Temperature transfer and fluid flow of nanofluids into the triangle shape enclosure have a wide applications in numerous industrial and engineering systems like heat exchangers, fire prevention, solar collectors, home ventilation systems, and refrigeration units etc.  -water,  -water, and  -water are very common nanofluids. These nanofluids are used widely for the augmentation of temperature transfer. Enhancement of warmed-up conductivity into nanofluids was studied by Choi and Eastman (1995). Manca et al. (2010) performed heat transfer in nanofluids. Influence of external magnetic field on natural convective flow into rectangle shape enclosure using warmed up and cooled neighbor walls was investigated by Ece and Buyuk (2006). Convective temperature disposal within a cavity heated partially was performed by Ozotop and Abu-Nada (2008). Free convective temperature flow value into nanofluids within an inclined was performed by Ghasemi and Aminossadati (2009). Impact of inclined angle using copper-water nanofluids within an enclosure was investigated Abu-Nada et al. (2009). Varol et al. (2007) investigated free convectional flow within a triangle shape cavity using isothermal heater. Numerical computations of FEM on convectional temperature flow into nanofluids within a rectangle shape enclosure was performed by Rahman et al. (2009). FEM numerical computations on free convection into nanofluids within a triangle shape cavity in account of both uniform and non-uniform warmed boundary condition was investigated by Basak et al. (2008). A comprehensive review on convectional flow was researched Kamiyo et al. (2010). Aydin and Yesiloz (2011) investigated usual convectional flow into a quadrantal enclosure using warmed and cooled neighbor walls. 

Magnetic field which creates an external force on the thermal and energy systems that highly attack the liquid flow and temperature flowing.  Sheikholeslami et al. (2014) researched about CuO-water nanofluid effect on convectional flow taking into account Lorentz force in flow domain. Computational study using Al2O3-water nanofluids on natural convectional flow was investigated by Rashmi et al. (2011). Bhardwaj and Dalal (2013) investigated convectional temperature flow as well as entropy generation into a right-angled shape triangle cavity. Natural convectional flow into a nanofluid using horizontal warmed plate was studied by Arani et al. (2011). Combined convectional flow and temperature flow characteristics into lid-driven square enclosure using heated blocks was investigated by Boulahia et al. (2016). Magneto hydro dynamics free convectional temperature transfer into nanofluid within isosceles triangle shape cavity was performed by Rahman et al. (2016). Convectional temperature flow into nanofluid within a triangular shape cavity was performed by Uddin et al. (2018). 

From the literature review, the principle intention is to investigate the temperature flow and fluid flow within a right-angled triangle shape cavity charged by copper-water nanofluid. The impact of Rayleigh number, volume fraction of nanoparticles and Hartmann number are performed numerically and discussed them from physical point of view.

Physical Model:

For present study, schematic spectacle for right-angled triangular enclosure has been presented in Fig 1. The length of vertical and horizontal wall of the enclosure is L. The vertical side of enclosure is heated at a low heat T=T c  , whilst its horizontal side is heated at temperature ,  and the inclined wall is insulated. The enclosure charged as water base fluid and copper ( ) HO for ( ) 2 Cu for nanoparticles. The gravitational acceleration g acts in downward approach towards yaxis and a uniformly external magnetic field including power ( ) 0 B is also imposed towards horizontal approach. The nanofluid is calculated as incompressible and laminar. Two dimensional time independent flow is processed for current study (Islam et al., 2020). The characteristics of nanoparticles and base fluid is given in Table 1. All solid boundaries are announced as no slip walls.

Mathematical Model:

The governing equations under aforementioned assumption in Cartesian co-ordinate system as follows

Boundary conditions: 

Dimensional Analysis: 

The dimensionless variables are listed below: 

Using the dimensionless variables in equations (1)-(4), the non-dimensional equations are: 

Dimensionless boundary conditions are: on horizontal wall: 

Local Nusselt number is defined towards horizontal hot wall of the cavity as

Average Nusselt number on horizontal hot side can be calculated: 

COMPUTATIONAL PROCEDURE:  

Table 2: Susceptibility of grid for 2 Cu HO  nanofluid when Ra=10⁴ . 

Governing dimensionless equations (7)-(10) including boundary conditions (11a)-(11c) are concluded numerically by using finite element technique of Galerkin weighted residual type that is very powerful tool to solve these kinds of nonlinear equations. This method is expressly explained by Zienkiewicz and Taylor (1991). The computational procedure for the present problem is accomplished using excellent PDEs solver Comsol Multi physics. For grid independent solution, a comprehensive mesh testing technique is calculated to guarantee the numerical solution when Ra=10⁴.

Code Validation: 

Fig 2:  Comparison between present study (bottom) and Aydin and Yesiloz (2011) (top) when Ra=10⁴

The subsequent five variant distinct grids designated as normal, fine, finer, extra fine, and extremely fine including of 1187, 1757, 5067, 13333 and 16773 elements in resolution field respectively. Average Nusselt number is considered for checking the grid delicacy for the aforementioned elements. Table 2 displays that average Nusselt number for 13333 elements exhibits less difference form compared to other elements. Consequently, grid size of 13333 elements is considered of the grid independent solution.  

We have compared our outcomes (bottom row) using streamline contours (left column) and isotherm contours (right column) along Yesiloz et al. (2013) (top row) for checking the validity of the present numerical scheme when Ra=10⁴ in Fig 2. The results show a strong agreement and gives us the confidence for using the current numerical scheme. 

The convectional temperature and fluid flow using 2 Cu HO nanofluids within right angled triangle shape enclosure is researched numerically by Galerkin weighted residual finite element method. 

The numerical results is presented and discussed for the parameters like the Rayleigh number (Ra), Hartmann number (Ha) and the volume fraction by the nanoparticles () using isotherms contours streamline contours and average Nusselt number.     

Impact of Rayleigh Number: 

Fig 3(a) demonstrates impact for Rayleigh number on streamline contours when . This figures illustrate that for low , the streamlines are not comprehendsively affected by the particles of the nanofluid though horizontal wall is heated uniform. This is a good indication of strong conduction. As Rayleigh number increases fluid within the enclosure becomes more decomposed on account of  temperature flow in that region quickly by convection. In addition, for the increase of Ra, compactness of streamline contours as well vortex strength within cavity enhances on account of predominance of free convection. Hence, convection is the dominant type of temperature flow for higher Ra. Fig 3(b) depicts the impact for Rayleigh number   on isotherm contours when . This figure displays that the streamlines are not superfluously impact within cavity for low Rayleigh number which announces that conduction is the original mode of temperature flow. As Rayleigh number improves , denseness of the isothermal contours at the center within cavity decreases.

This figure also show that the isothermal contours get close to each other and also collateral to each other and closely packed neighbor horizontal warmed as well as perpendicular cold walls as a result of the dominance of the free convection. Consequently, the temperature transfer value is enhanced with the increase of Rayleigh number (Ra ) caused by the potential convection mode. 

Impact of Hartmann Number   

Fig  4(a) depicts the results of the Hartmann number on streamline contours when . Fig  4(a) shows that the flow circulations are nearly total place within the enclosure as well as develop a clockwise rotational chamber for low Hartmann number ( Ha=0) .  

Without external magnetic impact (Ha= 0) , the adjacent fluid near the horizontal heated wall takes heat and these heated particles moves upwards on account of buoyant force whilst comparatively refrigerant liquid neighbor cold wall access to horizontal warmed wall consequently a smooth large circulation cell is created entire the cavity. As Ha increases ( 0) , the streamline patterns are changed and circulations power within the cavity decreases significantly. Because the external strong magnetic field  generates  Lorentz  force  inside  the enclosure that have a character of opposing of its propagation which weakens the streams within the enclosure. 

Fig 4(b) demonstrate Hartmann number  influence on isotherms for fixed . These figures illustrate that the isotherms patter are more and more pronounced for the low Hartmann number (Ha= 0) which indicates the strong convection happens for non-appearance of magnetic field. Isothermal strength decreases and less pronounce for increase of Ha . Because potential Lorentz force is produced from Hartmann number which reduce the temperature distribution.  

Therefore, Magnetic field can be used for controlling the flow domain as well as temperature transfer rate within enclosure. 

Impact of Nusselt number: 

Fig 5 demonstrates influence of Local Nusselt number using (a) Rayleigh number (Ra) and (b) Hartmann number (Ha) on horizontal heated wall. These figures represents, local Nusselt number increases for augmentation Ra and diminishes for the enhancement of Ha. These figures also represents that average temperature transfer value remains constant for low Ra (= 0)  which indicates that conduction is the principal mode of temperature flow. Fig 6 depicts outcomes of av Nu of (a) Rayleigh number ( ) Ra against Ha, and (b) nanoparticles volume fraction ( ) against  Ha on horizontal warmed wall. These figure narrate that av Nu increases with the increase of both Ra and 

Whereas diminishes for augmentation of Ha. The strong buoyancy force and external magnetic field change temperature transfer value significantly. Also, additional nanoparticles into the base fluid enhances remarkably temperature transfer value. This figures also show that the average rate of temperature flow remains constant for low Ra 4 ( 10 )  . 

We have investigated numerically free convectional flow as well as temperature flow of a right-angled triangle charged by cupper-water nanofluid considering with the help of Galerkin weighted residual finite element analysis. The influence of various parameter are presented using streamline contours, isotherm contours, and Nusselt number and interpreted. The following main findings are listed:

• Rayleigh number play significant roll on flow field and temperature transfer value.
• The fluid flow enhances for rising Rayleigh number whereas diminishes for rising Hartmann number.  
• Low Hartmann number and higher Rayleigh number conform better temperature transfer value. 
• Additional nanoparticles significantly improves temperature flow. 
• Average Nusselt number augments for upper Rayleigh number whereas diminishes for rising Hartmann number. 

NOMENCLATURE

Greek symbols

Subscript

The author is very grateful to the editor and reviewers. 

The authors declare no conflict of interest.