# Lagrangian description of fluid motion

1. Nov 16, 2015

### MartinKitty

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

2. Relevant equations
$$v(t,e_1,e_2)=\frac{d}{dt} x(t,e_1,e_2)$$
$$a(t,e_1,e_2)=\frac{d}{dt} v(t,e_1,e_2)$$
$$\psi(t,e_1,e_2)=\lambda x_1 x_2$$
3. The attempt at a solution
Calculations of velocity:
$$V_1(t,e_1,e_2)= e_1[(e^\lambda)^t\lambda cos\omega t-(e^\lambda)^t \omega sin\omega t]+e_2[(e^\lambda)^t\lambda sin\omega t+(e^\lambda)^t \omega cos\omega t]$$
$$V_2(t,e_1,e_2)= e_1[(e^-\lambda)^t\lambda sin\omega t-(e^-\lambda)^t \omega cos\omega t]+e_2[(-e^-\lambda)^t\lambda cos\omega t-(e^-\lambda)^t \omega sin\omega t]$$

2. Nov 21, 2015