$3.$ A steady velocity field is given by the expression: $\backslash (\backslash$vec$\{$V$\}=$x y$^\{2\}\backslash$hat$\{\backslash$imath$\}+2\backslash$hat$\{\backslash$jmath$\}\backslash ),$ where velocities are in $\backslash (\backslash$mathrm$\{$m$\}/\backslash$mathrm$\{$sec$\}\backslash ),$ and distances are measured in meters.

a$)$ Using the axes shown below, carefully plot the streamlines passing through points $\backslash ((1,0)\backslash )$ and $\backslash ((2,0)\backslash ).$ The steamlines should include arrows indicating the flow direction. Both streamlines should be drawn continuously over the range of $\backslash ($ x $\backslash )$ between $1$ and $4.($For the sake of neatness and accuracy, I$\text{'}$d strongly encourage the use of Excel or equivalent; i$.$e$.$ the graph indicated below is only provided to clarify the region to focus your attention.$)$

b$)$ At the location $\backslash ($ x$=3\mathrm{.}0\backslash ),$ carefully sketch the velocity vector on both streamlines, providing the u and v components, the velocity magnitude, as well as the flow angle, $\backslash (\backslash$theta $\backslash ),$ relative to the $\backslash ($ x $\backslash )-$direction.

c$)$ In the plane at $\backslash ($ x$=3\mathrm{.}0\backslash ),$ approximate what the velocity profile looks like between the two streamlines. You don't need an equation. Use the information obtained above for the two streamlines to approximate what the velocity profile will probably look like.

d$)$ Assuming the streamlines represent the flow of air, compute the mass flow rate between the streamlines in $\backslash (\backslash$mathrm$\{$kg$\}/\backslash$mathrm$\{$sec$\}\backslash ).($assume the depth into the paper is $1$ meter$).$