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Given an orthogonal projection $P$ and a free unitary Brownian motion $Y =(Y_t)_{t \geq 0}$ in a $W^{\star}$-non commutative probability space such that$Y$ and $P$ are $\star$-free in Voiculescu's sense, the main result of thispaper states that $P$ and $Y_tPY_t^{\star}$ are in general position at any time$t$. To this end, we study the dynamics of the unitary operator$SY_tSY_t^{\star}$ where $S = 2P-1$. More precisely, we derive a partialdifferential equation for the Herglotz transform of its spectral distribution,say $\mu_t$. Then, we provide a flow on the interval $[-1,1]$ in such a waythat the Herglotz transform of $\mu_t$ composed with this flow is governed byboth the Herglotz transforms of the initial ($t=0$) and the stationary ($t =\infty)$ distributions. This fact allows to compute the weight that $\mu_t$assigns to $z=1$ leading to the main result. As a by-product, the weight thatthe spectral distribution of the free Jacobi process assigns to $x=1$ followsafter a normalization. In the last part of the paper, we use combinatorics ofnon crossing partitions in order to analyze the term corresponding to theexponential decay $e^{-nt}$ in the expansion of the $n$-th moment of$SY_tSY_t^{\star}$.

General position of a projection and its image under a free unitary Brownian motion