Fluid–structure interaction analysis of free convection in an inclined square cavity partitioned by a flexible impermeable membrane with sinusoidal temperature heating S. A. M. Mehryan . A. J. Chamkha . M. A. Ismael . M. Ghalambaz Received: 30 May 2016 / Accepted: 17 February 2017 / Published online: 27 February 2017 Ó Springer Science+Business Media Dordrecht 2017 Abstract The problem of unsteady natural convec- tion inside an inclined square cavity partitioned by a flexible impermeable membrane is studied numeri- cally using the finite-element method along with the Arbitrary Lagrangian–Eulerian (ALE) approach. The bottom and top walls of the cavity are kept adiabatic. The left side wall is kept isothermal at a high temperature, while the right wall is cooled in a sinusoidal fashion. The cavity is provided by two eyelets to compensate volume changes due to the movement of the flexible membrane. The studied pertinent parameters are the Rayleigh number (in the range of 1E4–1E7), the amplitude of the sinusoidal wall temperature (A in the range of 0–1.0), the inclination angle of the cavity (in the range of -p/3 to p/3), and the body force parameter (Fv in the range of -1.64E-2 to ?1.64E-2) whereas the Prandtl number is fixed at 6.2. The results show that at a low Rayleigh number, the membrane shape is a function of the imposed body force. While at a high Rayleigh number, the buoyancy force becomes responsible for the membrane deflection. The natural convection is appreciably affected by the inclination angle of the cavity which in turn, affects the concave or convex shape of the membrane. Keywords Flexible membrane  Natural convection  Inclined cavity  Arbitrary Lagrangian– Eulerian method List of symbols A Amplitude of sinusoidal function d s Displacement vector E Dimensional Young’s modulus E s Non-dimensional elasticity modulus F v Body force vector g Gravitational acceleration vector L Cavity size P Pressure Pr Prandtl number Ra Thermal Rayleigh vector S. A. M. Mehryan Young Researchers and Elite Club, Yasooj Branch, Islamic Azad University, Yasooj, Iran e-mail: a.mansuri1366@gmail.com A. J. Chamkha Mechanical Engineering Department, Prince Mohammad Bin Fahd University, Al-Khobar 31952, Saudi Arabia e-mail: achamkha@pmu.edu.sa A. J. Chamkha Prince Sultan Endowment for Energy and Environment, Prince Mohammad Bin Fahd University, Al-Khobar 31952, Kingdom of Saudi Arabia M. A. Ismael Mechanical Engineering Department, Engineering College, University of Basrah, Basrah, Iraq e-mail: muneerismael@yahoo.com M. Ghalambaz (&) Department of Mechanical Engineering, Dezful Branch, Islamic Azad University, Dezful, Iran e-mail: m.ghalambaz@iaud.ac.ir 123 Meccanica (2017) 52:2685–2703 DOI 10.1007/s11012-017-0639-8