Help me, / Advanced Fluid dynamics Problems

[skyt3]

Help me, please / Advanced Fluid dynamics Problems

Homework Statement

Evaluate the radial component of the inertia term (u dot Del)u in cylindrical
coordinates using the following identities:

x=RcosΘ
y=RsinΘ
uEx+vEy = UrEr+UΘEΘ

and any other vector identities from Appendix A as required. Here R
and y are cylindrical coordinates, Ur and UΘ are the corresponding
velocity components, and Er, EΘ are the unit base vectors.

Homework Equations

Lagrangian derivative:

The Attempt at a Solution

 

[Valkarie]

The definition of the Lagrangian derivative is as follows: (u dot Del)u = (uEx + vEy) . (Del uE + Del vE) Using the identities given in the problem statement, this simplifies to: (UrEr + UΘEΘ).(Del UrEr + Del UΘEΘ) We can then use the following vector identities from Appendix A: Del UrEr = (Del Ur)Er + Ur(Del Er) and Del UΘEΘ = (Del UΘ)EΘ + UΘ(Del EΘ) Therefore, (u dot Del)u = (UrEr + UΘEΘ).((Del Ur)Er + Ur(Del Er) + (Del UΘ)EΘ + UΘ(Del EΘ)) Now, using the fact that (Del Er) and (Del EΘ) are both zero, we can simplify the equation to: (u dot Del)u = (Del Ur)Ur + (Del UΘ)UΘ The radial component of the inertia term is then: (u dot Del)uR = (Del Ur)Ur