Impact of a hot constructal tree-shaped fin on the convection flow of single wall carbon nanotube water nanofluid inside a sinusoidal enclosure

Heat transfer due to natural convection in different types of fins embedded closed enclosures are widely used in various fields including petrochemicals, solar collectors, heat exchangers, gas turbines, semiconductor devices, automobile radiators, etc. Herein, the natural convection flow of a water-based single-wall carbon nanotube (SWCNT) nanofluid inside a novel sinusoidal closed geometry is investigated. The horizontal bottom and top boundaries of the enclosure are presumed to be hot and thermally adiabatic, respectively. The left and right vertical sinusoidal boundaries are assumed to be cold. A hot constructal tree-shaped fin is placed at the center of the hot bottom wall. The impact of the simple constructal heated tree-shaped fin on steady, two-dimensional, laminar, and incompressible flow of the water-based SWCNT nanofluid inside the sinusoidal enclosure is analyzed. The mathematical formulation of water-based SWCNT nanofluid flow inside the enclosure is formulated using the Navier–Stokes equations under the Boussinesq approximation. A modified effective thermal conductivity model of carbon nanotubes including the radius of water molecules and carbon nanotubes is employed. Galerkin finite element simulations are conducted to study the flow and heat transfer characteristics via stream function, isotherms, local Nusselt number, and averaged Nusselt number graphs. The combined effect of the amplitude of the sinusoidal wall (A = 0.1, 0.15, and 0.2) with the Rayleigh number (R_a = 10⁴–10⁶), nanotube volume fraction (φ = 0.01–0.05), and the thickness of the tree-shaped fin (B = 0.02–0.06) are comprehensively studied. The temperature gradient decreases with increasing nanotube volume fraction and the sinusoidal wall amplitude, while it increases above the fin with the thickness of the tree-shaped fin. Moreover, the averaged Nusselt number increases with the thickness of the tree-shaped fin by 14.72%, 16.74%, and 19.6% for A = 0.1, 0.15, and 0.2 along the fin, respectively. The averaged Nusselt number along the fin is an increasing function of the Rayleigh number, nanotube volume fraction, thickness of the tree-shaped fin, and amplitude of waviness. Additionally, a novel correlation for the averaged Nusselt number along the fin is derived.