  We apply conformal flows of metrics restricted to the orthogonal distribution
$D$ of a foliation to study the question: Which foliations admit a metric such
that the leaves are totally geodesic and the mixed scalar curvature is
positive? Our evolution operator includes the integrability tensor of $D$, and
for the case of integrable orthogonal distribution the flow velocity is
proportional to the mixed scalar curvature. We observe that the mean curvature
vector $H$ of $D$ satisfies along the leaves the forced Burgers equation, this
reduces to the linear Schr\"{o}dinger equation, whose potential function is a
certain "non-umbilicity" measure of $D$. On order to show convergence of the
solution metrics $g_t$ as $t\to\infty$, we normalize the flow, and instead of a
foliation consider a fiber bundle $\pi: M\to B$ of a Riemannian manifold $(M,
g_0)$. In this case, if the "non-umbilicity" of $D$ is smaller in a sense then
the "non-integrability", then the limit mixed scalar curvature function is
positive. For integrable $D$, we give examples with foliated surfaces and
twisted products.
