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Chapter 6: Problem 97

Determine whether the Bernoulli equation can be applied between different radii for the vortex flow fields (a) \(\vec{V}=\omega r \hat{e}_{\theta}\) and (b) \(\vec{V}=\hat{e}_{\theta} K / 2 \pi r\)

Short Answer

Expert verified
For vortex flow fields: (a) Bernoulli's equation can be applied, as the conditions fulfill; (b) Bernoulli's equation cannot be applied, as the flow isn't irrotational even though it seems to be at first sight.

Step by step solution

01

Case (a): Evaluating the applicability of Bernoulli's equation

Given flow field for this case is \(\vec{V} = \omega r \hat{e}_{\theta}\). Here, the velocity field is irrotational as its curl is zero. Since the flow is purely rotational in the theta direction and there's no change in the radial direction or with time, it's steady. It doesn't depend on pressure, so it's incompressible. Based on these assessments, Bernoulli equation can be considered to be applicable.
02

Case (b): Evaluating the applicability of Bernoulli's equation

For this case, the flow field is \(\vec{V} = \hat{e}_{\theta} K / 2\pi r \). This flow field is azimuthal or circumferential, and inversely proportional to the radius \(r\). Even though it appears to be irrotational on first sight, on taking divergence it's found that its curl isn't zero, which indicates it is not irrotational. Hence, the use of the Bernoulli's equation wouldn't be correct in this scenario.

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