Solving Navier Stoke Problem with 3D Equation and Lubrication Approximation

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In summary, the paper uses the 3D Navier Stokes equation to calculate the lubrication coefficient. They obtain an imaginary value for w using the integral, but they obtain a real value using the mathematica Wolfram program. The paper states that w equals 1/2k ∂p/∂θ (ln r - 1/2) + C1 1/2 r + C2/r. Somebody could confirm this.
  • #1
iloc86
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hi every body, I am doing an analysis of a paper about a tool used in oil well drilling, they use 3D navier stoke equation and by lubrication aproximation the get this

-(1/r) ∂p/∂θ + k (∂( 1/r ∂(rw)/∂r )/∂r )=0

its necesary to get w so i use integral but i obtain an irreal i used mathematica wolfram. but in the paper they obtain

w= 1/2k ∂p/∂θ (ln r - 1/2) + C1 1/2 r + C2/r

somebody could confirm this
 
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  • #2
sorry imaginary
 
  • #3
-(1/r) ∂p/∂θ + k (∂( 1/r ∂(rw)/∂r )/∂r )=0
∂( 1/r ∂(rw)/∂r )/∂r = (1/k) (∂p/∂θ) (1/r)
1/r ∂(rw)/∂r = (1/k) (∂p/∂θ) ln(r) + C1
∂(rw)/∂r = (1/k) (∂p/∂θ) r ln(r) + C1 r
(rw) = (1/k) (∂p/∂θ) [(r²/2) ln(r)-r²/4] + C1 r²/2 + C2
w = (1/2k) (∂p/∂θ) [r ln(r)-(r/2)] + C1 r/2 + C2/r
 
  • #4
JJacquelin said:
-(1/r) ∂p/∂θ + k (∂( 1/r ∂(rw)/∂r )/∂r )=0
∂( 1/r ∂(rw)/∂r )/∂r = (1/k) (∂p/∂θ) (1/r)
1/r ∂(rw)/∂r = (1/k) (∂p/∂θ) ln(r) + C1
∂(rw)/∂r = (1/k) (∂p/∂θ) r ln(r) + C1 r
(rw) = (1/k) (∂p/∂θ) [(r²/2) ln(r)-r²/4] + C1 r²/2 + C2
w = (1/2k) (∂p/∂θ) [r ln(r)-(r/2)] + C1 r/2 + C2/r

I have issues with this. In particular, you have:

[tex]\int \left(\frac{1}{k} \frac{\partial p}{\partial \theta} \frac{1}{r}\right)dr=\int \partial\left(\frac{1}{r}\frac{\partial}{\partial r}(rw)\right)[/tex]

I do not see how you can integrate the left side with respect to r and get:

[tex]\int \left(\frac{1}{k} \frac{\partial p}{\partial \theta} \frac{1}{r}\right)dr=\frac{1}{k}\frac{\partial p}{\partial \theta} \ln(r)+c[/tex]

You're assuming [itex]\frac{\partial p}{\partial \theta}[/itex] is not a function of r and I do not think you can assume this. Also, wouldn't the constant of integration be an arbitrary function of theta in this case?
 
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  • #5
Hi every body thanks for the reply Jacqueline u obtain the same as the paper And jack in the doc they assume p doesn't change in r But as You say a c depent in theta What do u think?
 

1. What is the Navier-Stokes Problem?

The Navier-Stokes problem is a set of partial differential equations that describe the motion of fluid substances. These equations are used to understand the behavior of fluids in various contexts, including fluid mechanics, weather prediction, and aerodynamics.

2. What is the significance of using 3D equations in solving the Navier-Stokes problem?

3D equations are necessary to accurately model the complex behavior of fluids in three dimensions. In real-world scenarios, fluids rarely flow in simple, two-dimensional paths. Using 3D equations allows for more precise predictions and a better understanding of the fluid's behavior.

3. What is the role of the lubrication approximation in solving the Navier-Stokes problem?

The lubrication approximation is a simplification of the Navier-Stokes equations that is commonly used in boundary layer problems. It assumes that the fluid flow is mostly in one direction, allowing for a reduction in the complexity of the equations and making them easier to solve.

4. What challenges are faced when solving the Navier-Stokes problem with 3D equations and lubrication approximation?

Solving the Navier-Stokes problem with 3D equations and lubrication approximation can be a computationally intensive task. The equations are highly nonlinear and require advanced numerical methods to accurately solve. Additionally, the boundary conditions and initial conditions must be carefully chosen to ensure accurate results.

5. How is the Navier-Stokes problem with 3D equations and lubrication approximation used in practical applications?

The Navier-Stokes problem with 3D equations and lubrication approximation has many practical applications, including predicting fluid flow in pipes, designing aircraft wings, and understanding ocean currents. It is also used in various industries, such as aerospace, automotive, and energy, to optimize designs and improve efficiency.

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