The beam energy dependence of the elliptic flow,$v_2$, is studied in mid-central Au+Au collisions in the energy range of $3leq sqrt{s_{NN}} leq 30$ GeV within the microscopic transport model JAM. The results of three different modes of JAM are compared; cascade-,hadronic mean field-, and a new mode with modified equations of state, with a first order phase transition (1.O.P.T.) and with a crossover transition. The standard hadronic mean field suppresses $v_2$, while the inclusion of the effects of a 1.O.P.T. (and also of a crossover transition) does enhance $v_2$ at $sqrt{s_{NN}}<30$ GeV. The enhancement or suppression of the scaled energy flow, dubbed elliptic flowis understood as being due to out of plane- flow, i.e. $v_2<0$, dubbed out of plane - squeeze-out, which occurs predominantly in the early, compression stage. Subsequently, the in-plane flow dominates, in the expansion stage, $v_2 > 0$. The directed flow, dubbed bounce- off, is an independent measure of the pressure, which quickly builds up the transverse momentum transfer in the reaction plane. When the spectator matter leaves the participant fireball region, where the highest compression occurs, a hard expansion leads to larger $v_2$. A combined analysis of the three transverse flow coefficients, radial $v_0$-, directed $v_1$- and elliptic $v_2$- flow, in the beam energy range of $3leqsqrt{s_{NN}}leq10$ GeV, distinguishes the different compression and expansion scenarios: a characteristic dependence on the early stage equation of state is observed. The enhancement of both the elliptic and the transverse radial flow and the simultaneous collapse of the directed flow of nucleons offers a clear signature if 1.O.P.T. is realized at the highest baryon densities created in high energy heavy-ion collisions.
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