Explicit transitive flow on disc

$D_ntriangleq left{x in mathbb{R}^n:, |x|leq 1right}$ with its subspace topology. By a transitive flow on $D_n$ I mean a continuous function
$$
phi: [0,1]times D_nrightarrow D_n,
$$
which is continuously differentiable in its first argument, such that the set
$$
left{
phi(t,x): tin [0,1]
right}
$$
is dense in $D_n$ for some $x in D_n$.

Are there explicit examples, in closed-form, for such a function $phi$?

Note:
If we only require $phi$ to be continuous, then the Hahn–Mazurkiewicz guarantees the existence of such a surjective continuous function:
$$
psi:[0,1]rightarrow D_n.
$$
Taking $phi(t,x)triangleq psi(t)$ gives the existence of a continuous such function. However, this result doesn’t guarantee that $phi$ is smooth or give a closed-form expression in this case…