The current paper investigates the role of fluid-structure interaction (FSI) in the mixed convection inside a square cavity having two inlet and outlet openings. Flexible elastic fin is attached to the bottom wall of the cavity. The cavity is differentially heated by maintaining the vertical walls at two different temperatures. Equations govern the unsteady fields of fluid, thermal and stresses are solved numerically using the Galerkin Finite Element Method implemented in Arbitrary Lagrangian–Eularian (ALE) approach. The governing parameters of the present geometry are Cauchy number, which reflects the inertia to elastic forces, (Ca = 10−12 – 2 × 10−4), proximity of fin to the inlet opening (Xf = 0.2 – 0.8), Richardson number (Ri = 0.1 – 100) and the Reynolds number (Re = 50 – 250). The results show that the flexible fin enhances the Nusselt number better than the rigid fin. The shape of the fin and the Nusselt number reach the steady periodic state at higher values of Cauchy and Richardson numbers. It is also found that the average Nusselt number increases with Cauchy number. However, at very high values of Richardson number, the fin material should not be very elastic.