Energy transport analysis in natural convective flow of water:Ethylene glycol (50:50)-based nanofluid around a spinning down-pointing vertical cone

Introduction

In this industrialized world, the heat transfer process plays a significant role in upgrading the efficiency of industrial applications. To accomplish this global industrialization, in 1995 Choi devised a new progressive class of heat transfer fluids, known as nanofluids (Choi and Eastman, 1995), in which the characteristics of both nanoparticles and base fluid become efficient. Heat transfer in nanofluid is more proficient than in common fluids. Preparation of nanofluids is not simply the mixture of solids and liquids but requires beneficial methods, as elaborated in Xuan and Li (2000), which presents the procedure for nanofluid preparation. Choosing the nanoparticle and base fluid wisely leads to excellent results depending on the need, as discussed in Usri et al. (2015). Recent research works have been implemented using a novel category of fluids known as nanofluids, which have brought changes widely, including in the industrial, engineering, and medical fields (Vishnu Ganesh et al., 2014; Abdul Hakeem et al., 2017). Different geometrical shapes give different results, including cone and wedge (Anantha Kumar et al. (2018), rotating disk (Gholinia et al. (2019), and vertical cone geometric shapes (CemEce, 2005; Raju and Sandeep, 2016). Nanofluids are the best solutions for heat transfer fluids since they have good thermal performance. Therefore, researchers are proposing suitable models. In this regard, three methods are employed for improving thermal performance (Maleki et al., 2020). To adopt nanofluid applications in daily life, and to increase nanofluid’s performance in several applications, nanofluid stability is a critical factor discussed in (Chakraborty and Kumar Panigrahi, 2020). To enhance heat transfer, comparison among different nanofluids for different parameters has been conducted (Dinarvand and Pop, 2017; Aghamajidi et al., 2018). Nanofluid applications have been used in multidisciplinary research. There is a broad range of utilizations in the areas of microalgal cultivation, friction reduction, magnetic sealing, reactor–heat exchange, optical and biomedical applications, nanofluid detergent, electronics cooling, and heating buildings (Vargas-Estrada et al., 2020; Rafiq et al., 2021). With progress in nuclear energy, nanoparticles are also used as coolants in nuclear power plants (Hamidreza Arab BafraniNoori-kalkhoran et al., 2020), in enhancing oil recovery, nano-refrigerants, and nano-lubricants (de Carvalho et al., 2020; SahbanAlnarabiji and Husein, 2020; Salari and Seid Mahdi Jafari, 2020; Mallikarjuna et al., 2021), and in turning and grinding processes (SaswatKhatai et al., 2020). In addition to nanofluid, the flow of hybrid nanofluid across a stretched surface has recently been studied (Aly and Pop, 2019; Aly and Pop, 2020a; Aly and Ebaid, 2020; Aly and Pop, 2020b; Aly et al., 2021; Ahmad et al., 2022; Aly et al., 2022; Arafat et al., 2022; Reddy et al., 2022; Usafzai et al., 2022).

Using a water–ethylene glycol (50:50) combination as the base fluid and ${Al}_2{O}_3$ and ${Fe}_3{O}_4$ as the nanoparticles, we explored the natural convection flow around a heated vertical spinning cone under the influence of a magnetic field.

The aspect of the present work is listed below.

➢ Water–ethylene glycol (50:50) mixture is considered a base fluid with Pr = 29.86.

➢ ${Al}_2{O}_3$ and ${Fe}_3{O}_4$ are considered to be non-magnetic and magnetic nanoparticles, respectively, which are in thermal equilibrium with base fluid.

➢ The geometric cone is used for fluid flow as shown in Figure 1.

➢ The effect of viscous dissipation, the resistance heating effect of the fluid, and the slip effect are regarded as negligible.

FIGURE 1

FIGURE 1. Geometry of the problem.

Governing equations and problem formulation

A continuous two-dimensional flow of a combination of $H_{2}O-C_2{H}_6{O}_2$ (50:50) containing Al₂O₃ and Fe₃O₄ nanoparticles was studied, under the influence of a magnetic field. The flow was laminar, and the nanofluid was assumed to be incompressible.

The $y^{*}$ axis is the dimension normal to the cone’s surface, and the $x^{*}$ axis is the dimension toward the cone’s surface. The rotational angle is indicated by $\theta$, and it was assumed that cone spins with a constant angular velocity Ω.

The models that govern the phenomena are given below (Aghamajidi et al., 2018).

Continuity equation

${\left(r^{*}{u}^{*}\right)}_{x^{*}}+{\left(r^{*}{v}^{*}\right)}_{y^{*}}=0(1)$

Momentum equation in x direction

${\rho }_{nf}\left({\left(u^{*}\right)}_{x^{*}}{u}^{*}+{\left(u^{*}\right)}_{y^{*}}{v}^{*}-\frac{w^{{*}^2}}{x^{*}}\right)={\mu }_{nf}(u^{*}{y}^{*}{)}_{y^{*}}+{\left(\rho \beta \right)}_{nf}g\cos \gamma \left(T-T_o\right)-{\sigma }_{nf}{B}^2{u}^{*}(2)$

Momentum equation in y direction

${\rho }_{nf}\left({\left(w^{*}\right)}_{x^{*}}{u}^{*}+{\left(w^{*}\right)}_{y^{*}}{v}^{*}+\frac{{u^{*}w}^{*}}{x^{*}}\right)={\mu }_{nf}(w^{*}{y}^{*}{)}_{y^{*}}-{\sigma }_{nf}{B}^2{w}^{*}(3)$

Energy equation

$\left({\rho C}_p\right)\left(u^{*}{\left(T\right)}_{x^{*}}+v^{*}{\left(T\right)}_{y^{*}}\right)=k_{nf}{\left(T_{y^{*}}\right)}_{y^{*}}(4)$

Here, $(r^{*}{u}^{*}{)}_{x^{*}}$ denotes the partial derivative of $\left(r^{*}{u}^{*}\right)$ with respect to $x^{*}$.

The boundary conditions for the above governing equations are given below.

Prescribed surface temperature case:

$\mathrm{T}\left(x^{*},0\right)={\mathrm{T}}_0+\left({\mathrm{T}}_{\mathrm{r}}–{\mathrm{T}}_0\right)\frac{x^{*}}{L}\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{n}{y}^{*}=0(5)$

$u^{*}=0,v^{*}=0,w^{*}=r^{*}\Omega \mathrm{a}\mathrm{s}{\mathrm{y}}^{*}=0(6)$

$u^{*}\to 0,w^{*}\to 0,\mathrm{T}\to 0\mathrm{a}\mathrm{s}{y}^{*}\to \infty .(7)$

$\gamma$ is considered as half of the vertex angle, and the local radius of the cone is considered as r = x sin $\gamma$. Dimensional velocity components are denoted by $u^{*},v^{*},\mathrm{a}\mathrm{n}\mathrm{d}{w}^{*}$ in the $x^{*},y^{*},$ and $\theta$ directions, respectively.

The thermophysical properties are given in Table 1. The nanofluid properties are given by (Saranya et al., 2022)

$\mathrm{N}\mathrm{a}\mathrm{n}\mathrm{o}\mathrm{f}\mathrm{l}\mathrm{u}\mathrm{i}\mathrm{d}’\mathrm{s}\mathrm{k}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{v}\mathrm{i}\mathrm{s}\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{y},{\mathrm{\nu }}_{\mathrm{nf}}=\frac{{\mathrm{\mu }}_{\mathrm{f}}}{{\left(1\mathrm{-}\mathrm{\varphi }\right)}^{2.5}\left[\left(1\mathrm{-}\mathrm{\varphi }\right){\mathrm{\rho }}_{\mathrm{f}}+\mathrm{\varphi }{\mathrm{\rho }}_{\mathrm{s}}\right]}(8)$

$\mathrm{N}\mathrm{a}\mathrm{n}\mathrm{o}\mathrm{f}\mathrm{l}\mathrm{u}\mathrm{i}\mathrm{d}’\mathrm{s}\mathrm{d}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{y}{\mathrm{\rho }}_{\mathrm{n}\mathrm{f}}=\left(1-\mathrm{\varphi }\right){\mathrm{\rho }}_{\mathrm{f}}+\mathrm{\varphi }{\rho }_{\mathrm{s}}(9)$

$\mathrm{N}\mathrm{a}\mathrm{n}\mathrm{o}\mathrm{f}\mathrm{l}\mathrm{u}\mathrm{i}\mathrm{d}’\mathrm{s}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{m}\mathrm{a}\mathrm{l}\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{f}\mathrm{u}\mathrm{s}\mathrm{i}\mathrm{v}\mathrm{i}\mathrm{t}\mathrm{y},{\mathrm{\alpha }}_{\mathrm{nf}}=\frac{{\mathrm{k}}_{\mathrm{n}\mathrm{f}}}{{\left(\mathrm{\rho }{\mathrm{C}}_{\mathrm{p}}\right)}_{\mathrm{n}\mathrm{f}}},(10)$

$\mathrm{N}\mathrm{a}\mathrm{n}\mathrm{o}\mathrm{f}\mathrm{l}\mathrm{u}\mathrm{i}\mathrm{d}’\mathrm{s}\mathrm{h}\mathrm{e}\mathrm{a}\mathrm{t}\mathrm{c}\mathrm{a}\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{i}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{c}\mathrm{e},{\left(\mathrm{\rho }{\mathrm{C}}_{\mathrm{p}}\right)}_{\mathrm{nf}}=\left(1-\mathrm{\varphi }\right){\left(\mathrm{\rho }{\mathrm{C}}_{\mathrm{p}}\right)}_{\mathrm{f}}+\mathrm{\varphi }{\left(\mathrm{\rho }{\mathrm{C}}_{\mathrm{p}}\right)}_{\mathrm{s}},(11)$

$\mathrm{C}\mathrm{o}\mathrm{e}\mathrm{f}\mathrm{f}\mathrm{i}\mathrm{c}\mathrm{i}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{o}\mathrm{f}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{m}\mathrm{a}\mathrm{l}\mathrm{e}\mathrm{x}\mathrm{p}\mathrm{a}\mathrm{n}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n},{\left(\mathrm{\rho }\mathrm{\beta }\right)}_{\mathrm{nf}}=\left(1-\mathrm{\varphi }\right){\left(\mathrm{\rho }\mathrm{\beta }\right)}_f+\mathrm{\varphi }{\left(\mathrm{\rho }\mathrm{\beta }\right)}_{\mathrm{s}}(12)$

TABLE 1

TABLE 1. Thermophysical properties of $H_{2}O-C_2{H}_6{O}_2$ (50:50)-based fluid and ${Al}_2{O}_3/{Fe}_3{O}_4$ nanoparticles (Aghamajidi et al., 2018; Saranya et al., 2018; Saranya et al., 2022).

Nanofluid’s thermal conductivity and electrical conductivity are given by

$\frac{k_{nf}}{k_f}=\frac{\left(k_s+2k_f\right)-2\varphi \left(k_f-ks\right)}{\left(k_s+2k_f\right)+\varphi \left(k_f-ks\right)}\mathrm{a}\mathrm{n}\mathrm{d}\frac{{\sigma }_{nf}}{{\sigma }_f}=1+\frac{3\left(\sigma -1\right)\varphi }{\left(\sigma +2\right)-\left(\sigma -1\right)\varphi },\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}\sigma =\frac{{\sigma }_s}{{\sigma }_f},\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{l}\mathrm{y}(13)$

Here, $\varphi$ is the solid volume fraction. The subscripts sf, nf, and s denote the fluid, nanofluid, and solid, respectively.

For the current analysis, the following variables are utilized (Aghamajidi et al., 2018):

$r=\frac{r^{*}}{L},x=\frac{x^{*}}{L},y=\frac{y^{*}}{L}{Gr}^{1/4}(14)$

$u=\frac{u^{*}}{U},v=\frac{v^{*}}{U}{Gr}^{1/4},w=\frac{w^{*}}{\Omega L},\mathrm{\Theta }=\frac{T-T_0}{T_r-T_0}(15)$

where u, v, and w represent the velocity components x direction, y direction, and θ direction, respectively, and $\mathrm{\Theta }$ is the dimensionless temperature ratio. $r=x\sin \gamma$ is the radius of the cone and the magnetic field strength is B = $B_0\mathrm{b}\left(\mathrm{x}\right)/(r\sqrt{1-{r^{\prime }}^2)}.$

The Prandtl number (Pr), reference velocity (U), and Grashof number (Gr) are defined as

$\mathit{Pr}=\frac{\nu }{\alpha },U={\left[\mathrm{g}\mathit{cos}\gamma \beta L\left(T_r-T_0\right)\right]}^{1/2},Gr={\left(\frac{UL}{\nu }\right)}^2(16)$

where $\beta$ is the thermal expansion coefficient, L is the reference length, $\nu$ is kinematic viscosity, and $\alpha$ is thermal diffusivity. $T_r$ is any taken reference temperature unequal to the ambient temperature $T_0$.

The governing equations from Eq. 1 to Eq. 4 take the following non-dimensional form after substituting the dimensionless variables, as defined in Eqs 14–16:

${\left(ru\right)}_x+{\left(rv\right)}_y=0(17)$

$u{u}_x+v{u}_y-\frac{R{e}^2}{Gr}\frac{r^{\prime }}{r}{w}^2=\frac{1}{\left[\left(1-\varphi \right)+\varphi \left({\rho }_s/{\rho }_f\right)\right]}\left\{\frac{1}{{\left(1-\varphi \right)}^{2.5}}{\left(u_y\right)}_y+\left[\left(1-\varphi \right)+\varphi {\left(\rho \beta \right)}_s/{\left(\rho \beta \right)}_f\right]\mathrm{\Theta }-\frac{{\sigma }_{nf}}{{\sigma }_f}M{\mathrm{\Lambda }}^{2}u\right\}(18)$

$u{w}_x+v{w}_y+uw\frac{r^{\prime }}{r}=\frac{1}{\left(1-\varphi +\varphi {\rho }_s/{\rho }_f\right)}\left\{\frac{1}{{\left(1-\varphi \right)}^{2.5}}{\left(w_y\right)}_y-\left(\frac{{\sigma }_{nf}}{{\sigma }_f}\right)M{\Lambda }^{2}w\right\}(19)$

$u{\mathrm{\Theta }}_x+v{\mathrm{\Theta }}_y=\frac{1}{\left[\left(1-\varphi \right)+\varphi \left({\left(\rho C_p\right)}_s/{\left(\rho C_p\right)}_f\right)\right]}\left[\left(\frac{1}{\mathit{Pr}}\frac{K_{nf}}{K_f}\right){\left({\mathrm{\Theta }}_y\right)}_y\right](20)$

The boundary conditions are given below.

Prescribed surface temperature case:

$\mathrm{\Theta }=x\mathrm{a}\mathrm{s}\mathrm{y}=0(21)$

$u=0,v=0,w=r\mathrm{a}\mathrm{s}\mathrm{y}=0(22)$

$\mathrm{u}\to 0,\mathrm{w}\to 0,\mathrm{\Theta }\to 0\mathrm{a}\mathrm{s}y\to \infty (23)$

The rotational Reynolds number, the magnetic field function Λ, and the magnetic parameter are

$Re=\frac{\mathrm{\Omega }{L}^2}{{\nu }_f},\mathrm{\Lambda }=\frac{b\left(x\right)}{r\sqrt{1-{r^{\prime }}^2}},M=\frac{{\sigma }_f{B}_0^{2}L}{{U\rho }_f}.(24)$

The function $b\left(x\right)=r\sqrt{1-{r^{\prime }}^2}$ is the basis of the current analysis, which was conducted in a scenario in which the intensity of the magnetic field applied normal to the surface was uniform along the surface within the boundary layer, such that $\Lambda$ = 1:

$ru={\psi }_y,rv=-{\psi }_x,(25)$

where ψ is stream function and the boundary layer variables can be presented as

$\psi \left(x,y\right)=xrF\left(y\right),w=rG\left(y\right),\mathrm{\Theta }=xH\left(y\right),(26)$

Applying the boundary layer variables as written in Eq. 26, the non-dimensional governing Eqs 17–20 are converted into the system of Ordinary differential equations (ODEs).

Here F(y), G(y), and H(y) represent the tangential velocity profile, swirl velocity profile, and temperature profile, respectively.

$A_1{F}^{‴}+2F{F}^{″}-{F^{\prime }}^2+\epsilon G^2-A_2{A}_{5}M{\Lambda }^2{F}^{\prime }+A_{3}H=0(27)$

$A_1{G}^{″}+2F{G}^{\prime }-2G{F}^{\prime }-A_2{A}_{5}M{\Lambda }^{2}G=0(28)$

${\mathrm{A}}_4{\mathrm{H}}^{″}+\mathrm{Pr}\left[2\mathrm{F}{H}^{\prime }-{\mathrm{F}}^{\prime }\mathrm{H}\right]=0(29)$

where ${\mathrm{A}}_1=1/{\left(1-\mathrm{\varphi }\right)}^{2.5}\left(1-\mathrm{\varphi }+\mathrm{\varphi }\left({\mathrm{\rho }}_{\mathrm{s}}/{\mathrm{\rho }}_{\mathrm{f}}\right)\right)$, ${\mathrm{A}}_2=1/\left(1-\mathrm{\varphi }+\mathrm{\varphi }\left({\mathrm{\rho }}_{\mathrm{s}}/{\mathrm{\rho }}_{\mathrm{f}}\right)\right)$${\mathrm{A}}_3=\left(1-\mathrm{\varphi }+\mathrm{\varphi }{\left(\mathrm{\rho }\mathrm{\beta }\right)}_{\mathrm{s}}/{\left(\mathrm{\rho }\mathrm{\beta }\right)}_{\mathrm{f}}\right)/\left(1-\mathrm{\varphi }+\mathrm{\varphi }\left({\mathrm{\rho }}_{\mathrm{s}}/{\mathrm{\rho }}_{\mathrm{f}}\right)\right)$, ${\mathrm{A}}_4=\left({\mathrm{k}}_{\mathrm{n}\mathrm{f}}/{\mathrm{k}}_{\mathrm{f}}\right)/\left(1-\mathrm{\varphi }+\mathrm{\varphi }{\left(\mathrm{\rho }{\mathrm{C}}_{\mathrm{p}}\right)}_{\mathrm{s}}/{\left(\mathrm{\rho }{\mathrm{C}}_{\mathrm{p}}\right)}_{\mathrm{f}}\right)$${\mathrm{A}}_5={\mathrm{\sigma }}_{\mathrm{nf}}/{\mathrm{\sigma }}_{\mathrm{f}}=1+3\left(\mathrm{\sigma }-1\right)\mathrm{\varphi }/\left(\mathrm{\sigma }+2\right)-\left(\sigma -1\right)\varphi$, where $\mathrm{\sigma }={\mathrm{\sigma }}_{\mathrm{s}}/{\mathrm{\sigma }}_{\mathrm{f}}$$\epsilon ={\left(\mathrm{R}\mathrm{e}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{\gamma }\right)}^2$/Gr is the spin parameter.

The boundary conditions from Eqs 21 to 23 are reduced to

$F=0,F^{\prime }=0,G=1,H=1asy=0(30)$

$F^{\prime }\to 0,G\to 0,H\to 0asy\to \infty (31)$

Skin friction coefficient and local Nusselt number

The skin friction coefficient $C_f$ and the local Nusselt number (Nu), have been consequential in the engineering field and are defined as

${\mathrm{C}}_{\mathrm{f}}=\frac{2{\mathrm{\tau }}_{\mathrm{w}}}{{\mathrm{\rho }}_{\mathrm{f}}{\mathrm{U}}^2},\mathrm{N}\mathrm{u}=\frac{\mathrm{L}{\mathrm{q}}_{\mathrm{w}}}{{\mathrm{k}}_{\mathrm{f}}\left({\mathrm{T}}_{\mathrm{w}}-{\mathrm{T}}_{\mathrm{\infty }}\right)}(32)$

where ${\tau }_w$ is the skin friction and $q_w$ is the surface heat flux, written as

${\mathrm{\tau }}_{\mathrm{w}}={\mathrm{\mu }}_{\mathrm{n}\mathrm{f}}{\left(\frac{\partial {\mathrm{u}}^{*}}{\partial {\mathrm{y}}^{*}}\right)}_{{\mathrm{y}}^{*}=0},{\mathrm{q}}_{\mathrm{w}}=-{\mathrm{k}}_{\mathrm{n}\mathrm{f}}{\left(\frac{\partial \mathrm{T}}{\partial {\mathrm{y}}^{*}}\right)}_{{\mathrm{y}}^{*}=0}(33)$

Using the non-dimensional transformations, we obtain

${\mathrm{C}}_{\mathrm{f}}{\mathrm{G}\mathrm{r}}^{1/4}=2\left(\frac{{\mathrm{\mu }}_{\mathrm{n}\mathrm{f}}}{{\mathrm{\mu }}_{\mathrm{f}}}\right)x{\mathrm{F}}^{″}\left(0\right),\mathrm{N}\mathrm{u}{\mathrm{G}\mathrm{r}}^{-1/4}=-\left(\frac{{\mathrm{k}}_{\mathrm{n}\mathrm{f}}}{{\mathrm{k}}_{\mathrm{f}}}\right)x{\mathrm{H}}^{\prime }\left(0\right)(34)$

Numerical approach

Eqs 27–29 with boundary conditions (Eqs 30, 31) specify nonlinear ordinary differential equations. By using a fourth-order Runge–Kutta finite difference scheme and shooting approach, this model is solved numerically to examine the effects of M, $\in ,\varphi$ on F^I(y), G(y), H(y), C_f, and Nu.

We represent ${F=y}_1$, ${G=y}_4$, ${H=y}_6$ for our present problem and the important steps of the method as

$y_1^{\prime }=y_2$

$y_2^{\prime }=y_3$

$y_3^{\prime }=-1/A_1\left[2y_1{y}_3-{y_2}^2+\epsilon {y_4}^2-A_2{A}_{5}M{\mathrm{\Lambda }}^2{y}_2+A_3{y}_6\right]$

$y_4^{\prime }=y_5$

$y_5^{\prime }=-1/A_1\left[\left(2y_1{y}_5-2y_2{y}_4\right)-A_2{A}_{5}M{\mathrm{\Lambda }}^2{y}_4\right]$

$y_6^{\prime }=y_7$

$y_7^{\prime }=-1/A_4\left[\mathrm{Pr}\left(2y_1{y}_7-y_2{y}_6\right)\right]$

To authenticate our work, the outcomes were compared with the results of Ece (2006), Dinarvand (2011), and Aghamajidi et al. (2018), which are shown in Table 2. It is worth mentioning that the present outcomes have excellent compatibility with the solutions obtained by Ece (2006), Dinarvand (2011), and Aghamajidi et al. (2018) for the case ϕ = 0.

TABLE 2

TABLE 2. Comparison with the results of Ece (2006), Dinarvand (2011), and Aghamajidi et al. (2018) for regular fluid ($\varphi$ = 0), for the effect of M and $\in$ on F^II(0) and –H^l (0) with prescribed surface temperature, for Prandtl number =1.

Results and discussion

The effects of spin parameter $ϵ$, nanoparticle volume fraction $\varphi$, and magnetic parameter M on tangential velocity profile F^I(y), swirl velocity profile G(y), and temperature profile H(y) for the case of prescribed surface temperature were plotted.

The tangential velocity profile decreases as the range of magnetic parameters increases. This is because a magnetic field creates a drag force, known as the Lorentz force, in an electrically conducting fluid. There is a dip in the velocity profile due to this significant resistive force acting counter to the direction of fluid flow. As a result, as M becomes stronger, the hydrodynamic boundary layer thickness becomes thinner. ${Fe}_3{O}_4$ has higher tangential velocity than ${Al}_2{O}_3$. In the case of swirl velocity, as the magnetic parameter increases, swirl velocity decreases. ${Al}_2{O}_3$ has higher swirl velocity than ${Fe}_3{O}_4$, as described in Figure 2

FIGURE 2

FIGURE 2. Influence of M on F^I(y) and G(y) when ϕ = 0.01, $\in$ = 1, and Pr = 29.86.

Boundary layer behaviour for the case of prescribed surface temperature

To overcome drag force, the fluid must do some additional work, which is transformed into thermal energy and leads to an increase in the temperature of the fluid. ${Al}_2{O}_3$ has a higher temperature than ${Fe}_3{O}_4$. Therefore, as the strength of M increases, the thickness of the thermal boundary value rises, as described in Figure 3. From the physical point of view, as the nanoparticles transfer or dissipate heat, they cause a larger thermal boundary layer thickness and, finally, the intensification in the temperature of the fluid.

FIGURE 3

FIGURE 3. Influence of M on H(y) when ϕ = 0.01, $\in$ = 1, and Pr = 29.86.

As the spin parameter appears in the momentum equation, the effect of spin parameter is more in this equation, and increased values of the spin parameter dynamically promote tangential velocity. ${Fe}_3{O}_4$ has higher tangential velocity than ${Al}_2{O}_3$. Swirl velocity drops as spin parameter increases near the surface of the cone, as shown in Figure 4.

FIGURE 4

FIGURE 4. Characteristics of $\in$ on F^I(y) and G(y), when ϕ = 0.01, M = 1, and Pr = 29.86.

Temperature is reduced as spin parameter increases. Therefore, there is reduction in the thickness of the thermal boundary layer for varying magnitudes of spin parameter. Fe₃O₄ nanoparticles are compressed more toward the surface than are the Al₂O₃ nanoparticles as the spin parameter rises, as indicated in Figure 5.

FIGURE 5

FIGURE 5. Impact of $\in$ on H(y), when ϕ = 0.01, M = 1, and Pr = 29.86.

The influence of nanoparticle volume fraction on tangential and swirl velocity profile is depicted in Figure 6. As the value of ϕ rises, the tangential velocity of the flow increases. The opposite behavior is examined in the case of swirl velocity profile.

FIGURE 6

FIGURE 6. Variation of $\phi$ on F^I(y) and G(y) when $\in$ = 1, M = 1, and Pr = 29.86.

In Figure 7, the influence of the Al₂O₃ and Fe₃O₄ nanoparticle volume fraction for thermal distribution is plotted. Analysis of this plot showed that the temperature distribution builds up by enhancing the volume fraction of Al₂O₃.

FIGURE 7

FIGURE 7. Variation of $\mathrm{\varphi }$ on H(y) when $\in$ = 1, M = 1, and Pr = 29.86.

The coefficient of skin friction and Nusselt number

Table 3 shows the variation of skin friction coefficient and nusselt number for different values of solid volume fraction, spin parameter and magnetic parameter. It is found that magnetic nanoparticles have high skin friction and nusselt number values.

TABLE 3

TABLE 3. The coefficient of skin friction and Nusselt number.

Conclusion

The numerical solution was achieved using the fourth-order Runge–Kutta method combined with boundary conditions and shooting methods to the non-dimensional ODEs. The following graph-related points are noteworthy:

➢ Intensification of the extent of spin parameter dynamically promotes the tangential velocity, and ${Fe}_3{O}_4$ has a higher tangential velocity than ${Al}_2{O}_3$.

➢ Higher magnetic parameters decrease the momentum transport of hydrodynamic flow and accelerate thermal transport in the presence of $H_{2}O-C_2{H}_6{O}_2$ (50:50) mixture.

➢ Higher spin parameter reduces the thermal profile of nanofluid.

➢ The tangential velocity of Fe₃O₄ is shown to be greater than for the Al₂O₃ nanoparticle.

➢ It is worth mentioning that the present outcomes are highly compatible with solutions obtained in previous research for the special case.

Data availability statement

The original contributions presented in the study are included in the article/supplementary material; further inquiries can be directed to the corresponding author.

Author contributions

All authors listed have made a substantial, direct, and intellectual contribution to the work and approved it for publication.

Acknowledgments

The authors express their appreciation to “The Research Center for Advanced Materials Science (RCAMS)” at King Khalid University, Saudi Arabia, for funding this work under the grant number RCAMS/KKU/002-22. The authors would like to thank the Deanship of Scientific Research at Umm Al-Qura University for supporting this work by Grant Code (22UQU4331100DSR18).

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors, and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

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Nomenclature

k thermal conductivity, Wm⁻¹K⁻¹

H dimensionless fluid temperature

G dimensionless swirl velocity

C_p specific heat, Jkg⁻¹K⁻¹

M magnetic parameter

F dimensionless tangential velocity

g acceleration due to gravity, ms⁻²

L reference length, m

Nu_x local Nusselt number

C_f skin friction coefficient

B magnetic field intensity, kgs⁻²A⁻¹

${\mathbf{q}}_{\mathrm{w}}$ surface heat flux, Wm⁻²

r dimensionless radius

T_r reference temperature, K

Gr Grashof number

T_o temperature of the cone surface, K

Pr Prandtl number

u,v,w velocity component in the x,y,z direction respectively

T temperature, K

Re local Reynolds number

y dimensionless coordinate normal to the surface

U reference velocity, ms⁻¹

x dimensionless coordinate measured along the surface

β thermal expansion coefficient, K⁻¹

ϕ nanoparticle volume fraction

ε spin parameter

${\tau }_{\mathrm{w}}$ skin friction, Nm⁻²

γ half of vertex angle

$\rho$ density, kgm⁻³

$\psi$ dimensionless stream function

Ω angular velocity of the cone

$\theta$ angle of rotation

$\upsilon$ kinematic viscosity, m²s⁻¹

Θ dimensionless temperature ratio

$\sigma$ electrical conductivity, Sm⁻¹

$\alpha$ thermal diffusivity, m²s⁻¹

$\mu$ dynamic viscosity, kgm⁻¹s⁻¹