Understanding the Vector Laplacian in the Navier Stokes Equations

In summary, the Navier Stokes equations for fluid flow in vector form include a Laplacian term represented by \bigtriangleup or \bigtriangledown^2. This is equivalent to the gradient of the divergence of the vector minus the curl of the curl of the vector. These operations are not associative and should not have their parentheses removed.
  • #1
Brad_Ad23
502
1
I recently came across the vector version of the Navier Stokes equations for fluid flow.

[tex]\displaystyle{\frac{\partial \mathbf{u}}{\partial \mathbf{t}}} + ( \mathbf{u} \cdot \bigtriangledown) \mathbf{u} = v \bigtriangleup \mathbf{u} - grad \ p[/tex]

Ok, all is well until [tex]\bigtriangleup[/tex]. I know this represents the laplacian. What is the formulation of the Laplacian for this since it is a vector? Is it just simply the second partials dot product with the respective terms of the vector? Or is it something else?

edit: changed text where I say problem is [tex] \bigtriangledown[/tex] to the appropriate [tex]\bigtriangleup[/tex]
 
Last edited:
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  • #2
For whatever reason, I can't seem to use laTex ... have to do a bit more reading first.

But, are you sure that the delta you've picked out is the Laplacian ... looks like grad to me.
 
  • #3
Yes that is the Laplacian. Apparently they use that delta to represent it, it is also written as [tex]\bigtriangledown^2[/tex]
 
  • #4
[tex]\nabla^2\boldsymbol{v}=\nabla\left(\nabla\cdot\boldsymbol{v}\right)-\nabla\times\left(\nabla\times\boldsymbol{v}\right)[/tex]
 
  • #5
So let me make sure I have this straight.

[tex]\bigtriangledown^2 \ v = grad \ div \ v - curl \ curl \ v[/tex]
 
  • #6
I believe that is the correct interpretation.
 
  • #7
Well, that is an identity for the operators. But why don't you like the idea of the usual Laplacian acting on a vector? It's just a derivative operator, which is allowable on vectors as long as you remember that the basis vectors also have to be differentiated.

dhris
 
  • #8
Originally posted by Brad_Ad23
So let me make sure I have this straight.

[tex]\bigtriangledown^2 \ v = grad \ div \ v - curl \ curl \ v[/tex]

these operations are not associative, so you should not remove the parantheses.
 

1. What is the Navier-Stokes equation?

The Navier-Stokes equation is a mathematical formula that describes the motion of fluids, such as liquids and gases. It is named after the French mathematician and physicist Claude-Louis Navier and the Irish mathematician and physicist George Gabriel Stokes.

2. What is the significance of the Navier-Stokes equation?

The Navier-Stokes equation is considered one of the most important equations in fluid mechanics, as it can be used to model and predict the behavior of fluids in a wide range of situations. It is used in various fields, including engineering, meteorology, and oceanography.

3. What are the variables in the Navier-Stokes equation?

The variables in the Navier-Stokes equation include velocity, pressure, density, and viscosity. These variables represent the physical properties of the fluid and are used to calculate the rate of change of fluid motion.

4. What are the limitations of the Navier-Stokes equation?

The Navier-Stokes equation has several limitations, including assumptions made about the fluid, such as it being incompressible and having a constant viscosity. It also does not account for turbulent flow, which can significantly affect the behavior of fluids in certain situations.

5. How is the Navier-Stokes equation solved?

The Navier-Stokes equation is a set of partial differential equations and is typically solved using computational methods, such as finite element analysis or finite difference methods. These methods use numerical approximations to find solutions that closely match the behavior of real fluids.

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