Figure 1.
Physical flow scenario of ternary nanofluid past a wedge.
Citation: Vandana Gulati, Mansi Dass Singh, Pankaj Gulati. Role of mushrooms in gestational diabetes mellitus[J]. AIMS Medical Science, 2019, 6(1): 49-66. doi: 10.3934/medsci.2019.1.49
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| $ \overline{u}, \overline{v} $ | velocity along measure coordinates [m/s] |
| $ {\mu }_{ternnf} $ | enhanced dynamic viscosity [Pa s] |
| $ {\mu }_{hbnf} $ | enhanced bihybrid dynamic viscosity [Pa s] |
| $ {\rho }_{ternnf} $ | enhanced density [kg/m3] |
| $ {\rho }_{hbnf} $ | enhanced bihybrid density [kg/m3] |
| $ {\alpha }_{ternnf} $ | enhanced thermal diffusivity [m2/s] |
| $ {k}_{ternnf} $ | enhanced thermal conductivity [W/(m K)] |
| $ {k}_{hbnf} $ | enhanced bihybrid thermal conductivity [W/(m K)] |
| $ {\left(\rho {c}_{p}\right)}_{ternnf} $ | improved heat capacity [J/K] |
| $ {\left(\rho {c}_{p}\right)}_{hbnf} $ | improved bihybrid heat capacity [J/K] |
| $ \overline{T} $ | temperature [K] |
| $ {\overline{T}}_{\infty } $ | ambient temperature [K] |
| $ {\overline{T}}_{w} $ | wedge surface temperature [K] |
| $ {\overline{u}}_{w} $ | velocity at the surface [m/s] |
| $ {\overline{\omega }}_{1}, {\overline{\omega }}_{2}, {\overline{\omega }}_{3} $ | solid concentration of nanoparticles |
| $ {M}_{1} $ | magnetic number |
| $ {P}_{r} $ | Prandtl number |
| $ \eta $ | transformative variable |
| $ F{'} $ | dimensionless velocity |
| $ \beta $ | dimensionless temperature |
| $ {\lambda }_{1} $ | wedge parameter |
In the present time, promising characteristics of nanofluids and their hybrid types [1,2] attained much interest of the researchers and engineers. Thus, Kudenatti et al. [3] analyzed the dynamics of Power Law Nanofluid (PLN) over a non-static wedge. The authors emphasized on the importance of MHD and their role in the controlling motion of the fluid. Later on, Akcay et al. [4] reported the behaviour of shear drag along a moving wedge by engaging the multiple effects of the significant physical controlling parameters. Also, analysis of the velocity and temperature distribution over the surface was major focus of the authors. The study of non-transient heat transport playing key role in many of the applied research areas and change the fluid movement under the multiple parameter ranges Therefore, Jafar et al. [5] provided an in-depth analysis of steady magnetohydrodynamic [6] boundary layer flow and discussed variations in boundary layer region due to increasing values of the parameters.
The study of variety of nanoparticles with enhanced characteristics attracted the researchers and engineers. Therefore, extensive efforts have been made to investigate the nanofluids characteristics from various physical aspects. In 2013, Ellahi [7] discussed the potential effects of MHD on non-newtonian nanoliquid. The model developed using thermal dependent viscosity and the model solutions computed via analytical scheme and analyzed the dynamics of the model for multiple appeared parameters. Irreversibility analysis of power-law nanoliquid for electro osmotic CPF (Couette-Poiseuille flow) reported in [8]. To improve the heat capability of the basic fluid, the authors preferred Al2O3 nanoparticles and examined interesting variations in entropy generation. In 2022, Bhatti et al. [9] emphasized on the study of magneto-nanoliquid using bihybrid nanoparticles. Diminishes in the fluid movement and concentration boundary layer with increasing magnetic, slip and Schmidt effects were core findings of the study. Besides these, nanoparticles extensively contribute in drug delivery systems. In this regard, a useful analysis provided by Bhatti et al. [10] and concluded that the study would be advantageous for biomedical engineering and those who are striving to investigate the dynamics of blood in arteries.
In 2023, Yasir et al. [11] performed thermal enhancement in bihybrid nanoliquids. The components of the working liquid taken as GO, Ag, AA7072 and MoS2 and hybrid basic solvent H2O/EG 50%/50%. The numerical simulation of the model done and analyzed the results of the interest and reported that the performed analysis would be expedient for electronic equipment cooling and heat exchangers devices. Abbas et al. [12] introduced a model for Sutterby nanofluid (SNF) by incorporating the influence of magnetic induction and Darcy resistance. The authors inspected that an increasing Eckert number corresponded to greater kinetic energy of the particles and thus an increase in temperature. Similarly, Murad et al. [13], Nisar et al. [14], Alsharari and Mousa [15] described deep knowledge about the characteristics of Casson-Carreau fluid directed to a stagnation point, changes in the nanoliquid performance due to slippery surface, activation energy, and buoyancy effects on copper/water nanofluid in the presence of an insulated obstacle. The studies showed that nanoliquids possessing outstanding thermal characteristics that ultimately increase their applications spectrum.
The transient heat transfer with hydrogen based nanoparticles and function fluid has tremendous characteristics and is extensively uses in multiple engineering disciplines. Thus, Mahmood and Khan [16] and Guedri et al. [17] analyzed the micropolar nanoliquid model and the effects of Al2O3 and Cu nanoparticles and on the heat transfer, shear drag and thermal transmission using basic fluid with hydrogen effects. The study supported findings that increasing the quantity of nanoparticles intensifies the shear drag coefficient and enhances the temperature of the fluid. Those researchers who are interesting in the field of nanofluids (see [18,19,20,21,22]) and their applications in various engineering zone paved their attention in the development of new innovative thermal transmission models and reported comparative or individual analysis. Some of the most latest and potential studies on various nanofluids described by the various scientists (for instance see [23,24,25,26]) using variety of tiny particles and basic functional fluid.
A comparative study of Sakiadis and Blasius flow reported by Klazly et al. [27]. The study revealed that shear drags rises for Sakiadis case and it drops for Blasius case. After that, Kumar et al. [28] reported MHD Blasius/Sakiadis flow of radiated Williamson fluid under effects of variable fluid characteristics. Devi et al. [29] focused on the investigation of 2D transient flow due to static sheet. Further, the authors integrated the significant effects of quadratic type radiation in chemically reactive fluid and interpreted the model results. The steady laminar boundary layer flow in the presence of Rosseland radiation was discussed by Pantokratoras et al. [30]. The study non-Newtonian Carreau, fluid along a static and a moving sheet reported in [31], examining wall shear drag and velocity profile for both Blasius and Sakiadis scenarios. Bataller [32] made efforts to analyze the surface convection effects on the boundary layer with thermal radiations effects. Hady et al. [33] discussed the heat transmission in single phase nanoliquids using shape factors effects. The effects of MHD and convective on BSF (Blasius and Sakiadis Flow) by using Cattaneo-Christov flux model over a sheet investigated in [34]. Similarly, the brief study of BSF using the effects of surface conditions and nanoparticles properties was reported by Krishna et al. [35]. They concluded that the Nusselt number increases for Sakiadis flow. Further, the heat transport characteristics of BSF MHD Maxwell fluid was analyzed by Sekhar [36].
The study of electrically conducting non-Newtonian fluid under the variations of physical quantities investigated by Pantokratoras et al. [37]. The two-dimensional BSF flow of MHD radiated Williamson fluid with chemically reacting species of variable conductivity explored in [28]. The study of free convection for BSF through porous media investigated in [38]. Oyem et al. [39] performed the analysis of BSF of 2D incompressible fluid with soret/dufour effects. Pantokratoras et al. [40] discussed the BSF for Riga-surface and graphically analyzed the velocity profile as well as skin friction for both cases.
Heat transfer investigation in boundary layer flows attained huge attention of the researchers in all times. Such flows have potential applications in designing of airplane wings, nuclear thermal plants, aerodynamics, applied thermal engineering, chemical engineering and many other applied research areas. The researchers made efforts to analyze the heat transfer through boundary layer using simple, mono nano and bihybrid nanofluids under the influence of additional physical aspects. However, no attempt has be made to report the comparative thermal transmission under the influence of induced magnetic field in three types of nanofluids (mono nano, bihybrid and ternary nanofluids) which is important to discuss in the field. For this, reason the present study for boundary layer flow past a wedge (Falkner Skan flow) with pores at the surface is conducted. Modelling of the problem completed via transformative equations and new effective thermo physical characteristics of ternary nanofluids. The results of the model will be advantageous for various engineering applications where enhanced heat transfer is essential and the parametric ranges were considered for achieving better results.
Considered a steady incompressible and two-dimensional boundary layer flow through a wedge influenced by induced magnetic field. Let us supposed that, the velocities at the wedge surface and far from the wedge considered as $ {\overline{\mathrm{u}}}_{w}\left(x\right) = {\overline{U}}_{w}{x}^{m} $ and $ {\overline{\mathrm{u}}}_{e}\left(x\right) = {\overline{U}}_{\infty }{x}^{m} $, respectively. Here, $ {\overline{U}}_{w} $, $ {\overline{U}}_{\infty } $ are velocities and $ m $ is a constant with $ 0\le m\le 1 $. As Shown in Figure 1.
The governing equations according to the above considered model are as follows [41]:
| $ \frac{\partial \overline{u}}{\partial x}+\frac{\partial \overline{v}}{\partial y} = 0, $ | (1) |
| $ \frac{\partial {\overline{H}}_{1}}{\partial x}+\frac{\partial {\overline{H}}_{2}}{\partial y} = 0, $ | (2) |
| $ \overline{u}\frac{\partial \overline{u}}{\partial x}+\overline{v}\frac{\partial \overline{u}}{\partial y}-\frac{{\mu }_{ternnf}}{4\pi {\rho }_{f}}\left({\overline{H}}_{1}\frac{\partial {\overline{H}}_{1}}{\partial x}+{\overline{H}}_{2}\frac{\partial {\overline{H}}_{1}}{\partial y}\right) = \left({\overline{u}}_{e}\frac{{d\overline{u}}_{e}}{dx}-{\mu }_{ternnf}\frac{{\overline{H}}_{e}}{4\pi {\rho }_{f}}\frac{{d\overline{H}}_{e}}{dx}\right)+\frac{{\mu }_{ternnf}}{{\rho }_{ternnf}}\frac{{\partial }^{2}\overline{u}}{{\partial y}^{2}}, $ | (3) |
| $ \overline{u}\frac{\partial {\overline{H}}_{1}}{\partial x}+\overline{v}\frac{\partial {\overline{H}}_{1}}{\partial y}-{\overline{H}}_{1}\frac{\partial \overline{u}}{\partial x}-{\overline{H}}_{2}\frac{\partial \overline{u}}{\partial y} = {\mu }_{e}\frac{{\partial }^{2}{H}_{1}}{{\partial y}^{2}}, $ | (4) |
| $ \overline{u}\frac{\partial \overline{T}}{\partial x}+\overline{v}\frac{\partial \overline{T}}{\partial y} = {\alpha }_{ternf}\frac{{\partial }^{2}\overline{T}}{{\partial y}^{2}}. $ | (5) |
The applicable physical conditions for the current flow of ternary nanoliquid are considered as below:
|
$ \left\{¯u|y=0=¯uw(x),¯v|y=0=0,¯T|y=0=¯Tw,∂¯H1∂y|y=0=¯H2|y=0=0, \right. $
|
(6) |
|
$
\left\{¯u→¯ue(x)=¯U∞xm,¯T→¯T∞,¯H1|y→∞=¯He(x)=¯H0xm. \right. \text{when}~~ y\to \infty.
$
|
(7) |
Thermophysical properties of nanofluids [42,43] are of key interest in the analysis nanoliquids thermal transport. Therefore, the following properties (Table 1) of ternary nanofluid taken for estimation of enhanced nanoliquid characteristics.
| Dynamic viscosity | |
| Nanofluid | $ {\mu }_{nf}={\mu }_{f}{\left[1-{\varpi }_{1}\right]}^{-2.5}. $ |
| Hybrid nanofluid | $ {\mu }_{hbnf}={\mu }_{f}{\left[{\left(1-{\varpi }_{1}\right)}^{2.5}{\left(1-{\varpi }_{2}\right)}^{2.5}\right]}^{-1}. $ |
| Ternary nanofluid | $ {\mu }_{ternnf}={\mu }_{f}{\left[{\left(1-{\varpi }_{1}\right)}^{2.5}{\left(1-{\varpi }_{2}\right)}^{2.5}(1-{\varpi }_{3})\right]}^{-1}. $ |
| Density | |
| Nanofluid | $ {\rho }_{nf}=\left[\left(1-{\varpi }_{1}\right)+{\varpi }_{1}\frac{{\rho }_{p1}}{{\rho }_{f}}\right]{\rho }_{f}. $ |
| Hybrid nanofluid | $ {\rho }_{hbnf}=\left(1-{\varpi }_{2}\right)\left[\left(1-{\varpi }_{1}\right)+{\varpi }_{1}\frac{{\rho }_{p1}}{{\rho }_{f}}\right]+{\varpi }_{2}\frac{{\rho }_{p2}}{{\rho }_{f}}. $ |
| Ternary nanofluid | $ {\rho }_{ternf}=\left(1-{\varpi }_{3}\right)\left[\left\{\left(1-{\varpi }_{2}\right)\left[\left(1-{\varpi }_{1}\right)+{\varpi }_{1}\frac{{\rho }_{p1}}{{\rho }_{f}}\right]+{\varpi }_{2}\frac{{\rho }_{p2}}{{\rho }_{f}}\right\}\right]+{\varpi }_{3}\frac{{\rho }_{p3}}{{\rho }_{f}}. $ |
| Heat capacity | |
| Nanofluid | $ ({\rho {c}_{p})}_{nf}=\left[\left(1-{\varpi }_{1}\right)+{\varpi }_{1}\frac{{\left(\rho {c}_{p}\right)}_{p1}}{({\rho {c}_{p})}_{f}}\right]{\left(\rho {c}_{p}\right)}_{f}. $ |
| Hybrid nanofluid | $ ({\rho {c}_{p})}_{hbnf}=\left(1-{\varpi }_{2}\right)\left[\left(1-{\varpi }_{1}\right)+{\varpi }_{1}\frac{{\left(\rho {c}_{p}\right)}_{p1}}{({\rho {c}_{p})}_{f}}\right]+{\varpi }_{2}\frac{{\left(\rho {c}_{p}\right)}_{p2}}{({\rho {c}_{p})}_{f}}. $ |
| Ternary nanofluid | $ {\left(\rho {c}_{p}\right)}_{ternf}=\left(1-{\varpi }_{3}\right)\left[\left\{\left(1-{\varpi }_{2}\right)\left[\left(1-{\varpi }_{1}\right)+{\varpi }_{1}\frac{{\left(\rho {c}_{p}\right)}_{p1}}{({\rho {c}_{p})}_{f}}\right]+{\varpi }_{2}\frac{{\left(\rho {c}_{p}\right)}_{p2}}{({\rho {c}_{p})}_{f}}\right\}\right]+{\varpi }_{3}\frac{{\left(\rho {c}_{p}\right)}_{p3}}{({\rho {c}_{p})}_{f}}. $ |
| Thermal conductivity | |
| Nanofluid | $ {k}_{nf}=\frac{\left[\left({k}_{p1}+2{k}_{f}\right)-2{\varpi }_{1}\left({k}_{f}-{k}_{p1}\right)\right]}{[\left({k}_{p1}+2{k}_{f}\right)+{\varpi }_{1}({k}_{f}-{k}_{p1}\left)\right]}{k}_{f}. $ |
| Hybrid nanofluid | $ {k}_{hbnf}=\frac{\left[\left({k}_{p2}+2{k}_{nf}~~ \right)-2{\varpi }_{2}\left({k}_{nf}-{k}_{p2}\right)\right]}{[\left({k}_{p2}+2{k}_{nf}~~ \right)+{\varpi }_{2}({k}_{nf}-{k}_{p2}\left)\right]}{k}_{nf}. $ |
| Ternary nanofluid | $ {k}_{ternnf}=\frac{\left[\left({k}_{p3}+2{k}_{hbnf}~~~ \right)-2{\varpi }_{3}\left({k}_{hbnf}-{k}_{p3}\right)\right]}{[\left({k}_{p3}+2{k}_{hbnf}~~~ \right)+{\varpi }_{3}({k}_{hbnf}-{k}_{p3}\left)\right]}{k}_{hbnf}. $ |
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To acquire to desired heat transfer model, the following similarity equations were used along with mathematical operations:
|
$ \left.¯u=∂¯ψ∂y=¯U∞xmF′,¯v=−∂¯ψ∂x=−((¯m+1)νf¯U(x)2x)12(F+¯m−1¯m+1)ηF′,¯ψ=(2νfxU(x)(m+1))0.5F,β=¯T−¯T∞¯Tw−¯T∞,η=((m+1)U(x)2νfx)0.5,¯H1=¯H0xmG′,¯H2=−¯Ho(2νfx¯m−1(¯m+1)¯U∞)0.5{¯mG+0.5(¯m−1)ηG′}. \right\} $
|
(8) |
Finally, the following heat transport model obtained which includes the characteristics of ternary nanofluid.
| $ {F}^{{'}{'}{'}}+\frac{\frac{{\rho }_{ternnf}}{{\rho }_{f}}}{\frac{{\mu }_{ternnf}}{{\mu }_{f}}}\left[{F}^{{'}{'}}F+2m{\left(m+1\right)}^{-1}\left(1-{F}^{{'}2}\right)+\frac{{2\delta }_{1}}{\left(m+1\right)}\left(m{G}^{{'}2}-m{G}^{{'}{'}}G-m\right)\right] = 0, $ | (9) |
| $ {M}_{1}{G}^{{'}{'}{'}}+{G}^{{'}{'}}F-2m{\left(m+1\right)}^{-1}{F}^{{'}{'}}G = 0, $ | (10) |
| $ \frac{{k}_{ternnf}}{{k}_{f}}{\beta }^{{'}{'}}+\frac{Pr{\left(\rho {c}_{p}\right)}_{ternnf}}{{\left(\rho {c}_{p}\right)}_{f}}{\beta }^{{'}}F = 0, $ | (11) |
|
$ \frac{{k}_{ternnf}}{{k}_{f}} = \left[(kp1+2kf)−2ϖ1(kf−kp1)(kp1+2kf)+ϖ1(kf−kp1)∗(kp2+2knf )−2ϖ2(knf−kp2)(kp2+2knf )+ϖ2(knf−kp2)∗(kp3+2khbnf )−2ϖ3(khbnf−kp3)(kp3+2khbnf )+ϖ3(khbnf−kp3) \right]. $
|
The above-described model is supported by the following boundary conditions over the wedge:
| $ F = 0, {F}^{{'}} = {\lambda }_{1}, G = 0, {G}^{{'}{'}} = 0, \beta = 1~~ at~~ \eta = 0, $ | (12) |
| $ {F}^{{'}} = 1, G = 1, \beta \to 0~~when~~ \eta \to \infty . $ | (13) |
The embedded quantities are $ {\delta }_{1} = \frac{{\overline{\mu }}_{e}{{\overline{H}}_{0}}^{2}}{4\pi {\rho }_{f}{{\overline{U}}_{\infty }}^{2}}, Pr = {\nu }_{f}{\alpha }_{f}^{-1} $ and $ {\lambda }_{1} = \frac{{\overline{U}}_{w}}{{\overline{U}}_{\infty }} $. Moreover, the significant (skin friction and Nusselt number) formulas described by the following expressions (Table 2):
| Skin friction formulas | |
| Nanofluid | $ \frac{{\left[1-{\varpi }_{1}\right]}^{-2.5}}{\left[\left(1-{\varpi }_{1}\right)+{\varpi }_{1}\frac{{\rho }_{p1}}{{\rho }_{f}}\right]}{F}^{{'}{'}}\left(0\right). $ |
| Hybrid nanofluid | $ \frac{{\left[{\left(1-{\varpi }_{1}\right)}^{2.5}{\left(1-{\varpi }_{2}\right)}^{2.5}\right]}^{-1}}{\left(1-{\varpi }_{2}\right)\left[\left(1-{\varpi }_{1}\right)+{\varpi }_{1}\frac{{\rho }_{p1}}{{\rho }_{f}}\right]+{\varpi }_{2}\frac{{\rho }_{p2}}{{\rho }_{f}}}{F}^{{'}{'}}\left(0\right). $ |
| Ternary nanofluid | $ \frac{{\left[{\left(1-{\varpi }_{1}\right)}^{2.5}{\left(1-{\varpi }_{2}\right)}^{2.5}\left(1-{\varpi }_{3}\right)\right]}^{-1}}{\left(1-{\varpi }_{3}\right)\left[\left\{\left(1-{\varpi }_{2}\right)\left[\left(1-{\varpi }_{1}\right)+{\varpi }_{1}\frac{{\rho }_{p1}}{{\rho }_{f}}\right]+{\varpi }_{2}\frac{{\rho }_{p2}}{{\rho }_{f}}\right\}\right]+{\varpi }_{3}\frac{{\rho }_{p3}}{{\rho }_{f}}}{F}^{{'}{'}}\left(0\right). $ |
| Nusselt number formulas | |
| Nanofluid | $ \left|\left[\frac{\left({k}_{p1}+2{k}_{f}\right)-2{\varpi }_{1}\left({k}_{f}-{k}_{p1}\right)}{\left({k}_{p1}+2{k}_{f}\right)+{\varpi }_{1}\left({k}_{f}-{k}_{p1}\right)}\right]{\beta }^{{'}}\left(0\right)\right|. $ |
| Hybrid nanofluid | $ \left|\left[\frac{\left({k}_{p1}+2{k}_{f}\right)-2{\varpi }_{1}\left({k}_{f}-{k}_{p1}\right)}{\left({k}_{p1}+2{k}_{f}\right)+{\varpi }_{1}\left({k}_{f}-{k}_{p1}\right)}*\frac{\left({k}_{p2}+2{k}_{nf}~~ \right)-2{\varpi }_{2}\left({k}_{nf}-{k}_{p2}\right)}{\left({k}_{p2}+2{k}_{nf}~~ \right)+{\varpi }_{2}\left({k}_{nf}-{k}_{p2}\right)}\right]{\beta }^{{'}}\left(0\right)\right|. $ |
| Ternary nanofluid | $ \left|\left[(kp1+2kf)−2ϖ1(kf−kp1)(kp1+2kf)+ϖ1(kf−kp1)∗(kp2+2knf )−2ϖ2(knf−kp2)(kp2+2knf )+ϖ2(knf−kp2)∗(kp3+2khbnf )−2ϖ3(khbnf−kp3)(kp3+2khbnf )+ϖ3(khbnf−kp3) \right]{\beta }^{{'}}\left(0\right)\right|. $ |
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The current model is coupled and contains high nonlinearities, and an exact solution is not feasible. Therefore, the RKF-45 (see [44,45,46]) implemented for the analysis of the model using the software MATHEMATICA 13.0 and the impacts of physical constraints on the heat transport performances of mono, hybrid and ternary nanofluids. The adopted technique is applicable for initial value problems and the desired initial value problem was obtained after using the following transforms in the system:
| $ \left[{\zeta }_{1}, {\zeta }_{2}, {\zeta }_{3}, {\zeta }_{3}^{{'}}, {\zeta }_{4}, {\zeta }_{5}, {\zeta }_{6}, {\zeta }_{6}^{{'}}, {\zeta }_{7}, {\zeta }_{8}, {\zeta }_{8}{'}\right] = \left[F, {F}^{{'}}, {F}^{{'}{'}}, {F}^{{'}{'}{'}}, G, {G}^{{'}}, {G}^{{'}{'}}, {G}^{{'}{'}{'}}, \beta , {\beta }^{{'}}, {\beta }^{{'}{'}}\right]. $ | (14) |
The system was then arranged in the following way to reduce it into respective initial value problems.
| $ {F}^{{'}{'}{'}} = -\frac{\frac{{\rho }_{ternnf}}{{\rho }_{f}}}{\frac{{\mu }_{ternnf}}{{\mu }_{f}}}\left[{F}^{{'}{'}}F+2m{\left(m+1\right)}^{-1}\left(1-{F}^{{'}2}\right)+\frac{{2\delta }_{1}}{\left(m+1\right)}\left(m{G}^{{'}2}-m{G}^{{'}{'}}G-m\right)\right], $ | (15) |
| $ {{G}^{{'}{'}}}^{{'}} = -\frac{1}{{M}_{1}}\left[{G}^{{'}{'}}F-2m{\left(m+1\right)}^{-1}{F}^{{'}{'}}G\right], $ | (16) |
| $ {\beta }^{{'}{'}} = -\frac{1}{\frac{{k}_{ternnf}}{{k}_{f}}}\left[\frac{Pr{\left(\rho {c}_{p}\right)}_{ternnf}}{{\left(\rho {c}_{p}\right)}_{f}}{\beta }^{{'}}F\right]. $ | (17) |
Now, the system takes the below appropriate form for further computation:
| $ {\zeta }_{3}^{{'}} = -\frac{\frac{{\rho }_{ternnf}}{{\rho }_{f}}}{\frac{{\mu }_{ternnf}}{{\mu }_{f}}}\left[{\zeta }_{3}{\zeta }_{1}+2m{\left(m+1\right)}^{-1}\left(1-{\zeta }_{2}^{2}\right)+\frac{{2\delta }_{1}}{\left(m+1\right)}\left(m{\zeta }_{5}^{2}-m{\zeta }_{5}^{{'}}{\zeta }_{4}-m\right)\right], $ | (18) |
| $ {\zeta }_{6}^{{'}} = -\frac{1}{{M}_{1}}\left[{\zeta }_{6}{\zeta }_{1}-2m{\left(m+1\right)}^{-1}{\zeta }_{3}{\zeta }_{4}\right], $ | (19) |
| $ {\zeta }_{8}^{{'}} = -\frac{1}{\frac{{k}_{ternnf}}{{k}_{f}}}\left[\frac{Pr{\left(\rho {c}_{p}\right)}_{ternnf}}{{\left(\rho {c}_{p}\right)}_{f}}{\zeta }_{8}{\zeta }_{1}\right]. $ | (20) |
Accuracy of the technique is achieved by setting its tolerance up to $ {10}^{-6} $ and the step size $ 0.01 $. The authenticity of the results is subject to the asymptotic behavior of the temperature profile and it must satisfy the boundary conditions for the velocity and temperature distributions. These are obvious from the plotted results which gives the correctness of the model results.
The physical results of the model representing the flow of ternary, nano and bihybrid nanoliquids over a nonstationary wedge are demonstrated in this section. It is commendable to mention here that the three types of nanoparticles namely Al2O3, Cu and Ag taken for the formation of resultant nanoliquid and H2O is taken as working base solvent due its good solvent characteristics, density and high heat capacity. Further, $ {\overline{\omega }}_{1}, {\overline{\omega }}_{2} $ and $ {\overline{\omega }}_{3} $ are the associated concentration of the nanoparticles in the basic solvent. Further, the concentrations of nanoparticles were taken up to 20% and feasible value of Prandtl number for water is fixed at $ 6.2 $. This section further classified into three subsequent subsections which represent the model results for the velocity, ternary nanoliquid temperature, shear drag and Nusselt number. The study validation is also presented in subsection 4.4.
Figure 2 demonstrating the velocity $ F{'}\left(\eta \right) $ changes under potential effects of $ m, $ moving wedge number $ {\lambda }_{1} $ and magnetic number $ {\delta }_{1} $. The presentation of the velocity distribution shows that the fluid movement increases by enlarging the values of $ m $ and $ {\lambda }_{1} $. Physically, when the wedge moves then the neighboring fluid layer on the wedge surface gain the velocity of the wedge. As a consequence the particles moves rapidly. Then the frictional forces with the successive fluid layer reduces and the fluid velocity increases. After, $ \eta = 3.0 $ the velocity of the fluid reaches to its maximum speed and then moves with the speed of free stream. These results elucidated in Figures 2(a) and 2(b), respectively. On the other side, the magnetic number reduces the fluid motion and almost negligible variations observed in Figure 2(c).
Figures 3 and 4 organized to analyze the tangential velocity distribution $ G{'}\left(\eta \right) $ for increasing $ {\lambda }_{1}, {M}_{1} $ and $ {\delta }_{1} $. The tangential velocity of the fluid varies in very interesting way while the wedge moves and the reciprocal Hartmann parameter $ {M}_{1} $ enlarges. It is inspected that $ G{'}\left(\eta \right) $ declines prominently near the wedge surface (Figure 3a); however, these variations become slow and finally reaches to the its maximum limit i.e., $ {G}^{{'}}\left(\eta \right) = 1 $ after which the fluid moves with constant velocity. Figure 3(b) indicates that the reciprocal Hartmann parameter (Figure 3b) is effective in increasing the velocity component $ G{'}\left(\eta \right) $ and is maximum fluctuation is observed around $ \eta = 0 $. The 3D representation of the results in Figures 3(a) and 3(b) given in Figures 3(c) and 3(d), respectively. Figure 4 demonstrating the impacts of $ {\delta }_{1} $ which shows that the velocity drops when the parameter $ {\delta }_{1} $ enhances and maximum drop is noticed for nanoliquids. Figure 4(d) supports the 3D pictorial view of the $ {\delta }_{1} $ variations on the profile of $ G{'}\left(\eta \right) $.
This subsection represents the comparative enhanced heat transfer for Al2O3/H2O (mono nanoliquid), (Al2O3-CuO)/H2O (bihybrid nanoliquid) and (Al2O3-CuO-Ag)/H2O (ternary nanoliquid) due to variations in $ {\delta }_{1}, {\lambda }_{1} $ and $ {M}_{1} $. For this, Figure 5 furnished.
Figure 5(a) indicates that when the magnetic forces enlarges, the temperature of the fluids also increases. However, higher temperatures are observed for ternary nanoliquid than bihybrid and mono nanoliquids. Physically, ternary nanoliquid (Al2O3-CuO-Ag)/H2O comprises the thermal conductivity and heat capacity of three particles which increase thermal conductivity of ternary nanoliquid which enhance it heat transport property. Thus, in ternary nanoliquid the heat transfer is greater than in the other nanofluids. Similarly, the temperature decreases with increasing $ {\lambda }_{1} $ (Figure 5b) but the behavior is greater in the ternary nanofluid. Further, thermal boundary layer thickness increase for larger $ {\delta }_{1} $ and no significant contribution of the reciprocal Hartmann number (Figure 5c) was noticed in thermal enhancement of the nanofluids.
The study of shear drags and thermal gradient (Nusselt number) in nanofluids is essential from various engineering field and industrial purposes. Therefore, this subsection fill this requirement under the variations of the model parameters. For this, Figures 5 and 6 plotted for the shear drags and Nu trends.
Figures 6(a) and 6(b) present the shear drag in the mono nanofluid, bihybrid nanofluid and ternary nanofluid against $ {\delta }_{1} $ and $ {\lambda }_{1} $, respectively. In both the cases, the greatest shear drag effects were in the ternary nanoliquids. Physically, composite density of (Al2O3-CuO-Ag) nanoparticles make the resultant fluid denser than that of mono (Al2O3) and bihybrid (Al2O3-CuO) nanoparticles. Due to this reason maximum shear drag observed on the wedge surface.
The analysis of thermal gradient at the surface is important for heating/cooling purposes. Thus, the set of Figures 7(a) and 7(b) show the trends of Nusselt number on the surface for increasing values of $ {\delta }_{1} $ and $ {\lambda }_{1} $, respectively. For both parametric variations, the Nusselt number is greatest in the ternary nanoliquid. Physically, the sum of thermal conductivities of the nanoparticles boosts the thermal conductivity of ternary nanofluid. Therefore, maximum heat transmitted on the wedge surface. However, for bihybrid and mono nanoliquids, the heat transport mechanism is slow due to weak thermal conductivity.
The study validation successfully performed and is presented in this subsection. In order make the current model compatible with the existing models in open science literature, it is essential to set the values $ {\phi }_{j} $ for $ j = \mathrm{1, 2}, 3 $, $ {\lambda }_{1} $ and $ {\delta }_{1} $ such that these tends to zero. Then, the model results for shear drag $ F{'}{'}\left(0\right) $ computed at multiple stages of the parameter $ m $. These results given in Table 3 and it is cleared that the current results aligned with the results of Ahmed et al. [47], Watanabe [48] and Adnan et al. [49]. This shows that the results obtained from the study are correct and can be replicated in future studies.
| $ \boldsymbol{m} $ | Results from various studies for $ \boldsymbol{F}\boldsymbol{{'}}\boldsymbol{{'}}\left(0\right) $ | |||
| present results | Ahmed et al. [47] | Watanabe [48] | Adnan et al. [49] | |
| 0.0 | 0.4695999960 | 0.46959 | 0.46960 | 0.469590 |
| 0.0141 | 0.5046143299 | 0.504614 | ----- | 0.504614 |
| 0.0435 | 0.5689777817 | 0.568977 | 0.56898 | 0.568977 |
| 0.0909 | 0.6549788596 | 0.654978 | 0.65498 | 0.654978 |
| 0.1429 | 0.7319985706 | 0.731998 | 0.73200 | 0.731998 |
| 0.2000 | 0.8021256343 | 0.802125 | 0.80213 | 0.802125 |
| 0.3333 | 0.9276536249 | 0.927653 | 0.92765 | 0.927653 |
| 0.5000 | 1.0389035229 | 1.038903 | 1.03890 | 1.038903 |
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The study of thermal enhancement in ternary nanofluids over a moving wedge amid an induced magnetic field is presented. The basic model was transformed into a simplified form via defined transformative rules and improved properties of the ternary nanofluid. To investigate the influence of the model parameters on the dynamics of fluid, numerical analysis was conducted and the results were portrayed multiple parametric values. It is concluded that:
• The fluids movement (nanofluid, hybrid nanofluid and ternary nanofluid) can be intensified by increasing the values of $ m $ and $ {\lambda }_{1} $ as $ 0.1, 0.2, 0.3 $ and $ 1.0, 1.2, 1.3 $, respectively.
• The magnetic field strongly opposed the fluid velocity over the wedge surface.
• The tangential velocity $ G{'}\left(\eta \right) $ diminishes rapidly when $ {\lambda }_{1} $ and $ {\delta }_{1} $ increase as $ 1.0, 1.2, 1.3 $.
• The ternary nanofluid has high capacity to transmit the heat with increasing $ {\delta }_{1} $, while the transmission in the nanofluid and hybrid nanoliquid is slow.
• The thermal gradient in ternary nanoliquid was 65%, for the hybrid nanofluid, it was 45% and for the common nanofluid it was 35% which shows that ternary nanofluids are excellent for thermal transport applications.
The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through large group Research Project under grant number RGP2/16/44.
The authors declare no conflict of interest.
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| 5. | Walid Aich, Hamad Almujibah, Sherzod Shukhratovich Abdullaev, Mutasem Z. Bani-Fwaz, Ahmed M Hassan, Thermal performance of radiated annular extended surface using advanced nanomaterials influenced by various physical controlling parameters for nucleate boiling case, 2023, 51, 2214157X, 103524, 10.1016/j.csite.2023.103524 | |
| 6. | Khaleeq ur Rahman, Nidhish Kumar Mishra, Mutasem Z. Bani-Fwaz, Thermal study of Darcy–Forchheimer hybrid nanofluid flow inside a permeable channel by VIM: features of heating source and magnetic field, 2023, 148, 1388-6150, 14385, 10.1007/s10973-023-12611-5 | |
| 7. | Taysir Mhedheb, Walid Hassen, Abdallah Mhimid, Mohammed A. Almeshaal, Muapper Alhadri, Lioua Kolsi, Parametric analysis of a solar parabolic trough collector integrated with hybrid-nano PCM storage tank, 2023, 51, 2214157X, 103652, 10.1016/j.csite.2023.103652 | |
| 8. | Khaleeq Ur Rahman, Zafar Mahmood, Sami Ullah Khan, Aatif Ali, Zhixiong Li, Iskander Tlili, Enhanced thermal study in hybrid nanofluid flow in a channel motivated by graphene/Fe3O4 and Newtonian heating, 2024, 21, 25901230, 101772, 10.1016/j.rineng.2024.101772 | |
| 9. | Aneesa Nadeem, Haitham A Mahmoud, Aatif Ali, Sayed M Eldin, Significance of Koo-Kleinstreuer-Li model for thermal enhancement in nanofluid under magnetic field and thermal radiation factors using LSM, 2023, 15, 1687-8132, 10.1177/16878132231206906 | |
| 10. | Lotfi Ben Said, Warisha Gul, Zafar Mahmood, Mutasem Z. Bani-Fwaz, Hijaz Ahmad, Sami Ullah Khan, Walid Aich, Melting thermal process in buoyancy driven radiated flow of (MoS2-SiO2-Au)/H2O near the stagnant point under mixed convection, 2024, 60, 2214157X, 104615, 10.1016/j.csite.2024.104615 | |
| 11. | Kamel Smida, Muhammad Umer Sohail, Iskander Tlili, Asma Javed, Numerical thermal study of ternary nanofluid influenced by thermal radiation towards convectively heated sinusoidal cylinder, 2023, 9, 24058440, e20057, 10.1016/j.heliyon.2023.e20057 | |
| 12. | Obai younis, Houssem Laidoudi, Aissa Abderrahmane, Abdeldjalil Belazreg, Naef A.A. Qasem, Raad Z. Homod, Muhyaddine Rawa, Ahmed M. Hassan, Thermal pattern of nano-encapsulated PCM in a lid-driven cavity with presence of a heated body, magnetic field and limited permeability, 2023, 50, 2214157X, 103469, 10.1016/j.csite.2023.103469 | |
| 13. | Aneesa Nadeem, Nejla Mahjoub Said, LSM analysis of thermal enhancement in KKL model-based unsteady nanofluid problem using CCM and slanted magnetic field effects, 2023, 1388-6150, 10.1007/s10973-023-12801-1 | |
| 14. | Waseem Abbas, Nejla Mahjoub Said, Nidhish Kumar Mishra, Zafar Mahmood, Muhammad Bilal, Significance of coupled effects of resistive heating and perpendicular magnetic field on heat transfer process of mixed convective flow of ternary nanofluid, 2024, 149, 1388-6150, 879, 10.1007/s10973-023-12723-y | |
| 15. | Azhar Rasheed, Sami Ullah Khan, Muhammad Bilal, Dennis Ling Chuan Ching, Lotfi Ben Said, Ahmed Mir, Lioua Kolsi, Ilyas Khan, Investigation of comparative entropy in different nanofluids inspired by solar radiations and unsteady effects: Model analysis for permeable channel, 2024, 17, 16878507, 101158, 10.1016/j.jrras.2024.101158 | |
| 16. | Naef A.A. Qasem, Aissa Abderrahmane, Abdeldjalil Belazreg, Obai Younis, Yacine Khetib, Kamel Guedri, Investigation of phase change heat transfer in a rectangular case as function of fin placement for solar applications, 2024, 54, 2214157X, 103996, 10.1016/j.csite.2024.103996 | |
| 17. | Bowen Liu, Chunming Wang, Numerical simulation for the evolution in surface morphology of titanium alloy by nanosecond pulsed laser ablation, 2024, 315, 02540584, 128997, 10.1016/j.matchemphys.2024.128997 | |
| 18. | C. Pownraj, A. Karthik, S. Suresh, A. Valan Arasu, Jitendra Kumar Katiyar, Investigation of tribo-thermal properties of inter-mixed AYSZ nanoceramic composite/SAE20W40 nanolubricant, 2023, 148, 1388-6150, 12411, 10.1007/s10973-023-12606-2 | |
| 19. | Nidhish Kumar Mishra, Khalid Abdulkhaliq M. Alharbi, Khaleeq ur Rahman, Sayed M. Eldin, Mutasem Z. Bani-Fwaz, Investigation of improved heat transport featuring in dissipative ternary nanofluid over a stretched wavy cylinder under thermal slip, 2023, 48, 2214157X, 103130, 10.1016/j.csite.2023.103130 | |
| 20. | Waqas Ashraf, Study of heat transfer in ternary nanofluid between parallel plates influenced by different physical parameters, 2023, 148, 1388-6150, 10645, 10.1007/s10973-023-12418-4 | |
| 21. | Md. Mahadul Islam, Md. Mamun Molla, Mixed convection and sensitivity analysis of impinging jet flow of engine oil on rotating heated cylinder with high Prandtl numbers, 2024, 64, 2214157X, 105375, 10.1016/j.csite.2024.105375 | |
| 22. | Mutasem Z Bani-Fwaz, Adnan , Zafar Mahmood, Muhammad Bilal, Adel A EI-Zahhar, Ilyas Khan, Shafiullah Niazai, Computational investigation of thermal process in radiated nanofluid modulation influenced by nanoparticles (Al2O3) and molecular (H2O) diameters, 2024, 11, 2288-5048, 22, 10.1093/jcde/qwae011 | |
| 23. | Walid Aich, Ghulfam Sarfraz, Nejla Mahjoub Said, Muhammad Bilal, Ahmed Faisal Ahmed Elhag, Ahmed M. Hassan, Significance of radiated ternary nanofluid for thermal transport in stagnation point flow using thermal slip and dissipation function, 2023, 51, 2214157X, 103631, 10.1016/j.csite.2023.103631 | |
| 24. | Naim Ben Ali, Zafar Mahmood, Mutasem Z. Bani-Fwaz, Sami Ullah Khan, Iskander Tlili, Thermal efficiency of radiated nanofluid through convective geometry subject to heating source, 2024, 15, 20904479, 102947, 10.1016/j.asej.2024.102947 | |
| 25. | Subhalaxmi Dey, Surender Ontela, Pradyumna Kumar Pattnaik, Satyaranjan Mishra, Performance of carbon nanotubes (CNTs) on the development of radiating hybrid nanofluid flow through an stretching cylinder, 2023, 2397-7914, 10.1177/23977914231207947 | |
| 26. | K. Vishnu Ram, Aeswin Lawrance, R. Harish, Impact of nanoparticle shape on the thermal performance of eco-friendly soybean-based nanofluids in cooling titanium alloys, 2024, 63, 2214157X, 105309, 10.1016/j.csite.2024.105309 | |
| 27. | Walid Aich, Waseem Abbas, Muhammad Bilal Riaz, M.A. Ahmed, Lotfi Ben Said, Sami Ullah Khan, Impacts of nanoscaled metallic particles on the dynamics of ternary Newtonian nanofluid laminar flow through convectively heated and radiated surface, 2024, 53, 2214157X, 103969, 10.1016/j.csite.2023.103969 | |
| 28. | Ghulam Haider, Naveed Ahmed, Unsteady Ohmic dissipative flow of ZnO-SAE50 nanofluid past a permeable shrinking cylinder, 2023, 34, 0957-4484, 455401, 10.1088/1361-6528/aced57 | |
| 29. | Sumaira Fayyaz, Zafar Mahmood, Sami Ullah Khan, Mohammed A. Tashkandi, Lioua Kolsi, Thermal study of single phase nanofluid model using radiative γAl2O3 nanomaterial under Hall current and momentum slip phenomena, 2024, 0958-305X, 10.1177/0958305X241270239 | |
| 30. | Mutasem Z. Bani-Fwaz, Sumaira Fayyaz, Nidhish Kumar Mishra, Zafar Mahmood, Sami Ullah Khan, Muhammad Bilal, Investigation of unsteady nanofluid over half infinite domain under the action of parametric effects and EPNM, 2024, 1388-6150, 10.1007/s10973-024-13121-8 | |
| 31. | Badreddine Ayadi, Sadia Karim, Zafar Mahmood, Sami Ullah Khan, Mutasem Z. Bani-Fwaz, Muhammad Bilal, Wajdi Rajhi, Lotfi Ben Said, Study of ZnO-SAE50 over a radiated permeable exponentially elongating curved device subject to non-uniform thermal source and Newtonian heating, 2024, 63, 2214157X, 105275, 10.1016/j.csite.2024.105275 | |
| 32. | Nidhish Kumar Mishra, Muhammed Umer Sohail, Mutasem Z. Bani-Fwaz, Ahmed M. Hassan, Thermal analysis of radiated (aluminum oxide)/water through a magnet based geometry subject to Cattaneo-Christov and Corcione’s Models, 2023, 49, 2214157X, 103390, 10.1016/j.csite.2023.103390 | |
| 33. | Chetan Kumar, Vashista Ademane, Vasudeva Madav, Experimental study of convective heat transfer distribution of non-interacting wall and perpendicular air jet impingement cooling on flat surface, 2024, 60, 2214157X, 104532, 10.1016/j.csite.2024.104532 | |
| 34. | Shabbir Ahmad, Kashif Ali, Hafiz Humais Sultan, Fareeha Khalid, Moin-ud-Din Junjua, Farhan Lafta Rashid, Humberto Garcia Castellanos, Yashar Aryanfar, Tamer M. Khalaf, Ahmed S. Hendy, A Breakthrough in Penta-Hybrid Nanofluid Flow Modeling for Heat Transfer Enhancement in a Spatially Dependent Magnetic Field: Machine Learning Approach, 2025, 46, 0195-928X, 10.1007/s10765-024-03467-4 | |
| 35. | Praveen Kumar Kanti, Prabhu Paramasivam, V. Vicki Wanatasanappan, Seshathiri Dhanasekaran, Prabhakar Sharma, Experimental and explainable machine learning approach on thermal conductivity and viscosity of water based graphene oxide based mono and hybrid nanofluids, 2024, 14, 2045-2322, 10.1038/s41598-024-81955-1 | |
| 36. | Noura Alsedais, Mohamed Ahmed Mansour, Abdelraheem Mahmoud Aly, Sara I. Abdelsalam, Artificial neural network validation of MHD natural bioconvection in a square enclosure: entropic analysis and optimization, 2025, 41, 0567-7718, 10.1007/s10409-024-24507-x | |
| 37. | Mutasem Z. Bani-Fwaz, Sami Ullah Khan, Zafar Mahmood, Yasir Khan, A. M. Obalalu, Mohammad Khalid Nasrat, Entropy performance of nonlinear mathematical hydrocarbon based model using ternary alloys (AA7072/AA7075/Ti6AI4V) under influential solar radiations and convective effects, 2024, 14, 2045-2322, 10.1038/s41598-024-81901-1 | |
| 38. | Ghulfam Sarfraz, Sami Ullah Khan, Zafar Mahmood, Yasir Khan, Tadesse Walelign, Significance of Hamilton–Crosser’s model for thermal improvement in transient ternary nanofluid problem from multiple physical aspects, 2024, 0217-9849, 10.1142/S0217984925500903 | |
| 39. | Latif Ahmad, Assmaa Abd-Elmonem, Saleem Javed, Muhammad Yasir, Umair Khan, Yalcin Yilmaz, Aisha M. Alqahtani, Dissipative disorder analysis of Homann flow of Walters B fluid with the applications of solar thermal energy absorption aspects, 2025, 15, 2190-5487, 10.1007/s13201-024-02335-8 | |
| 40. | Mutasem Z. Bani-Fwaz, Sadia Karim, Sami Ullah Khan, Muhammad Bilal, Yasir Khan, Iskander Tlili, Significance of Fe3O4/MnZnFe2O4 nanoparticles for thermal efficiency of ethylene glycol: Analysis for single-phase nanofluid flow using Cattaneo Christov theory, 2025, 0217-9849, 10.1142/S0217984925501052 | |
| 41. | Shabir Ahmad, Ikram Ullah, Saira Shukat, Murtaza Ali, Lorentz force and Brownian motion features on entropy optimization in nanofluid swirling flow through porous configuration, 2025, 8, 2520-8160, 10.1007/s41939-024-00711-0 | |
| 42. | Mutasem Z. Bani-Fwaz, Sami Ullah Khan, B. Shankar Goud, Tadesse Walelign, Kanayo Kenneth Asogwa, Iskander Tlili, Thermal performance of Falkner Skan model (FSM) for (GOMoS2)/(C2H6O2-H2O) 50:50% nanofluid under radiation heating source, 2025, 15, 2045-2322, 10.1038/s41598-025-86470-5 | |
| 43. | Nidhish Kumar Mishra, Khaleeq ur Rahman, Mutasem Z. Bani-Fwaz, Yasir Khan, Investigation of micropolar nanofluid using AA7072 alloys under intrinsic polarities: model-based computational analysis, 2025, 1388-6150, 10.1007/s10973-024-13986-9 | |
| 44. | Warisha Gul, Zafar Mahmood, Sami Ullah Khan, Muhammad Bilal, A. M. Obalalu, Yasir Khan, Iskander Tlili, Multiscale parametric influence on stagnation point Darcy-Forchheimer single phase ternary nanofluid problem using solar radiations: design for vertical sheet, 2025, 8, 2520-8160, 10.1007/s41939-025-00747-w | |
| 45. | Mutasem Z. Bani‐Fwaz, Ishtiaque Mahmood, Numerical MHD heat transport in C12H26‐C15H32 using modified Tiwari–Das model inspired by nanoparticles characteristics, 2025, 105, 0044-2267, 10.1002/zamm.70071 | |
| 46. | Khaleeq ur Rahman, Syed Zulfiqar Ali Zaidi, Refka Ghodhbani, Dana Mohammad Khidhir, Muhammad Asad Iqbal, Iskander Tlili, Thermomechanics of radiated hybrid nanofluid interacting with MHD and heating source: Significance of nanoparticles shapes, 2025, 18, 16878507, 101544, 10.1016/j.jrras.2025.101544 |
Vandana Gulati, Mansi Dass Singh, Pankaj Gulati. Role of mushrooms in gestational diabetes mellitus[J]. AIMS Medical Science, 2019, 6(1): 49-66. doi: 10.3934/medsci.2019.1.49
| Dynamic viscosity | |
| Nanofluid | $ {\mu }_{nf}={\mu }_{f}{\left[1-{\varpi }_{1}\right]}^{-2.5}. $ |
| Hybrid nanofluid | $ {\mu }_{hbnf}={\mu }_{f}{\left[{\left(1-{\varpi }_{1}\right)}^{2.5}{\left(1-{\varpi }_{2}\right)}^{2.5}\right]}^{-1}. $ |
| Ternary nanofluid | $ {\mu }_{ternnf}={\mu }_{f}{\left[{\left(1-{\varpi }_{1}\right)}^{2.5}{\left(1-{\varpi }_{2}\right)}^{2.5}(1-{\varpi }_{3})\right]}^{-1}. $ |
| Density | |
| Nanofluid | $ {\rho }_{nf}=\left[\left(1-{\varpi }_{1}\right)+{\varpi }_{1}\frac{{\rho }_{p1}}{{\rho }_{f}}\right]{\rho }_{f}. $ |
| Hybrid nanofluid | $ {\rho }_{hbnf}=\left(1-{\varpi }_{2}\right)\left[\left(1-{\varpi }_{1}\right)+{\varpi }_{1}\frac{{\rho }_{p1}}{{\rho }_{f}}\right]+{\varpi }_{2}\frac{{\rho }_{p2}}{{\rho }_{f}}. $ |
| Ternary nanofluid | $ {\rho }_{ternf}=\left(1-{\varpi }_{3}\right)\left[\left\{\left(1-{\varpi }_{2}\right)\left[\left(1-{\varpi }_{1}\right)+{\varpi }_{1}\frac{{\rho }_{p1}}{{\rho }_{f}}\right]+{\varpi }_{2}\frac{{\rho }_{p2}}{{\rho }_{f}}\right\}\right]+{\varpi }_{3}\frac{{\rho }_{p3}}{{\rho }_{f}}. $ |
| Heat capacity | |
| Nanofluid | $ ({\rho {c}_{p})}_{nf}=\left[\left(1-{\varpi }_{1}\right)+{\varpi }_{1}\frac{{\left(\rho {c}_{p}\right)}_{p1}}{({\rho {c}_{p})}_{f}}\right]{\left(\rho {c}_{p}\right)}_{f}. $ |
| Hybrid nanofluid | $ ({\rho {c}_{p})}_{hbnf}=\left(1-{\varpi }_{2}\right)\left[\left(1-{\varpi }_{1}\right)+{\varpi }_{1}\frac{{\left(\rho {c}_{p}\right)}_{p1}}{({\rho {c}_{p})}_{f}}\right]+{\varpi }_{2}\frac{{\left(\rho {c}_{p}\right)}_{p2}}{({\rho {c}_{p})}_{f}}. $ |
| Ternary nanofluid | $ {\left(\rho {c}_{p}\right)}_{ternf}=\left(1-{\varpi }_{3}\right)\left[\left\{\left(1-{\varpi }_{2}\right)\left[\left(1-{\varpi }_{1}\right)+{\varpi }_{1}\frac{{\left(\rho {c}_{p}\right)}_{p1}}{({\rho {c}_{p})}_{f}}\right]+{\varpi }_{2}\frac{{\left(\rho {c}_{p}\right)}_{p2}}{({\rho {c}_{p})}_{f}}\right\}\right]+{\varpi }_{3}\frac{{\left(\rho {c}_{p}\right)}_{p3}}{({\rho {c}_{p})}_{f}}. $ |
| Thermal conductivity | |
| Nanofluid | $ {k}_{nf}=\frac{\left[\left({k}_{p1}+2{k}_{f}\right)-2{\varpi }_{1}\left({k}_{f}-{k}_{p1}\right)\right]}{[\left({k}_{p1}+2{k}_{f}\right)+{\varpi }_{1}({k}_{f}-{k}_{p1}\left)\right]}{k}_{f}. $ |
| Hybrid nanofluid | $ {k}_{hbnf}=\frac{\left[\left({k}_{p2}+2{k}_{nf}~~ \right)-2{\varpi }_{2}\left({k}_{nf}-{k}_{p2}\right)\right]}{[\left({k}_{p2}+2{k}_{nf}~~ \right)+{\varpi }_{2}({k}_{nf}-{k}_{p2}\left)\right]}{k}_{nf}. $ |
| Ternary nanofluid | $ {k}_{ternnf}=\frac{\left[\left({k}_{p3}+2{k}_{hbnf}~~~ \right)-2{\varpi }_{3}\left({k}_{hbnf}-{k}_{p3}\right)\right]}{[\left({k}_{p3}+2{k}_{hbnf}~~~ \right)+{\varpi }_{3}({k}_{hbnf}-{k}_{p3}\left)\right]}{k}_{hbnf}. $ |
DownLoad:
CSV
| Skin friction formulas | |
| Nanofluid | $ \frac{{\left[1-{\varpi }_{1}\right]}^{-2.5}}{\left[\left(1-{\varpi }_{1}\right)+{\varpi }_{1}\frac{{\rho }_{p1}}{{\rho }_{f}}\right]}{F}^{{'}{'}}\left(0\right). $ |
| Hybrid nanofluid | $ \frac{{\left[{\left(1-{\varpi }_{1}\right)}^{2.5}{\left(1-{\varpi }_{2}\right)}^{2.5}\right]}^{-1}}{\left(1-{\varpi }_{2}\right)\left[\left(1-{\varpi }_{1}\right)+{\varpi }_{1}\frac{{\rho }_{p1}}{{\rho }_{f}}\right]+{\varpi }_{2}\frac{{\rho }_{p2}}{{\rho }_{f}}}{F}^{{'}{'}}\left(0\right). $ |
| Ternary nanofluid | $ \frac{{\left[{\left(1-{\varpi }_{1}\right)}^{2.5}{\left(1-{\varpi }_{2}\right)}^{2.5}\left(1-{\varpi }_{3}\right)\right]}^{-1}}{\left(1-{\varpi }_{3}\right)\left[\left\{\left(1-{\varpi }_{2}\right)\left[\left(1-{\varpi }_{1}\right)+{\varpi }_{1}\frac{{\rho }_{p1}}{{\rho }_{f}}\right]+{\varpi }_{2}\frac{{\rho }_{p2}}{{\rho }_{f}}\right\}\right]+{\varpi }_{3}\frac{{\rho }_{p3}}{{\rho }_{f}}}{F}^{{'}{'}}\left(0\right). $ |
| Nusselt number formulas | |
| Nanofluid | $ \left|\left[\frac{\left({k}_{p1}+2{k}_{f}\right)-2{\varpi }_{1}\left({k}_{f}-{k}_{p1}\right)}{\left({k}_{p1}+2{k}_{f}\right)+{\varpi }_{1}\left({k}_{f}-{k}_{p1}\right)}\right]{\beta }^{{'}}\left(0\right)\right|. $ |
| Hybrid nanofluid | $ \left|\left[\frac{\left({k}_{p1}+2{k}_{f}\right)-2{\varpi }_{1}\left({k}_{f}-{k}_{p1}\right)}{\left({k}_{p1}+2{k}_{f}\right)+{\varpi }_{1}\left({k}_{f}-{k}_{p1}\right)}*\frac{\left({k}_{p2}+2{k}_{nf}~~ \right)-2{\varpi }_{2}\left({k}_{nf}-{k}_{p2}\right)}{\left({k}_{p2}+2{k}_{nf}~~ \right)+{\varpi }_{2}\left({k}_{nf}-{k}_{p2}\right)}\right]{\beta }^{{'}}\left(0\right)\right|. $ |
| Ternary nanofluid | $ \left|\left[(kp1+2kf)−2ϖ1(kf−kp1)(kp1+2kf)+ϖ1(kf−kp1)∗(kp2+2knf )−2ϖ2(knf−kp2)(kp2+2knf )+ϖ2(knf−kp2)∗(kp3+2khbnf )−2ϖ3(khbnf−kp3)(kp3+2khbnf )+ϖ3(khbnf−kp3) \right]{\beta }^{{'}}\left(0\right)\right|. $ |
DownLoad:
CSV
| $ \boldsymbol{m} $ | Results from various studies for $ \boldsymbol{F}\boldsymbol{{'}}\boldsymbol{{'}}\left(0\right) $ | |||
| present results | Ahmed et al. [47] | Watanabe [48] | Adnan et al. [49] | |
| 0.0 | 0.4695999960 | 0.46959 | 0.46960 | 0.469590 |
| 0.0141 | 0.5046143299 | 0.504614 | ----- | 0.504614 |
| 0.0435 | 0.5689777817 | 0.568977 | 0.56898 | 0.568977 |
| 0.0909 | 0.6549788596 | 0.654978 | 0.65498 | 0.654978 |
| 0.1429 | 0.7319985706 | 0.731998 | 0.73200 | 0.731998 |
| 0.2000 | 0.8021256343 | 0.802125 | 0.80213 | 0.802125 |
| 0.3333 | 0.9276536249 | 0.927653 | 0.92765 | 0.927653 |
| 0.5000 | 1.0389035229 | 1.038903 | 1.03890 | 1.038903 |
DownLoad:
CSV
| Dynamic viscosity | |
| Nanofluid | $ {\mu }_{nf}={\mu }_{f}{\left[1-{\varpi }_{1}\right]}^{-2.5}. $ |
| Hybrid nanofluid | $ {\mu }_{hbnf}={\mu }_{f}{\left[{\left(1-{\varpi }_{1}\right)}^{2.5}{\left(1-{\varpi }_{2}\right)}^{2.5}\right]}^{-1}. $ |
| Ternary nanofluid | $ {\mu }_{ternnf}={\mu }_{f}{\left[{\left(1-{\varpi }_{1}\right)}^{2.5}{\left(1-{\varpi }_{2}\right)}^{2.5}(1-{\varpi }_{3})\right]}^{-1}. $ |
| Density | |
| Nanofluid | $ {\rho }_{nf}=\left[\left(1-{\varpi }_{1}\right)+{\varpi }_{1}\frac{{\rho }_{p1}}{{\rho }_{f}}\right]{\rho }_{f}. $ |
| Hybrid nanofluid | $ {\rho }_{hbnf}=\left(1-{\varpi }_{2}\right)\left[\left(1-{\varpi }_{1}\right)+{\varpi }_{1}\frac{{\rho }_{p1}}{{\rho }_{f}}\right]+{\varpi }_{2}\frac{{\rho }_{p2}}{{\rho }_{f}}. $ |
| Ternary nanofluid | $ {\rho }_{ternf}=\left(1-{\varpi }_{3}\right)\left[\left\{\left(1-{\varpi }_{2}\right)\left[\left(1-{\varpi }_{1}\right)+{\varpi }_{1}\frac{{\rho }_{p1}}{{\rho }_{f}}\right]+{\varpi }_{2}\frac{{\rho }_{p2}}{{\rho }_{f}}\right\}\right]+{\varpi }_{3}\frac{{\rho }_{p3}}{{\rho }_{f}}. $ |
| Heat capacity | |
| Nanofluid | $ ({\rho {c}_{p})}_{nf}=\left[\left(1-{\varpi }_{1}\right)+{\varpi }_{1}\frac{{\left(\rho {c}_{p}\right)}_{p1}}{({\rho {c}_{p})}_{f}}\right]{\left(\rho {c}_{p}\right)}_{f}. $ |
| Hybrid nanofluid | $ ({\rho {c}_{p})}_{hbnf}=\left(1-{\varpi }_{2}\right)\left[\left(1-{\varpi }_{1}\right)+{\varpi }_{1}\frac{{\left(\rho {c}_{p}\right)}_{p1}}{({\rho {c}_{p})}_{f}}\right]+{\varpi }_{2}\frac{{\left(\rho {c}_{p}\right)}_{p2}}{({\rho {c}_{p})}_{f}}. $ |
| Ternary nanofluid | $ {\left(\rho {c}_{p}\right)}_{ternf}=\left(1-{\varpi }_{3}\right)\left[\left\{\left(1-{\varpi }_{2}\right)\left[\left(1-{\varpi }_{1}\right)+{\varpi }_{1}\frac{{\left(\rho {c}_{p}\right)}_{p1}}{({\rho {c}_{p})}_{f}}\right]+{\varpi }_{2}\frac{{\left(\rho {c}_{p}\right)}_{p2}}{({\rho {c}_{p})}_{f}}\right\}\right]+{\varpi }_{3}\frac{{\left(\rho {c}_{p}\right)}_{p3}}{({\rho {c}_{p})}_{f}}. $ |
| Thermal conductivity | |
| Nanofluid | $ {k}_{nf}=\frac{\left[\left({k}_{p1}+2{k}_{f}\right)-2{\varpi }_{1}\left({k}_{f}-{k}_{p1}\right)\right]}{[\left({k}_{p1}+2{k}_{f}\right)+{\varpi }_{1}({k}_{f}-{k}_{p1}\left)\right]}{k}_{f}. $ |
| Hybrid nanofluid | $ {k}_{hbnf}=\frac{\left[\left({k}_{p2}+2{k}_{nf}~~ \right)-2{\varpi }_{2}\left({k}_{nf}-{k}_{p2}\right)\right]}{[\left({k}_{p2}+2{k}_{nf}~~ \right)+{\varpi }_{2}({k}_{nf}-{k}_{p2}\left)\right]}{k}_{nf}. $ |
| Ternary nanofluid | $ {k}_{ternnf}=\frac{\left[\left({k}_{p3}+2{k}_{hbnf}~~~ \right)-2{\varpi }_{3}\left({k}_{hbnf}-{k}_{p3}\right)\right]}{[\left({k}_{p3}+2{k}_{hbnf}~~~ \right)+{\varpi }_{3}({k}_{hbnf}-{k}_{p3}\left)\right]}{k}_{hbnf}. $ |
| Skin friction formulas | |
| Nanofluid | $ \frac{{\left[1-{\varpi }_{1}\right]}^{-2.5}}{\left[\left(1-{\varpi }_{1}\right)+{\varpi }_{1}\frac{{\rho }_{p1}}{{\rho }_{f}}\right]}{F}^{{'}{'}}\left(0\right). $ |
| Hybrid nanofluid | $ \frac{{\left[{\left(1-{\varpi }_{1}\right)}^{2.5}{\left(1-{\varpi }_{2}\right)}^{2.5}\right]}^{-1}}{\left(1-{\varpi }_{2}\right)\left[\left(1-{\varpi }_{1}\right)+{\varpi }_{1}\frac{{\rho }_{p1}}{{\rho }_{f}}\right]+{\varpi }_{2}\frac{{\rho }_{p2}}{{\rho }_{f}}}{F}^{{'}{'}}\left(0\right). $ |
| Ternary nanofluid | $ \frac{{\left[{\left(1-{\varpi }_{1}\right)}^{2.5}{\left(1-{\varpi }_{2}\right)}^{2.5}\left(1-{\varpi }_{3}\right)\right]}^{-1}}{\left(1-{\varpi }_{3}\right)\left[\left\{\left(1-{\varpi }_{2}\right)\left[\left(1-{\varpi }_{1}\right)+{\varpi }_{1}\frac{{\rho }_{p1}}{{\rho }_{f}}\right]+{\varpi }_{2}\frac{{\rho }_{p2}}{{\rho }_{f}}\right\}\right]+{\varpi }_{3}\frac{{\rho }_{p3}}{{\rho }_{f}}}{F}^{{'}{'}}\left(0\right). $ |
| Nusselt number formulas | |
| Nanofluid | $ \left|\left[\frac{\left({k}_{p1}+2{k}_{f}\right)-2{\varpi }_{1}\left({k}_{f}-{k}_{p1}\right)}{\left({k}_{p1}+2{k}_{f}\right)+{\varpi }_{1}\left({k}_{f}-{k}_{p1}\right)}\right]{\beta }^{{'}}\left(0\right)\right|. $ |
| Hybrid nanofluid | $ \left|\left[\frac{\left({k}_{p1}+2{k}_{f}\right)-2{\varpi }_{1}\left({k}_{f}-{k}_{p1}\right)}{\left({k}_{p1}+2{k}_{f}\right)+{\varpi }_{1}\left({k}_{f}-{k}_{p1}\right)}*\frac{\left({k}_{p2}+2{k}_{nf}~~ \right)-2{\varpi }_{2}\left({k}_{nf}-{k}_{p2}\right)}{\left({k}_{p2}+2{k}_{nf}~~ \right)+{\varpi }_{2}\left({k}_{nf}-{k}_{p2}\right)}\right]{\beta }^{{'}}\left(0\right)\right|. $ |
| Ternary nanofluid | $ \left|\left[(kp1+2kf)−2ϖ1(kf−kp1)(kp1+2kf)+ϖ1(kf−kp1)∗(kp2+2knf )−2ϖ2(knf−kp2)(kp2+2knf )+ϖ2(knf−kp2)∗(kp3+2khbnf )−2ϖ3(khbnf−kp3)(kp3+2khbnf )+ϖ3(khbnf−kp3) \right]{\beta }^{{'}}\left(0\right)\right|. $ |
| $ \boldsymbol{m} $ | Results from various studies for $ \boldsymbol{F}\boldsymbol{{'}}\boldsymbol{{'}}\left(0\right) $ | |||
| present results | Ahmed et al. [47] | Watanabe [48] | Adnan et al. [49] | |
| 0.0 | 0.4695999960 | 0.46959 | 0.46960 | 0.469590 |
| 0.0141 | 0.5046143299 | 0.504614 | ----- | 0.504614 |
| 0.0435 | 0.5689777817 | 0.568977 | 0.56898 | 0.568977 |
| 0.0909 | 0.6549788596 | 0.654978 | 0.65498 | 0.654978 |
| 0.1429 | 0.7319985706 | 0.731998 | 0.73200 | 0.731998 |
| 0.2000 | 0.8021256343 | 0.802125 | 0.80213 | 0.802125 |
| 0.3333 | 0.9276536249 | 0.927653 | 0.92765 | 0.927653 |
| 0.5000 | 1.0389035229 | 1.038903 | 1.03890 | 1.038903 |
