answer problem $3$

Assume a steady, $2$D flow and one velocity component is $v_x(x,y)=C{x}^m$

Where $C,m$ are constants.

a$)$ Assume the fluid is incompressible and $v_y(x,0)=0$; determine $v_y(x,y).$

b$)$ For what values of $m$ will the flow be irrotational? $($hint: see the definition of

irrotational flow$)$

Consider the flow$-$field $u=\frac{x}{t+t_0},v=v_0,W=0,$ Find:

$($i$)$ Streamline: equation of a streamline passing through a point $(x_0,y_0,z_0)$ at time

$t=t_0($the subscript $\text{'}0\text{'}$ is used symbolically and doesn't necessary mean the values are

zero$)$

$($ii$)$ Streakline: Equation of a coloured line at time, that is visible as a consequence of

dye injection at $(x_0,y_0,z_0),$ starting from $t=t_0$ and continuing till $t=2t_0$

$($iii$)$ Pathline: Locus of a fluid particle that is at $(x_0,y_0,z_0)$ at time $t=t_0$

Consider a velocity field $u_{?}=x{e}_{{?}_{?}}(x)-y{e}_{{?}_{?}}(y).$

a$)$ Determine the equation for streamlines and type of flow $($i$.$e$.$ translational,

extensional, and rotational$)$

b$)$ Find the relationship between stream function and vorticity