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
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
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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
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