What is the role of control volume in the Navier-Stokes equations?

In summary, the Navier-Stokes smoothness problem has made progress with showing that smooth conditions exist for all time in a domain, thanks to Penny Smith's proof. However, a flaw was found in the mathematics, which is currently being resolved. The problem of finding actual solutions to the equations still remains. The Navier-Stokes equations have four forms, including Lagrangian and Eulerian in both differential and integral approaches. The control volume for an integral approach is simply V.
  • #1
verdigris
119
0
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 mathematics - hopefully it can be sorted out soon.There's still the problem though of finding actual solutions to the equations!

Thanks for the info on control volume.I was just wondering if in reality there
is a real,if very small size,to the differential element.
 
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  • #2
It's not a control volume but a differential element (infinitely small). Imagine a cube with sides that measure, in cartesian coordinates [tex]\delta x[/tex], [tex]\delta y[/tex] and [tex]\delta z[/tex].
 
  • #3
Actually Fred, the NS equations have 4 forms:

Lagrangian (moving frame of reference)
Eulerian (stationary)
..and then...
Differential
Integral

So for example, looking simply at continuity (sorry again for my lack of latex):
Differential Lagrangian:
Dp/Dt + rho*del•V =0
Where D/Dt is the substantial derivative with respect to time

Differential Eulerian:
dp/dt + del•(rho*V) = 0
(note this form is now strongly conservative as all variables are inside of a derivative)

Integral Lagrangian:
D/Dt [Volume Integral] rho dV = 0

Integral Eulerian:
d/dt [Volume Integral] rho dV + [surface integral] rho*V dS = 0

The entire equations can be derived any of the four ways. It's easiest (at least for me) to remember one form, and then how to go from one form to another.

To answer the question, the control volume for an integral approach is simply V, there is no need to know anything else besides that.
 

1. What is a Navier-Stokes control volume?

A Navier-Stokes control volume is a mathematical concept used in fluid dynamics to analyze the flow of a fluid through a specific region. It involves creating a control volume or imaginary boundary around a certain portion of the fluid and studying the flow of mass, momentum, and energy within that volume.

2. How is a Navier-Stokes control volume different from a control surface?

A control surface is a 2-dimensional boundary that is used to study the flow of a fluid in a specific direction, while a Navier-Stokes control volume is a 3-dimensional boundary that allows for a more comprehensive analysis of flow within a specific region. Control volumes also take into account the effects of pressure, gravity, and other forces on the fluid flow.

3. What are the applications of Navier-Stokes control volumes?

Navier-Stokes control volumes are used in a variety of fields, including aerodynamics, hydraulic engineering, and weather forecasting. They are also used in the design and analysis of pumps, turbines, and other fluid flow systems.

4. What are the assumptions made in the Navier-Stokes control volume analysis?

The Navier-Stokes equations assume that the fluid is incompressible, viscous, and Newtonian (follows Newton's law of viscosity). It also assumes that the fluid is in a steady state and that there are no external forces acting on the control volume.

5. What are the limitations of Navier-Stokes control volume analysis?

Navier-Stokes control volume analysis can become complex and computationally intensive for more complex flow systems. It also has limitations in analyzing certain types of flows, such as turbulent flows. In some cases, other numerical methods, such as computational fluid dynamics, may be more suitable for studying fluid flow.

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