Question

find the stream function(x,y,t)for the given velocity field x=ut,v=x
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Answer 1

Step-by-step explanation:

Let us consider a velocity field flow where velocity vector components are given by u and v . To satisfy the previous condition, the following equation holds:

d⃗ s=λV⃗

Which translates to :

d⃗ s∧V⃗ =0⃗

or :

dxu=dyv

Hence, to get the velocity vector components, one just need to perform derivation. The u component will be given by :

u=∂ψ∂y

While the v component will be given by :

v=−∂ψ∂x

Let me draw out an example. Consider the streamline function :

ψ=myx2+y2

We can fairly deduce the following for velocity vector u component :

u=∂ψ∂y=mx2−y2(x2+y2)2

and also for velocity vector v component :

v=−∂ψ∂x=m2xy(x2+y2)2 .