Significances of magnetic dipole flow and melting heat transfer over suspended dust particles along with carbon nanotubes

The current research letter is materialized to examine the combined effect of thermophoretic dust particle diffusion and magnetic dipole flow toward a stretchable sheet. The slip effect is used in the development of the momentum equation. Water is considered as a base fluid, and single-walled carbon nanotubes and multiwalled carbon nanotubes are nanoparticles. The aforementioned rheological model is formulated in the way of nonlinear partial differential equations (PDEs). The structure of nonlinear biformed ordinary differential equations (ODEs) is obtained from the modeled nonlinear PDEs by incorporating valid sameness transformations. The system of ODEs is then figured out by employing the tool of MATLAB function bvp4c. The significant nature of the contributing parameter for both the single-wall and multiwall cases is presented via graphs. From the results, we conclude that an impact of the melting effect, mounting of $Pr$, reported a thinner thermal boundary layer thickness. Furthermore, a clear rise of nanofluid temperature is noticed for the greater melting $M_1$ parameter. The slip parameter plays a significant role in the enhancement of the velocity profile.

Introduction

Nanofluids may be used to improve the performance of different electronic and mechanical devices regarding heat transfer due to their high thermal conductivity. Therefore, researchers have suggested many proposals to upgrade the capability of transfer of heat in the fluids. Nanofluids are an advanced way to enhance the heat-transfer rate in liquids. They contain nanoparticles homogeneously dissolved in simple fluids, for example, oil, water, and glycerin. There are many kinds of nanoparticles, such as aluminum, silver, gold, titanium, copper, the oxides of these metals, and carbon nanotubes. These fluids have several applications in industries, like vehicle cooling, heat exchangers, electronic cooling, nuclear reactors, cooling thermal insulation, cancer therapy, energy production, and so on.

Moreover, nanofluids have several applications, like micromachines in microreactors, cars, computer chips, microelectronics, and cooling agents in aeroplanes. Nanofluid properties may depend on different parameters such as particle dimension, particle distribution, shapes, size, and so on. Choi, 1995 was the first to observe that adding nanoparticles to a host fluid boosts its thermic strength. Later, Choi et al., 2001 showed through experiments that the enhancement in fluid thermic strength could also be achieved by taking oil as the base fluid. After the pioneering work of Choi, several scientists start working on nanofluids. Sheikholeslami et al., 2018 examined the nanofluid flow over a permeable channel in addition to Lorentz forces. Chen et al., 2015 inspected the nanofluid flow over a vertical symmetrical heated parallel plate with magnetic effects. Hayat and Nadeem, 2017 collectively worked on silver and copper oxide-based hybrid nanofluids to boost the heat-transfer rate. Dogonchi and Ganji (Dogonchi and Ganji, 2016) discussed the thermal radiation effect and flow of magnetohydrodynamic nanofluids between the contracting walls. Bilal et al. (Bilal et al., 2018) investigated the nanofluid flow across the contracting cylinder. Hayat et al., 2016 hypothesized the stagnation point flow with mass flux and variable heat conductivity, where a linear stretched surface subordinates the flow. Recent studies regarding the slip motion can be found in Rizwan Ul Haq et al., 2017; Ul Haq et al., 2018; Rizwan et al., 2020; and Ul Haq et al., 2020.

In the presence of an electrically conducting fluid, the phenomenon is called magnetohydrodynamics (MHD) when the magnetic field is applied across the moving fluid. MHD plays an essential role in many technological and industrial sectors such as nuclear fuels, cancer tumor treatment, blood pump machines, MHD generators, flow meters, and accelerator pumps. In the existence of the magnetic field, convection problems of the electrically conducting fluid have significant results because of its broad uses in astrophysics, missile technology, geophysics, and plasma physics. It is also frequently used in biomedicine industries. It is noticed that the combined impact of thermal radiation and MHD is an essential factor in the study of fluid dynamics owing to their extensive applications in furnace design, nuclear reactor safety, fire spreads, glass production, modern energy transformation devices, turbid water bodies, natural gas-fired and open cycle coal, photochemical reactors, natural convection in cavities, and many others. Metallurgy industries, petroleum, and chemical engineering are some specific processes where the importance of MHD along with thermal radiation can be observed. Qayyum et al. (Fakour et al., 2015) examined the nonlinear chemical reactions in MHD flow above the stretched sheet having a variable thickness. They also examined heat absorption/generation and nonlinear convection. Pourmehran et al. (Qayyum et al., 2017) considered the bouncy effects bygone a vertical sheet for investigating the MHD convective nanofluid. Shiekholeslami (Mahanthesh et al., 2017) used the Lattice Boltzmann technique as a mesoscopic methodology to inspect the convective flow of MHD nanofluids in a porous cubic container. Zhao et al. (Pourmehran et al., 2016) focused on the thermally generated flow of a nanofluid (water) through porous nanotubes and discussed heat-transfer features. Hayat et al., 2018 designed the magnetic-micropolar flow transverse to a curved stretching surface with homogeneous/heterogeneous interactions through an analytical technique. Khan et al. (Sheikholeslami, 2017) explained the flow of an unsteady MHD Newtonian fluid with coupled mass and heat transfer where wall shear stress is applied to the fluid. Using the vertical microchannel having conducting and non-conducting walls, Jha and Aina (Zhao et al., 2019) focused on studying applied and induced magnetic fields and natural convective flow. Hayat et al. (Hayat et al., 2017a) investigated the fluid flow rate in the existence of thermal surface boundary conditions. Hashemi et al. (Reddy et al., 2018) discussed the dynamics of a micropolar nanofluid in a radiative porous medium coupled with MHD natural convection and incorporation of an elliptical heat source. Alkanhal et al. (Alkanhal et al., 2019) studied the thermal management of MHD nanofluids within a porous medium enclosed in a wavy shaped cavity with a square obstacle in the presence of a radiation heat source. M.G. Reddy et al. (Reddy et al., 2022) discussed the effect of thermal conductivity on Blasius–Rayleigh–Stokes flow and heat transfer over a moving plate by considering the magnetic dipole moment.

The no-slip condition is also an essential part of Navier Stoke’s equations. For most of the non-Newtonian fluids, the no-slip condition is inadequate as few polymers melt frequently, showing the microscopic wall slip, which has a modulating impact and has a nonlinear relationship between traction and slip velocity. Partial velocity slip occurs when the liquid flows over the polish surface or through different processes such as suspension, emulsions, polymer solutions, and foams. The Navier velocity slip condition has typical importance while observing the slip phenomena. Solutal slip and thermal slip conditions may also appear in different industrial procedures. The boundary layer slip flow problem occurs in internal cavities and the polishing of heart valves. Many researchers and scientists have investigated the behavior of flow under no-slip conditions. However, in real-life applications, it is not mandatory in every case because there are different circumstances in which velocities of body and fluid particles are distinct. The behavior of flow in microstructures/nanostructures such as hard drives, microvalves, micropumps, and micronozzles is indicated by the slip boundary conditions. Recent studies regarding the slip motion can be found in Hayat et al., 2013; Mukhopadhyay, 2013; Shen et al., 2015; Hayat et al., 2017b; Ul Haq et al., 2017; Hashemi et al., 2018; and Hayat et al., 2019.

This article examines the magnetic dipole flow and melting heat transfer over suspended dust particles with carbon nanotubes and the slip effect. To compose a nanofluid, the water base fluid is mixed up with two types of carbon nanotubes [single-walled carbon nanotubes (SWCNTs) and multiwalled carbon nanotubes (MWCNTs)]. By applying significant similarity transformation, nonlinear partial differential equations (PDEs) are being modified into a system of ordinary differential equations (ODEs); employing the MATLAB function bvp4c, we were able to get numerical solutions to these nonlinear ordinary differential equations. Tables and graphs are used to demonstrate the impact of different physical characteristics.

Mathematical formulation

Presumptions of the flow:

• Assume the 2D flow of dusty carbon nanotubes in an elaborated sheet where the flow is time-reliant as considered.

• $U_w\left(x\right)=bx$ is the velocity of the stretched sheet, where the stretching rate is $b>0$ (Figure 1).

• The stretching surface as measured with the $x$-coordinate.

• The fluid utilized here is a dusty base fluid with the composition of water and carbon nanotubes (SWCNT/MWCNT).

• Equilibrium is assumed to be established between carbon nanotubes and the dust base fluid and the slip amid them.

• The magnetic dipole field induced is ignored as the Reynolds number assumed is the least.

FIGURE 1

FIGURE 1. Physical model of the flow and coordinate system.

After using boundary layer approximation Bernoulli’s equation in the free stream, the system of governing equations in the form of PDEs is as follows (Ramesh et al., 2012; Kumar et al., 2019):

Fluid phase

$\frac{\partial \tilde{u}}{\partial x}+\frac{\partial \tilde{v}}{\partial y}=0\text{,}(1)$

$\tilde{u}\frac{\partial \tilde{u}}{\partial x}+\tilde{v}\frac{\partial \tilde{v}}{\partial y}=\frac{{\mu }_{nf}}{{\rho }_{nf}}\frac{{\partial }^2\tilde{u}}{\partial y^2}+\frac{KN}{{\rho }_{nf}}\left({\tilde{u}}_p-\tilde{u}\right)-\frac{{\lambda }_{0}M}{{\rho }_{nf}}\frac{\partial H}{\partial x},(2)$

$\begin{array}{l}{\left(\rho c_p\right)}_{nf}\left[\tilde{u}\frac{\partial T}{\partial x}+\tilde{v}\frac{\partial T}{\partial y}\right]\\ =k_{nf}\frac{{\partial }^{2}T}{\partial y^2}+\frac{N{c}_{pf}}{{\tau }_t}\left(T_p-T\right)+\frac{N}{{\tau }_v}{\left({\tilde{u}}_p-\tilde{u}\right)}^2-\left[\tilde{u}\frac{\partial H}{\partial x}+\tilde{v}\frac{\partial H}{\partial y}\right]{\mu }_f{\lambda }_{0}T\frac{\partial M}{\partial T},\end{array}(3)$

Dust phase

$\frac{\partial {\tilde{u}}_p}{\partial x}+\frac{\partial {\tilde{v}}_p}{\partial y}=0,(4)$

${\tilde{u}}_p\frac{\partial {\tilde{u}}_p}{\partial x}+{\tilde{v}}_p\frac{\partial {\tilde{u}}_p}{\partial y}=\frac{K}{m}\left(\tilde{u}-{\tilde{u}}_p\right)(5)$

${\tilde{u}}_p\frac{\partial T_p}{\partial x}+{\tilde{v}}_p\frac{\partial T_p}{\partial y}=\frac{c_{pf}}{c_{mf}{\tau }_t}\left(T-T_p\right)(6)$

The corresponding boundary conditions are given by

$\tilde{u}=U_w+k_1\frac{\partial \tilde{u}}{\partial y},v=0,T=T_m\text{ at }y=0\text{,}$

$\tilde{u}\to 0,{\tilde{u}}_p\to 0,{\tilde{v}}_p\to \tilde{v},T\to T_m,T_p\to T_{\infty }\text{as y}\to \infty \text{.}(7)$

The expressions for viscosity, density, specific heat, and thermal conductivity of the carbon nanotube are as follows:

${\overline{\rho }}_{nf}={\rho }_f\left(1-\varphi +\varphi \left(\frac{{\rho }_{CNT}}{{\rho }_f}\right)\right),{\tilde{\mu }}_{nf}=\frac{1}{{\left(1-\varphi \right)}^{2.5}},{\tilde{\alpha }}_{nf}=\left(1-\varphi \right){\left({\rho }_f\right)}_f+\varphi {\left({\rho }_s\right)}_{CNT},$

${\left(\rho c_p\right)}_{nf}=\left(1-\varphi \right){\left(\rho c_p\right)}_f+\varphi {\left(\rho c_p\right)}_{CNT},\frac{k_{nf}}{k_f}=\frac{1-\varphi +2\varphi \left(\frac{k_{CNT}}{k_{CNT}-k_f}\right)\ln \left(\frac{k_{CNT}+k_f}{2k_f}\right)}{1-\varphi +2\varphi \left(\frac{k_f}{k_{CNT}-k_f}\right)\ln \left(\frac{k_{CNT}+k_f}{2k_f}\right)}\text{.}$

Due to the magnetic dipole, the assumed liquid flow is affected by the magnetic field, whose magnetic scalar potential is given by (Everts et al., 2020; Reddy et al., 2022)

${\varphi }_1=\frac{\gamma }{2\pi }\left[\frac{x}{x^2+{\left(y+a\right)}^2}\right],H_x=-\frac{\partial {\varphi }_1}{\partial x}=\frac{\gamma }{2\pi }\left\{\frac{x^2-{\left(y+a\right)}^2}{{\left[x^2-{\left(y+a\right)}^2\right]}^2}\right\}\mathrm{a}\mathrm{n}\mathrm{d} H_y=-\frac{\partial {\varphi }_1}{\partial x}=\frac{\gamma \ast }{2\pi }\left\{\frac{2x\left(y+a\right)}{{\left[x^2-{\left(y+a\right)}^2\right]}^2}\right\}.$

where

$H={\left[{\left(\frac{\partial {\varphi }_1}{\partial x}\right)}^2+{\left(\frac{\partial {\varphi }_1}{\partial y}\right)}^2\right]}^{\frac{1}{2}}(9)$

From Eq. 9, one can get

$\frac{\partial }{\partial x}H=-\frac{\gamma }{2\pi }\left(\frac{2x}{{\left(y+a\right)}^4}\right)\text{,}$

$\frac{\partial }{\partial x}H=-\frac{\gamma }{2\pi }\left(\frac{2}{{\left(y+a\right)}^3}+\frac{4x^2}{{\left(y+a\right)}^5}\right)$

We assume that the applied field $H$ is sufficiently strong to saturate the assumed fluid and the variation of magnetization M with temperature T is approximated by the linear equation

$M=K\left(T_c-T\right).$

where $K$ indicates the gyromagnetic coefficient. We establish the below transformations to lessen the dimensions of the above equation to 1

$\tilde{u}=bx{f}^{\prime }\left(\eta \right),\tilde{\nu }=-\sqrt{b{\nu }_f}f\left(\eta \right),{\tilde{u}}_p=bx{F}^{\prime }\left(\eta \right),{\tilde{\nu }}_p=-\sqrt{b{\nu }_f}F\left(\eta \right)\text{,}$

$\eta =\sqrt{\frac{b}{v_f}}y,\theta \left(\eta \right)=\frac{T-T_w}{T_w-T_{\infty }},{\theta }_p\left(\eta \right)=\frac{T_p-T_w}{T_w-T_{\infty }}\text{.}(10)$

On employing the above, Eqs 1, 4 are satisfied identically, and Eqs 2, 3, 5, 6 reduce as

Fluid phase:

$\begin{array}{l}{f}^{‴}+\frac{1}{{\left(1-\varphi \right)}^{2.5}\left(1-\varphi +\varphi \left(\frac{{\left({\rho }_s\right)}_{CNT}}{{\rho }_f}\right)\right)}\left(f{f}^{″}-f^{\prime 2}\right)\\ -\frac{2\beta \theta }{{\rho }_f{(\eta +\gamma )}^4}+\left[{\left(1-\varphi \right)}^{2.5}\right]\left[l{\beta }_v\left(F^{\prime }-f^{\prime }\right)\right]=0\end{array}(12)$

$\begin{array}{l}\frac{k_{hnf}}{k_f}{\theta }^{″}+\mathrm{Pr}\left(1-\varphi +\varphi \left(\frac{{\left(\rho c_p\right)}_{CNT}}{{\left(\rho c_p\right)}_f}\right)\right)f{\theta }^{\prime }-\frac{2\beta \left(\theta -\epsilon \right)f}{{\left(\eta +\gamma \right)}^3}\\ +\frac{\mathrm{Pr}l{\beta }_t}{m}\left[{\theta }_p-\theta \right]+\mathrm{Pr}l{\beta }_{v}Ec{\left[F^{\prime }-f^{\prime }\right]}^2=0,\end{array}(13)$

Dust phase:

$F{F}^{″}-F^{\prime 2}+{\beta }_v\left[f^{\prime }-F^{\prime }\right]=0,(14)$

$F{\theta }_p^{\prime }+\gamma {\beta }_t\left[\theta -{\theta }_p\right]=0(15)$

Reduced boundary conditions:

$\mathrm{Pr}f\left(0\right)+M_1\theta ,\left(0\right)=0,f^{\prime }\left(0\right)=1+A{f}^{″}\left(0\right),\theta \left(0\right)=0$

$f^{\prime }\left(\infty \right)=0,F^{\prime }\left(\infty \right)=0,F\left(\infty \right)=f\left(\infty \right),\theta \left(\infty \right)=0,{\theta }_p\left(\infty \right)=0.(16)$

The skin friction coefficient and Nusselt number are defined as

$C_f=\left(1+\frac{1}{\beta }\right)\frac{{\tau }_w}{{\rho }_f{u}_w^2},N{u}_x=\frac{q_w{u}_w}{b{k}_{nf}\left(T{}_f-T_{\infty }\right)}\frac{\partial T}{\partial y}\text{,}$

The transformed form of the above one is

$\sqrt{{\mathrm{Re}}_x}{C}_f=\left(1+\frac{1}{\beta }\right)\frac{1}{{\left(1-\varphi \right)}^{2.5}}{f}^{″}\left(0\right),\frac{N{u}_x}{\sqrt{{\mathrm{Re}}_x}}=-\frac{k_{nf}}{k_f}{\theta }^{\prime }\left(0\right),$

The following are the dimensionless parameters that exist in the problem:

$\beta =\frac{\gamma }{2\pi }\frac{{\mu }_{0}K\left(T_c-T_w\right)\rho }{{\mu }^2},{\beta }_{\upsilon }=\frac{1}{b{\tau }_{\upsilon }},\gamma =\sqrt{\frac{Sp{c}^2}{\mu }},\mathrm{Pr}=\frac{1}{b{\tau }_t},Ec=\frac{{\tilde{U}}_w^2}{c_p\left(T_w-T_{\infty }\right)},M_1=\frac{C_f\left(T_{\infty }-T_m\right)}{\lambda +C_s\left(T_m-T_0\right)},A=k_1\sqrt{\frac{b}{\nu }} \mathrm{a}\mathrm{n}\mathrm{d},{\mathrm{Re}}_x=\frac{u_w^2}{b{v}_f}.$

Numerical procedure

The reformed system of coupled nonlinear ODEs (Eqs 12–15) in conjunction with boundary conditions (Eq. 16) is worked out for various parameters by employing the tool of MATLAB function bvp4c. It is an efficient method used by several authors to tackle such problems. A fair amount of computations are executed by using the tool mentioned above, so to deal with it, we considered $[0;{\text{η}}_{\text{max}}]$ as a domain in place of $[0;1)$, where ${\text{η}}_{\text{max}}$ is an appropriate real number, as the results are not altered for $\text{η}>{\text{η}}_{\text{max}}$. Authentication of the accuracy of the numerical procedure used was done by comparison first; computations for $f″\left(0\right)$ are carried out for the viscous fluid for various values of $\beta$ and compared with the available published results as shown in Table 1, and they establish an excellent agreement.

TABLE 1

TABLE 1. Comparison of present numerical results with previous ones.

Fluid phase

$f=y_1, f^{\prime }=y_2, f^{″}=y_3, F=y_4, F^{\prime }=y_5$

$\theta =y_6, {\theta }^{\prime }=y_7, {\theta }_p=y_8$

$y_3^{\prime }=-\left[\frac{1}{{\left(1-\varphi \right)}^{2.5}\left(1-\varphi +\varphi \left(\frac{{\left({\rho }_s\right)}_{CNT}}{{\rho }_f}\right)\right)}\left(y_1{y}_3-y_2^2\right)-\frac{2\beta \theta }{{\rho }_f{(\eta +\gamma )}^4}+\left[{\left(1-\varphi \right)}^{2.5}\right]\left[l{\beta }_v\left(y_5-y_2\right)\right]\right](17)$

$\begin{array}{l}{y}_7^{\prime }=-\frac{1}{\frac{k_{hnf}}{k_f}}\left[\mathrm{Pr}\left(1-\varphi +\varphi \left(\frac{{\left(\rho c_p\right)}_{CNT}}{{\left(\rho c_p\right)}_f}\right)\right)y_1{y}_7-\frac{2\beta \left(y_6-\epsilon \right)y_1}{{\left(\eta +\gamma \right)}^3}+\frac{\mathrm{Pr}l{\beta }_t}{m}\left[y_8-y_6\right]\right]\\ -\frac{1}{\frac{k_{hnf}}{k_f}}\left[\mathrm{Pr}l{\beta }_{v}Ec{\left[y_5-y_2\right]}^2\right]\end{array}(18)$

Dust phase

$y_5^{\prime }=\frac{1}{y_4}\left[y_5^{\prime }-{\beta }_v\left[y_2-y_5\right]\right](19)$

$y_8^{\prime }=-\frac{1}{y_4}\left[\gamma {\beta }_t\left[y_6-y_8\right]\right](20)$

Reduced boundary conditions:

$\mathrm{Pr}{\mathrm{y}}_1\left(0\right)+M_1{\mathrm{y}}_6\left(0\right)=0,{\mathrm{y}}_2\left(0\right)=1+A{\mathrm{y}}_3\left(0\right),{\mathrm{y}}_6\left(0\right)=0$

${\mathrm{y}}_2\left(\infty \right)=0,{\mathrm{y}}_5\left(\infty \right)=0,{\mathrm{y}}_4\left(\infty \right)={\mathrm{y}}_1\left(\infty \right),{\mathrm{y}}_6\left(\infty \right)=0,{\mathrm{y}}_8\left(\infty \right)=0(21)$

Results and discussion

In the present article, the investigation is carried out on the significance of magnetic dipole flow and melting heat transfer over suspended dust particles and carbon nanotubes. Table 2 represents the mathematical form of transport characteristics of the nanofluid. The computation for the various profiles is deliberated via graphs, whereas the rate quantities’ numerical results are displayed in a tabular form (Table 3).

TABLE 2

TABLE 2. Thermophysical properties of water and nanotubes (Reddy and Kumar, 2021).

TABLE 3

TABLE 3. Numerical values of $R{e}_x^{1/2}{C}_f$ and $R{e}_x^{1/2}N{u}_x$ for various parameters.

Velocity profile (${\mathit{f}}^{\prime }\left(\mathit{\eta }\right)$ and ${\mathit{F}}^{\prime }\left(\mathit{\eta }\right)$)

The magnetic field strength ($\gamma$) parameter has a significant influence on the induced magnetism, as shown in Figures 2, 3. It is perceived in Figure 2 that an expansion in the field strength of the magnetic ($\gamma )$ parameter leads to a decline in the velocity of the fluid phase $\left(f^{\prime }\left(\eta \right)\right)$. This is because an increment in $\gamma$ is directly proportional to Lorentz forces at the surface. This is a resistive type of force that declines the fluid velocity and the magnetic field intensity. Magnetic field strength ($\text{γ}$) parameters have a terrific control on the velocity profile for the dusty phase $\left[F^{\prime }\left(\eta \right)\right]$, as illustrated in Figure 4. This numerical rise in $\gamma (=0.1, 0.3, 0.5, 0.7)$ represents a consequent reduction in the velocity profile for the dusty phase. Consequently, the inner thickness of the boundary layer also scales back for larger $\gamma$. Furthermore, we can observe that the performance velocity profile in the MWCNT case is more efficient than that of the SWCNT case.

FIGURE 2

FIGURE 2. Impact of $\gamma$ on $f^{\prime }\left(\eta \right)$.

FIGURE 3

FIGURE 3. Impact of $\gamma$ on $F^{\prime }\left(\eta \right)$.

FIGURE 4

FIGURE 4. Impact of the volume fraction parameter on $f^{\prime }\left(\eta \right)$.

In Figures 4, 5, the behavior of $\varphi$ on the velocity of both $f^{\prime }\left(\eta \right)$ and $F^{\prime }\left(\eta \right)$ phases was highlighted. A higher amount of nanoparticles sufficiently enhances the velocity of the fluid for both $f^{\prime }\left(\eta \right)$ and $F^{\prime }\left(\eta \right)$ phases. Besides, the complementary circumscription layer has been appreciated for strengthening the numerical values of $\varphi$. Additionally, the fluid flow of the MWCNT.case is higher than the velocity of the fluid flow of SWCNT. The upshot of $\beta$ (hydrodynamic interaction parameter) on velocity distribution on both $f^{\prime }\left(\eta \right)$ and $F^{\prime }\left(\eta \right)$ cases are observed in Figures 6, 7. A significant escalation in the $\beta$ parameter causes a decrement in both $f^{\prime }\left(\eta \right)$ and $F^{\prime }\left(\eta \right)$ phases, which leads to scale down the fluid at the circumscription layer. Furthermore, the complementary circumscription momentum layer also dismisses both $f^{\prime }\left(\eta \right)$ and $F^{\prime }\left(\eta \right)$ phases for esteemed values of $\beta$. Additionally, the fluid flow velocity in MWCNT is much more than that of the fluid flow velocity of SWCNT. Figures 8, 9 illustrate the dimensionless velocity profiles for both $f^{\prime }\left(\eta \right)$ and $F^{\prime }\left(\eta \right)$ cases over variant values of emerging parameter $A$. These figures show that both $f^{\prime }\left(\eta \right)$ and $F^{\prime }\left(\eta \right)$ enhance booming values of $A$. Likewise, the complementary circumscription thermal layer rises for both $f^{\prime }\left(\eta \right)$ and $F^{\prime }\left(\eta \right)$ by thicker values of $A$.

FIGURE 5

FIGURE 5. Impact of the volume fraction parameter on $F^{\prime }\left(\eta \right)$.

FIGURE 6

FIGURE 6. Impact of the hydrodynamic interaction parameter on $f^{\prime }\left(\eta \right)$.

FIGURE 7

FIGURE 7. Impact of the hydrodynamic interaction parameter on $F^{\prime }\left(\eta \right)$.

FIGURE 8

FIGURE 8. Impact of the slip parameter on $f^{\prime }\left(\eta \right)$.

FIGURE 9

FIGURE 9. Impact of the slip parameter on $F^{\prime }\left(\eta \right)$.

Temperature profile $\left(\mathit{\theta }\right(\mathit{\eta })$ and ${\mathit{\theta }}_{\mathit{p}}\left(\mathit{\eta }\right))$

Figures 10, 11 portray the performance of the Prandtl number on the thermal gradient for both $\theta \left(\eta \right)$ and ${\theta }_p\left(\eta \right)$ phases. It shows an improvement in the temperature distribution of both $\theta \left(\eta \right)$ and ${\theta }_p\left(\eta \right)$ for bulker numerical values of $Pr$. This is the consequence of enhanced thermal layer thickness. The heat-transfer rate increases because of the temperature gradient at the boundary. Additionally, the complementary circumscription thermal layer is also predominant for greater values of $Pr$. Here, we can observe that the efficiency of fluid temperature is more in the SWCNT case than in the MW CNT case. Figures 12, 13 represent the analogous behaviors of $\beta$ (hydrodynamic interaction parameter) on temperature fields for both $\theta \left(\eta \right)$ and ${\theta }_p\left(\eta \right)$ phases. They are both critical functions of $\beta$. Meanwhile, the thickness of boundary layers of temperature becomes thinner for amplified $\beta$. Additionally, the MWCNT case dominates the fluid temperature more than the SWCNT case. The thermal gradient structure expands with advancing values of ${\text{M}}_1$ as showcased in Figures 14, 15 for both $\text{θ}\left(\text{η}\right)$ and ${\text{θ}}_{\text{p}}\left(\text{η}\right)$ cases. Here, the melting parameter has a direct relation with the thermal gradient and for preponderant values of $M_1$ heightens the temperature gradient for both $\left(\text{η}\right)$ and ${\text{θ}}_{\text{p}}\left(\text{η}\right)$ cases. Likewise, the complementary circumscription thermal layer also has an upsurge for hiking values of ${\text{M}}_1$. Meanwhile, the SWCNT case has a higher impact on fluid temperature when treated with the MWCNT case. The temperature gradient nature is portrayed in Figures 16, 17 for the various $\text{γ}$ parameters. It is noted that the assessable rise in ($\gamma$) shows a remarkable decrement in the temperature distribution for both $\theta \left(\eta \right)$ and ${\theta }_p\left(\eta \right)$ cases. Likewise, there is circumscription thermal layer shrinkage for hiking values of $\gamma$. Meanwhile, the MWCNT case has a higher impact on fluid temperature when treated with the SWCNT case.

FIGURE 10

FIGURE 10. Impact of Prandtl number on $\theta \left(\eta \right)$.

FIGURE 11

FIGURE 11. Impact of Prandtl number on ${\theta }_p\left(\eta \right)$.

FIGURE 12

FIGURE 12. Impact of the hydrodynamic interaction parameter on $\theta \left(\eta \right)$.

FIGURE 13

FIGURE 13. Impact of the hydrodynamic interaction parameter on ${\theta }_p\left(\eta \right)$.

FIGURE 14

FIGURE 14. Impact of the melting parameter on $\theta \left(\eta \right)$.

FIGURE 15

FIGURE 15. Impact of the melting parameter on ${\theta }_p\left(\eta \right)$.

FIGURE 16

FIGURE 16. Impact of magnetic field strength on $\theta \left(\eta \right)$.

FIGURE 17

FIGURE 17. Impact of magnetic field strength on ${\theta }_p\left(\eta \right)$.

Numerical outcomes of $R{e}_x^{1/2}{C}_f$ and $R{e}_x^{1/2}N{u}_x$ friction factors are shown in Table 3. In the table, we analyzed that when enlarging the values of parameters $\beta , {\beta }_1, {\varphi }_1$ and $M_1$, $R{e}_x^{1/2}{C}_f$ shows thicker behavior, while the hiking values of the exact parameter show opposite behavior for $R{e}_x^{1/2}N{u}_x$. Furthermore, $R{e}_x^{1/2}N{u}_x$ heightens for booming values of $Pr$ and ${\beta }_t$.

Conclusion

This research work inspected the significance of magnetic dipole flow and melting heat transfer over suspended dust particles. The impact of melting is also discussed in the energy equation. Fluid flow and the heat-transfer rate are discussed through a mathematical model of second-order, nonlinear ODEs, which is analyzed numerically. The effectiveness of appropriate parameters is elaborated graphically. The impacts of essential parameters are reported.

• The velocity field declined against the impact of $\gamma$ parameter.

• The velocity of the fluid increases with higher estimations of $\varphi$.

• The fluid temperature decays with a bulkier assessment of $\beta$.

• Because of melting, the thermal field is more for the rising assessment of $Pr$.

• On enlarging the values of parameters $\beta$ and $M_1$, $R{e}_x^{1/2}{C}_f$ shows thicker behavior.

• ${\text{Re}}_{\text{x}}^{1/2}{\text{Nu}}_{\text{x}}$ heightens for booming values of $\text{Pr}$ and ${\text{β}}_{\text{t}}$.

• The velocity field is dramatically shortened with higher $\beta$.

• The slip parameter plays a significant role in enhancing the velocity profile.

Data availability statement

The raw data supporting the conclusion of this article will be made available by the authors, without undue reservation.

Author contributions

HA: conceptualization, methodology, validation, investigation, resources, data curation, writing—original draft, writing—review and editing, visualization, and supervision.

Funding

This work was supported by the Deanship of Scientific Research, Vice Presidency for Graduate Studies and Scientific Research, King Faisal University, Saudi Arabia (Project No. GRANT639).

Conflict of interest

The author declares 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

$A$ velocity slip parameter

$\beta$ Hydrodynamic interaction parameter

${\beta }_v$ fluid–particle interaction for velocity

${\beta }_t$ fluid–particle interaction for temperature

$c_{pf}$ specific heat of fluid particles

$c_{mf}$ specific heat of dust particles

$Ec$ Eckert number

$k_1$ slip constant

${\lambda }_0$ magnetic permeability

MWCNT multiwall carbon nanotube

${\mu }_{nf}$ viscosity of carbon nanotubes

$M_1$ melting parameter

$N$ density of dust particles

$\mathrm{Pr}$ Prandtl number

$\varphi$ volume fraction of the nanoparticle parameter

${\mathbf{Re}}_{\mathbf{x}}$ local Reynolds number

${\rho }_{nf}$ density of carbon nanotubes

SWCNT single-wall carbon nanotube

$T,T_p$ temperature of the fluid and dust phases

${\tau }_v$ relaxation time for the dust particle

$T_w$ wall temperature

$T_{\infty }$ ambient fluid temperature

$\tilde{u},{\tilde{u}}_p$ velocity components along the $x$-axis

$U_w$ wall velocity

$\tilde{v},{\tilde{v}}_p$ velocity components along the $y$-axis

$\gamma$ magnetic field strength