Transformation of Reynolds Equation from Cartesian to cylindrical

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SUMMARY

The discussion focuses on the transformation of the Reynolds Equation from Cartesian to cylindrical coordinates, specifically addressing the mathematical representation of fluid dynamics. The equations presented include the original Cartesian form and its cylindrical counterpart, highlighting the importance of understanding the derivation process. Participants emphasize reviewing the steps in the Cartesian derivation before attempting the cylindrical transformation to ensure accuracy and comprehension.

PREREQUISITES
  • Understanding of fluid dynamics principles
  • Familiarity with the Reynolds Equation
  • Knowledge of partial differential equations
  • Basic skills in coordinate transformations
NEXT STEPS
  • Study the derivation of the Reynolds Equation in Cartesian coordinates
  • Learn about coordinate transformations in fluid dynamics
  • Explore applications of the Reynolds Equation in cylindrical coordinates
  • Investigate numerical methods for solving partial differential equations
USEFUL FOR

Students and professionals in fluid dynamics, mathematicians focusing on differential equations, and engineers working with cylindrical coordinate systems will benefit from this discussion.

rakan
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TL;DR
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)
 
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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.
 

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