Transformation of Reynolds Equation from Cartesian to cylindrical

In summary, the Reynolds equation is a partial differential equation used to describe the motion of a viscous fluid. It is important in fluid mechanics for predicting fluid behavior in engineering applications. The equation can be transformed from Cartesian to cylindrical coordinates, which simplifies the mathematics and allows for analysis of flow in circular or cylindrical geometries. The advantages of using cylindrical coordinates include reduced complexity and better representation of rotational or swirling flows. However, there may be limitations to using cylindrical coordinates, such as in cases where another coordinate system may be more suitable or when the transformation introduces error.
  • #1
rakan
2
0
TL;DR Summary
How to convert the Reynolds equation from Cartesian coordinate (equation1) to cylindrical coordinate (equation2)? by showing the steps?
∂/∂x ((ρh^3)/12μ ∂p/∂x) + ∂/∂z ((ρh^3)/12μ ∂p/∂z) = ∂/∂x (ρh (U_1+U_2)/2) + ∂/∂z (ρh (W_1+W_2)/2) + (∂(ρh))/∂t (1)

1/r ∂/∂r (r (ρh^3)/12μ ∂p/∂r) + 1/r ∂/∂θ ((ρh^3)/12μ ∂p/r∂θ) = rω/2 ∂(ρh)/r∂θ + (∂(ρh))/∂t (2)
 
Engineering news on Phys.org
  • #2
Rather than attempt a conversion, why not review the steps in the Cartesian derivation, and then attempt to repeat the equivalent steps in cylindrical coordinates? I think that is the way I would approach this problem.
 

FAQ: Transformation of Reynolds Equation from Cartesian to cylindrical

What is the Reynolds Equation?

The Reynolds Equation is a mathematical equation used to describe the flow of a viscous fluid between two surfaces in relative motion. It takes into account factors such as fluid viscosity, surface roughness, and pressure gradients.

Why is it important to transform the Reynolds Equation from Cartesian to cylindrical coordinates?

Transforming the Reynolds Equation from Cartesian to cylindrical coordinates allows for a more accurate representation of fluid flow in systems with cylindrical symmetry, such as pipes or bearings. It also simplifies the equation and makes it easier to solve.

How is the transformation from Cartesian to cylindrical coordinates done?

The transformation from Cartesian to cylindrical coordinates involves converting the variables in the Reynolds Equation from x and y to r and θ, where r is the distance from the center axis and θ is the angle around the axis. This is done using mathematical formulas and substitutions.

What are the benefits of using cylindrical coordinates in the Reynolds Equation?

Using cylindrical coordinates in the Reynolds Equation allows for a more intuitive understanding of fluid flow in systems with cylindrical symmetry. It also simplifies the equation and makes it easier to solve, as well as providing more accurate results for these types of systems.

Are there any limitations to using cylindrical coordinates in the Reynolds Equation?

While cylindrical coordinates are useful for systems with cylindrical symmetry, they may not be suitable for more complex systems with irregular shapes. In these cases, other coordinate systems may need to be used or additional equations may need to be incorporated into the analysis.

Similar threads

Thermodynamics: Enthelpy and energy change of a column of air
Replies
1
Views
2K
I Cylindrical coordinates -Curvilinear
Replies
1
Views
1K
I Generating Divergence Equation In Cylindrical Coordinates
Replies
3
Views
1K
Normal Stress Balance at a Fluid-fluid Interface
2
Replies
51
Views
7K
I Schwarzschild in Cartesian: Tricks for Transformation
Replies
18
Views
3K
Trouble with using the command WhenEvent in Mathematica
Replies
2
Views
1K
Show proof of point C in the given problem that involves Polar equation
Replies
3
Views
609
B Thoughts about Galilean transformations
Replies
13
Views
1K
MHB Change from cartesian coordinates to cylindrical and spherical
Replies
2
Views
2K
I Atmospheric Density vs Altitude in an O'Neil Cylinder
Replies
1
Views
1K

Hot Threads

Recent Insights

Back
Top