What is Navier stokes equation: Definition and 31 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. H

    Coordinate transformation of the Navier Stokes equation

    i have successfully transformed the continuity equation using coordinate transform,but having trouble with the momentum equation . can someone kindly provide the transformation of the right hand sight of equation of the image i have attached.
  2. chandrahas

    A Why are the Navier-Stokes equations inconsistent in this case?

    Consider the case of a one-dimensional incompressible, non-viscous fluid flowing down a vertical pipe under the influence of gravity. Since we assume the flow is constant along the cross section of the pipe from the one dimensional assumption, let us denote the velocity of the fluid down the...
  3. 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...
  4. 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.
  5. 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)...
  6. 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.
  7. 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...
  8. 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...
  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. 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...
  11. 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/
  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. 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...
  15. 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...
  16. 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...
  17. 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...
  18. 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.
  19. 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...
  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. 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'...
  22. 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...
  23. MdAsher

    Understanding Fluid Dynamics of a Flying Frisbee

    Hi All, I'm hoping to work on deriving the governing fluid flow equations for a frisbee in flight theoretically and then to test it on a wind tunnel, and compare results. If u could please help on how do i apply/derive the necessary equations. Respectful Regards
  24. 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...
  25. 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...
  26. 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?
  27. 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##...
  28. 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...
  29. 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...
  30. B

    Understanding the Vector Laplacian in the Navier Stokes Equations

    I recently came across the vector version of the Navier Stokes equations for fluid flow. \displaystyle{\frac{\partial \mathbf{u}}{\partial \mathbf{t}}} + ( \mathbf{u} \cdot \bigtriangledown) \mathbf{u} = v \bigtriangleup \mathbf{u} - grad \ p Ok, all is well until \bigtriangleup. I know...