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.
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...
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...
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...
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...
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...
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...
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...
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...
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...
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 :)
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...
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...
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...
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...
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...
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=10.1.1.374.5979&rep=rep1&type=pdf]
and he mentions after time discretization he ends...
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...
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...
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...
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...
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...
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?
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...
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...
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
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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?
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.
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...