Dynamical Stability in presence of non-minimal derivative dependent coupling of $k$-essence field with a relativistic fluid

In this paper we investigate a non-minimal, space-time derivative dependent, coupling between the $k$-essence field and a relativistic fluid using a variational approach. The derivative coupling term couples the space-time derivative of the $k$-essence field with the fluid 4-velocity via an inner product. The inner product has a coefficient whose form specifies the various models of interaction. By introducing a coupling term at the Lagrangian level and using the variational technique we obtain the $k$-essence field equation and the Friedmann equations in the background of a spatially flat Friedmann-Lemaitre-Robertson-Walker (FLRW) metric. Explicitly using the dynamical analysis approach we analyze the dynamics of this coupled scenario in the context of two kinds of interaction models. The models are distinguished by the form of the coefficient multiplying the derivative coupling term. In the simplest approach we work with an inverse square law potential of the $k$-essence field. Both of the models are not only capable of producing a stable accelerating solution, they can also explain different phases of the evolutionary universe.