$(1)$Two$-$Dimensional Stream Function

Given a single scalar function $\backslash$psi $,$ the components of a vector quantity, U$=$ u$$i $+$ v $$j$,$ can be calculated as

follows:

u $=\backslash$psi $/$y $(1)$

v $=\backslash$psi $/$x $(2)$

$(2)$ Different Ideal Flow Patterns

$(2\mathrm{.}1)$ Flow Impinging on a Flat Surface

Calculate $\backslash$psi $,$ u$,$ and v on a plane, where xmin $=$ ymin $=10$ and xmax $=$ ymax $=10.$ Use different grid spacing$$s$,$ save your results into a file and visualize the results.

The $\backslash$psi function is given by:

$\backslash$psi $($x$,$ y$)=2$ A x y $(3)$

where A $=1.$

$(2\mathrm{.}2)$ Flow Around An Ideal Vortex

Calculate $\backslash$psi $,$ u$,$ and v on a plane, where xmin $=$ ymin $=10$ and xmax $=$ ymax $=10.$ Use different grid spacing$$s$,$ save your results into a file and visualize the results.

The $\backslash$psi function is given by:

$\backslash$psi $($x$,$ y$)=(\backslash$Gamma $/2\backslash$pi $)$ ln p sqr$(($x$^2)+($y$^2))(4)$

where $\backslash$Gamma $=1.$

$(2\mathrm{.}3)$ Flow Around A Two$-$Dimensional Half$-$Body

Calculate $\backslash$psi $,$ u$,$ and v on a plane, where xmin $=20,$ ymin $=10$ and xmax $=20,$ ymax $=10.$ Use different grid spacing$$s$,$ save your results into a file and visualize the results.

The $\backslash$psi function is given by:

$\backslash$psi $($x$,$ y$)=$ U y $+($m$/2\backslash$pi $)$tan$^1($y$/$x$)(5)$

where U $=1,$ and m $=1.$ Solve in python code with comments