Solving the Biharmonic Equation in Polar Coordinates: Tips and Techniques

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In summary, the conversation is about solving the biharmonic equation in polar or rectangular coordinates. The person asking has already searched for information but didn't understand the sources. They are questioning if separation of variables is a valid method for solving this equation. Another person recommends checking example 4.2-1 in the book "Transport Phenomena" for further insight. The original person thanks them for the suggestion, as this topic is related to their inquiry.
  • #1
member 428835
hey pf!

does anyone here have a link (or perhaps would care to share some info) on how to solve the biharmonic equation in polar coordinates (or, at least rectangular coordinates): $$ \nabla^4 \psi = 0$$ where [itex] \psi = f(r,\theta) [/itex]

i should say i have already done the obvious searches but didnt understand wikipedia and wolfram simply gave the answer, which is great but curiosity still has me. does separation of variables really work here?

thanks!
 
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  • #2
joshmccraney said:
hey pf!

does anyone here have a link (or perhaps would care to share some info) on how to solve the biharmonic equation in polar coordinates (or, at least rectangular coordinates): $$ \nabla^4 \psi = 0$$ where [itex] \psi = f(r,\theta) [/itex]

i should say i have already done the obvious searches but didnt understand wikipedia and wolfram simply gave the answer, which is great but curiosity still has me. does separation of variables really work here?

thanks!
See Transport Phenomena, by Bird, Stewart, and Lightfoot, example 4.2-1, p. 122.
Creeping Flow around a sphere.
 
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thanks chestermiller! ironically enough creeping flow around a sphere is the reason for my inquiry! i appreciate it!
 

Related to Solving the Biharmonic Equation in Polar Coordinates: Tips and Techniques

1. What is the Biharmonic Equation in Polar Coordinates?

The Biharmonic Equation in Polar Coordinates is a partial differential equation that describes the relationship between the Laplacian of a function and its fourth derivatives. It is commonly used in engineering and physics to solve problems involving elasticity, fluid mechanics, and electrostatics.

2. Why is solving the Biharmonic Equation in Polar Coordinates important?

Solving the Biharmonic Equation in Polar Coordinates allows us to understand and predict the behavior of physical systems with circular or cylindrical symmetry. It also provides a more accurate solution in certain situations compared to Cartesian coordinates.

3. What are some tips for solving the Biharmonic Equation in Polar Coordinates?

One tip is to use separation of variables to break down the equation into simpler parts. Another tip is to utilize symmetry to reduce the number of variables in the equation. Additionally, using appropriate boundary conditions and simplifying the equation through substitutions can also be helpful.

4. What techniques can be used to solve the Biharmonic Equation in Polar Coordinates?

Some techniques include using Fourier series, Green's functions, and integral transforms such as the Hankel transform. Numerical methods such as finite difference or finite element methods can also be used to approximate solutions.

5. Can the Biharmonic Equation in Polar Coordinates be applied to real-world problems?

Yes, the Biharmonic Equation in Polar Coordinates has many practical applications in various fields such as structural and material engineering, fluid dynamics, and electromagnetics. It can be used to model and solve problems involving circular or cylindrical objects or systems.

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