# Mixed Convection of Silica–Molybdenum Disulphide/Water Hybrid Nanoliquid over a Rough Sphere

^{1}

^{2}

^{*}

## Abstract

**:**

_{2}) and molybdenum disulphide (MoS

_{2}) nanoadditives which are added in H

_{2}O to form MoS

_{2}–SiO

_{2}/H

_{2}O hybrid nanoliquid. The partial differential equations describing the boundary layer flow characteristics are reduced into non-dimensional form with appropriate non-similar reduction. It should be noted that the governing equations have been written using the conservation laws of mass, momentum and energy. These considered equations allow simulating the analyzed phenomenon using numerical techniques. Implicit finite difference approximation and technique of Quasilinearization are utilized to work out the dimensionless control equations. The influence of various physical characteristics included in this challenge, such as the velocity fields and temperature patterns, is investigated. The study of border gradients is performed, which deals with the skin friction and energy transport strength. The plots of computational outcomes are considered, which ascertain that velocity distribution reduces, whilst coefficient of friction at the surface, energy transport strength and temperature distribution augment for enhancing values of hybrid nanofluid. For enhancing magnitude of combined convection parameter, dimensionless velocity distribution, surface drag coefficient and energy transport strength enhance, while temperature distribution diminishes. High impact of hybrid nanofluid on energy transport strength and the surface friction compared to the host liquid and mono nanofluid in presence/absence of surface roughness is shown. Velocity distribution enhances for rising values of velocity ratio parameter. Enhancing values of frequency parameter rise the friction at the surface and energy transport strength. It is also examined that the hybrid nanofluid has a maximum temperature for the blade-shaped nanoparticles and has a low temperature for the spherical-shaped nanoparticles.

## 1. Introduction

_{2}, CuO-MgO, Cu-Al

_{2}O

_{3}and MWCNTs-Fe

_{2}O

_{4}-SWCNTs, etc. [9,10,11,12,13]. Since, many authors such as Maraj et al. [14], Ghadikolaei et al. [15], Khan et al. [16], etc., have studied the water containing Molybdenum disulphide and Silica nanoparticles, considering the either of one or both nanoparticles. Chemical attributes of molybdenum disulfide including capabilities of lubrication, heat conductivity and thermal capacitance are employed in various engineering systems [17]. In addition, Silicon oxide (silica) is widely applied in energy transport applications and it can be successfully use as nano-inclusions [18]. The work of these authors has motivated us to perform the present research and therefore, we have investigated the current problem.

_{2}is characterized by large band gaps structure which is related to the structure of grapheme and as a result, MoS

_{2}is employed in different engineering systems [19,20,21,22,23,24,25,26]. Moreover, chemical attributes of MoS

_{2}can be applied in various mechanical systems [23]. MoS

_{2}particles have been coated with copper to improve their combination with the copper particles for the sintering [24]. Further, silicon is widely spread element on earth after oxygen [25]. Silicon has very useful attributes that illustrates many different applications of this element in various fields. Silicon chips have essentially changed different technical systems including electronics, space and aviation industry, and others. Silicon is used in photovoltaic panels that characterize the high efficiency and suitability. Silicon oxide (silica) is widely employed in energy transport applications and it can be successfully applied as a nanoadditive [18].

_{2}O

_{3}/H

_{2}O nanoliquid past a surface and their results reveal that the surface drag coefficient lessens for opposing buoyancy flow, while it boosts for assisting buoyancy flow. Further, Muhammad et al. [35] have analyzed combined convection over a curved surface with MWCNT-Cu/H

_{2}O hybrid nanofluid and the obtained outcomes ascertain that the fluid velocity enhances for combined convection parameter. Bognar et al. [36] have studied the nanofluid flow over a uniformly moving plate considering different nanoparticles and their outcomes reveal that the values of velocity and temperature are greater for alumina nanoparticles compared to titania and magnetite particles. In addition, Bognar and Hriczo [37] have studied the ferrofluids behavior over a flat sheet under the variable magnetic field influence. They have revealed that a rise of the ferromagnetic parameter characterizes the velocity reduction and temperature augmentation. Many researchers have also worked on the combination of silica and molybdenum nanoparticles by considering various base fluids such as water, ethylene glycol, etc. [38,39].

- –
- combined convective flow over a rough sphere;
- –
- impact of molybdenum-silica/water hybrid nanofluid on flow structures and heat patterns;
- –
- effect of surface roughness on hydro- and thermodynamics.

## 2. Mathematical Analysis

_{w}and T

_{∞}be the temperature at the border and away from the surface. The geometry and the configuration of the flow system are schematically presented in Figure 1. Here x is the curvilinear coordinate over the sphere border, while coordinate y is perpendicular it. The section r(x) explains the contour of the considered object.

_{hnf}, μ

_{hnf}, (ρc

_{p})

_{hnf}, (ρβ)

_{hnf}and k

_{hnf}denote the density, viscosity, thermal capacitance, heat expansion coefficient and heat conductivity of the hybrid nanosuspension. Following Khan et al. [36], Devi and Devi [46], Rostami et al. [5] and Waini et al. [7], the chemophysical attributes for mono and hybrid nanosuspensions are provided in Table 1.

_{1}and φ

_{2}, respectively where φ

_{1}= φ

_{2}= 0 is meant to be the regular liquid. The hybrid nanosuspension includes 0.1 volume fraction of SiO

_{2}(i.e., φ

_{1}) and the concentration of MoS

_{2}is changed from 0 to 0.1 $\left(0\le {\varphi}_{2}\le 0.1\right)$. Table 2 demonstrates the chemical attributes of the nanoadditives and the host liquid (see Maraj et al. [14], Rostami et al. [5], Ghadikolae et al. [15] and Khan et al. [16]).

_{2}and MoS

_{2}are denoted by the subscripts s

_{1}and s

_{2}, respectively. The subscripts hnf, f and nf and denotes hybrid nanoliquid, base liquid and nanoliquid respectively.

## 3. Method of Analysis

_{∞}.

## 4. Results and Discussion

_{1}) is included to the host liquid (H

_{2}O) and accordingly molybdenum disulphide (φ

_{2}) is introduced with numerous concentrations to obtain MoS

_{2}-SiO

_{2}/water hybrid nanosuspension. By considering the different combination of nanoparticles, the similar approach has been investigated by many researchers [46,49,50]. The surface roughness is modeled in terms of large frequency and low amplitude sinusoidal waves. The frequency characteristic (n) and low variable (α) are employed for 5 ≤ n ≤ 100 and 0 ≤ α ≤ 0.3. The smooth border is denoted using α = 0, whilst the rough border is denoted using α ≠ 0 [44,51]. As curvilinear $\overline{x}$ and y coordinates are perpendicular one another, the angle between these coordinates is taken as π/2. The magnitude ε = 0 characterizes the no-slip condition at the border, i.e., smooth border. The empirical shape factor s for nanoparticles is calculated using the formula s = ψ/3, where ψ is the sphericity. As per the Hamilton and Crosser approach [52], sphericity is the proportion of sphere area to the border area of physical particles with the same volumes. In the numerical computation, we have considered Pr = 7 that demonstrates the water as the host liquid, and it is constant throughout the investigation [30,53].

_{2}) on temperature distribution $\left\{G\left(\overline{x},\eta \right)\right\}$. For enhancing values of combined convection characteristic, the temperature of the liquid diminishes. The difference between the wall temperature and ambient flow temperature increases, for increasing values of Ri, which reduces the fluid temperature. Further, for escalating magnitude of volume fraction (φ

_{2}), the temperature of the liquid augments. The liquid heat conductivity enhances for enhancing magnitude of volume fraction (φ

_{2}) and rises the capacity of the hybrid nanofluid to conduct more heat, which enhances the temperature of the liquid.

_{2}) on velocity profile $\left\{F\left(\overline{x},\eta \right)\right\}$. For enhancing magnitudes of concentration (φ

_{2}) the velocity of the liquid diminishes. It is clear that the velocity of the fluid is larger in absence of nanoparticles compared its presence, for each corresponding value of ε. For increasing values of φ

_{2}the hybrid nanofluid becomes denser and concentrated. Moreover, velocity of the fluid enhances with enhancing values of velocity ratio characteristic. This is caused because the larger value of velocity ratio characteristic indicates more slip at the border that raises the liquid velocity.

_{2}) on temperature profile $\left\{G\left(\overline{x},\eta \right)\right\}$. Temperature of the liquid diminishes with enhance in this small parameter. As we seen that the fluid temperature diminishes more in rough surface (α ≠ 0) case compared to smooth surface (α = 0). Such phenomenon can be explained by the packets of fluid that are captured in the troughs formed owing to the surface roughness. It experiences slip at the wall of the sphere, when the liquid flows about the rough sphere and this enhances the velocity of the liquid. The growth of the liquid velocity yields more fluid movement across the wall of the sphere and this diminishes the liquid’s temperature. In addition, the liquid’s temperature enhances with rise in the volume fraction (φ

_{2}). It is very clear that hybrid nanofluid has greater thermal conductivity than that of base fluid. Physically, it means that the nanoparticles with high thermal conductivity allow intensifying the heat transference and an inclusion of such nanoparticles enhances energy transport with a rise of the temperature.

_{2}) on surface drag coefficient $\left({\mathrm{Re}}^{1/2}{C}_{f}\right)$ and energy transport strength $\left({\mathrm{Re}}^{-1/2}Nu\right)$. Surface drag coefficient and energy transport strength are enhanced for rising values of combined convection characteristic. The positive magnitudes of Ri illustrate the essential impact of free convection over the forced one. Thus, the buoyancy force inside the liquid intensifies the liquid motion and as a result it augments frictions at the surface. In addition, the drop between the border temperature and free stream temperature enhances for rising values of Ri, which increases the energy transport strength from wall to fluid. Further, for enhancing values of volume fraction (φ

_{2}), the friction at the surface and the energy transport strength enhance. This is because addition of nanoparticles in the host liquid leads to intensifying its temperature and as a result one can find an intensification of the energy transport.

_{2}) on energy transport strength $\left({\mathrm{Re}}^{-1/2}Nu\right)$. The energy transport strength boosts with enhancing nanoparticle volume fraction (φ

_{2}). The physical reason is that these nanoparticles which are characterized by large heat conductivity display the augmentation of the energy transport coefficient because this parameter is directly proportional to heat conductivity. Further, heat transfer rate enhances with enhance in the small parameter. The sinusoidal waves are observed in presence of surface roughness. This is because the sphere has more area owing to the presence of roughness and thereby intensifies the heat transfer in a larger quantity from the sphere into the ambient fluid.

_{2}-SiO

_{2}/H

_{2}O), mono nanofluid (SiO

_{2}/H

_{2}O) and base liquid (H

_{2}O) with various values of small parameter (α) for the surface drag coefficient $\left({\mathrm{Re}}^{1/2}{C}_{f}\right)$ and energy transport strength $\left({\mathrm{Re}}^{-1/2}Nu\right)$. Energy transport strength and coefficient of friction at the border are enhanced for hybrid nanosuspension compared to mono nanofluid and regular liquid. In addition, the friction and rate of heat transfer high for rough surface (α ≠ 0) case compared to smooth surface (α = 0) and sinusoidal waves are seen in presence of rough surface (α ≠ 0). Hence, we can conclude that the hybrid nanofluid has more thermal conductivity than that of nanofluid and base fluid due to the rising magnitudes of nanoparticles concentration fundamentally designate the quicker heat transmission from surface of the cylinder to ambient fluid, this boosts the thermal conductivity and consequently temperature of hybrid nanofluid, which causes the intensive enhance in the magnitudes of temperature profile and rate of heat transfer coefficient, significantly.

## 5. Conclusions

- –
- the hybrid nanofluid augments the temperature, as well as Nusselt number at the wall;
- –
- velocity distribution is reduced, while friction at the surface enhances in the case of hybrid nanofluid;
- –
- slip exists near the surface of the sphere owing to roughness, which yields steep jump in the liquid velocity close to the border;
- –
- for enhancing magnitudes of small parameter, the friction at the surface and the Nusselt number are enhanced;
- –
- skin-friction parameter and the Nusselt number are boosted for hybrid nanoliquid on comparison with mono nanofluid and the base fluid;
- –
- skin-friction parameter and the Nusselt number have the highest values for blade-shaped nanoadditives and lowest value is found for spherical-formed nanoparticles.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

F | dimensionless velocity |

Nu | Nusselt number |

G | non-dimensional temperature |

g | gravity acceleration |

n | frequency parameter |

v | y-velocity projection |

r(x) | radius of the section normal to axis of the sphere |

Pr | Prandtl number |

U_{0} | reference velocity |

f | dimensionless stream function |

R | radius of the sphere |

T_{w} | temperature at the wall |

Re | Reynolds number |

Gr | Grashof number |

Ri | combined convection parameter |

T | temperature |

T_{∞} | ambient temperature |

u | x-velocity projection |

U_{∞} | free stream velocity constant |

x, y | Cartesian coordinates |

U_{e} | free stream velocity |

## Greek symbols

α | small parameter |

$\Delta \overline{x},\text{}\Delta \eta $ | step size in $\overline{x}$ and η directions |

ε | velocity ratio parameter |

$\overline{x}$, η | transformed variables |

ν_{f} | kinematic viscosity |

φ_{1} | volume fraction of silica nanoparticle |

φ_{2} | volume fraction of molybdenum disulphide nanoparticle |

ψ | dimensionless stream function |

## Subscripts

f | base fluid |

hnf | hybrid nanofluid |

nf | mono nanofluid |

s1 | solid component for silica |

s2 | solid component for molybdenum disulphide |

$\overline{x}$, η | variables of the partial derivatives |

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**Figure 2.**Impact of combined convection characteristic (Ri) and velocity ratio characteristic (ε) on $\left\{F\left(\overline{x},\eta \right)\right\}$ when $\overline{x}=1.5$, φ

_{1}= 0.1, φ

_{2}= 0.1, n = 10 and α = 0.1.

**Figure 3.**Impact of combined convection characteristic (Ri) and volume fraction (φ

_{2}) on $\left\{G\left(\overline{x},\eta \right)\right\}$ when $\overline{x}=0.5$, φ

_{1}= 0.1, ε = 0.15, n = 10 and α = 0.1.

**Figure 4.**Influence of velocity ratio characteristic (ε) and volume fraction (φ

_{2}) on $\left\{F\left(\overline{x},\eta \right)\right\}$ when $\overline{x}=1.5$, φ

_{1}= 0.1, ε = 0.15, n = 10 and α = 0.1.

**Figure 5.**Impact of small parameter (α) and volume fraction of (φ

_{2}) on $\left\{G\left(\overline{x},\eta \right)\right\}$ when $\overline{x}=0.5$, Ri = 10, ε = 0.15, n = 10 and φ

_{1}= 0.1.

**Figure 6.**Impact of combined convection characteristic (Ri) and volume fraction (φ

_{2}) on $\left({\mathrm{Re}}^{1/2}{C}_{f}\right)$ when φ

_{1}= 0.1, ε = 0.15, n = 10 and α = 0.1.

**Figure 7.**Impact of combined convection characteristic (Ri) and volume fraction (φ

_{2}) on $\left({\mathrm{Re}}^{-1/2}Nu\right)$ for φ

_{1}= 0.1, ε = 0.15, n = 10 and α = 0.3.

**Figure 8.**Impact of small parameter (α) and volume fraction (φ

_{2}) on $\left({\mathrm{Re}}^{-1/2}Nu\right)$ for Ri = 10, ε = 0.15, n = 10 and φ

_{1}= 0.1.

**Figure 9.**Impact of small parameter (α) on $\left({\mathrm{Re}}^{1/2}{C}_{f}\right)$ when Ri = 10, ε = 0.15 and n = 10.

**Figure 10.**Impact of small parameter (α) on $\left({\mathrm{Re}}^{-1/2}Nu\right)$ when Ri = 10, ε = 0.15 and n = 10.

**Figure 11.**Impact of α and n on $\left({\mathrm{Re}}^{1/2}{C}_{f}\right)$ when φ

_{1}= 0.1, φ

_{2}= 0.1, Ri = 10, ε = 0.15.

**Figure 12.**Impact of α and n on $\left({\mathrm{Re}}^{-1/2}Nu\right)$ when φ

_{1}= 0.1, φ

_{2}= 0.1, Ri = 10, ε = 0.15.

**Figure 13.**Impact of shape factor on $\left\{G\left(\overline{x},\eta \right)\right\}$ when φ

_{1}= 0.1, φ

_{2}= 0.1, Ri = 10, n = 10, α = 0.3, $\overline{x}=0.5$, ε = 0.15.

**Figure 14.**Impact of shape factor on $\left({\mathrm{Re}}^{-1/2}Nu\right)$ when φ

_{1}= 0.1, φ

_{2}= 0.1, Ri = 10, n = 10, ε = 0.15.

**Figure 15.**Impact of shape factor on $\left\{F\left(\overline{x},\eta \right)\right\}$ when φ

_{1}= 0.1, φ

_{2}= 0.1, Ri = 10, n = 10, $\overline{x}=0.5$, ε = 0.15.

**Figure 16.**Impact of non-similarity variable $\left(\overline{x}\right)$ and combined convection parameter (Ri) on $\left\{F\left(\overline{x},\eta \right)\right\}$ and $\left\{G\left(\overline{x},\eta \right)\right\}$ when φ

_{1}= 0.1, φ

_{2}= 0.1, α = 0.25, n = 10, ε = 0.1.

**Figure 17.**Angular patterns of the skin friction parameter for Pr = 0.7 compared to outcomes of Rajakumar et al. [32] obtained by choosing all parameter values to zero.

Properties | Nanofluid | Hybrid Nanofluid |
---|---|---|

Dynamic viscosity | ${\mu}_{nf}=\frac{{\mu}_{f}}{{\left(1-{\varphi}_{1}\right)}^{2.5}}$ | ${\mu}_{hnf}=\frac{{\mu}_{f}}{{\left(1-{\varphi}_{1}\right)}^{2.5}{\left(1-{\varphi}_{2}\right)}^{2.5}}$ |

Density | ${\rho}_{nf}=\left(1-{\varphi}_{1}\right){\rho}_{f}+{\varphi}_{1}{\rho}_{{s}_{1}}$ | ${\rho}_{hnf}=\left[{\varphi}_{2}{\rho}_{{s}_{2}}+\left(1-{\varphi}_{2}\right)\left\{\begin{array}{l}{\varphi}_{1}{\rho}_{{s}_{1}}+\\ \left(1-{\varphi}_{1}\right){\rho}_{f}\end{array}\right\}\right]$ |

Thermal conductivity | ${k}_{nf}={k}_{f}\frac{{k}_{{s}_{1}}+2{k}_{f}-2{\varphi}_{1}\left({k}_{f}-{k}_{{s}_{1}}\right)}{{k}_{s1}+2{k}_{f}+{\varphi}_{1}\left({k}_{f}-{k}_{{s}_{1}}\right)}$ | ${k}_{hnf}={k}_{nf}\frac{{k}_{{s}_{2}}+2{k}_{nf}-2{\varphi}_{2}\left({k}_{nf}-{k}_{{s}_{2}}\right)}{{k}_{{s}_{2}}+2{k}_{nf}+{\varphi}_{2}\left({k}_{nf}-{k}_{{s}_{2}}\right)}$ with ${k}_{nf}={k}_{f}\frac{{k}_{{s}_{1}}+2{k}_{f}-2{\varphi}_{1}\left({k}_{f}-{k}_{{s}_{1}}\right)}{{k}_{s1}+2{k}_{f}+{\varphi}_{1}\left({k}_{f}-{k}_{{s}_{1}}\right)}$ |

Thermal expansion coefficient | ${\left(\rho \beta \right)}_{nf}=\left(1-{\varphi}_{1}\right){\left(\rho \beta \right)}_{f}+{\varphi}_{1}{\left(\rho \beta \right)}_{{s}_{1}}$ | ${\left(\rho \beta \right)}_{hnf}=\left[\begin{array}{l}{\varphi}_{2}{\left(\rho \beta \right)}_{{s}_{2}}+\\ \left(1-{\varphi}_{2}\right)\left\{\begin{array}{l}{\varphi}_{1}{\left(\rho \beta \right)}_{{s}_{1}}+\\ \left(1-{\varphi}_{1}\right){\left(\rho \beta \right)}_{f}\end{array}\right\}\end{array}\right]$ |

Heat capacitance | ${\left(\rho {c}_{p}\right)}_{hnf}=\left(1-{\varphi}_{1}\right){\left(\rho {c}_{p}\right)}_{f}+{\varphi}_{1}{\left(\rho {c}_{p}\right)}_{{s}_{1}}$ | ${\left(\rho {c}_{p}\right)}_{hnf}=\left[\begin{array}{l}{\varphi}_{2}{\left(\rho {c}_{p}\right)}_{{s}_{2}}+\\ \left(1-{\varphi}_{2}\right)\left\{\begin{array}{l}{\varphi}_{1}{\left(\rho {c}_{p}\right)}_{{s}_{1}}+\\ \left(1-{\varphi}_{1}\right){\left(\rho {c}_{p}\right)}_{f}\end{array}\right\}\end{array}\right]$ |

Properties | Water | SiO_{2} | MoS_{2} |
---|---|---|---|

c_{p} (J·kg^{−1}·K^{−1}) | 4179 | 730 | 397.746 |

ρ (kg·m^{−3}) | 997.1 | 2650 | 5060 |

k (W·m^{−1}·K^{−1}) | 0.613 | 1.5 | 34.5 |

Β × 10^{−5} (K^{−1}) | 21.0 | 42.7 | 2.8242 |

**Table 3.**Type of nanoadditives with their geometry parameter values [3].

Nanoparticles Type | Geometry Parameter |
---|---|

Spherical | 3.0 |

Bricks | 3.7 |

Cylindrical | 4.9 |

Platelets | 5.7 |

Blade | 8.6 |

**Table 4.**Comparison of hybrid nanofluid (φ

_{1}= 0.1 and φ

_{2}= 0.1) and mono nanofluid (φ

_{1}= 0.1 and φ

_{2}= 0.0) for different magnitudes of combined convection coefficient (Ri) at smooth and rough surfaces, for skin-friction coefficient and energy transport strength when $\overline{x}=0.5$, n = 10, ε = 0.1.

Combined Convection Parameter (Ri) | Mono Nanofluid (φ _{1} = 0.1 and φ_{2} = 0.0) | Hybrid Nanofluid (φ _{1} = 0.1 and φ_{2} = 0.1) | |||
---|---|---|---|---|---|

Re^{1/2}C_{f} | Re^{−1/2}Nu | Re^{1/2}C_{f} | Re^{−1/2}Nu | ||

Ri = −2 | smooth surface (α = 0) | 1.91924 | 1.73737 | 2.76707 | 2.07747 |

Ri = 0 | 2.68659 | 1.85759 | 3.54184 | 2.18376 | |

Ri = 3 | 3.71230 | 1.99284 | 4.60356 | 2.30996 | |

Ri = 6 | 4.64655 | 2.10531 | 5.58725 | 2.41753 | |

Ri = 10 | 5.79680 | 2.22851 | 6.81020 | 2.54115 | |

Ri = −2 | rough surface (α = 0.25) | 115.67549 | 2.11400 | 150.75397 | 2.46100 |

Ri = 0 | 116.33356 | 2.19692 | 151.43079 | 2.53750 | |

Ri = 3 | 117.24895 | 2.30142 | 152.38640 | 2.63768 | |

Ri = 6 | 118.10291 | 2.39003 | 153.28743 | 2.72483 | |

Ri = 10 | 119.17278 | 2.49174 | 154.42511 | 2.82653 |

**Table 5.**Comparison values of energy transport strength (Re

^{−1/2}Nu) with published outcomes for different magnitudes of $\overline{x}$ and Ri for Pr = 0.7.

Nazar et al. [54] | Mohamed et al. [55] | Present Results | |||||||
---|---|---|---|---|---|---|---|---|---|

$\overline{\mathit{x}}$ | Ri = −1 | Ri = 0 | Ri = 1 | Ri = −1 | Ri = 0 | Ri = 1 | Ri = −1 | Ri = 0 | Ri = 1 |

0^{0} | 0.7870 | 0.8162 | 0.8463 | 0.7858 | 0.8150 | 0.8406 | 0.7918 | 0.8180 | 0.8426 |

10^{0} | 0.7818 | 0.8112 | 0.8371 | 0.7809 | 0.8103 | 0.8362 | 0.7778 | 0.8139 | 0.8385 |

20^{0} | 0.7669 | 0.7974 | 0.8239 | 0.7615 | 0.7967 | 0.8232 | 0.7729 | 0.8008 | 0.8254 |

30^{0} | 0.7422 | 0.7746 | 0.8024 | 0.7719 | 0.7741 | 0.8018 | 0.7497 | 0.7792 | 0.8054 |

40^{0} | 0.7076 | 0.7429 | 0.7725 | 0.7074 | 0.7425 | 0.7721 | 0.7134 | 0.7475 | 0.7737 |

50^{0} | 0.6624 | 0.7022 | 0.7345 | 0.6624 | 0.7032 | 0.7354 | 0.6754 | 0.7081 | 0.7377 |

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**MDPI and ACS Style**

Patil, P.M.; Shankar, H.F.; Sheremet, M.A.
Mixed Convection of Silica–Molybdenum Disulphide/Water Hybrid Nanoliquid over a Rough Sphere. *Symmetry* **2021**, *13*, 236.
https://doi.org/10.3390/sym13020236

**AMA Style**

Patil PM, Shankar HF, Sheremet MA.
Mixed Convection of Silica–Molybdenum Disulphide/Water Hybrid Nanoliquid over a Rough Sphere. *Symmetry*. 2021; 13(2):236.
https://doi.org/10.3390/sym13020236

**Chicago/Turabian Style**

Patil, Prabhugouda M., Hadapad F. Shankar, and Mikhail A. Sheremet.
2021. "Mixed Convection of Silica–Molybdenum Disulphide/Water Hybrid Nanoliquid over a Rough Sphere" *Symmetry* 13, no. 2: 236.
https://doi.org/10.3390/sym13020236