(dx/dt)=(R-Rc)X-aX^3 arises in the study of the transition from laminar flow to
ID: 3345600 • Letter: #
Question
(dx/dt)=(R-Rc)X-aX^3 arises in the study of the transition from laminar flow to turbulent flow. This differential equation is often called the Landau equation, after L.D. Landau.
Often in physical problems, some observable quantity depends on a parameter that, when increased, can reach a critical value at which the quantity suddenly changes character. In the above DE, we will consider R to be that changing parameter and let a and Rc be positive constants. Keep in mind that x remains a dependent variable of time.
a. when R<Rc, explain why there is only one equilibrium solution that is stable.
b. in the case R>Rc, why there are equilibrium solutuions at x=0 and x= +/- sqrt( (R-Rc)/a) ?
(You can find them algebraically- "use what you know!") Are there any others?
c. The special case, R=Rc(called a bifurcation point). if you sketch a rough graph in the R-x plane showing the equilibrium solutions you found above, You'll see what the word "bifurcation" means. Your graph should look like a pitchfork and your graph will NOT be a function!
Explanation / Answer
ANSWER IS AT
www.thermalfluidscentral.org/e-books/book-viewer.php?b=41&s..
Related Questions
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.