Divergence of the Navier-Stokes Equation

In summary, the Navier-Stokes equation is a governing equation for fluid motion that takes into account material derivative, kinematic viscosity, and modified pressure gradient. However, the velocity field must be non-solenoidal for this equation to hold. When taking the divergence of the equation, the terms in blue cannot come from the blue section of the equation as they are non-linear in velocity. It is advised to write out the equation for each component and take partial derivatives to ensure accuracy.
  • #1
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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 non-solenoidal (∇⋅U ≠ 0).

How, then, can we take the divergence of this equation and get the following result?:

upload_2016-5-2_15-2-40.png


I can follow all of the terms other than the one underlined in blue. I know that it comes from the blue section of the Navier-Stokes written above, since I can easily get all the other terms.

Thanks in advance
 
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  • #2
FluidStu said:
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 non-solenoidal (∇⋅U ≠ 0).

How, then, can we take the divergence of this equation and get the following result?:

View attachment 100112

I can follow all of the terms other than the one underlined in blue. I know that it comes from the blue section of the Navier-Stokes written above, since I can easily get all the other terms.

Thanks in advance
The terms in blue can not come from the blue section of the NS. They are non-linear in velocity, and blue section of the NS equation is linear in velocity. The blue terms in your relationship must come from the left side of the NS equation.

My advice to you is to write out the NS equation for each of the three components. Then take the partial of the x component with respect to x, the partial of the y component with respect to y, and the partial of the z component with respect to z. Then add the resulting 3 equations. This will be bulletproof.

chet
 

1. What is the Navier-Stokes equation and why is it important in fluid mechanics?

The Navier-Stokes equation is a set of partial differential equations used to describe the motion of fluids, such as air and water. It is important because it allows scientists and engineers to model and predict the behavior of fluids in various scenarios, such as in pipes, around objects, or in weather systems.

2. What does it mean for the Navier-Stokes equation to diverge?

When the Navier-Stokes equation diverges, it means that the solutions to the equations become unstable and unpredictable. This can happen when there are extreme changes in the flow of the fluid, such as sudden changes in velocity or pressure.

3. Why is the divergence of the Navier-Stokes equation a challenging problem in fluid mechanics?

The divergence of the Navier-Stokes equation is a challenging problem because it is a non-linear equation, meaning that small changes in the initial conditions can lead to large changes in the predicted outcomes. This makes it difficult to accurately model and predict fluid behavior, especially in complex scenarios.

4. What are some current research efforts to address the challenges of the divergence of the Navier-Stokes equation?

Scientists and mathematicians are constantly working on improving numerical methods and algorithms to solve the Navier-Stokes equation more accurately and efficiently. Some approaches include using higher-order methods, adaptive mesh refinement, and incorporating data from experiments to improve the accuracy of the models.

5. How does the divergence of the Navier-Stokes equation impact our daily lives?

The divergence of the Navier-Stokes equation can have a significant impact on various aspects of our daily lives, such as weather prediction, air and water flow in pipes and channels, and the design of vehicles and structures. Understanding and accurately modeling fluid behavior is crucial in many industries, from aviation and engineering to environmental and climate science.

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