Introduction

Convective fluid flows in cavities have garnered a great deal of attention in recent years due to its industrial and engineering applications, such as solar energy technology, refrigeration, heat exchangers and the cooling of electronic devices [1,2,3]. However, the efficiency of these heat transfer systems are limited by the low thermal conductivity of traditional coolants such as water, air, ethylene glycol and engine oil. Thus, the use of nanofluids for the enhancement of heat transfer performance in liquid-cooled systems has been extensively investigated in the literature [2, 4].

The study of fluid flow phenomena with heated or heat-generating obstacles is important in the design of compact cooling systems for modern electronic devices [5]. For this reason, many papers have been published in recent year on convective heat transfer in cavities or channels with internal heated solid objects [3, 5,6,7,8]. Job and Gunakala [6] considered the mixed convective flow of SWCNT-water and Au-water nanofluids through a grooved channel with two internal heat-generating solid cylinders. The influence of many important parameters such as the groove geometry, cylinder radius, groove area, and solid volume fraction on nanofluid flow and heat transfer was investigated. The authors determined that the nanofluid temperature is increased with an increase in the cylinder size. Moreover, the trend in the heat transfer rate on the cylinder surface with increased cylinder size is dependent on the range of values of the cylinder radius. The mixed convective flows of water-based Cu, alumina and titania nanofluids in a square lid-driven cavity with isothermally heated and cooled solid objects was explored by Garoosi and Rashidi [7] using Manninen’s two phase approach. The influence of parameters such as the thermal conductivity ratio, number of heated obstacles, and nanoparticle volume fraction, size and type on the convective heat transfer phenomena and concentration distribution was analyzed. In particular, it was found that the heat transfer rate decreases and the nanoparticle distribution is increasingly non-uniform with increased nanoparticle diameter. Job and Gunakala [5] examined the unsteady MHD convective flows of Cu-water and Ag-water nanofluids within an L-shaped channel with a porous inner layer and four internal heat-generating solid components. The authors found that heat transfer enhancement occurs when the concentration of nanoparticles increase. Also, it was found that the removal of heat from the heat-generating components is more effective for the Ag-water nanofluid than the Cu-water nanofluid. Shulepova et al. [8] numerically investigated the radiative and mixed convective flow of alumina-water nanofluid in a lid-driven cavity containing a solid heat-generating square object with sinusoidally varying heat flux. It was revealed in this study that the addition of nanoparticles reduces the heater temperature and the convective flow rate. Bozorg and Siavashi [3] used the Eulerian two-phase approach to investigate the mixed convection flow of a power-law nanofluid within a square cavity with internal rotating isothermally heated and cooled cylinders. The authors explored the influence of parameters such as the angular velocity of the rotating cylinders, power-law index of the nanofluid and the nanoparticle volume fraction on the heat transfer and entropy generation within the cavity. Their results showed that the influence of natural convection or forced convection on the rate of heat transfer is dependent on the direction of rotation of the cylinders and the non-Newtonian power law index.

It has been identified in several studies that the heat transfer performance in cavities or channels can be enhanced by incorporating corrugations on the walls of the physical system [2, 4]. Job and Gunakala [2] studied the time-dependent convective flow of SWCNT-water and alumina-water nanofluids in a sinusoidally corrugated trapezoidal cavity in the presence of an applied magnetic field. In their study, viscous and Joule dissipations were considered, and the governing equations were solved using a stabilized mixed finite element method with equal-order elements. The effects of the corrugated wall amplitude, nanoparticle solid volume fraction, time and Hartmann number on streamlines, isotherms and heat transfer rate were examined. The research conducted in Job and Gunakala [2] was extended in the work of Job et al. [4], in which the combined effects of Rayleigh number and Hartmann number on the unsteady MHD flow and heat transfer of SWCNT-water and alumina water nanofluids in a corrugated trapezoidal cavity were analyzed. Alsabery et al. [9] conducted a numerical study on the two-phase convective flow of a hybrid Cu-alumina/water nanofluid within a corrugated lid-driven rectangular cavity containing a solid heat-conducting block. In their study, the effects of thermophoresis and Brownian motion were considered, and the governing equations were solved using the penalty finite element method. The authors examined the effects of varying the Richardson number, wall corrugation wavelength, nanoparticle volume fraction, and the size and position of the solid block on the streamlines, isotherms and nanoparticle distribution. The findings indicate that using the Cu-alumina/water hybrid nanofluid leads to greater heat transfer rates compared to usual nanofluids based on Cu or alumina. The study showed that the heat transfer rate increases as the Richardson number and volume fraction of hybrid nanoparticles are increased, and as the size of the solid block decreases. It was also determined that the concentration distribution of nanoparticles is influenced by the investigated parameters.

To the best of the authors’ knowledge, there is no existing study on the mixed convective MHD flow of hybrid nanofluids in a trapezoidal cavity with corrugated walls and an internal heat-generating object. Therefore in the present work, we examine the unsteady MHD convective flow of water based hybrid Ag-Al\(_2\)O\(_3\) nanofluid in a corrugated trapezoidal cavity with a rotating heat-generating solid cylinder.

Problem formulation

In this study, we consider the unsteady incompressible laminar flow of hybrid nanofluid with velocity \(\mathbf{v} =u\mathbf{i} +v\mathbf{j} \) in a symmetrical trapezoidal cavity with height h. The hybrid nanofluid is comprised of water as the base fluid, and spherical Ag and Al\(_2\)O\(_3\) nanoparticles with solid volume fractions \(\phi \) and \(\varphi \) respectively. Initially (at time \(t=0\)), the nanofluid is at rest and has constant temperature \(T_0\). For time \(t>0\), the corrugated upper and lower walls of the cavity are isothermally cooled with temperature \(T_0\); the geometries of these walls are described (respectively) by the functions

$$\begin{aligned} f_L(x)= & {} Ah\,sin \biggl (\frac{2\pi (x+h)}{\lambda h}\biggr ),\,\,-h\le x\le h \end{aligned}$$
(1)
$$\begin{aligned} f_U(x)= & {} h\biggl [1+A\,sin \biggl (\frac{\pi (2x+h)}{\lambda h}\biggr )\biggr ],\,\,-\frac{h}{2}\le x\le \frac{h}{2},\nonumber \\ \end{aligned}$$
(2)

where A and \(\lambda \) are the amplitude and wavelength of the corrugations. The left and right walls are adiabatic and have a \(45^\circ \) inclination angle to the vertical axis (see Fig. 1). The convective nanofluid flow is induced by the rotation of a heat-generating solid cylinder that has angular velocity \(\varvec{\omega }=\omega \mathbf{k} \), radius \(r_\mathrm{cyl}\) and volumetric heat flux q. A horizontal magnetic field with strength \(B_0\) is applied to the trapezoidal cavity (see Fig. 1).

Fig. 1
figure 1

Schematic diagram for the problem

We assume that the hybrid nanofluid is Newtonian and has a sufficiently low concentration of Ag and Al\(_2\)O\(_3\) nanoparticles so that interactions among these nanoparticles are negligible. Thermal radiation and viscous and Joule dissipations are neglected, and the magnetic Reynolds number is very small. The temperature and heat flux are considered to be continuous at the cylinder surface. Moreover, the Boussinesq approximation is applied in describing the action of buoyancy forces on the system. Hence, the governing equations [10,11,12] and the corresponding initial conditions and boundary conditions are as follows:

Continuity equation

$$\begin{aligned} \frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0. \end{aligned}$$
(3)

X-momentum equation

$$\begin{aligned} \rho _{hnf}\bigg (\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}\bigg )=-\frac{\partial p}{\partial x}+\mu _{hnf}\bigg (\frac{\partial ^2 u}{\partial x^2}+\frac{\partial ^2 u}{\partial y^2}\bigg ).\qquad \end{aligned}$$
(4)

Y-momentum equation

$$\begin{aligned} \rho _{hnf}\bigg (\frac{\partial v}{\partial t}+u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}\bigg )= & {} -\frac{\partial p}{\partial y}+\mu _{hnf}\bigg (\frac{\partial ^2 v}{\partial x^2}+\frac{\partial ^2 v}{\partial y^2}\bigg ) \nonumber \\&+(\rho \beta )_{hnf}g(T-T_0)-\sigma _{hnf}B_0^2v.\nonumber \\ \end{aligned}$$
(5)

Energy equation for nanofluid

$$\begin{aligned}&(\rho c_p)_{hnf}\bigg (\frac{\partial T}{\partial t}+u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}\bigg )\nonumber \\&\qquad =\frac{\partial }{\partial x}\bigg (\kappa _{hnf}\frac{\partial T}{\partial x}\bigg )+\frac{\partial }{\partial y}\bigg (\kappa _{hnf}\frac{\partial T}{\partial y}\bigg ). \end{aligned}$$
(6)

Energy equation for heat-generating cylinder

$$\begin{aligned} (\rho c_p)_\mathrm{cyl}\frac{\partial T_\mathrm{cyl}}{\partial t}=\kappa _\mathrm{cyl}\bigg (\frac{\partial ^2 T_\mathrm{cyl}}{\partial x^2}+\frac{\partial ^2 T_\mathrm{cyl}}{\partial y^2}\bigg )+q. \end{aligned}$$
(7)

Initial conditions

$$\begin{aligned} u=v=0\,\,and \,\,T=T_0\,\,at \,\,t=0. \end{aligned}$$
(8)

Boundary conditions

$$\begin{aligned}&u=v=0\,\,and \,\,T=T_0\,\,on the upper and lower walls \nonumber \\ \end{aligned}$$
(9)
$$\begin{aligned}&u=v=\frac{\partial T}{\partial \mathbf{n} }=0\,\,on the left and right side walls \end{aligned}$$
(10)
$$\begin{aligned}&\frac{1}{r_{cyl}^2}{} \mathbf{r} \times \mathbf{v} =\varvec{\omega },\,\,\mathbf{v} \cdot \mathbf{n} =0,\,\,\kappa _{hnf}\frac{\partial T}{\partial \mathbf{n} }=\kappa _\mathrm{cyl}\frac{\partial T_\mathrm{cyl}}{\partial \mathbf{n} }\nonumber \\&and \,\,T=T_\mathrm{\mathrm cyl}\,\,on the surface of the solid cylinder ,\quad \end{aligned}$$
(11)

where \(\mathbf{r} \) is the position vector on the surface of the cylinder and \(\mathbf{n} \) is the unit outward normal vector to the boundaries of the trapezoidal region and the cylinder.

The density \(\rho _{hnf}\), heat capacity \((\rho c_p)_{hnf}\), thermal expansion coefficient \(\beta _{hnf}\), thermal conductivity \(\kappa _{hnf}\), viscosity \(\mu _{hnf}\) and electrical conductivity \(\sigma _{hnf}\) of the hybrid nanofluid are described as follows [9, 13, 14]:

$$\begin{aligned} \rho _{hnf}= & {} \rho _\mathrm{Ag}\phi +\rho _{\mathrm{Al}_2\mathrm{O}_3}\varphi +(1-\phi -\varphi )\rho _f \end{aligned}$$
(12)
$$\begin{aligned} (\rho c_p)_{hnf}= & {} \rho _\mathrm{Ag}(c_p)_\mathrm{Ag}\phi +\rho _{\mathrm{Al}_2\mathrm{O}_3}(c_p)_{\mathrm{Al}_2\mathrm{O}_3}\varphi +(1-\phi -\varphi )\rho _f(c_p)_f \end{aligned}$$
(13)
$$\begin{aligned} (\rho \beta )_{hnf}= & {} \rho _\mathrm{Ag}\beta _\mathrm{Ag}\phi +\rho _{\mathrm{Al}_2\mathrm{O}_3}\beta _{\mathrm{Al}_2\mathrm{O}_3}\varphi +(1-\phi -\varphi )\rho _f\beta _f \end{aligned}$$
(14)
$$\begin{aligned} \frac{\kappa _{hnf}}{\kappa _f}= & {} 1+4.4Re_{np}^{0.4}Pr^{0.66}\bigg (\frac{T}{T_{fr}}\bigg )^{10}\big (\kappa _\mathrm{Ag}^{0.03}\phi ^{0.66}+\kappa _{\mathrm{Al}_2\mathrm{O}_3}^{0.03}\varphi ^{0.66}\big )\kappa _f^{-0.03} \end{aligned}$$
(15)
$$\begin{aligned} \frac{\mu _{hnf}}{\mu _f}= & {} \frac{1}{1-34.87d_f^{0.3}\big (d_\mathrm{Ag}^{-0.3}\phi ^{1.03}+d_{\mathrm{Al}_2\mathrm{O}_3}^{-0.3}\varphi ^{1.03}\big )} \end{aligned}$$
(16)
$$\begin{aligned} \frac{\sigma _{hnf}}{\sigma _f}= & {} 1+\frac{3(\sigma _\mathrm{Ag}\phi +\sigma _{\mathrm{Al}_2\mathrm{O}_3}\varphi -\sigma _f)(\phi +\varphi )}{\sigma _\mathrm{Ag}\phi +\sigma _{\mathrm{Al}_2\mathrm{O}_3}\varphi +2\sigma _f-(\sigma _\mathrm{Ag}\phi +\sigma _{\mathrm{Al}_2\mathrm{O}_3}\varphi -\sigma _f)(\phi +\varphi )}, \end{aligned}$$
(17)

where \(Re_{np}=\frac{2\rho _fk_bT(\phi +\varphi )}{\pi \mu _f^2(d_\mathrm{Ag}\phi +d_{\mathrm{Al}_2\mathrm{O}_3}\varphi )}\) is the nanoparticle Reynolds number, \(Pr=\frac{\mu _f(c_p)_f}{\kappa _f}\) is the Prandtl number, \(T_{fr}=273.15K\) is the freezing point of water, \(k_b=1.38\times 10^{-23}J/K\) is the Boltzmann constant, and \(d_\mathrm{Ag}\), \(d_{\mathrm{Al}_2\mathrm{O}_3}\) and \(d_f=0.385nm \) are the diameters of Ag nanoparticles, Al\(_2O_3\) nanoparticles and water respectively.

The corresponding thermophysical properties Ag and Al\(_2\)O\(_3\) nanoparticles and water [9, 15,16,17] are provided in Table 1.

Table 1 Thermophysical properties of pure water, Ag and Al\(_2O_3\)
Table 2 Grid-independence test values for \(Re=1\), \(Pr=6.2\), \(r_\mathrm{cyl}=0.2\), \(\phi =0.02\), \(\varphi =0.01\), \(Q=1\), \(Ri=1\), \(Ha=1\), \(d_\mathrm{Ag}=d_\mathrm{Al_2O_3}=15\) nm, \(T_0=298.15K\), \(\alpha _\mathrm{rat}=3\); \(A=0.05\), \(\lambda =0.5\), \(N=50\) and \(t=1\)

By substituting the dimensionless variables

$$\begin{aligned}&\bigg (\hat{x},\hat{y},\hat{f}_L(\hat{x}),\hat{f}_U(\hat{x}),\hat{r}_\mathrm{cyl}\bigg ) =\frac{1}{h}\bigg (x,y,f_L(x),f_U(x),r_\mathrm{cyl}\bigg ),\nonumber \\&\quad \hat{t}=\omega t,\,\,(\hat{u},\hat{v})=\frac{1}{h\omega }(u,v), \nonumber \\&\quad \hat{p}=\frac{p}{\rho _fh^2\omega ^2},\,\,\hat{T} =\frac{T-T_0}{T_0},\,\,\hat{T}_\mathrm{cyl}=\frac{T_\mathrm{cyl}-T_0}{T_0} \end{aligned}$$
(18)

in the governing Eqs. (3)–(7) and dropping the “hat” notation for convenience, we obtain the following non-dimensionalized governing equations:

$$\begin{aligned}&\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0 \end{aligned}$$
(19)
$$\begin{aligned}&\frac{\rho _{hnf}}{\rho _f}\bigg (\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}\bigg )=-\frac{\partial p}{\partial x}+\frac{1}{Re}\frac{\mu _{hnf}}{\mu _f}\bigg (\frac{\partial ^2 u}{\partial x^2}+\frac{\partial ^2 u}{\partial y^2}\bigg )\nonumber \\ \end{aligned}$$
(20)
$$\begin{aligned}&\frac{\rho _{hnf}}{\rho _f}\bigg (\frac{\partial v}{\partial t}+u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}\bigg )=-\frac{\partial p}{\partial y}+\frac{1}{Re}\frac{\mu _{hnf}}{\mu _f}\bigg (\frac{\partial ^2 v}{\partial x^2}+\frac{\partial ^2 v}{\partial y^2}\bigg ) \nonumber \\&\quad +\frac{(\rho \beta )_{hnf}}{\rho _f\beta _f}Ri\,T-\frac{\sigma _{hnf}}{\sigma _f}\frac{Ha^2}{Re}v \end{aligned}$$
(21)
$$\begin{aligned}&\frac{(\rho c_p)_{hnf}}{\rho _f(c_p)_f}\bigg (\frac{\partial T}{\partial t}+u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}\bigg )=\frac{1}{RePr\kappa _f}\bigg [\frac{\partial }{\partial x}\bigg (\kappa _{hnf}\frac{\partial T}{\partial x}\bigg )\nonumber \\&+\frac{\partial }{\partial y}\bigg (\kappa _{hnf}\frac{\partial T}{\partial y}\bigg )\bigg ] \end{aligned}$$
(22)
$$\begin{aligned}&\frac{\partial T_\mathrm{cyl}}{\partial t}=\frac{\alpha _{rat}}{RePr}\bigg (\frac{\partial ^2 T_\mathrm{cyl}}{\partial x^2}+\frac{\partial ^2 T_\mathrm{cyl}}{\partial y^2}+Q\bigg ), \end{aligned}$$
(23)

where \(Re=\frac{\rho _f\omega h^2}{\mu _f}\), \(Ha=B_0h\sqrt{\frac{\sigma _f}{\mu _f}}\), \(Ri=\frac{\beta _fgT_0}{\omega }\), \(Pr=\frac{\mu _f(c_p)_f}{\kappa _f}\) and \(\alpha _\mathrm{rat}=\frac{\rho _f(c_p)_f\kappa _\mathrm{cyl}}{\kappa _f(\rho c_p)_\mathrm{cyl}}\) are (respectively) the Reynolds number, Hartmann number, Richardson number, Prandtl number and thermal diffusivity ratio. The dimensionless initial conditions are

$$\begin{aligned} u=v=T=0\,\,at \,\,t=0 \end{aligned}$$
(24)

whereas the non-dimensionalized boundary conditions are

$$\begin{aligned}&u=v=T=0 \,\,on the upper and lower walls \end{aligned}$$
(25)
$$\begin{aligned}&u=v=\frac{\partial T}{\partial \mathbf{n} }=0\,\,on the left and right side walls \end{aligned}$$
(26)
$$\begin{aligned}&\frac{1}{r_\mathrm{cyl}^2}{} \mathbf{r} \times \mathbf{v} =\mathbf{k} ,\,\,\mathbf{v} \cdot \mathbf{n} =0,\,\,\frac{\kappa _{hnf}}{\kappa _f}\frac{\partial T}{\partial \mathbf{n} }=\kappa _\mathrm{rat}\frac{\partial T_\mathrm{cyl}}{\partial \mathbf{n} }\,\,and \nonumber \\&T=T_\mathrm{cyl}\,\,on the surface of the solid cylinder , \end{aligned}$$
(27)

where \(\kappa _\mathrm{rat}=\frac{\kappa _\mathrm{cyl}}{\kappa _f}\) is the thermal conductivity ratio.

Fig. 2
figure 2

Comparison plots of Alsaberry et al. [9] and the present algorithm

The average Nusselt number \(Nu_\mathrm{ave}\) (heat transfer rate) on the surface \(\Gamma \) of the heat-generating solid cylinder is given by

$$\begin{aligned} Nu_\mathrm{ave}=\frac{1}{2\pi r_\mathrm{cyl}}\int _{\Gamma }\frac{\partial T}{\partial \mathbf{n} }\,dl \end{aligned}$$
(28)

Solution methodology, grid-independence and validation

We obtain a numerical solution to the non-dimensionalized governing Eqs. (19)–(23) with initial and boundary conditions (24)–(27) using the penalty finite element method. According to this method [18], the nanofluid pressure is expressed as

$$\begin{aligned} p=-\frac{1}{\varepsilon }\bigg (\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}\bigg ), \end{aligned}$$
(29)

where \(\varepsilon \) is a small positive number. We substitute (29) into the momentum equations (19) and (20) and construct a penalized variational formulation of the problem with suitable solution spaces. A semi-discrete formulation is then created with a triangulation of quadratic triangular elements [19]. Using the Crank–Nicolson method for time discretization with N time steps, the governing equations are reduced to a system of nonlinear equations; these equations are solved iteratively in FreeFem++ [20] with penalty parameter \(\varepsilon =10^{-8}\) using the fixed-point method with a relative error tolerance of \(10^{-4}\).

To obtain a suitable finite element mesh for the present work, we compute the average Nusselt number \(Nu_\mathrm{ave}\) on the surface of the solid cylinder using five different meshes with 2036, 3360, 5452, 8066 and 10564 quadratic triangular elements (Table 2). The criterion used for grid independence is that the \(Nu_\mathrm{ave}\) relative error between two successive finite element meshes is less than 0.5%. Using this criterion, a mesh containing 5452 elements was chosen for this numerical study.

The present finite element algorithm is validated by making comparisons of streamlines and isotherms with the numerical study of Alsabery et al. [9] on the mixed convective Cu-\(Al_2O_3\)/water hybrid nanofluid flow in a lid-driven wavy square cavity with a heat-conducting solid block (Fig. 2). We determine that there is good agreement between the compared results; thus, the present algorithm is useful for obtaining an accurate numerical solution to the problem investigated herein.

Table 3 \(Nu_\mathrm{ave}\) for different t and varying \(d_\mathrm{Ag}\) and \(d_\mathrm{Al_2O_3}\)

Results and discussion

Numerical results for streamlines, isotherms and average Nusselt number are presented for different values of the nanoparticle diameters \(d_\mathrm{Ag}\), \(d_\mathrm{Al_2O_3}\) and solid cylinder radius \(r_\mathrm{cyl}\). Unless otherwise stated, the following numerical values were used: \(Re=1\), \(Pr=6.2\), \(r_\mathrm{cyl}=0.2\), \(\phi =0.02\), \(\varphi =0.01\), \(Q=1\), \(Ri=1\), \(Ha=1\), \(d_\mathrm{Ag}=d_\mathrm{Al_2O_3}=15\) nm, \(T_0=298.15\) K, \(\alpha _\mathrm{rat}=3\); \(A=0.05\), \(\lambda =0.5\) and \(t=1\).

Figures 3 and 4 show the influence of varying \(d_{Ag}\) and \(d_\mathrm{Al_2O_3}\) on streamlines and isotherm plots. From Fig. 3, we observe flow circulations along the walls of the trapezoidal cavity and the surface of the internal solid cylinder, and crescent-shaped circulations between the cylinder and the side walls of the cavity. These circulations are generated by the combined effects of free convection due to buoyancy forces acting on the hybrid nanofluid and forced convection due to the anticlockwise rotation of the solid cylinder. We also see that there are slight reductions in the sizes of the crescent-shaped flow circulation regions as either \(d_\mathrm{Ag}\) or \(d_\mathrm{Al_2O_3}\) is reduced from 15nm to 2nm. This indicates a decrease in the velocity of the hybrid nanofluid due to increased fluid viscosity as the Ag and Al\(_2\)O\(_3\) nanoparticle diameters are reduced. Figure 4 reveals that the temperature within the trapezoidal cavity is highest near the centre of the heat-generating solid cylinder and lowest on the upper and lower wavy walls. This temperature was found to decrease with a reduction in either \(d_\mathrm{Ag}\) or \(d_\mathrm{Al_2O_3}\). An increase in the diameters of Ag and \(Al_2O_3\) nanoparticles enhances the thermal conductivity of the hybrid nanofluid [13]. This increase in heat conduction leads to the observed reduction in the hybrid nanofluid temperature.

The average Nusselt number \(Nu_\mathrm{ave}\) on the surface of the heat-generating cylinder over time t was displayed in Table 3 for different values of \(d_\mathrm{Ag}\) and \(d_\mathrm{Al_2O_3}\). \(Nu_\mathrm{ave}\) increases with time as a result of enhanced heat transfer from the solid cylinder to the hybrid nanofluid. Furthermore, when either \(d_\mathrm{Ag}\) or \(d_\mathrm{Al_2O_3}\) is decreased, \(Nu_\mathrm{ave}\) increases for each value of t due to an increase in the hybrid nanofluid thermal conductivity.

Fig. 3
figure 3

Streamline plots for varying \(d_\mathrm{Ag}\) and \(d_\mathrm{Al_2O_3}\)

Fig. 4
figure 4

Isotherm plots for varying \(d_\mathrm{Ag}\) and \(d_\mathrm{Al_2O_3}\)

Figures 5 and 6 display the effects of the solid cylinder radius \(r_\mathrm{cyl}\) on streamlines and isotherms. We notice in Fig. 5 that when \(r_\mathrm{cyl}=0.1\), the crescent-shaped circulations of fluid between the cylinder and side walls are absent due to weak mixed convection within the hybrid nanofluid. However, these crescent-shaped flow circulations are present when \(r_\mathrm{cyl}\ge 0.2\) and increase in size with increased \(r_\mathrm{cyl}\) as a result of enhancements in mixed convection associated with greater heat generation and angular velocity on the surface of the increasingly large cylinder. From Fig. 6, we found that the temperature within the cavity increases as \(r_\mathrm{cyl}\) increases; this occurs as a result of a larger heat-generating cylinder at the centre of the cavity. Moreover, the isotherms within the cylinder are more elongated with increasing \(r_\mathrm{cyl}\), since the impact of side wall inclination on heat transfer from the heat-generating cylinder to the hybrid nanofluid is more pronounced as the cylinder increases in size.

The impact of the solid cylinder radius \(r_\mathrm{cyl}\) on the average Nusselt number \(Nu_\mathrm{ave}\) over time t is depicted in Fig. 7. It was identified that for each value of t, \(Nu_\mathrm{ave}\) is increased with an increased in \(r_\mathrm{cyl}\). This occurs due to enhanced heat transfer from the cylinder to the surrounding fluid as the cylinder size is increased.

Fig. 5
figure 5

Streamline plots for varying \(r_\mathrm{cyl}\)

Fig. 6
figure 6

Isotherm plots for varying \(r_\mathrm{cyl}\)

Fig. 7
figure 7

Plot of \(Nu_\mathrm{ave}\) vs. t for varying \(r_\mathrm{cyl}\)

Conclusions

The present study focused on the MHD convective flow of silver-alumina/ water hybrid nanofluid within a wavy trapezoidal cavity containing an internal heat-generating and rotating cylinder. The influence of the diameters of silver and alumina nanoparticles and solid cylinder radius on streamlines, isotherms and average Nusselt number (heat transfer rate) on the surface of the cylinder was examined.

It was revealed in this work that the flow circulation regions near the rotating cylinder increase in size when the diameters of silver and alumina nanoparticles and the cylinder radius increase. Furthermore, the temperature of the hybrid nanofluid can be increased by reducing the silver or alumina nanoparticle diameters and increasing the radius of the solid cylinder. The study also determined that the heat transfer rate on the surface of the solid cylinder can be increased by increasing the cylinder radius and decreasing the nanoparticle diameters.

The findings of this study contribute to an improved understanding of hybrid nanofluid flows and convective heat transfer phenomena in cavities with irregular geometries. In particular, by investigating the role of silver and alumina nanoparticle diameters and the size of the rotating and heat-generating solid cylinder in heat transfer enhancement, liquid cooling systems used in engineering and industrial applications can be designed and operated in a more efficient manner.