow (g = 0). The comparison will involve obtaining the velocity at a point in the flow field and the ΔP/L term. For example, you can compare the...
  23. gfd43tg

    Navier Stokes thin film on infinite wall

    Homework Statement Consider steady, incompressible, parallel, laminar flow of a film of oil falling down an infinite vertical wall (Figure P-1). The oil film thickness is “h” and gravity acts in the negative Z-direction (downward on the figure). There is no applied pressure driving the flow –...
  24. M

    Understanding Navier-Stokes Derivation and Momentum Flow | PF Discussion

    hey pf! so when deriving navier stokes we have, from Newton's second law, \sum \vec{F} = m\frac{d \vec{V}}{dt} when deriving the full navier stokes (constant density) the acceleration term can be thought of as two pieces: a body change of velocity within the control volume and a mass flow...
  25. N

    Navier Stokes Equation: Examples & Explanation

    Hi guys. Can somebody give mé examples for what the navier stokes equation normally are uset for.? I do not understand or Can see, what we use it for.? And Can someone help mé understand the equation? Like what each term is?
  26. G

    Biological Fluid Dynamics - Approximate Navier Stokes Equations Solns

    Homework Statement Good evening. First post on this forum! The problem I wish to state would take too long to write by hand so I thought it best to do so via attachment. The question I am stuck on is part d and, in fact, part e also. Homework Equations All relevant equations are given...
  27. M

    Fluid mechanics navier stokes flow around geometry

    hey pf! i am studying fluid mechanics and was wondering if any of you are familiar with a flow around some geometry? for example, perhaps a 2-D fluid flowing around a circle? if so please reply, as i am wondering how to model the navier-stokes equations. i'll be happy to post the equations...
  28. M

    Fluid mechanics study navier stokes

    hey pf! i am studying fluid mechanics and we are reviewing navier/stokes equations. we have gone over a few problems, but i could definitely use practice on more. do you all have any suggestions that include solutions, not just answers, so if I am stuck i can see how to solve? problems...
  29. M

    Is my derivation of the Navier Stokes equation using Newton's second law valid?

    hey pf! can you tell me if this derivation sounds reasonable for the navier stokes equation, from Newtons second law into a partial differential equation. i'm really just concerned with one part. specifically, i start the derivation with \Sigma F = ma. I am comfortable with the force term...
  30. E

    Deriving navier stokes in polar

    Can anyone point me to a derivation of the navier stokes equations in polar? I don't see where the single derivative in theta terms are coming from in the first 2 components.
  31. Saladsamurai

    Navier Stokes EQs: Some Insight Please

    I have been doing some serious review of fluids in order to prep for some CFD. I have been re-deriving the NS Equations in all of their various forms. Something seems to have cropped up that I have worked myself in circles about. Let's take the momentum equation in Conservative Integral form...
  32. 0

    Mathematica Reduced navier stokes in mathematica help please

    reduced navier stokes in mathematica urgent help please ok I am modelling airflow in the upper airway for application i obstructive sleep apnoea, but I have hit a brick wall with mathematica. I have a system of 3 differential equations with boundary conditions, and I need to solve to find 3...
  33. B

    Weak form of Navier Stokes Equation

    Homework Statement Folks, determine the weak form given Navier Stokes eqns for 2d flow of viscous incompressible fluids ##\displaystyle uu_x+vu_y=-\frac{1}{\rho} P_x+\nu(u_{xx}+u_{yy})## (1) ##\displaystyle uv_x+vv_y=-\frac{1}{\rho} P_y+\nu(v_{xx}+v_{yy})## ##\displaystyle u_x+v_y=0##...
  34. B

    Finding Clear Explanations for Navier Stokes Equations

    Hi I am currently revising for an exam taking place in 3 days. I am finding it difficult understanding Navier Stokes Equations. It won't necessarily involve difficult questions on Navier Stokes Equations, i myself just find it difficult revising from academic books which explain the problems...
  35. F

    Multivalued Navier Stokes Solution

    I was messing around with the Navier-Stokes equations a while ago and I found a time dependent 2D solution. The force I used was periodic, bounded, and smooth. The question I have is with regards to the time functions in the solution. The solution is spatially periodic and has the form: u =...
  36. O

    Exploring Navier Stokes Equations for General Fluid Flow Analysis

    Does anyone know of a program that can give a good approximation of fluid flow based on the Navier Stokes equations? I know there are FEA programs out there that do linear flow, like in pipes, but what I'm looking for is general flow, for applications that aren't constrained to a pipe. Does such...
  37. C

    Solve Navier Stokes Eq for Vo | Don't Get Prof's Answer

    I need to solve 0=u[(d/dr)((1/r)*(d/dr)(r*Vo))] for Vo the prof gets Vo=Co*r/2+C1/r I don't get the same answer as him, does anyone know how to do this?
  38. Saladsamurai

    Navier Stokes EQ: Derivation in Integral Form

    Hello! :smile: I am doing some review and it has occurred to me that I always confuse myself when I derive the the momentum equation in integral form. So I figure I will try to hammer through it here and ask questions as I go in order to clarify certain points. I know that there are many...
  39. M

    NS Millennium Problem: Uniqueness & Solutions Explained

    I recently came across the NS millennium problem and I read that uniqueness for the NS equations is unknown. I have two questions. First question, if solutions are found to be non-unique, would the NS equations have to be corrected? Second question, since uniqueness is unknown, if someone...
  40. R

    Navier Stokes Equation Derivation and Inertial Forces

    Hi I was reading Introduction to Fluid Mechanics by Nakayama and Boucher and I got lost in their derivation of the Navier Stokes Theorem. They basically started out with a differential of fluid with dimensions dx, dy, and b. Then they say that the force acting on it F = (F_x, F_y) is F_x...
  41. H

    Variable viscosity in the Navier Stokes equations

    Hi, If the viscosity in the NS equations was a variable, what extra equation is used to solve the NS equations? Thanks.
  42. C

    Spatially averaging the Navier Stokes equations on a multiply connected domain

    Hi all, The problem at hand is a bubbly flow in a cylinder: I'm using an FEM to identify how the walls effect the drag on bubbles in a flow. To test my results I want to set up an infinite cylinder with randomly distributed spheres and then average the Navier-Stokes equations over the entire...
  43. D

    Navier Stokes: Homework Q&A on Solution & Equations

    Homework Statement Solution and question are here: http://i51.tinypic.com/bg63qb.png Homework Equations Equations listed in image. The Attempt at a Solution I made several assumptions and there's one that I made that isn't correct but I don't understand why. My textbook lists an...
  44. V

    Navier Stokes Equation to create a Velocity Profile

    Homework Statement An incompressible, viscous fluid is placed between horizontal, infinite, parallel plates. The two plates move in opposite directions with constant velocities U1 and U2. The pressure gradient in the x-direction is zero and the only body force is due to the fluid weight. Use...
  45. N

    Navier stokes equeations, shear term

    Hey!, I was repeating for myself a course I had from a earlier year, fluid mechanics. I looked at the derivation of the navier stokes equations, and there is one term that does not give meaning to me. Take a look at the x-momentum equation here...
  46. Saladsamurai

    How to Derive 2D Navier-Stokes Equations for Steady Flow Using a Control Volume?

    Homework Statement I need to derive the 2D N-S equations for steady, incompressible, constant viscosity flow in the xy-plane. I need to use a control volume approach (as opposed to system approach) on a differential control volume (CV) using the conseervation of linear momentum...
  47. M

    Fluids - Using Navier Stokes

    Incompressible fluid of kinematic viscosity v and density p flows in the x direction between two parallel planes at y = +-h, under the action of an unsteady unidirectional pressure gradient -p(G +Qcosnt), where G, Q, n are constants. Verify that unidirectional motion is possible and that...
  48. H

    Navier stokes and pressure on objects

    Hi every one, I am having a few problems with some research I am doing. I put this in the PDE section as it seams related, but it is for a specific application and I am not sure that it wouldn't be better suited to the mechanical engineering section. I am wanting to find the pressure...
  49. K

    Incompressible Navier Stokes - Short Question

    Hello! The incompressible Navier Stokes equation consists of the two equations and Why can't i insert the 2nd one into the first one so that the advection term drops out?! \nabla\cdotv = v\cdot\nabla = 0 => (v\cdot\nabla)\cdotv = 0