Consider the 2D symmetric separable solutions Φn(r,φ) in cylindrical coordinates Identify the separable solution that satisfies the solid-wall boundary conditions on the cylindrical surface of radius rc and the asymptotic condition Φ≃r^2 as r→∞. Show that at large distances (as r/rc→∞), this solutions approaches the extensional flow Use the complex potential W(z)=Φ(r,φ)+iΨ(r,φ) of this flow and hence its streamfunction W(r,φ). Sketch few streamlines of this flow to illustrate its properties.
Consider the 2D symmetric separable solutions Φn(r,φ) in cylindrical coordinates Identify the separable solution that satisfies the solid-wall boundary conditions on the cylindrical surface of radius rc and the asymptotic condition Φ≃r^2 as r→∞. Show that at large distances (as r/rc→∞), this solutions approaches the extensional flow Use the complex potential W(z)=Φ(r,φ)+iΨ(r,φ) of this flow and hence its streamfunction W(r,φ). Sketch few streamlines of this flow to illustrate its properties.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
Consider the 2D symmetric separable solutions Φn(r,φ) in cylindrical coordinates
Identify the separable solution that satisfies the solid-wall boundary conditions on the cylindrical surface of radius rc and the asymptotic condition
Φ≃r^2 as r→∞.
Show that at large distances (as r/rc→∞), this solutions approaches the extensional flow
Use the complex potential W(z)=Φ(r,φ)+iΨ(r,φ) of this flow and hence its streamfunction W(r,φ). Sketch few streamlines of this flow to illustrate its properties.
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