The motion of a very viscous fluid in the two-dimensional (wedge) region −α < φ can be described, in (ρ, φ) coordinates, by the (biharmonic) equation together with the boundary conditions ∂ψ/∂φ = 0 at φ = ±α, which represent the fact that there is no radial fluid velocity close to either of the bounding walls because of the viscosity, and ∂ψ/∂ρ = ±ρ at φ = ±α, which impose the condition that azimuthal flow increases linearly with r along any radial line. Assuming a solution in separated-variable form, show that the full expression for ψ is ∇^2∇^2φ\:≡\:∇^4φ\:=\:0

Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
icon
Related questions
Question

3. The motion of a very viscous fluid in the two-dimensional (wedge) region −α < φ can be described, in (ρ, φ) coordinates, by the (biharmonic) equation together with the boundary conditions ∂ψ/∂φ = 0 at φ = ±α, which represent the fact that there is no radial fluid velocity close to either of the bounding walls because of the viscosity, and ∂ψ/∂ρ = ±ρ at φ = ±α, which impose the condition that azimuthal flow increases linearly with r along any radial line. Assuming a solution in separated-variable form, show that the full expression for ψ is ∇^2∇^2φ\:≡\:∇^4φ\:=\:0 

Expert Solution
steps

Step by step

Solved in 3 steps

Blurred answer
Knowledge Booster
Radiation
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, mechanical-engineering and related others by exploring similar questions and additional content below.
Similar questions
  • SEE MORE QUESTIONS
Recommended textbooks for you
Elements Of Electromagnetics
Elements Of Electromagnetics
Mechanical Engineering
ISBN:
9780190698614
Author:
Sadiku, Matthew N. O.
Publisher:
Oxford University Press
Mechanics of Materials (10th Edition)
Mechanics of Materials (10th Edition)
Mechanical Engineering
ISBN:
9780134319650
Author:
Russell C. Hibbeler
Publisher:
PEARSON
Thermodynamics: An Engineering Approach
Thermodynamics: An Engineering Approach
Mechanical Engineering
ISBN:
9781259822674
Author:
Yunus A. Cengel Dr., Michael A. Boles
Publisher:
McGraw-Hill Education
Control Systems Engineering
Control Systems Engineering
Mechanical Engineering
ISBN:
9781118170519
Author:
Norman S. Nise
Publisher:
WILEY
Mechanics of Materials (MindTap Course List)
Mechanics of Materials (MindTap Course List)
Mechanical Engineering
ISBN:
9781337093347
Author:
Barry J. Goodno, James M. Gere
Publisher:
Cengage Learning
Engineering Mechanics: Statics
Engineering Mechanics: Statics
Mechanical Engineering
ISBN:
9781118807330
Author:
James L. Meriam, L. G. Kraige, J. N. Bolton
Publisher:
WILEY