Solving a Simple ODE from the Navier-Stokes

In summary, the conversation discusses solving an ODE derived from the Navier Stokes equations using a power law solution. However, the resulting solution does not match the expected radial profile, leading to further exploration and the use of a new function. The final solution involves an exponential function and the integral of an error function, with the angle dependence determined by boundary conditions.
  • #1
Peregrine
22
0
I've reduced a portion of the Navier Stokes to solve a flow problem, and am left with the following ODE:

[tex]u (\frac {\partial^2 Vz} {\partial^2 r}) + \frac {r} {u}\frac {\partial Vz} {\partial r} = 0 [/tex]

I tried to solve this equation by assuming a power law solution with
[tex]Vz = Cr^n [/tex]

Which yields

[tex] n(n-1)r^{n-2} + nr^{n-2} = 0 [/tex],

Thus n^2 - n + n = 0

Which seems to indicate [tex] n^2 =0 [/tex] => n = 0

So Vz = C. But, I don't think this is what physically happens, Iam expecting a radial profile and not a flat profile, so I'm looking for where I took a wrong turn. Any ideas? Thanks.
 
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  • #2
HINT: Substitute [itex] \frac{\partial V_{z}}{\partial r} [/itex] by a new function, let's call it [itex] f(r,...) [/itex]. Then see what you get.

Daniel.
 
  • #3
So, using the hint I get:

uf''(r) + (u/r)f'(r) = 0

Assuming an exponential function,
[tex]f(r) = e^{nr}[/tex]
[tex]f'(r) = ne^{nr}[/tex]
[tex]f''(r) = n^2e^{nr}[/tex]

Thus

[tex]un^2e^{rn} + (u/r)ne^{rn} = 0[/tex]
[tex]u(n^2+1/rn) = 0[/tex]
[tex]un(n+1/r) = 0[/tex]
[tex]n = 0, -1/r[/tex]

So:
[tex]Vz = Ae^{0} + Be^{-1/r} [/tex]

Is that the correct methodology? Thanks again.
 
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  • #4
I get the eq for "f"

[tex] u\frac{df}{dr}+\frac{r}{u}f=0 [/tex]

with the solution

[tex] f(r)=Ce^{-\frac{r^{2}}{2u^{2}}} [/tex]

and then finally

[tex] V(r,\vartheta)=C\int Ce^{-\frac{r^{2}}{2u^{2}}} \ dr + g(\vartheta) [/tex]

The integral brings in the erf function, while the angle dependence should be determined by boundary conditions.

Daniel.
 

Related to Solving a Simple ODE from the Navier-Stokes

1. What is an ODE and how does it relate to the Navier-Stokes equations?

An ODE, or ordinary differential equation, is a mathematical equation that describes the relationship between a function and its derivatives. In the context of fluid mechanics, the Navier-Stokes equations are a set of partial differential equations that govern the motion of fluid particles. By solving a simple ODE from the Navier-Stokes equations, we can gain insight into the behavior of fluids in various scenarios.

2. How do I solve a simple ODE from the Navier-Stokes equations?

To solve a simple ODE from the Navier-Stokes equations, you will first need to specify the initial and boundary conditions for the problem. Next, you can use various analytical or numerical methods to solve the ODE, such as separation of variables, substitution, or Euler's method. It is important to note that the complexity of the ODE and the accuracy of the solution method will vary depending on the specific problem.

3. What are some real-world applications of solving ODEs from the Navier-Stokes equations?

The Navier-Stokes equations and their resulting ODEs have many practical applications, including predicting the flow of air around airplanes, designing efficient pipelines for transporting fluids, and understanding the behavior of ocean currents. These equations are also crucial in the study of turbulence, which has implications in weather forecasting, aerodynamics, and many other fields.

4. Are there any limitations to solving ODEs from the Navier-Stokes equations?

While ODEs from the Navier-Stokes equations provide valuable insights into fluid behavior, they also have some limitations. These equations are based on certain assumptions and simplifications, such as the fluid being incompressible and the flow being steady. In reality, these assumptions may not always hold, and more complex equations, such as the full Navier-Stokes equations or computational fluid dynamics, may be needed to accurately describe fluid motion.

5. What are some future developments in solving ODEs from the Navier-Stokes equations?

There is ongoing research and development in the field of solving ODEs from the Navier-Stokes equations, with the goal of improving accuracy and efficiency. Some advancements include the use of machine learning techniques to better predict turbulent flows and the development of new numerical methods for solving the equations. Additionally, there is ongoing work on extending the Navier-Stokes equations to incorporate additional factors, such as heat transfer or chemical reactions, for more comprehensive modeling of fluid behavior.

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