# 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.
19. ### 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.
20. ### 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...
21. ### 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...
22. ### 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'...
23. ### 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...
24. ### 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
25. ### 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...
26. ### 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...
27. ### 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?
28. ### 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##...
29. ### 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...
30. ### 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...
31. ### 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...