**Authors:**
V. Batishchev,
V. Getman

**Abstract:**

We study the self-similar fluid flows in the Marangoni layers with the axial symmetry. Such flows are induced by the radial gradients of the temperatures whose distributions along the free boundary obey some power law. The self-similar solutions describe thermo-capillar flows both in the thin layers and in the case of infinite thickness. We consider both positive and negative temperature gradients. In the former case the cooling of free boundary nearby the axis of symmetry gives rise to the rotation of fluid. The rotating flow concentrates itself inside the Marangoni layer while outside of it the fluid does not revolve. In the latter case we observe no rotating flows at all. In the layers of infinite thickness the separation of the rotating flow creates two zones where the flows are directed oppositely. Both the longitudinal velocity and the temperature have exactly one critical point inside the boundary layer. It is worth to note that the profiles are monotonic in the case of non-swirling flows. We describe the flow outside the boundary layer with the use of self-similar solution of the Euler equations. This flow is slow and non-swirling. The introducing of an outer flow gives rise to the branching of swirling flows from the non-swirling ones. There is such the critical velocity of the outer flow that a non-swirling flow exists for supercritical velocities and cannot be extended to the sub-critical velocities. For the positive temperature gradients there are two non-swirling flows. For the negative temperature gradients the non-swirling flow is unique. We determine the critical velocity of the outer flow for which the branching of the swirling flows happens. In the case of a thin layer confined within free boundaries we show that the cooling of the free boundaries near the axis of symmetry leads to the separating of the layer and creates two sub-layers with opposite rotations inside. This makes sharp contrast with the case of infinite thickness. We show that such rotation arises provided the thickness of the layer exceed some critical value. In the case of a thin layer confined within free and rigid boundaries we construct the branching equation and the asymptotic approximation for the secondary swirling flows near the bifurcation point. It turns out that the bifurcation gives rise to one pair of the secondary swirling flows with different directions of swirl.

**Keywords:**
free surface,
rotation,
fluid flow,
bifurcation,
boundary layer,
Marangoni layer

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