Control of macrosegregation phenomena and deformation-related defects is a main issue in steel continuous casting. Numerical simulation could help industrial engineers to master these defects. However, as a first step, it is essential to achieve a concurrent computation of fluid flow in the bulk liquid and stress-strain evolution in the already solidified regions. With this aim in view, a new specific partitioned solver has been developed to model the liquid flow, essentially induced by the inlet jet distributed by the submerged nozzle, as well as the thermal deformation of the solid shell. The solver procedure allows simulating the transient regime, up to convergence to the steady-state regime. For this purpose, the computational finite element mesh moves and grows continuously. Within this evolving mesh, three different zones are defined: the solid shell as a pure Lagrangian zone, the liquid nozzle region as a pure Eulerian zone, and an intermediate Eulerian-Lagrangian zone. Conservation equations (energy, mass, and momentum) are solved in a general arbitrary Lagrangian-Eulerian framework, with a level-set formulation to track the free surface evolution at the meniscus. The article is composed of two parts. In the first part, the model is detailed with the resolution steps involved in the coupled resolution approach. In the second part, a simple verification test case is firstly proposed, followed by a more relevant and practical application to model an industrial pilot continuous casting process.