How can I break down the Navier Stokes equation? (momentum equation)

In summary, the conversation discusses a homework assignment involving the Navier-Stokes equation and its derivation. The equation is derived using the approach found in Grainger's textbook and the conversation also mentions a Medium article on the topic. The concept of breaking down the equation is also brought up, with clarification on the momentum equation and its relation to Newton's 2nd law of motion. The dot product of the velocity vector with the gradient of the stress tensor is also mentioned.
  • #1
Carbon273
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Homework Statement
So I am trying to break down the complexity of my homework assignment where I need to perform the operation: (velocity vector)*(momentum equation) as a step to define the transport of kinetic energy equation. For the sake of academic integrity I wish to fully understand the concept by breaking it down. So my first question is, how do I gain understanding of the momentum equation in this context? How can I understand to unpack this equation. I have a feeling this is the navier stokes equation in a very condensed manner. If it is, how do I break it down, especially with the tensor embedded in there.
Relevant Equations
Equation shown below:
1567347965796.png
 
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  • #3
Sorry if this comes too close to the line for self-promotion, but a few months ago I wrote a Medium article about the derivation of the Navier-Stokes equation that mostly followed the approach in Grainger's textbook:



(Yes, I know that it says "part 1" in the subtitle, implying the existence of a "part 2". I'll get to it eventually.)
 
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  • #4
What do you mean by "break down" exactly?
 
  • #5
The momentum equation (aka, the equation of motion) is a differential version of Newton's 2nd law of motion applied to a fluid. Is that what you were asking? You are supposed to take the dot product of the velocity vector with the momentum equation. Are you asking how one mathematically takes the dot product of the velocity vector with the gradient of the stress tensor?
 

FAQ: How can I break down the Navier Stokes equation? (momentum equation)

1. What is the Navier Stokes equation and why is it important?

The Navier Stokes equation, also known as the momentum equation, is a fundamental equation in fluid mechanics that describes the motion of a fluid. It is important because it allows scientists and engineers to predict and understand the behavior of fluids, which is crucial in many applications such as designing aircrafts and analyzing weather patterns.

2. What are the components of the Navier Stokes equation?

The Navier Stokes equation has three main components: the convective term, which describes the acceleration of fluid particles due to changes in velocity; the pressure term, which accounts for the forces exerted by the fluid on its surroundings; and the viscous term, which accounts for the frictional forces within the fluid.

3. Can the Navier Stokes equation be simplified?

Yes, the Navier Stokes equation can be simplified depending on the specific problem being studied. For example, in some cases, the convective term may be negligible and can be ignored. Additionally, certain assumptions such as the fluid being incompressible or the flow being steady can also simplify the equation.

4. How can I break down the Navier Stokes equation to understand it better?

Breaking down the Navier Stokes equation can be done by understanding each term and its physical meaning. The convective term represents the acceleration of fluid particles, the pressure term represents the forces within the fluid, and the viscous term represents the frictional forces. It can also be helpful to visualize the equation in terms of vectors to better understand the direction and magnitude of each term.

5. Are there any tools or methods to solve the Navier Stokes equation?

Yes, there are several tools and methods to solve the Navier Stokes equation. These include analytical solutions, which involve solving the equation using mathematical equations and techniques; numerical solutions, which involve discretizing the equation and solving it using computational methods; and experimental methods, which involve conducting physical experiments to observe and analyze fluid behavior. Each method has its own advantages and limitations, and the choice of method depends on the specific problem being studied.

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