Consider the planar velocity field

vec$(U)=(f(x,y),-Bx{y}^3)$

with the constant $B>0.$

a$)$ Determine the function $f(x,y)$ such that the flow is incompressible. Choose the simplest

solution $($integration constant equals zero$).$

b$)$ Calculate the curl grad$$vec$(U)$ of this velocity field.

c$)$ Determine the region in the xy$-$plane where the curl rotvec$(U)=0.$ What is the velocity vec$(U)=0$ in

this region?

d$)$ Determine, up to an integration constant, the equation $y=y(x)$ of the streamlines.

e$)$ Determine the integration constants in the streamline equation for two streamlines, the first

passing through the point $(1,4)$ and the second through the point $(8,4).$

f$)$ Sketch qualitatively these two streamlines.

g$)$ Calculate the velocity gradient tensor $L_{ij}.$

h$)$ Determine the acceleration vector for this velocity field.

i$)$ In which regions of the flow is the shear deformation equal to zero?

j$)$ How can the rotation tensor ${}_{ij}$ be exprescod for the regions where the shear deformation is

equal to zero?