Group classification of one-dimensional equations of capillary fluids where the specific energy is a function of density, density gradient and entropy

•Complete group classification of the equations describing non-isentropic flows of fluids with internal inertia is presented.•The algebraic approach for the group classification is applied.•The group classification separates all models into 82 different classes according to the admitted Lie group.

An application of group analysis provides a regular procedure for mathematical modeling by classifying differential equations with respect to arbitrary elements. This paper presents the group classification of one-dimensional equations of fluids where the internal energy is a function of the density, the gradient of the density and the entropy. The group classification separates all models into 83 different classes according to the admitted Lie group. Some invariant solutions are studied.