Steady, incompressible flow consist of following velocity components:

$u=\dfrac{cx}{y} \\ v=c \bullet ln(xy)$

Obtain a relevant stream function for the fluid flow.

Given that the above velocity components represent a Steady and Incompressible flow,

$u=\dfrac{cx}{y} \\ v=c∙ln(xy)$

We know that the relation between the stream function and the velocity functions are as follows,

$u=\dfrac{∂ψ}{∂y} \\ v=-\dfrac{∂ψ}{∂x}$

Now,

$\dfrac{∂ψ}{∂y}=\dfrac{cx}{y}$

Integrating with respect to y treating x as a constant,

$ψ=cx∙ln⁡(y)+fn(x).............. (1)$

Also,

$\dfrac{∂ψ}{∂x}=-c∙ln(xy)$

Integrating with respect to x treating y as a constant,

$ψ=∫-c∙ln(xy)dx \\ ψ=∫-c∙[ln(x)+ln⁡(y)]dx \\ ψ=-c\{∫ln(x)dx+x∙ln⁡(y)\}+fn(y)$

Using Integration by parts,

$$ψ=-c \bigg\{ln(x)∙x-∫ \dfrac 1x x dx+x∙ln⁡(y)\bigg\}+fn(y)$$

$ψ=-c\bigg\{ln(x)∙x-x+x∙ln⁡(y)\bigg\}+fn(y) \hspace{1cm}\bigg\{∵∫uv dx=u∫v dx-∫\bigg[\dfrac{du}{dx} ∫v dx \bigg]\bigg\} \\ ψ=-cx \{ln(x)+ln⁡(y)-1 \}+fn(y).........… (2)$

Comparing equation (1) and (2)

$ψ=cx\{1-ln(xy)\}$

…this is the required stream function.