Fluid Dynamics, Linearization Question

In summary: This equation shows that the density perturbations are advected by the background fluid flow. In summary, we have shown that the pressure and density perturbations, \delta p and \delta \rho, obey the linearized equations of motion for an ideal fluid in hydrostatic equilibrium. These equations describe how small amplitude pressure waves are driven by density perturbations and advected by the fluid flow.
  • #1
bakra904
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Homework Statement


Consider an ideal fluid large enough to experience its own gravitational attraction. If the fluid is initially at hydrostatic equilibrium with density [tex]\rho_{0} (r)[/tex] and pressure [tex]p_{0}(r)[/tex] , it can develop small amplitude pressure waves which may be analyzed as follows.

Show that [tex]\delta p = p - p_{0}[/tex] and [tex]\delta\rho = \rho - \rho_{0}[/tex] obey the linearized equations of motion

[tex]\frac{\partial\rho}{\partial t} = - \nabla . ( \rho_{0} v)[/tex]

[tex]\rho_{0} \frac{\partial v}{\partial t} = -\nabla\delta p + \delta\rho g [/tex]



Homework Equations



The relevant equations are

[tex]\rho\frac{\partial v}{\partial t} + \rho (v . \nabla) v = - \nabla p[/tex] + g
[tex]\frac{\partial\rho}{\partial t} + \nabla (\rho .v) = 0 [/tex]


The Attempt at a Solution



For the solution I began writing
[tex] p = p_{0} + \delta p[/tex] (1)
[tex] \rho = \rho_{0} + \delta\rho[/tex] (2)

and assumed that [tex]\delta\rho <<< \rho_{0}[/tex] (3)
and [tex]\delta p <<< p_{0} [/tex] (4)

I then substituted (3) and (4) into (1) and (2) and only kept the lowest order terms (since [tex]\delta\rho[/tex] is small compared to [tex]\rho[/tex] ).

So then I get [tex]\rho_{0}\frac{\partial v}{\partial t} = - \nabla\delta p + \delta\rho g[/tex]
and [tex]\frac{\partial \delta\rho}{\partial t} + \rho_{0}\nabla .v = 0 [/tex]

If anyone could help from here that'd be much appreciated!
 
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  • #2


Hello, thank you for your post. It seems like you are on the right track with your solution. Let's continue from where you left off.

We have the equations:
\rho_{0}\frac{\partial v}{\partial t} = - \nabla\delta p + \delta\rho g
and \frac{\partial \delta\rho}{\partial t} + \rho_{0}\nabla \cdot v = 0

We can simplify the first equation by using the continuity equation:
\nabla \cdot (\rho_0 v) = 0
which implies that
\rho_0 \nabla \cdot v = - v \cdot \nabla \rho_0

Substituting this into the first equation, we get:
\rho_0 \frac{\partial v}{\partial t} = - v \cdot \nabla \rho_0 - \nabla \delta p + \delta \rho g

We can now use the ideal gas law to express \delta p in terms of \delta \rho:
\delta p = c_s^2 \delta \rho
where c_s is the speed of sound in the fluid.

Substituting this into our equation, we get:
\rho_0 \frac{\partial v}{\partial t} = - v \cdot \nabla \rho_0 - c_s^2 \nabla \delta \rho + \delta \rho g

This is the linearized equation of motion for the fluid. We can also write it in a more compact form as:
\rho_0 \frac{\partial v}{\partial t} = - \nabla \cdot (\rho_0 v) - c_s^2 \nabla \delta \rho + \delta \rho g

This equation shows that the pressure waves are driven by the density perturbations and the gravitational acceleration.

For the second equation, we can use the continuity equation again to get:
\frac{\partial \delta \rho}{\partial t} + \nabla \cdot (\rho_0 v) = 0

Substituting our previous result for \nabla \cdot (\rho_0 v), we get:
\frac{\partial \delta \rho}{\partial t} + \rho_0 v
 
  • #3




Your solution so far looks good. In order to show that these equations obey the linearized equations of motion, we need to compare them to the relevant equations you provided.

First, let's look at the equation for continuity:

\frac{\partial\rho}{\partial t} + \nabla (\rho .v) = 0

If we substitute in our expressions for \rho and v from (2) and (3), we get:

\frac{\partial (\rho_{0} + \delta\rho)}{\partial t} + \nabla [(\rho_{0} + \delta\rho) v] = 0

Expanding this out and keeping only the lowest order terms, we get:

\rho_{0}\frac{\partial v}{\partial t} + \nabla (\rho_{0} v) = 0

This is exactly the same as the second equation you provided, so we can see that our expression for continuity is consistent with the linearized equation.

Next, let's look at the equation for momentum conservation:

\rho\frac{\partial v}{\partial t} + \rho (v . \nabla) v = - \nabla p + g

Substituting in our expressions for \rho and v from (2) and (3), we get:

(\rho_{0} + \delta\rho)\frac{\partial v}{\partial t} + (\rho_{0} + \delta\rho) (v . \nabla) v = - \nabla (\rho_{0} + \delta p) + g

Expanding this out and keeping only the lowest order terms, we get:

\rho_{0}\frac{\partial v}{\partial t} + \rho_{0} (v . \nabla) v = - \nabla \delta p + \delta\rho g

This is exactly the same as the first equation you provided, so we can see that our expression for momentum conservation is also consistent with the linearized equation.

Therefore, we have shown that the expressions for \delta p and \delta\rho do indeed obey the linearized equations of motion for an ideal fluid. Great job!
 

1. What is fluid dynamics?

Fluid dynamics is the study of the movement and behavior of fluids, such as liquids and gases. This includes the analysis of how fluids flow, their pressure, and how they interact with their surroundings.

2. What is a linearization question in fluid dynamics?

A linearization question in fluid dynamics involves approximating a nonlinear system or equation into a simpler linear form. This simplifies the analysis and allows for easier calculations and predictions.

3. Why is linearization important in fluid dynamics?

Linearization is important in fluid dynamics because many real-world systems are nonlinear and difficult to solve. By approximating them into a linear form, we can gain a better understanding of their behavior and make more accurate predictions.

4. How do you linearize a system in fluid dynamics?

To linearize a system in fluid dynamics, you first need to identify the nonlinear equations or systems that need to be approximated. Then, you can use techniques such as Taylor series expansion or small perturbations to simplify the equations into a linear form.

5. What are some applications of linearization in fluid dynamics?

Linearization has many applications in fluid dynamics, such as in the study of aerodynamics for aircraft design, the analysis of fluid flow in pipes and channels, and the prediction of weather patterns and ocean currents. It is also used in the development of control systems for fluid-based technologies, such as pumps and turbines.

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