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Consider a velocity field $u = x\hat{i}+2y\hat{j}$. To find its stream function,$$\frac{\partial\psi}{\partial y}=x$$ $$\psi = xy + f(x)$$ and $$\frac{\partial\psi}{\partial x}=-2y$$ $$\psi=-2xy+g(y)$$ I am stuck at this point because both $\psi$ do not agree with each other and arbitrary functions $f$ and $g$ are not single variable if we proceed further. Please help in proceeding further to find the stream function of this velocity field.

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The given velocity field does not correspond to incompressible flow since the continuity equation is not satisfied, i.e.,

$$\nabla \cdot \mathbf{u} = \frac{\partial u }{\partial x} + \frac{\partial v}{\partial y} = \frac{\partial}{\partial x} (x) + \frac{\partial }{\partial y} (2y) = 3 \neq 0$$

The streamfunction does not exist in this case.

A proof of the existence of a streamfunction for two-dimensional, incompressible flow satisfying $\psi_y = u, \, \psi_x = -v$ is given here.

Aside

This velocity field is irrotational with

$$\nabla \times \mathbf{u} = \frac{\partial v }{\partial x} - \frac{\partial u}{\partial y} = 0,$$

and can be expressed in terms of a poatential

$$\mathbf{u} = - \nabla \phi,$$

where $\phi = -x^2/2 - y^2 +C$.

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