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.
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...
1.
Homework Statement
I'm having trouble using equation 2.1 or 2.2 in the article to find the curl of the navier-stokes equation. I understand how to find curl, but can't make sense of the explanation/steps in the document provided by the professor. Homework Equations
All relavent...
Can anyone point me to some publications or archives which feature developments in solving the N-S existence and smoothness problem? Basically, I'd like to read up about how far people have gone towards solving the problem, e.g. a new method to analyze the equations.
Also, what fields of...
I have a pressure flow problem where I'm trying to understand the velocity profile of a fluid in an annular space between a stationary exterior cylinder and a rotating, longitudinally advancing cylinder at its center.
So the boundary conditions a zero velocity at the exterior surface and a...
The no-slip boundary value constraint for Navier-Stokes solutions was explained in my fluid dynamics class as a requirement to match velocities at the interfaces.
So, for example, in a shearing flow where there is a moving surface, the fluid velocity at the fluid/surface interface has to...
Homework Statement
C dimensionless solute concentration
C0 constant
Grc solutal Grashof number
Grh thermal Grashof number
Le Lewis number
N buoyancy ratio N = Grc/Grh
Pr Prandtl number
r dimensionless radial coordinate
t dimensionless time coordinate
z dimensionless axial coordinate...
In layman's terms, can someone explain what the Navier-Stokes Equations express?
And also, can someone explain to me, what the real problem is in trying to understand the Navier-Stokes problem/Turbulence?
I am currently trying to provide a mathematical model that describes the flow through a diverging square pipe. I am trying to simplyfy the navier stokes equations by usings assumptions but am unsure if my current progress is correct.
The problem is as follows:
Fluid enters a section of a...
Homework Statement
In [1], problem 2.3 (ii), we are asked to show that for a pipe with circular cross section r = a and constant pressure gradient P = -dp/dz one has
u_z = \frac{P}{4 \mu}\left( a^2 - r^2 \right)
u_r = 0
u_\theta = 0
References:
[1] D.J. Acheson. <em>Elementary fluid...
I was given an assignment in my Fluid Mechanics Module with the title:
Application of the Navier-Stokes Equations in Tribology Applications
Yet the lecture has given no starting point and we haven't yet done anything to do with Navier-Stokes
Can anyone help me on where to start?
The energy-momentum tensor for a perfect fluid is T^{ab}=(\rho_0+p)u^au^b-pg^{ab} (using the +--- Minkowski metric).
Using the conservation law \partial_bT^{ab}=0, I'm coming up with (\rho+\gamma^2p) [\frac{\partial\mathbb{u}}{{\partial}t}+ (\mathbb{u}\cdot\mathbb{\nabla})\mathbb{u}]=...
Hello, I have Navier stokes in 1D
\rho\left(\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}\right)=-\frac{\partial p}{\partial x}+\mu\frac{\partial^2u}{\partial x^2}
Condition of imcompressibility gives
\frac{\partial u}{\partial x}=0
So I have Navier stokes...
Hello everyone
I'm for the first time trying to model using the Navier-Stokes equations.
I want to model a 2D problem where I have an incompressible, non viscous fluid. I have a region (a segment of line) where a force is applied to the fluid.
For example: a rectangular box with size 2L x L. In...
Notation "(v*gradient operator)v" in Navier-Stokes
What does \left( \textbf{v} \cdot \nabla \right) \textbf{v} mean, assuming knowledge of the gradient operator? And, specifically, how would that be expanded? In general, I'm ignorant of the notation \left( f \left( y, \frac{d}{dx}...
First post; I am starting to read the official problem description of the http://www.claymath.org/millennium/Navier-Stokes_Equations/navierstokes.pdf" and am having trouble understanding the units involved in the first equation :rolleyes:
The equation, verbatim, is
\begin{equation}...
The Clay Math. Navier-Stokes problem is solved in the peer-reviewed journal paper Electron. J. Diff. Equ. vol 2010(2010), No. 93., pp. 1-14. http://ejde.math.txstate.edu
A group of students in reddit.com tried to break the proof...
Navier-Stokes (Low-Re Flow past a Sphere)
My first time posting..
I am looking for guidance in how to solve this question : "Show that the pressure is a HARMONIC FUNCTION (\nabla^2 P = 0 I did that), and that the following solution P = P_{\infty} - \mu\nabla\cdot (U/r) where P_{\infty} is the...
It has been a while since I've had calculus. I am working on a fluid mechanics problem:
I have reduced the Navier-Stokes equation and this is what I have:
mu [d/dr (1/r d/dr (r v(angular)))] = 0
How do I solve for v(angular) ?
Hi, I'm trying to find expressions for fluid fluid in boundary layer for the case of N-S in a cylinder with permeable walls. Fluid is forced into the cylinder by external pressure, but the bulk of the flow in the cylinder will be longitudinal. High flow rate along the cylinder, turbulent. Sounds...
I'm looking to get a full solution to the Navier-Stokes equation to describe fluid flow through a pipe with moving surfaces.
For now I am just concerned with a two dimensional system. The upper and lower boundaries are parallel to the x-axis. The surfaces of the boundaries move sinusoidally...
I was wondering if someone could help me this Navier-Stokes Equation.
f[(δv/δt) + v.Dv] = -DP + Dt + f
Could someone maybe explain the symbols and what it means.
I'm not sure but I think Navier-Stokes equations describe fluid motion.
(The P could be ρ. I'm not too sure)
Thanks
Hello all,
Still at my frisbee modeling program, I started to ask myself how I could get better approximations of stuff like COP versus angle-of-attack, drag/lift coefficients, etc. I've been checking out the Navier-Stokes equations because I understand they can be used to model fluid flow...
Do the navier-stokes equations inlude the seven that have not been solved and, if you successfully solve them, you get a prize of $1 Million per equation?
Thank you.
Hey guys,
Just trying to non-dimensionalise the navier stokes equation. We were taught how to do it when you scale x,y,z with one reference length L...just wondering how to do it if I scale x,y,z with a,b,c respectively.
Edit - this is what I already know...
I've been trying to figure out how I can start with the Navier-Stokes equation and end up at the Reynolds Transport Theorem. Could anyone provide a link to a derivation of this? or some advice of some sort?
As show below the Fourier Transform of Naiver-Stokes equation. I wonder if the pressure should be in the Fourier transform? In the below transformation there is no pressure.
N-S
\frac{\partial\vec{u}}{\partial t}\+(\vec{u}\bullet\nabla)\vec{u}=-\frac{\nabla P}{\rho}+\nu\nabla^{2}
N-S in...
I'm trying to find a simply derivation of the incompressible navier-stokes equations, as stated in the official problem description at the cmi website, or in "The Millenium Problems", by Keith Devlin:
\frac{\partial u}{\partial t}+(u\cdot\nabla)u=f-\nabla p+\nu\Delta u
\nabla\cdot u=0
I...
navier-stokes smoothness problem almost solved
Penny Smith has made progress with showing that smooth conditions exist for all time in a domain for the Navier Stokes equations
http://notes.dpdx.net/2006/10/06/penny-smiths-proof-on-the-navier-stokes-equations/
However a flaw was found in the...
I've reduced a portion of the Navier Stokes to solve a flow problem, and am left with the following ODE:
u (\frac {\partial^2 Vz} {\partial^2 r}) + \frac {r} {u}\frac {\partial Vz} {\partial r} = 0
I tried to solve this equation by assuming a power law solution with
Vz = Cr^n
Which...
I am surprised that nobody has posted yet of the hottest hews in PDE's, Penny Smith's proposed proof that smooth, "immortal" solutions of the N-S equations exist. If it pans out, this will collect one of the famous Clay Millennium Prizes, a cool million. Smith says that unlike Perelman, she'd...
i'm revising for my exams, and i didn't go to many of my fluids lectures, now I'm well confused. in the navier-stokes equation for viscous fluid flow, there is a term:
v(del squared)u
where v is the kinematic viscosity and u is the velocity field of the fluid. at this point in my notes, the...
The Clay Institute wants a proof that an initially smooth flowing fluid stays free of turbulence in the long run.
Can the Navier-Stokes equations, which describe a fluid that initially has no turbulence in it, be equivalent to a set of equations describing vortices,
with the vortices...
Help me find solutions for the Navier-stokes equations and you could get rich.
The problem is this:
A fluid enters a pipe and flows through it smoothly at the outset.
Will it keep flowing smoothly?
Sounds easy to solve but it isn't because nobody has
won the million dollars yet.
Here is...