Heatline visualization of MHD natural convection heat transfer of nanofluid in a prismatic enclosure

Temperature transfer by virtue of natural convection for visualizing heat transport characteristics through heatline method within a prismatic cavity filled with Cu-H2O nanofluid considering two different temperature boundary conditions is performed numerically. Two top inclined walls are warmed-up at low temperature whilst the bottom wall is heated two different heated conditions such as uniform temperature condition and linear temperature condition. Two vertical walls are insulated. Finite element technique of Galerkin weighted residual form is employed for solving nonlinear partial differential equations for numerical calculation. Heatlines, isotherm contours, streamline contours, and Nusselt number are employed for displaying numerical simulated results for the model parameters entitled nanoparticles volume fraction, Hartmann number and Rayleigh number. The outcomes indicate that heat transfer rate has a significant impact on thermal boundary condition and shape of the nanoparticles. The temperature transfer value enhances significantly for higher Rayleigh number as well as nanoparticles volume fraction. Hartmann number has a positive impact on fluid flow and temperature transport. The characteristics of heat transport using heatlines method are also performed for predicting the better energy transform compared to isotherm contours. In addition, different types of nanofluids are also employed to examine the best heat transport performance.

Finite element technique of Galerkin weighted residual form is employed for solving nonlinear partial differential equations for numerical calculation. Heatlines, isotherm contours, streamline contours, and Nusselt number are employed for displaying numerical simulated results for the model parameters entitled nanoparticles volume fraction, Hartmann number and Rayleigh number. The outcomes indicate that heat transfer rate has a significant impact on thermal boundary condition and shape of the nanoparticles. The temperature transfer value enhances significantly for higher Rayleigh number as well as nanoparticles volume fraction. Hartmann number has a positive impact on fluid flow and temperature transport. The characteristics of heat transport using heatlines method are also performed for predicting the better energy transform compared to isotherm contours. In addition, different types of nanofluids are also employed to examine the best heat transport performance. Nanofluids consist nanoparticles such as Co, Zn, Ag, Cu, Al 2 O 3 , TiO 2 , and Fe 3 O 4 etc. and conventional fluid named water, engine oil, and kerosene etc. These nanofluids are widely used as noble fluid in many engineering and industrial applications. The main objectives of the investigation of nanofluids is that finding of height feasible solutions using low cost. In last few years, the interdisciplinary investigations of nanofluids are increased for the wonderful real life engagement in industries and engineering. For instance, solar water heating, heat exchangers, fuel cell, diesel combustion, vehicle's engines, nuclear systems, domestic refrigerator, and lubrications. Choi 1 investigated about the technology of nanofluids for its present and future enrollment. Buongiorno 2 performed convective temperature transport in nanofluids. Jou et al. 3 investigated numerically about free conventional temperature augmentation within a rectangle shape cavity using nanofluids. Ece et al. 4 performed about magnetic field influence on free convective flow using warm-up and cold adjoining walls within a rectangle shape enclosure. Das et al. 5 investigated nonofluids and its application in science and technology. Oztop et al. 6 studied about convective flow within partly warmed-up rectangle shape cavity using nanofluids. Temperature transference by buoyancy introduced flowing becomes a monumental engineering wonderment now-a-days. Natural convection heat transfer and cooling systems in closed enclosures are very significant wonderment in geophysical, geothermal reservoirs, heat exchanger design, mechanical and electronic industries. Many scientists and researchers are highly motivated to investigate the distribution of temperature transport, heat flow and flow pattern. For instance, Ghasemi et al. 7 performed regarding free convective temperature flow using CuO-water nanofluids within the dangling cavity. Rahmam et al. 8 investigated about combined convective flow within a rectangle shape enclosure. Wong et al. 9 investigated about present as well as future enrolments of nanofluids. Seleh et al. 10 investigated about free convectional temperature flow employing nanofluids within trapezoid shape cavity. Arani et al. 11 investigated as regards natural convectional flow into nanofluids within a square enclosure using warmed blade. Basak et al. 12 investigated regarding the analysis of heatlines upon free convectional flow employing nanofluids within a square enclosure for several warmed-up boundary systems. Cheikh et al. 13 investigated regarding fee convectional flow of nanofluids within a square cavity employing a non-uniform warmed-up conditions. Free convectional temperature flow into nanofluids using constant heat flux was researched by Seyyedi et al. 14 . Salma et al. 15 investigated about free convectional flow of nanofluids within a prismatic enclosure.

List of symbols
In two dimensional convection heat transfer process, heatline is the best method of understanding and analyzing the heat flow. The real path of convection heat transfer or the heat recovery system is visualized by heatline method. The streamlines completely illustrate the flow of the fluid whereas isotherm contours narrate mere the distributions of temperature which are not adequate for visualizing of temperature transportation. The heatline method is similar with streamlines that is important for analyzing fluid motion. Heatline techniques is the best measures for the finding of temperature recovery systems as well as actual temperature transfer way by the virtue of convection. Heatlines typically illustrate temperature functions that satisfy energy equation whilst stream function satisfy conservation of mass equations. Heatline is associated with Nusselt number that depends on non-dimensional form of conversion. The heat flux lines are represented by heatlines which illustrate the trajectory of heat flow that are normal to the isothermal lines in convection temperature transport. The heatlines are represented mathematically by heat function and each heat function corresponds a constant function. Kimura and Bejan 16 was first introduced the concept of heatline. Salma et al. 17 further studied about heatlines visualization on magneto-hydrodynamics combined convective flow including a heated block. Uddin et al. 18 researched about the fundamentals concepts as well as the applications of nanofluids. Alam et al. 19 investigated regarding the heatlines exploration upon free convective flowing as well as temperature transfer within prismatic cavity employing FEM. Alsabery et al. 20 investigated about the experiment of heatlines analysis over convectional flow within a square enclosure using sinusoidally temperature mutations. Sheikha et al. 21 Fig. 1 showed a schematic spectacle of geometry attached to the Coordinate system and thermophysical properties of nanoparticles including base fluids are presented in Table 1.  Boundary conditions. On the bottom wall: Physical and thermal properties of nanofluids. For enhancing the thermal performance of nanofluids, the thermal and physical properties of nanofluids are important. The following physical and thermal characteristics of nanofluids are taken into consideration and listed as viscosity, density, thermal diffusivity, heat capacitance, thermal conductivity, thermal expansion coefficient and electrical conductivity, respectively (see Kalbani et al. 29 ):

Dimensional analysis
We introduce the following non-dimensional variables for reducing the Eqs. (1)-(4) into non-dimensional form including boundary conditions (5a)-(5d): The governing equations in dimensionless form are expressed as:

Computational process
The governing non-dimensional Eqs. (14)- (17) including boundary conditions (18a)-(18d) are performed applying finite element analysis of Galerkin weighted residual form. Zienkiewicz et al. 30 described the numerical technique elaborately. Firstly, the non-uniform triangular element are formed by discretizing the domain of the solution into limited grid numbers. Six node triangular elements are applied for the progression of this method. Then, integral equations are formed from governing PDEs equations using Galerkin weighted residual technique. Gauss's quadrature technique is also employed in every part of the integral equations. These equations are also modified using boundary conditions. Newton-Raphson iteration technique is imposed into these algebraic equations for solving these algebraic equations in matrix form. The convergence criterion of this method is imposed as M n+1 − M n ≤ 10 −5 , where M represent U, V , θ as dependent variables and n is iteration number.

Grid independence test.
A comprehensive grid independence experiment is employed to ensure the grid independent solution for the present problem of the prismatic enclosure when Ra = 10 5 , φ = 0.04 and n = 3 (spherical shape of nanoparticles). Five different non-uniform grid systems within resolution are employed named normal, fine, finer, extra fine, and extremely fine including elements 1365, 2116, 5699, 14,514, and 20,584. For the above mentioned elements, the average Nusselt number (Nu av ) is calculated for understanding the grid fineness that is shown in Fig. 2. The elements 14,514 represents a little difference with the outcomes obtained for the elements 20,584. Hence, elements size 14,514 are employed for numerical calculations to obtain required grid independent solution.
Code validation. The present outcomes are compared with Ghasemi et al. 24 to check-out the accuracy using streamline contours (left column) and isotherm contours (right column) when φ = 0.02, Ra = 10 5 and Ha = 30 , which is shown in Fig. 3. The present numerical code also validate through numerical data with Ghasemi et al. 31 for different Rayleigh number and nanoparticles volume fraction considering two dimensional natural convection flow of Al 2 O 3 -water nanofluid under the presence of magnetic effect ( Ha = 30 ). The present numerical code also compared with Wan et al. 32 considering Pr = 0.07 and square cavity filled with air. The cur- On the left and right vertical walls:    www.nature.com/scientificreports/ rent code is transferred for air filled square cavity for comparison. The outcomes represents a good compliance using for employing current numerical code in Table 2.

Result and discussion
The obtained outcomes are presented employing streamline contours, isotherm contours and heatlines for analyzing free convection temperature flow as well as fluid flow within a prismatic enclosure. The spherical shape copper nanoparticles ( n = 3) are treated into water base fluid. The numerical results have been discussed for various parameters named nanoparticles volume fraction (φ) , Hartmann number (Ha) and Rayleigh number (Ra) upon liquid flow, heat transfer as well as temperature transport characteristics exercising isotherm lines, streamlines, heatlines, local and average Nusselt number using two distinct temperature boundary conditions along horizontal wall. Figure 4 displays the impact of Rayleigh number Ra(= 10 3 − 10 6 ) when φ = 0.04 and Ha = 10 on the streamlines, isotherm contours as well as heatlines using uniform warmed-up condition on horizontal wall. Figure 4(a) shows that two symmetric counter rotating rolls to the central vertical line are formed within cavity with every Ra . Also, vortex eyes are situated neighbor the center of half cross-sectional of the cavity though there www.nature.com/scientificreports/ is uniformly heated bottom. The left-hand chamber circulates anti-clockwise direction whereas the right-hand chamber circulates clockwise direction within the enclosure. The physical meaning behind this that the density of cold liquid near the upper inclined walls is higher compare to warmed-up fluid neighbor the bottom wall within enclosure. The weighty fluid travel below whilst comparatively low frequent liquid turns upwards. Then, the bulky liquid thrust into thermal boundary layer neighbor bottom hot wall and accomplishes rotating pattern. For increases of Ra , the two recirculation cells grow. This introduces that the warmed-up liquid are accelerated more by virtue of buoyancy effect. The temperature transfer mode (conduction or convection) as well as the utility of temperature are detected by isotherm contours. Figure 4(b) shows, as low Rayleigh number, Ra(= 10 3 , 10 4 ) , isotherm contours are almost parallel neighbor the bottom hot wall that indicate that convection is weaker within the cavity. Conduction is the principle mode of temperature flow within the enclosure on account of uniform warmed-up condition on bottom wall. The low compactness of the isotherm contours are also observed in the cavity center that represents poorly convective temperature flow. For the increasing values of Ra , isotherm contours turn out over distorted as well as disappeared in the cavity center manufacturing a particular shape as mushroom. For Ra = 10 5 , 10 6 , the particular pattern of isotherms like mushroom indicates that the temperature energy flows within cavity fluid from bottom warmed-up side by virtue of potential buoyancy effects. Figure 4(c) demonstrates that the heatlines are unadulterated and approximately parallel with perpendicular walls resulting the transformation of temperature as a result of conduction. The heatlines consistency neighbor the warmed-up wall enhance as well as pervert for higher buoyancy driven parameter Ra . The compactness of the heatlines increases at middle of the enclosure due to the strong convection. Furthermore, for higher Ra , two symmetric small vortices to the central vertical line are created inside the enclosure, which is consistent with the stream function pattern. Therefore, the convection is dominant form of temperature flow at upper buoyancy driven parameter.
Rayleigh number (Ra = 10 3 − 10 6 ) effects on the streamline contours, isotherm contours and heatlines when φ = 0.04 and Ha = 10 for linearly warmed condition on bottom wall are displayed in Fig. 5, respectively. Figure 5(a) illustrates that the streamline contours are not comprehensively affected with the particles of the nanofluid within the enclosure. The temperature of horizontal side is upper compared to dangling sides thereby neighboring liquid of bottom wall receives temperature and thereafter these fluid turn out upward to the colder fluid consequently a circulation cell is created consequently temperature transfer value enhances on colder walls. Physical meaning for this is, for the enough temperature differences between warmed up and cooled walls, a potential buoyancy force is constructed within the enclosure. For low Ra , convection impacts are little conspicuous because of the insignificant performance of inertia force in temperature flow mechanisms. For upper values of Ra(= 10 5 , 10 6 ) , streamline contours become decomposed and one potential propagation into the fluid within the cavity is appeared for the increases of the intensity of convective temperature flowing and resulting a secondary circulation is developed on the upper corners neighbor the inclined walls. Figure 5(b) shows, for less Ra (= 10 3 , 10 4 ) , parallel isotherm contours neighbor warmed-up wall are observed because of potential conduction form of temperature distributions. For upper Rayleigh number, exceeding distorted isotherm contours are also noticed at the cavity center because of the increases of the convection within the enclosure. For Ra = 10 6 , the compactness and high dense of the isotherm pattern is noticed neighbor horizontal warmed-up and dangling roof cold walls that represents high temperature gradient at that regions.
To represents the temperature flow within the enclosure, conductive temperature flux − ∂θ ∂X , ∂θ ∂Y is employed for heat function whereas convective temperature flux (Uθ, V θ ) is employed for displaying temperature flow. Heatlines represent the way of temperature flowing that arises form warmed-up regime whereas finish at cold regime. Figure 5(c) illustrates that the heatlines are distributed almost indiscriminately according clockwise and counterclockwise circulations for less Ra(= 10 3 ) from horizontal warmed-up side to upper inclined cold sides due to low strength of fluid flow as well as temperature flowing mainly by conduction.
As increases Ra , the flow intensity increases significantly, the pattern of the heatlines are gradually decomposed and an additional tiny rolling chamber is created in upper corner within enclosure. The intense of temperature flow is higher neighbor the bottom warmed wall compared with upper cold wall. Therefore, for the determination of temperature flow as well as the analysis of the temperature transfer, heatlines are sufficient mechanisms. In addition, heatlines appearances as streamline contours for higher Ra. Figure 6, respectively, demonstrates the outcomes of nanoparticles volume fraction (φ = 0, 0.025, 0.05, 0.1) when Ra = 10 5 upon streamline contours, isotherm contours and heatlines for uniform bottom warmed-up wall. Figure 6(a) depicts that two opposite symmetrical counter rotational chambers to central vertical line are formed within the cavity for uniform heated wall. The left circulation is clockwise sense and the other cell is clockwise sense within the enclosure. This particular form be the outcome of uniform warmed-up boundary condition on horizontal side. The physical reason behind this particular form of streamline contours already narrate. For the increases of φ , the pattern of streamlines are nearly similar kind of unadulterated conduction. Furthermore, adding nanoparticles into base fluid within the cavity, a very little effects of nanofluids on convection is observed. Figure 6(b) illustrates isothermal contours are discomposed at center plane within cavity manufacturing a particular form as mushroom. This depicts that the energy is streaming within liquid from horizontal warmedup side. This figure also show that the isotherm contours are nearly parallel neighbor warmed-up and cold walls. The fluid neighbor horizontal warmed-up wall accepts temperature and moves upwards into colder fluid. Consequently, temperature flow enhances at cold wall. This figure depict that there is a dense of the isotherm pattern near the warmed-up and cooled walls which indicate the high temperature gradient in that regions. That circumstance stay same for the addition of nanoparticles within the enclosure.  Figure 7(a) demonstrate the streamlines are parallel to each other neighbor warmed-up wall because of the conduction as well as a secondary vortex is developed on the upper corners neighbor inclined wall within the cavity. For higher value of nanoparticles the isotherms are distorted and one massive vortex is noticed at the middle of the enclosure. Figure 7(b) narrates that the parallel isotherms neighbor warmed-up and cooled walls. We have already observed this similar particular form. Figure 7  By applying an external magnetic field, a stronger field works over moving fluid which has a magnetic impressionability that weaken the flow circulation inside the enclosure. In addition, the Lorentz force generated by applying magnetic field that has a nature to oppose the varying its generation in case of the movement of the liquid and consequently, this force field weakens the streams inside the enclosure. Figure 8(b) shows that the isothermal lines are more and more distorted inside the cavity and denser near the bottom heated wall with the absence of Hartmann number. At low Hartmann number (Ha = 0) , higher temperature gradient is observed near bottom warmed wall. The denseness of the streamlines reduces with the influence of stronger magnetic field. Moreover, the isothermal lines moves upward more near the middle part of the bottom heated wall which indicates higher heat transport region. A large circulation cell with a small tube in top corned of the enclosure is noticed in Fig. 9(a) for linear temperature condition. This rotating cell become smaller with rising influence of Hartmann number that indicates the decoction velocity by virtue of the effect of the Hartmann number. This happens due to the effect of magnetic field which retards the flow of the liquid. Figure 8(c) shows the heat transport characteristics of heatline method  www.nature.com/scientificreports/ Hartmann number for uniform temperature system than linear temperature system. In addition, by changing the value of nanoparticles volume fraction, the Local Nusselt number is significantly affected. Figure 13 shows the average Nusselt number enhances monotonically by higher Ra . It is interesting observed that uniform heated condition on bottom wall provides highest temperature flow within the enclosure than linear temperature condition. Figures 14, 15 and 16 narrate that average Nusselt number increases with increases of Rayleigh number and addition of nanoparticles into the base fluid whereas decreases with the increase of Hartmann number. The rate of heat transport increase significantly with the increases of nanoparticles volume fraction. This is due to the higher thermal conductivity of nanofluids than base fluid. In addition, at low Ra, the influence of volume fraction of nanoparticles is more pronounced on heat transfer. Figure 16 depicts that rate of heat transfer is significantly higher for blade shape of nanoparticles than spherical shape of nanoparticles. This is due to the fact that less sphericity of the blade shape of nanoparticles. Moreover, the average Nusselt number is more apparent for higher nanoparticles volume fraction. Moreover, the average Nusselt number is higher for balde shape of nanoparticles compared to other all shape of nanoparticles such as spherical, brick, cylinder and platelet. www.nature.com/scientificreports/ The parameter nanoparticles volume fraction is the key factor of the studying how nanoparticles affect the fluid flow as well as temperature transport of nanofluids. The types of nanoparticles is also a key factor for the enhancement of heat transfer. Table 3 illustrates average Nusselt number (Nu av ) along bottom heated wall of the cavity for different value of nanoparticles volume fractions and different types of nanoparticles of base fluids such as water (H 2 O), kerosene and ethylene glycol (EG) with four different types of nanoparticles such as Cu, Co, Al 2 O 3 , and TiO 2 when Ra = 10 5 , Ha = 15, and n = 3. It is observed that the average Nusselt number increases with the increases of nanoparticles volume fraction for all types of nanofluids. This is due to the higher thermal conductivity of nanoparticles. The Table depicts that highest heat transfer for kerosene based nanofluids compared to water-based and ethylene glycol based nanofluids. Although, Cu nanoparticles has higher thermal conductivity than Co, Al 2 O 3 , and TiO 2 , Co-kerosene nanofluids show higher rate of heat transfer. The lowest heat transfer is observed for water based nanofluids due to the lower thermal conductivity and higher dynamical viscosity of the based fluid.

Conclusions
The convective flow and temperature into cupper-water nanofluid within a prismatic enclosure to visualize the heatline has been investigated considering two various heated boundary conditions. The numerical methods finite element analysis of Galerkin weighted residual form is executed. The outcomes for model parameters on flow field as well as temperature flow are presented using streamline contours, isothermal contours, heatlines, local and average Nusselt numbers and interpreted. The main outcomes are listed below: • At low Rayleigh number, the elementary form of temperature flow is conduction and influence nanoparticles are more pronounced on heat transfer. • Convective temperature flow enhances significantly for the augmentation of Rayleigh number.
• Flow field are strongly affected within the cavity for higher Rayleigh number.
• Better temperature flow is performed for upper Rayleigh number through convection. • The average temperature transport rate increases with additional nanoparticles volume fraction.
• The uniform mode of thermal boundary condition represents the highest temperature transfer rate for copper-water nanofluid. • The increasing intensity of Hartmann number reduces fluid velocity as well as heat flow.
• Effective heat transport can be obtained with the existence of nanoparticles. The kerosene based nanoparticles shows highest heat transfer rate compared to other nanofluids. • The shape of nanoparticles is a key factor of heat transport. Higher rate of heat transport is observed for blade shape of nanoparticles.