Marangoni convection in a C-shape enclosure with partially heated walls

A numerical study was carried out to investigate Marangoni convection of nanofluid in a C-shape cavity with partially heated walls. The opposite sides of the walls are cooled at constant temperature while the rest of the partitions are kept adiabatic. The governing equations and boundary conditions are then introduced to describe the fluid flow and temperature distribution within the enclosure before the equations are non-dimensionalised and solved using the finite element method. The solutions, presented as streamlines, isotherms, local Nusselt and average Nusselt for varying Marangoni number, Rayleigh number, and depth, are then discussed.


Introduction
With the ever-increasing dependence on technology and machines in most, if not all industries, the demand for better cooling solutions and fluids that are capable of transferring heat at a high rate has never been greater.One such option is the Marangoni convection (Ma), which has been a topic of interest for the past few decades.Some of its applications include electronic parts cooling, solar water heating and atmospheric fallout.Several investigations and experiments have been conducted that considered the surface tension temperature gradient under various conditions and cavities, including C-shape enclosures.Mansour et al. [1] investigated natural convection fluid circulation and heat transfer inside a C-shape enclosure filled with nanofluid.It was discovered that the heat transfer rate improved as the aspect ratio of the cavity decreased.Mohebbi et al. [2] also conducted a study of natural convection in a C-shape enclosure, though the cavity contained a heat source that was considered in various locations inside the enclosure.They concluded that when the Rayleigh number (Ra) is high and that when the heat source was in a vertical wall, the Nusselt numbers (Nu) achieved its highest values.
Haq et al. [3] investigated heat transfer inside a C-shape cavity with partially heated walls filled with Single Wall Carbon Nanotubes.It was found that increasing the Ra helped enhance the stream flow and isotherms behaviour, though increasing the Hartmann number (Ha) had the opposite effect.Bakier [4] studied natural convection in a partially heated C-shape enclosure with an open end.The cavity was filled with nanofluid.He concluded that the presence of nanoparticles (ϕ) helped improve heat and mass transfer through the open section when the Ra is small.In addition to that, the temperature of the cavity increased as the heat source length became longer.
Nia et al. [5] investigated natural convection heat transfer of nanofluid in an L-shape enclosure with curved boundaries.They concluded that the curved L-shape models increased the Nu values significantly.Lugarini et al. [6] numerically studied natural convection and surface radiation in a heated wall with a C-shape fracture.It was concluded that heat transfer decreased when radiation was present.Yildiz et al. [7] investigated natural convection of nanofluid in a U-shape enclosure with cold ribs where

Mathematical formulation
The two-dimensional representation of Ma inside a nanofluid C-shape enclosure as shown in Figure 1 was considered with height (H) and length (L) where in this study both are assumed to be equal.The distance AD is denoted as d, while a is the length of BC and GF.The heated walls are denoted by Th while the cold walls, Tc.Marangoni convection transpires at the top wall GH while the rest of the partitions are adiabatic (BC and FG).The nanofluid inside the cavity is water-based nanofluid with Al2O3 nanoparticles (H2O-Al2O3).The thermo-physical properties are shown in Table 1.

Physical Property H2O Al2O3
[ ⋅  −3 ] 997.1 3600 For this study fluid flow is laminar, steady, and incompressible with viscous dissipation being absent.This assumption is made by assuming the fluid flows in infinitesimal parallel layers with no disruption between them.By using the following substitutions, The governing equations and boundary conditions based on these assumptions are, 0, UV XY with boundary conditions, 0, for walls AB, BC, CD, DE, EF and HA, 0, for walls CB and GF, 0, for wall GH, for walls HA and AB, 0, for walls CD, DE and EF, Ma 0, for wall GH, The definition of other parameters used in this investigation are depicted in Table 2, While, the Nuav is then computed by taking the integral of, ( )

Results and discussion
In this investigation, COMSOL 5.3a was utilised to solve the governing equations ( 1) -( 5) with respect to its boundary conditions (6) incorporated with the Finite Element Method (FEM) explained in Taylor and Hood [8] and Dechaumphai [9].The current study was investigated numerically for the values of 0.3 ≤ d ≤ 0.7, 0 ≤ Ma ≤ 10 3 and 10 3 ≤ Ra ≤ 10 4 with fixed Prandtl number, Pr = 0.052 and 0.03  = .The solutions obtained are compared with Saleh and Hashim [10] with the absolute error is less than 10% as tabulated in Table 3. Figure 2 illustrate fluid flow inside the enclosure, when Ma = 0, a single cell circulates counterclockwise inside the cavity and the strength of flow increases as d increases.When Ma ≠ 0, the single cell splits into two with the top one rotating clockwise and the bottom, counterclockwise.Furthermore, the top cell circulates faster and gets bigger as d increases.For Ra = 10 4 , the cells for all parameters oscillate at a higher rate and when d = 0.7, the cell retains its shape rather than splitting itself up.Increasing Ma, its produces two separate cells with the bottom one having a slightly weaker flow for the value of d = 0.3, whereas the bottom cell for d = 0.7 oscillates faster compared to the one closer to where Ma takes place.

Conclusion
Numerical study was conducted for Ma inside a C-shape cavity filled with nanofluid.The governing equations and boundary conditions based on the problem are non-dimensionalised and then solved using FEM before the results were collected and presented as streamlines, isotherms, local Nusselt and average Nusselt for various values of Ma, Ra and d.It can be concluded that, • the strength of the flow increases with Ma, Ra and d, • the highest values of Nuloc and Nuav were around d = 0.3 due to the hot and cold walls being closer, and • Ma has no effect on the horizontal heated wall.

Figure 1 .
Figure 1.Two-dimensional physical model of the C-shaped enclosure

Table 2 .
Parameters definition

Table 3 .
Comparisons of