Transverse expansion of (1 + 2) dimensional magneto-hydrodynamics flows with longitudinal boost invariance

In the present work, we investigate the effects of the magnetic field on expanding hot and dense nuclear matter as an ideal fluid. We consider QGP, in the particular case of a (1 + 2) dimensional longitudinally boost-invariant fluid expansion, in the background of an inhomogeneous magnetic field that is generated by external sources. We assume the magnetic field points in the direction perpendicular to the reaction plane, follows the power-law decay in proper time, and has two components on the transverse plane. To simplify our calculation, we suppose the investigated fluid has azimuthal symmetry, and magneto-hydrodynamic equations are described in a polar coordinate system on the transverse plane of reaction. Our results depict the space-time evolution of the transverse expansion of the fluid in the presence of an inhomogeneous external magnetic field. Moreover, we show when the magnetic field decays in proper time τ with a power-law τn/2(n<1), two distinct solutions can be found depending on the values of n. We show that n<−2 gives rise to the physical scenario whereas the n>−2 leads to non-physical results.