Lagrangian description of fluid motion

  • #1

Homework Statement


Find velocity, acceleration, stream function and vorticity. Prove that velocity is equal to the acceleration. Functions given:
[tex]X_1(t,e_1,e_2)= (e^\lambda)^t[e_1cos\omega t+e_2sin(\omega t)][/tex]
[tex]X_2(t,e_1,e_2)= (e^-\lambda)^t[-e_1sin\omega t+e_2cos(\omega t)][/tex]

Homework Equations


[tex]v(t,e_1,e_2)=\frac{d}{dt} x(t,e_1,e_2)[/tex]
[tex]a(t,e_1,e_2)=\frac{d}{dt} v(t,e_1,e_2)[/tex]
[tex]\psi(t,e_1,e_2)=\lambda x_1 x_2[/tex]

The Attempt at a Solution


Calculations of velocity:
[tex]V_1(t,e_1,e_2)= e_1[(e^\lambda)^t\lambda cos\omega t-(e^\lambda)^t \omega sin\omega t]+e_2[(e^\lambda)^t\lambda sin\omega t+(e^\lambda)^t \omega cos\omega t][/tex]
[tex]V_2(t,e_1,e_2)= e_1[(e^-\lambda)^t\lambda sin\omega t-(e^-\lambda)^t \omega cos\omega t]+e_2[(-e^-\lambda)^t\lambda cos\omega t-(e^-\lambda)^t \omega sin\omega t][/tex]
 

Answers and Replies

  • #2
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