1 Introduction

Nowadays, the major need for modern technological industries is high-performance cooling for standard production. By the submersion of nanoparticles to the base fluid, high-performance cooling can be achieved. The major drawbacks of conventional fluids in heat transfer processes are due to the lack of proper thermal conductivity. Nanofluids can overcome this problem. One of the important applications of nanofluid in thermal sciences was developed by Eastman et al. [1]. They pioneered the idea of using nanoparticles in thermal engineering. An important role of heat transfer in most industrial applications is to dissipate heat to the surroundings from the high-temperature surfaces of different devices. Hence, convection plays a major role to dissipate this heat. Free or natural convection is caused primarily due to the temperature difference between different surfaces [2]. Many investigations have been published so far to improve the free convection heat transfer such as changing the flow surface geometry, altering the boundary values, using additives in fluids, using extended surfaces and application of different external forces in the surface geometry [2]. A lot of computational studies have been carried out in different geometries using different nanofluids. Tiwari and Das [3] studied a two-sided lid-driven differentially heated square cavity to analyse the behaviour of Cu–water nanofluid using SIMPLE algorithm finite volume approach. Adjacent two walls were moving at a constant temperature, while the other two walls were insulated. It was concluded that the Richardson Number and the direction of the fluid flow affect the heat transfer and fluid flow. Iwatsu et al. [4] investigated a square cavity with a temperature gradient between the top and bottom walls having a viscous fluid. Numerical analysis was performed to predict the thermal and velocity fields. The results showed that the heat transfer was intensified when Gr/Re2 ≪ 1. Sharifpur et al. [5] investigated the heat transfer in a rectangular cavity filled by ethylene glycol (EG)–water as the nanofluid. Two walls were externally heated, and the analysis was performed using ANSYS Fluent 15 to predict the local Nusselt number distribution. Koo et al. [6,7,8] studied the impact of different micro-scale parameters such as surface roughness on the heat transfer of micro-heat sink systems. Dogonchi et al. [9, 10] investigated the influence of Brownian motion on the thermal conductivity of nanofluids in non-parallel stretching walls. They concluded that the velocity, temperature, Nusselt number boosts with increase in stretching parameter. They also considered the effects of the magnetic parameter, volume fraction and radiation parameter of the nanofluid on velocity and temperature profiles. Mastianiet al. [11] studied Cu–water-based nanofluid in a laminar mixed convection flow for different lid-driven cavities. They had taken different volume fractions for both Boussinesq and non-Boussinesq approximations to analyse the streamlines, isotherms, mid-plane velocities, mid-plane temperature and the average Nusselt (Nu) number in different boundary conditions. It was concluded that the Nu number increases as the volume fraction of the nanoparticles grows. The effect of buoyancy forces and shear forces on the Nusselt number had also been studied. Mahapatra et al. [12] studied the effect of the magnetic field in a two-dimensional natural convection flow in an inclined enclosure. They considered the effect of thermal radiation and heat generation on the flow pattern and temperature field. In the present study, we are trying to focus on natural convection nanofluid flow in an annulus between a circular cylinder and a rhombus enclosure. The novelty of the modelled problem is based on the following aspects.

  1. 1.

    Natural convective nanofluid flows in an annulus between a circular cylinder and a rhombus enclosure is scrutinized.

  2. 2.

    Hydromagnetics characteristics are addressed firstly in this communication.

  3. 3.

    Computations are carried out through the novel CVFEM approach.

2 Problem description and basic equations

CuO–water nanofluid natural convection in an annulus between a circular cylinder and a rhombus enclosure under magnetic field is investigated (Fig. 1). The continuity, momentum and energy equations for two-dimensional steady-state natural convection flow in the enclosure can be defined as:

$$\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0$$
(1)
$$\rho_{\text{nf}} \left( {u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y}} \right) = - \frac{\partial p}{\partial x} + \mu_{\text{nf}} \left( {\frac{{\partial^{2} u}}{{\partial x^{2} }} + \frac{{\partial^{2} u}}{{\partial y^{2} }}} \right) + \sigma_{\text{nf}} \left( {B_{y} B_{x} v - B_{y}^{2} u} \right)$$
(2)
$$\rho_{\text{nf}} \left( {u\frac{\partial v}{\partial x} + v\frac{\partial v}{\partial y}} \right) = - \frac{\partial p}{\partial y} + \mu_{\text{nf}} \left( {\frac{{\partial^{2} v}}{{\partial x^{2} }} + \frac{{\partial^{2} v}}{{\partial y^{2} }}} \right) + \rho_{\text{nf}} \beta_{\text{nf}} g\left( {T - T_{\text{c}} } \right) + \sigma_{\text{nf}} \left( {B_{y} B_{x} u - B_{x}^{2} v} \right)$$
(3)
$$u\frac{\partial T}{\partial x} + v\frac{\partial T}{\partial y} = \frac{{k_{\text{nf}} }}{{\left( {\rho C_{p} } \right)_{\text{nf}} }}\left( {\frac{{\partial^{2} T}}{{\partial x^{2} }} + \frac{{\partial^{2} T}}{{\partial y^{2} }}} \right)$$
(4)

where \(B_{x} = B_{0} \cos \gamma\), \(B_{y} = B_{0} \sin \gamma\).

Fig. 1
figure 1

a Physical model and coordinate system b grid distribution

Full size image

ρnf, (ρCp)nf, (ρβ)nf and σnf are defined as follows:

$$\rho_{\text{nf}} = \left( {1 - \phi } \right)\rho_{\text{f}} + \phi \rho_{\text{s}}$$
(5)
$$\left( {\rho C_{\text{p}} } \right)_{\text{nf}} = \left( {1 - \phi } \right)\left( {\rho C_{\text{p}} } \right)_{\text{f}} + \phi \left( {\rho C_{\text{p}} } \right)_{\text{s}}$$
(6)
$$\left( {\rho \beta } \right)_{\text{nf}} = \left( {1 - \phi } \right)\left( {\rho \beta } \right)_{\text{f}} + \phi \left( {\rho \beta } \right)_{\text{s}}$$
(7)
$$\frac{{\sigma_{\text{nf}} }}{{\sigma_{\text{f}} }} = 1 + \frac{{3\left( {\frac{{\sigma_{\text{s}} }}{{\sigma_{\text{f}} }} - 1} \right)\phi }}{{\left( {\frac{{\sigma_{\text{s}} }}{{\sigma_{\text{f}} }} + 2} \right) - \left( {\frac{{\sigma_{\text{s}} }}{{\sigma_{\text{f}} }} - 1} \right)\phi }}$$
(8)

The KKL (Koo–Kleinstreuer–Li) correlations for knf and μnf by taking into account the Brownian motion are as [6,7,8,9,10]:

$$k_{\text{nf}} = k_{\text{static}} + k_{\text{Brownian}}$$
(9)
$$\mu_{\text{nf}} = \mu_{\text{static}} + \mu_{\text{Brownian}} = \mu_{\text{static}} + \frac{{k_{\text{Brownian}} }}{{k_{\text{f}} }} \times \frac{{\mu_{\text{f}} }}{\Pr }$$
(10)

In which

$$\frac{{k_{\text{static}} }}{{k_{\text{f}} }} = \frac{{k_{\text{s}} + \left( {m - 1} \right)k_{\text{f}} - \left( {m - 1} \right)\phi \left( {k_{\text{f}} - k_{\text{s}} } \right)}}{{k_{\text{s}} + \left( {m - 1} \right)k_{\text{f}} + \phi \left( {k_{\text{f}} - k_{\text{s}} } \right)}}$$
(11)
$$\mu_{\text{static}} = \frac{{\mu_{\text{f}} }}{{\left( {1 - \phi } \right)^{2.5} }}$$
(12)
$$k_{\text{Brownian}} = 5 \times 10^{4} \phi \left( {\rho C_{\text{p}} } \right)_{\text{f}} \sqrt {\frac{{k_{\text{b}} T}}{{\rho_{\text{s}} d_{\text{s}} }}} \beta^{\prime}\left( {T,\phi ,d_{\text{s}} } \right)$$
(13)

Here, m is the shape factor. The shape factor for various particle shapes is given in Table 1 [13,14,15]. In addition, the nanofluid thermo-physical features are portrayed in Table 2 [16,17,18].

Table 1 Shape factor values for various nanoparticle shapes [13,14,15]
Full size table
Table 2 Thermo-physical features of nanoparticle and water [16,17,18]
Full size table

In Eq. (14), the empirical βʹ-function is defined as follows:

$$\begin{aligned} \beta^{\prime}\left( {T,\phi ,d_{\text{s}} } \right) = \left( {a_{1} + a_{2} \ln \left( {d_{\text{s}} } \right) + a_{3} \ln \left( \phi \right) + a_{4} \ln \left( \phi \right)\ln \left( {d_{\text{s}} } \right) + a_{5} \ln \left( {d_{\text{s}} } \right)^{2} } \right)\ln \left( T \right) \hfill \\ \, + \left( {a_{6} + a_{7} \ln \left( {d_{\text{s}} } \right) + a_{8} \ln \left( \phi \right) + a_{9} \ln \left( \phi \right)\ln \left( {d_{\text{s}} } \right) + a_{10} \ln \left( {d_{\text{s}} } \right)^{2} } \right) \hfill \\ \end{aligned}$$
(14)

where the coefficients ai (i = 0.0.10) for CuO-H2O nanofluid are portrayed in Table 3.

Table 3 The coefficient values of CuO–water nanofluid [9, 10]
Full size table

The vorticity and stream function are defined as follows:

$$v = - \frac{\partial \psi }{\partial x}, \, \;u = \frac{\partial \psi }{\partial y}, \, \; \, \omega = \frac{\partial v}{\partial x} - \frac{\partial u}{\partial y}$$
(15)

We define these dimensionless variables:

$$X = \frac{x}{L},\; \, Y = \frac{y}{L}, \, \;\varPi = \frac{{\omega L^{2} }}{{\alpha_{\text{f}} }}, \, \; \, \varPsi = \frac{\psi }{{\alpha_{f} }}, \, \;U = \frac{uL}{{\alpha_{\text{f}} }}, \, \;V = \frac{vL}{{\alpha_{\text{f}} }}, \, \;\theta = \frac{{T - T_{\text{c}} }}{{T_{\text{h}} - T_{\text{c}} }}$$
(16)

Considering Eq. (16), the governing equations change to following form:

$$\begin{aligned} \frac{\partial \varPsi }{\partial Y}\frac{\partial \varPi }{\partial X} - \frac{\partial \varPsi }{\partial X}\frac{\partial \varPi }{\partial Y} & = \frac{{\mu_{\text{nf}} /\mu_{\text{f}} }}{{\rho_{\text{nf}} /\rho_{\text{f}} }}\Pr \left( {\frac{{\partial^{2} \varPi }}{{\partial X^{2} }} + \frac{{\partial^{2} \varPi }}{{\partial Y^{2} }}} \right) + \frac{{\beta_{\text{nf}} }}{{\beta_{\text{f}} }}{\text{Ra}}\Pr \frac{\partial \theta }{\partial X} \\ & \quad + \frac{{\sigma_{\text{nf}} /\sigma_{\text{f}} }}{{\rho_{\text{nf}} /\rho_{\text{f}} }}{\text{Ha}}^{2} \Pr \left( {B_{y}^{\prime 2} \frac{\partial U}{\partial Y} + B_{x}^{\prime } B_{y}^{\prime } \frac{\partial U}{\partial X} - B_{x}^{\prime 2} \frac{\partial V}{\partial X} - B_{x}^{\prime } B_{y}^{\prime } \frac{\partial V}{\partial Y}} \right) \\ \end{aligned}$$
(17)
$$\frac{\partial \varPsi }{\partial Y}\frac{\partial \theta }{\partial X} - \frac{\partial \varPsi }{\partial X}\frac{\partial \theta }{\partial Y} = \frac{{k_{nf} /k_{f} }}{{\left( {\rho C_{p} } \right)_{nf} /\left( {\rho C_{p} } \right)_{f} }}\left( {\frac{{\partial^{2} \theta }}{{\partial X^{2} }} + \frac{{\partial^{2} \theta }}{{\partial Y^{2} }}} \right)$$
(18)
$$\frac{{\partial^{2} \varPsi }}{{\partial X^{2} }} + \frac{{\partial^{2} \varPsi }}{{\partial Y^{2} }} = - \varPi$$
(19)

Subject to the boundary conditions:

$$\begin{array}{*{20}l} {\theta = 1} \hfill & {\quad {\text{on}}\;{\text{the}}\;{\text{local}}\;{\text{heater}}} \hfill \\ {\theta = 0} \hfill & {\quad {\text{on}}\;{\text{the}}\;{\text{outer}}\;{\text{circular}}\;{\text{wall}}} \hfill \\ {\varPsi = 0} \hfill & {\quad {\text{on}}\;{\text{the}}\;{\text{all}}\;{\text{walls}}} \hfill \\ \end{array}$$
(20)

where \(\Pr = \nu_{\text{f}} /\alpha_{\text{f}}\) is the Prandtl number, \({\text{Ra}} = g\beta_{\text{f}} \left( {T_{\text{h}} - T_{\text{c}} } \right)L^{3} /\alpha_{\text{f}} \nu_{\text{f}}\) is the Rayleigh number and \({\text{Ha}} = B_{0} L\sqrt {\sigma_{\text{f}} /\mu_{\text{f}} }\) is the Hartmann number.

The local and average Nusselt numbers along the cold wall can be defined as:

$${\text{Nu}}_{{{\text{loc}}.}} = - \frac{{k_{\text{nf}} }}{{k_{\text{f}} }}\frac{\partial \theta }{\partial n}$$
(21)
$${\text{Nu}}_{{{\text{ave}}.}} = \frac{1}{2\pi }\int_{0}^{2\pi } {{\text{Nu}}_{{{\text{loc}}.}} {\text{d}}\zeta }$$
(22)

where n is the direction normal to the outer walls.

3 Numerical solution and validation

Here, the governing equations of this new type of mathematical model are solved via the control volume finite element method (CVFEM) [19,20,21,22,23,24]. Table 4 shows the comparison between present results and other works for the average Nusselt number (Nuavg). The results in this table show good agreement with previously published results. To attain mesh independence, various mesh sizes are examined (see Table 5). It is found that the grid of 41 × 431 must be chosen for the present problem.

Table 4 Comparison between present results and other works for the average Nusselt number (Nuavg)
Full size table
Table 5 Influence of grid size on Nuavg when Ra = 105, Ha = 50 and m = 3
Full size table

4 Results and discussion

Figures 2, 3 and 4 display the effect of Ha on streamlines and isotherms for different values of Ra from 103 to 105 when the volume fraction of the nanofluid (ϕ) is zero and shape factor of nanoparticles (m) is 3. Figure 2 describes the streamlines and isotherms for different Ra in accordance with the Ha = 0. Two similar shapes of eddies are formed for any value of Ra within this range, but the directions of the streamlines for different eddies are different. Clockwise and anticlockwise flows are shown via negative and positive signs of stream functions, respectively. Again, it can be shown that by increasing Rayleigh number from 103 to 105, the modulus values of the different directed eddies at the centre will increase. So, the |Ψmax|nf increases with increase in the values of Ra from 103 to 105. Moreover, the Ra is substantially higher the circular eddies are disturbed and streamlines are more predominant in the outer circular periphery. With higher |Ψmax|nf, Ra increases and the nature of streamlines is perturbed. The decreasing nature of the heat transfer gradient is observed with higher |Ψmax|nf. At lower |Ψmax|nf, the isotherms are concentrated near the heated obstacles walls, whereas at higher |Ψmax|nf the isotherms are dispersed from the heated walls to the outer circular periphery walls.

Fig. 2
figure 2

Streamlines and isotherms for different values of Ra when Ha = 0, \(\phi = 2\%\) and m = 3

Full size image
Fig. 3
figure 3

Streamlines and isotherms for different values of Ra when Ha = 25, \(\phi = 2\%\) and m = 3

Full size image
Fig. 4
figure 4

Streamlines and isotherms for different values of Ra when Ha = 50, \(\phi = 2\%\) and m = 3

Full size image

In Fig. 3, the streamlines and isotherms are depicted for different values of Ra from 103 to 105 when Ha = 25. The streamlines are observed to be more circular by increasing the fluid velocity. Eddies formed are smaller with higher values of Ra. When the Ra is low, the isotherms are formed near the boundary of the heated obstacle surfaces. With the increase in Ra, the isotherms diverge towards the outer circular periphery. The temperature gradient is higher at the edge boundaries and decreases thereafter in the radial direction at lower values of Ra. The other nature of streamlines and isotherms is same, as shown in Fig. 2. Figure 4 describes the streamlines and isotherms when the Ha is further increased to 50 for the same range of Ra. From this figure, it can be seen that with increasing Ra, the circular nature of streamlines is more predominating. So, the streamlines are more concentrated for the increasing value of Ra. The other nature of streamlines and isotherms is similar, as shown in Fig. 2.

Figure 5 shows the comparison of streamlines and isotherms of various nanofluids for different ϕ and Ra when Ha = 25, m = 3. By decreasing the Rayleigh number from 105 to 103, the streamlines are more dispersed. This effect is more predominant while the volume fraction is further increased. Thereby, |Ψmax|nf value increases with the increase in the volume fraction for each value of Ra. Also when the volume fractions are fixed, and the Ra is increased from 103 to 105, there is a considerable increase in |Ψmax|nf. When Ra = 103, the isotherms are observed near all the four obstacle walls. The temperature gradient is also uniform. But with the increase in Ra, this nature diminishes. The isotherms are concentrated more towards the outer periphery from the heated wall.

Fig. 5
figure 5

Comparison of the streamlines and isotherms between various nanofluids (dashed lines—\(\phi = 4\%\) and solid lines—\(\phi = 2\%\)) for different values of Ra when Ha = 25 and m = 3

Full size image

Figure 6 displays the dependency of local Nusselt number (Nuloc) with different values of Ha, Ra and ϕ. At Ra = 103 and Ha = 25, the Nuloc increases with increase in the ϕ up to a certain range of ζ, after which the same nature is observed with decreasing value of Nuloc and the same patterns are observed again forming a sinusoidal curve. But when Ha = 50, the maximum values of Nuloc are lesser than the maximum values of Nuloc when Ha = 25 for any value of ϕ. Other nature of Nuloc remains unchanged. This shows that the increasing value of Ha decreases the heat transfer rate of the nanofluid. Again, there will be no such changes in Nuloc for the increasing value of ϕ when Ra = 104 and Ha = 25. But as ζ boosts up to a certain range, Nuloc forms a pick for any value of ϕ. But, when Ha = 50, Nuloc grows with the decreasing value of ϕ and it coincides at a point with forming a pick for the lower value of ζ. For a higher value of ζ, the curve pattern of Nuloc is same as we described before but the maximum value of Nuloc is less compared to the higher value of Ha. When Ra = 105 and Ha = 25, the Nuloc remains unchanged for any value of ϕ, but as it reaches a pick, different values are found for different ϕ. When Ha = 50, the Nuloc is slightly different only at a pick with a different value of ϕ. From Fig. 6, it can be concluded that the Nuloc lessens with increasing values of Ra for any value of Ha. Also, it can be seen that the lowest value of Nuloc occurs within a short range of ζ = 270°. From Fig. 7, it is obvious that by increasing the nanoparticle shape factor the Nuloc also goes up for the fixed value of Ra, Ha and ϕ = 2% when Nuloc maintaining the same pattern of heat transfer rate, as shown in Fig. 6 for specific Ra and Ha. From Fig. 7, it is seen that by increasing Rayleigh number Nuloc augments for a fixed value of Ha when ϕ = 2%. In addition, it can be seen that the lowest value of Nuloc occurs within a short range of ζ = 270°.

Fig. 6
figure 6

Local Nusselt number (Nuloc.) for different values of Ra, Ha and ϕ when m = 3

Full size image
Fig. 7
figure 7

Local Nusselt number (Nuloc.) for different values of m at different conditions when \(\phi = 2\%\)

Full size image

Figure 8 depicts the average Nusselt number (Nuave) for different values of Ra, Ha and ϕ when m = 3. From this figure, it can be seen that Nuave is decreasing with the increasing value of ϕ from 2 to 4% for any fixed value of Ra and Ha. Also Nuave enhances with an increase in the value of Ra from 103 to 105 for the fixed value of Ha and ϕ. For the fixed value of Ra and ϕ, Nuave is maximum when Ha boosts. Figure 9 shows the Nuave with respect to the different values of Ra, Ha and m when ϕ = 2%. Increasing nanoparticle shaping factor increases Nuave for any fixed value of Ha and Ra. Similarly, with an increase in Ra, Nuave also grows for any fixed value of Ha and m. It can be also seen that Nuave rapidly increases within the range of 104 and 105. Also by increasing Hartmann number, the maximum Nuave diminishes for any fixed value of m and Ra.

Fig. 8
figure 8

Average Nusselt number (Nuave.) for different values of Ra, Ha and \(\phi\) when m = 3

Full size image
Fig. 9
figure 9

Average Nusselt number (Nuave.) for different values of m at different conditions when \(\phi = 2\%\)

Full size image

At high Ha and low Ra, average Nu has a lesser gradient, which implies that it grows at a lower rate with an increment in Ra at high Ha. But at low Ra number and low Ha, Nu boosts at a higher proportion as compared to the former. This concludes that average Nu has an inverse relation with Ha at low Ra, but at higher Ra, the average Nu relationship with Ha is not maintained.

5 Conclusions

CuO–water nanofluid is taken in a 2D circular geometry with a rhombus-shaped obstacle maintaining a constant temperature of the upper two adjacent walls. The streamlines and isotherms have been resulted using the control volume finite element method by applying KKL model for the simulation of nanofluid. Two similar shapes but differently directed eddies are formed for any value of Ra in streamlines. Again, |Ψmax|nf increases with an increase in the values of Ra from 103 to 105. The results have confirmed that isotherms were concentrated at the adiabatic obstacle walls at a low value of the Rayleigh number in the nanofluid flow, and also the temperature gradient was less in the radial direction in this same condition. It can be concluded that the heat transfer is affected for a low value of Rayleigh number. Similarly, eddies are formed near the adiabatic walls of the obstacle for low value of Rayleigh number. By increasing the volume fraction of the nanoparticles, more eddies are formed and also it diverges near the outer periphery.