Answer
Plot $II$
Work Step by Step
Let us set $v$ as constant, say $k$.
$x=sinu$, $y=cosu sink$, $z=sink$
Let us name $sink$ as $a$
$x=sinu$, $y=acosu$, $z=a$
This is an ellipse parallel to $xy$ plane.
Only Plot $II$ has such grid curves.