Duel Solutions in Hiemenz Flow of an Electro-Conductive Viscous Nanofluid Containing Elliptic Single-/Multi-Wall Carbon Nanotubes With Magnetic Induction Effects

Stagnation (Hiemenz) flows feature in numerous applications in engineering sciences including, laser materials processing [1], coating dynamics [2], and vapor deposition [3]. Such flows are characterized by the fluid impinging on either a plane (orthogonal) [4] or oblique (non-orthogonal) surface [5] and a symmetric or asymmetric distribution in flow in the vicinity of the stagnation point which can be modeled as a flow toward an infinite flat plate. In thermal materials processing operations [6], liquids are frequently stretched or contracted along the impingement surface. This is achieved by stretching the surface or shrinking it, and this may be done at linear, quadratic, exponential [7,8], or other rates. In simulating such flows, the boundary-layer theory provides a solid framework. Sakiadis [9] initiated the study of viscous flows from a continuously moving solid surface. Crane [10] extended the Sakiadis model to linear stretching sheets. Many subsequent studies have been communicated including heat transfer and representative works include those of Dandapat et al. [11] (on surface tension effects), Tsai et al. [12] (on non-uniform heat generation effects) and Bhattacharyya et al. [13] (on hydrodynamic wall slip in stagnation flow).

A key focus in thermal engineering and materials manufacturing in recent years has been the desire to improve in thermal conductivity of heat transfer fluids. Nanotechnology has been a significant aspect of this focus. Among the many nanomaterials developed, “nanofluids” have become increasingly popular owing to their demonstrated versatility in multiple technologies ranging from energy and medicine to environment and transportation [14–17]. Introduced by Choi et al. [18], heat transfer rates have been shown to be significantly elevated via doping conventional base fluids (water, oils, etc.) with a variety of nanoparticles (metallic, carbon-based, etc.). Further experimental corroboration of this behavior has been confirmed by many researchers [19–22]). These studies have shown that even with a small solid volume fraction of nanoparticles (usually less than 5%), the thermal conductivity of nanofluids can be enhanced by 10–50%. Extensive discussion of these trends is documented in the studies by Trisaksri and Wongwives [23], Wang and Majumdar [24], Eastman et al. [25], among others. Mehmood et al. [26] investigated the oblique stagnation flow of nanofluids containing metallic nanoparticles with radiative flux and wall slip effects. Carbon nanotubes (CNTs) are an alternative nanofluid suspension, which have special thermal properties with very high thermal conductivities for their cylindrical carbon molecular origin. Several experimental and theoretical studies of CNT nanofluid dynamics have been reported in recent years. Ding et al. [27] investigated the heat transfer behavior of CNT nanofluids flowing through a horizontal tube, observing that convective heat transfer is enhanced and dependent on the Reynolds number and solid volume fraction of CNTs. Volder et al. [28] presented a detailed appraisal of the commercial and industrial usage of carbon nanotubes. Akbar et al. [29] considered the stagnation point flow of CNT nanofluid from a stretching sheet, highlighting that due to the high density and thermal conductivity of single-wall carbon nanotubes (SWCNTs), the Nusselt number is higher relative to multi-wall carbon nanotubes (MWCNTs). Akbar et al. [30] have also described a second law analysis of CNT nanofluid metachronal pumping and convection.

Magnetohydrodynamics (MHD) [31] involves the interaction of viscous, electrically conducting fluids via the application of external magnetic fields which may be static or oscillating. As such MHD can be utilized very effectively in controlling heat, momentum, and mass transfer in materials processing and can enable significant modifications of material constitution and subsequent performance. Magnetized nanofluid flows have therefore received considerable attention in recent years. Some examples of such works include those of Bég et al. [32] (on transient mixed magneto-convective nanofluid exponentially stretching sheet flow in porous media), Seth and Mandal [33] (on stagnation flow of electromagnetic nanofluids in coating systems), and Thumma et al. [34] (on magnetic nanomaterial thermal stretching on a tilted substrate). However, these studies did not consider CNT-based nanofluids. In recent years, significant progress has been made in modifying CNTs to achieve electromagnetic properties. Magnetic CNT nanofluids can be synthesized via the chemical combination of magnetic nanoparticles or nanocrystals and CNTs in order to obtain nanohybrid structures, through encapsulation of magnetic molecules inside the carbon nanotubes (endohedral functionalization) or grafting/decorating CNTs on their surface (exohedral functionalization) by bioconjugation chemistry or electrochemical deposition. It has been found [35,36] that the thermal conductivity (TC) heat transfer nanofluids can be fabricated which feature both CNTs and magnetic-field-sensitive nanoparticles (e.g., Fe₂O₃). In such functional systems, the Fe₂O₃ particles generate aligned chains under an external magnetic field which assist in connecting nanotubes and effectively boost the thermal conductivity. This behavior is however very sensitive to the duration of magnetic field application since excessive exposure may result in agglomeration of the ferrite nanoparticles and clustering of CNTs which then reduces the thermal conductivity of the nanofluid. A promising additional method of achieving magnetic CNT nanofluids includes manipulation of the chemical structure of, for example, Mn₁₂Ac single-molecule magnets (SMM) and encapsulation within the carbon nanotube. Many further excellent studies of magnetic CNT nanofluids are available in Refs. [37–43]. A further important phenomenon in MHD nanofluid flows is electromagnetic induction. This requires the inclusion of a further balance equation for induced magnetic field and generates more complex effects compared with conventional MHD flows in which a Lorentzian magnetic drag alone is present. Although extensive work has been communicated in conventional viscous MHD flows with induction effects [44–46], relatively sparse attention has been directed at magnetic induction nanofluid transport phenomena. Metallic aqueous nanofluid flows from stretching sheets have been analyzed by Bég et al. [47,48] for a variety of nanoparticles including copper, titanium oxide, silver, and aluminum oxide. Uddin et al. [49] have considered slip stagnation point flow of a magnetic nanofluid over a stretching/shrinking wedge with micro-organism doping. More recently, CNT-based magnetic nanofluids have been scrutinized in the scientific literature. Iqbal et al. [50] used numerical quadrature to compute the CNT nanofluid stagnation point flow with the induced magnetic field, considering both single (SWCNT) and multi-walled (MWCNT) configurations and water- and kerosene-oil-base fluids. They showed that induced magnetic field is an increasing function of solid nanoparticles volumetric fraction and that MWCNTs achieve a more prominent enhancement in the induced magnetic field compared with SWCNTs for both water- and kerosene-oil-based fluids.

Motivated by expanding the understanding of magnetized CNT nanofluid materials processing fluid dynamics with electromagnetic induction included, in the present study, a rigorous examination of plane stagnation flow of such fluids from a stretching surface is described for both SWCNT and MWCNT suspensions. Both water- and kerosene base fluids are considered, and more attention is devoted to the modifications in magnetic induction, temperature, and momentum distributions than previously reported. Computational solutions are presented using the efficient bvp4c solver in matlab. Validation with earlier studies is included. Computations of reduced skin friction and reduced wall heat transfer rate (Nusselt number) are also presented in order to establish the critical parameter values for the possible existence of dual solutions, which is another novelty of the current work. A detailed parametric study of the influence of all nanoscale and thermophysical on non-dimension velocity, induced magnetic field, and temperature profiles is conducted. Extensive validation with simpler models in the literature is also included.

The steady two-dimensional stagnation point flow of an electrically conducting incompressible viscous CNT nanofluid toward a linear stretching sheet is considered in a Cartesian coordinate system $(x¯,y¯)$⁠. The sheet surface is parallel to the $x¯−$ axis and the nanofluid occupies the region $y¯>0$⁠. For comparative analysis, two types of carbon nanotubes are considered, i.e., single- and multiple-wall CNTs. Moreover, water and kerosene oil are taken as the base fluids. The freestream velocity is taken as $U¯e(x)=ax¯$⁠, and velocity of stretching sheet is $U¯w(x)=cx¯$⁠, where a and c are the positive constants. Additionally, an induced magnetic field vector $(H¯)$ is presented by $He¯(x)=H0x¯$ with two parallel and normal components say $H¯1$ and $H¯2$⁠, where the uniform magnetic field H₀ is imposed parallel to the plate and external to the boundary layer in the direction of $x¯$-axis. The induced magnetic field of normal components $H¯2$ vanishes at the wall with the parallel components $H¯1$ and approaching the imposed magnetic field value of H₀ at the edge of the boundary layer. T_w is the prescribed surface temperature and T_∞ is the ambient temperature. Moreover, φ_w and φ_∞ are the nano particle volume fraction at the surface and ambient values of nanoparticle volume fraction, respectively. The physical flow model is presented in Fig. 1.

Here, $u¯,v¯,H¯1$⁠, and $H¯2$ are velocity and magnetic field components along the $x¯−$ and $y¯−$ directions, respectively. Here, μ denotes the magnetic permeability and μ_e denotes the magnetic diffusivity where $μe=14πσμ$⁠. T is the fluid temperature; ρ_nf, μ_nf are density and dynamic viscosity, respectively, of the magnetic CNT nanofluid. φ is the particle volume fraction parameter; ρ_f and ρ_CNT are density of fluid and carbon nanotube (CNT), respectively. (c_p)_nf, (c_p)_f, and (c_p)_CNT are specific heat of nanofluid, base fluid, and CNTs. k_nf, k_f, and k_CNT are thermal conductivities of nanofluid, base fluid, and the CNTs respectively.

Here, α is the ratio of thermal conductivity.

In the current study, the Xue model [56] is employed to evaluate the thermal conductivity and the dimensionless heat transfer rates of CNT nanofluids.

Here, ν_f refers to kinematic viscosity of the base fluid where υ_f = μ_f/ρ_f.

Here, $Rex=U¯wx¯νf$ is the local Reynolds number.

Numerical calculations have been carried out for different values of the control parameters. Thermophysical properties of base fluids and CNTs are given in Tables 1 and 2 providing the data for all relevant thermophysical properties of nanofluids with solid volume fraction of CNTs.

The range of nanoparticle volume fraction φ is considered from 0 to 0.2 where φ = 0 corresponding to a regular viscous fluid [29]. The stretching sheet ratio parameter, A, is prescribed values of A = 0.5, 0.8, 1.2, 1.5 [50]. The value of reciprocal of magnetic Prandtl number is λ = 0.5, 1, 5, 10 [52]. The value of magnetic parameter β is taken equal to β = 0.1, 0.2, 1, 1.5, 2.5 [63]. The case β = 0 corresponds to the flow of nanofluid in the absence of induced magnetic field.

The validation of the matlab code is conducted by comparing it with previously published studies. Here, the accuracy of the results is represented by Table 3 which compares the reduced skin friction f″(0) for the range 0.1 ≤ A ≤ 1.0 and φ = β = 0 with the homotopy solutions obtained by Hayat et al. [64], Hayat et al. [65] and Iqbal et al. [50]. Very good agreement is achieved testifying to the accuracy of the matlab bvp4c solutions.

Table 4 shows the comparison of skin friction coefficient, and Table 5 documents the comparisons for heat transfer coefficient for both water- and kerosene-oil-based SWCNT- and MWCNT nanofluids for the range of parameters 0 ≤ φ ≤ 0.2.

The impact of magnetic parameter, β, on the reduced skin friction for water- and kerosene-based SWCNT with keeping other parameters fixed is visualized in Figs. 2–5. These figures show that dual solutions exist when A > A_c for both water- and kerosene-based SWCNT and MWCNT. However, no solutions are found to exist when A < A_c, indicating that there is boundary-layer separation, and therefore, laminar boundary-layer approximations are not physically realizable. Figures 2 and 3 display the variation in reduced skin friction f″(0) of water-based SWCNT nanofluid and MWCNT nanofluid, respectively, as a function of stretching rate ratio parameter (A) for representative values of β when Pr = 6.2, λ = 0.5, and φ = 0.2.

Figure 2 shows that for magnetic parameter β = 0.2, the critical value determined is A_c = −1.02. It is observed that when the magnetic parameter β increases, i.e., β = 0.5, the range of A_c values where solutions exist is shrinking (A > A_c = −0.82) for water-based SWCNT nanofluid. For further increment in the magnetic parameter β = 1, the range becomes A > A_c = −0.46. So the study implies that when the magnetic effect occurring at the boundary increases, the range of A_c values for the existence of the solutions becomes smaller. Evidently, with the increase in magnetic parameter β the reduced skin friction strongly decreases (values are more negative) for water-based SWCNT. In Fig. 3, the value of the magnetic parameter β = 0.2 produces a critical value A_c = −1.05, and when the magnetic parameter β increases from 0.2 to 0.5, the range of A_c values where solutions exist is again shrank (A > A_c = −0.83) for water-based MWCNT-nanofluid. For further increment in the magnetic parameter β = 1, the range becomes A > A_c = −0.49.

Therefore, the reduced skin friction decreases (values are increasingly negative) for water-based SWCNT-nanofluid with elevation in magnetic parameter β due to an increase of magnetic effect again due to modification in the magnetic force acting, leading to deceleration. It is concluded that SWCNT-nanofluid has higher reduced skin friction as compared to MWCNT-nanofluid for water.

Figures 4 and 5 display the reduced skin friction f″(0) of kerosene-oil-based SWCNT and MWCNT, respectively, as a function of the stretching rate ratio parameter (A) for representative values of β when Pr = 21, λ = 0.5, and φ = 0.2. In Fig. 4, for the value of magnetic parameter β = 0.2, a critical value A_c = −0.43 is computed. It is observed that when the magnetic parameter β increases, i.e, β = 0.5, the range of A_c values where solutions exist is compressed (A > A_c = −0.41) for kerosene-based SWCNT-nanofluid.

For further increment in the magnetic parameter β = 1, the range is decreased and becomes A > A_c = −0.39. Clearly with an elevation in magnetic parameter β, the reduced skin friction is strongly modified for kerosene-oil-based SWCNT. In Fig. 5 for a value of magnetic parameter β = 0.2, the critical value computed is A_c = −0.80, and when the magnetic parameter β increases from 0.2 to 0.5, the range of A_c values where solutions exist becomes reduced (A > A_c = −0.78) for kerosene-oil-based MWCNT. For further enhancement in the magnetic parameter β = 1, the range becomes A > A_c = −0.75. So, it is apparent from the figures that the reduced skin friction decreases for kerosene-oil-based MWCNT with an increase in magnetic parameter β. So, the analysis further indicates that when the magnetic effect occurs at the boundary, the resistance between the fluid and the plate decreases, accelerating the boundary-layer separation, and it happens in the range where the dual solution exists. The boundary layer accelerates more with the growing magnetic effect. Moreover, from these figures, it is noticed that the existence of a dual solution takes place when A > A_c, which indicates a physically unrealizable boundary-layer separation and boundary-layer approximation.

Again, this is due to adjustment in magnetic body force with the stronger magnetic field, as simulated in the induction modified term, −β(g^′2 − gg″ − 1) appearing in the momentum Eq. (27). Overall, SWCNT-nanofluid, however, achieves higher magnitudes of reduced skin friction as compared to MWCNT-nanofluid for kerosene oil. On the other hand, kerosene oil CNT nanofluids attain higher skin friction as compared to water-based CNT nanofluids.

The influence of reciprocal magnetic Prandtl number λ on reduced heat transfer, i.e., Nusselt number function, −θ′(0), for water- and kerosene-oil-based SWCNT and MWCNT-nanofluids as a function of stretching rate ratio parameter (A) for representative values of β when Pr = 6.2, λ = 0.5, and φ = 0.2 is depicted in Figs. 6 and 7. These figures indicate that dual solutions exist when A > A_c for both water- and kerosene-based SWCNT and MWCNT nanofluids. However, no solutions are found to exist when A < A_c, and indicating the presence of boundary-layer separation for which the laminar boundary-layer approximations break down. Figure 6 shows that for reciprocal magnetic Prandtl number λ = 1, the critical value obtained is A_c = 0.351. With a higher reciprocal magnetic Prandtl number λ = 5, the range of A_c values where solutions exist are shrinking (A > A_c = 0.398 ) for water-based SWCNT-nanofluid. For further increment in the reciprocal magnetic Prandtl number λ = 10, the range compresses to A > A_c = 0.451. With elevation in reciprocal magnetic Prandtl number λ, the reduced heat transfer increases for water-based SWCNT. $λ=μeνf$ and expresses the ratio of magnetic diffusion rate to viscous diffusion rate. The magnetic Reynolds number in the regime is adequately large to invoke magnetic induction. Magnetic Prandtl number quantifies the distortion in the induced magnetic field by the flow field. For cases where the magnetic Reynolds number is extremely small, there is no distortion in the magnetic field. For a reciprocal magnetic Prandtl number λ = 1, the actual magnetic Prandtl number is also 1 and viscous and magnetic diffusivities are equal. For λ = 5, the magnetic Prandtl number is 0.2, and therefore, the magnetic diffusion rate is five times greater than the viscous diffusion rate. This produces a much greater sensitivity in the magnetic induction field.

In Fig. 7, the value of reciprocal magnetic Prandtl number λ = 1 corresponds to a critical value of the sheet stretching rate parameter (A) of A_c = 0.350, and when the reciprocal magnetic Prandtl number increases to λ = 5, the range of A_c values where solutions exist is compressed (A > A_c = 0.369 ) for water-based MWCNT-nanofluids. For further enhancement in the reciprocal magnetic Prandtl number λ = 10, (i.e., actual magnetic Prandtl number of 0.1), the range becomes A > A_c = 0.40. With an increase in reciprocal magnetic Prandtl number λ, the reduced heat transfer is slightly depleted for water-based MWCNT. Reduced heat transfer rates are marginally greater for water SWCNT-nanofluids compared with water MWCNT-nanofluids.

Figures 8 and 9 display the reduced heat transfer −θ′(0) for kerosene-oil-based SWCNT and MWCNT-nanofluids, respectively, as a function of stretching rate ratio parameter (A) for representative values of reciprocal magnetic Prandtl number λ when Pr = 21, λ = 0.5, and φ = 0.2. From Fig. 8, the value of reciprocal magnetic Prandtl number λ = 1 produces a critical stretching parameter ratio value of A_c = 0.301. It is observed that when the magnetic Prandtl number λ increases, i.e., λ = 5, the range of A_c values where solutions exist becomes reduced (A > A_c = 0.325) for kerosene-oil-based SWCNT. For further increment in the magnetic Prandtl number λ = 10, the range contracted to A > A_c = 0.398.

Elevation in reciprocal magnetic Prandtl number λ results in reduced heat transfer decreasing for kerosene-oil-based SWCNT-nanofluids, since with stronger magnetic diffusivity there is greater conversion to heat in the boundary layer and an associated depletion in heat transfer to the wall. In Fig. 9, for a value of reciprocal magnetic Prandtl number λ = 1, we obtain a critical value A_c = 0.35 and for a magnetic Prandtl number of λ = 5 the range of A_c values where solutions exist is contracted to (A > A_c = 0.40) for kerosene-oil-based MWCNT. A subsequent boost in reciprocal of magnetic Prandtl number to λ = 10 further reduces the critical range of stretching sheet ratio parameters to A > A_c = 0.45. Again, there is a suppression in reduced heat transfer with greater reciprocal of magnetic Prandtl number λ for kerosene-oil-based MWCNT-nanofluids. Evidently, SWCNT-nanofluid attains higher heat transfer rates than MWCNT-nanofluids, and this is probably attributable to the superior thermal conductivity of SWCNTs as compared to MWCNTs. It is also demonstrated that kerosene-oil-based CNTs have higher heat transfer than water-based CNTs. The different critical values of the stretching sheet ratio parameter, A_c with representative values of the magnetic parameter for water- and kerosene-oil-based CNTs are summarized in Table 6.

Table 7 displays the behavior of wall skin friction $Rex12Cf$ and heat transfer rate $Rex−12Nux$ with increasing solid volume fraction parameter and magnetic parameter for both water-based SWCNT and MWCNT-nanofluids. Shear stress is clearly enhanced with an increase in solid volume fraction parameters for both SWCNT and MWCNT cases, i.e., greater doping of the nanofluid produces flow acceleration. On the other hand, a decay is observed in shear stress with an increasing magnetic parameter which is due to the retardation in the flow with a stronger magnetic field. Heat transfer rates are consistently elevated with solid volume fraction parameter and magnetic parameter for both CNT cases.

Table 8 presents values for wall shear stress and heat transfer rates with increasing solid volume fraction parameter and magnetic parameter for kerosene-based SWCNT and MWCNT-nanofluids. A boost in shear stress is induced with a higher solid volume fraction parameter for both SWCNT and MWCNT. On the other hand, a reduction in shear stress (i.e., retardation) is produced with increasing magnetic parameters. Heat transfer rate is again accentuated with solid volume fraction parameter and magnetic parameter for both CNT cases. Tables 7 and 8 imply that kerosene oil achieves higher shear stress and heat transfer rates than water as the base fluid.