Laplacian term in Navier-Stokes equation

In summary, the conversation discusses deriving part of the Navier-Stokes equations and the discrepancy between getting 2\eta\nabla^2\mathbf{u} and \eta\nabla^2\mathbf{u}. The individual mentions noticing the repeated subscript in the equations and the assumption that \nabla\mathbf{u}^T=\nabla\mathbf{u}. The conversation ends with the individual asking for clarification on the rules they may be missing.
  • #1
Hypatio
151
1
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 [itex]\eta\nabla^2\mathbf{u}[/itex]. However, when I try to get this from Eq. 1, I get [itex]2\eta\nabla^2\mathbf{u}[/itex]. I am getting [itex]2\eta[/itex] because

[itex]\eta\left(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}\right)=\eta(\nabla \mathbf{u}+\nabla \mathbf{u}^T)=2\eta\nabla \mathbf{u}[/itex]
and then the taking the divergence gives [itex]\nabla\cdot 2\eta\nabla\mathbf{u}=2\eta\nabla^2\mathbf{u}[/itex] for constant viscosity [itex]\eta[/itex]

I suspect that my second step in the first line is wrong. But I don't get it.

Thanks in advance.
 
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  • #2
You might want to keep the divergence and the terms in the parentheses together to remind yourself that you only sum over the j subscript (which is repeated).

You should notice that the derivatives of the second term in the parentheses add up to the divergence of u.
 
  • #3
I don't see a repeated subscript. Do you mean that [itex]\nabla\mathbf{u}^T=\nabla\mathbf{u}[/itex]? That is in fact my assumption and thus why I get 2*eta, not eta. Is this not true?:

[itex]\eta(\nabla\mathbf{u}+\nabla\mathbf{u}^T)=2\eta\nabla\mathbf{u}[/itex]

if so, what happened to the 2?

What rules am I missing??
 
  • #4
Hypatio said:
I don't see a repeated subscript.
In the original Eq. 1:
$$\frac{\partial} {\partial x_j} \left[ \eta\left(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}\right) + \ldots \right]$$ has the subscript ##j## both outside and inside the parentheses.
 
  • #5
The i-th component of the force (rewriting equation 1 with [itex]\lambda=0[/itex] ) is

[itex]F_i=\eta\sum\limits_{j}{\frac{\partial^2{u_i}}{\partial^2{x_j}}}+\eta \frac{\partial}{\partial{x_i}}\sum\limits_{j}\frac{\partial{u_j}} {\partial{x_j}} [/itex].

The second term is zero cause the divergence of u is zero hence all that is left is [itex]\eta\sum\limits_{j}{\frac{\partial^2{u_i}}{\partial^2{x_j}}}=\eta\nabla^2u_i[/itex].
 
Last edited:

1. What is the Laplacian term in the Navier-Stokes equation and what does it represent?

The Laplacian term in the Navier-Stokes equation is a mathematical term that represents the diffusion of momentum in a fluid. It takes into account the changes in velocity and pressure over small distances within the fluid.

2. How does the Laplacian term affect the overall behavior of the Navier-Stokes equation?

The Laplacian term plays a crucial role in describing the flow of fluids. It helps to accurately model the diffusion of momentum, which is essential for understanding the behavior of fluids in real-world scenarios such as in pipes, pumps, and turbines.

3. What is the physical meaning of the Laplacian operator in the Navier-Stokes equation?

The Laplacian operator is a differential operator that calculates the rate of change of a quantity over a small distance. In the Navier-Stokes equation, it represents the change in velocity and pressure over small distances within the fluid.

4. How do researchers and engineers use the Laplacian term in the Navier-Stokes equation?

Researchers and engineers use the Laplacian term to simulate and study the behavior of fluids in various situations. This term allows them to accurately model the diffusion of momentum and make predictions about the flow of fluids under different conditions.

5. Can the Laplacian term be neglected in certain cases in the Navier-Stokes equation?

In some cases, the Laplacian term can be neglected if the fluid flow is highly turbulent or if the fluid has a relatively low viscosity. However, in most practical scenarios, this term is essential for accurately predicting the behavior of fluids.

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