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The purpose of the present paper is to model a cavity, which is equally divided vertically by a thin, flexible membrane. The membranes are inevitable components of many engineering devices such as distillation systems and fuel cells. In the present study, a cavity which is equally divided vertically by a thin, flexible membrane is model using the fluid–structure interaction (FSI) associated with a moving grid approach.

The cavity is differentially heated by a sinusoidal time-varying temperature on the left vertical wall, while the right vertical wall is cooled isothermally. There is no thermal diffusion from the upper and lower boundaries. The finite-element Galerkin technique with the aid of an arbitrary Lagrangian–Eulerian procedure is followed in the numerical procedure. The governing equations are transformed into non-dimensional forms to generalize the solution.

The effects of four pertinent parameters are investigated, i.e., Rayleigh number (104 = Ra = 107), elasticity modulus (5 × 1012 = E_T = 1016), Prandtl number (0.7 = Pr = 200) and temperature oscillation frequency (2p = f = 240p). The outcomes show that the temperature frequency does not induce a notable effect on the mean values of the Nusselt number and the deformation of the flexible membrane. The convective heat transfer and the stretching of the thin, flexible membrane become higher with a fluid of a higher Prandtl number or with a partition of a lower elasticity modulus.

The authors believe that the modeling of natural convection and heat transfer in a cavity with the deformable membrane and oscillating wall heating is a new subject and the results have not been published elsewhere.

This paper investigates a numerical treatment to steady mixed convection in a lid-driven square cavity with arc-shaped moving wall or lid. The horizontal walls are thermally insulated. The vertical left wall is kept isothermally at high temperature, while the right arc-shaped moving wall is kept isothermally at low temperature.

Finite difference method in Cartesian coordinates with the upwind scheme is used in numerical solution. The irregular curved boundary has been treated by invoking non-uniform mesh grid with the ability to generate boundary fitted nodes. Jensen’s formulas of Neumann’s boundary condition have derived for the non-uniform mesh grid. The arc-shaped moving wall is considered as a segment of a rotating cylinder; thus, the studied pertinent parameters are the rotational speed of the arc-shaped wall in both aiding and opposing directions ω = −1,000-1,000, the arc-wall radius Ro = 0.5099-1.534 which is governed by its center (X₀, Y₀) = (1.1, 0.5)-(2.45, 0.5) and the Rayleigh number Ra = 10³ − 10⁶.

The results have shown that for low Rayleigh numbers, the rotational speed enhances heat transfer irrespective to the direction of rotation, while for high Rayleigh numbers, the aiding anticlockwise rotation (negative ω) enhances the heat transfer, while the opposing clockwise rotation (positive ω) manifests a retardation effect on the heat transfer. For a motionless arc-wall, its radius is ineffective for aiding heat transfer, while for non-zero arc-shaped wall speed, the heat transfer is an increasing function of its radius.

The arc-shaped moving wall has never been investigated until now. Therefore, the originality of this paper is due to studying the mixed convection in a lid-driven cavity with moving arc-shaped wall and inspecting the effect of its curvature and rotational speed in both directions on the flow and thermal fields.