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Discuss under what conditions do both the stream function ψ and the velocity potential ϕ exist...

Discuss under what conditions do both the stream function ψ and the velocity potential ϕ exist for a flow field? When does one exist but not the other?

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When the flow is 2 dimensional irrotational then both (potential function and stream function) will exist.

When the flow is 2 dimensional rotational then only stream function will exist and potential function will not exist (because potential function does not exist in rotational flow)

When the flow is 3 dimensional irrotational then potential function will exist but stream function will not exist.

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