Model study of the energy dependence of the correlation between anisotropic flow and the mean transverse momentum in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>Au</mml:mi><mml:mo>+</mml:mo><mml:mi>Au</mml:mi></mml:mrow></mml:math> collisions

A hybrid model that employs the hadron-string transport UrQMD and (3+1)D relativistic viscous hydrodynamic code vHLLE, is used to investigate beam energy dependence of correlation coefficient $\rho(v^{2}_{2},[p_{T}])$ between average transverse momentum $[p_{T}]$ hadrons emitted in an event square anisotropic flow $v_2^2$. For Au+Au collisions, predicts characteristic patterns for event-shape variances $v_n^2$ (${\rm Var}([p_T])$ ${\rm Var}(v_2^2)$), covariance cov}(v_{2}^{2},[p_T])$), consistent with attenuation effects specific shear viscosity $\eta/s$. In contrast, predicted be insensitive but sensitive initial-state geometry collisions. These observations suggest a precise set measurements Var}([p_T])$, Var}(v_2^2)$, cov}(v_{2}^{2},[p_T])$ as function beam-energy event-shape, could aid precision extraction temperature baryon chemical-potential $\eta/s$ from wealth data obtained RHIC scan.