# Impact of Buoyancy and Stagnation-Point Flow of Water Conveying Ag-MgO Hybrid Nanoparticles in a Vertical Contracting/Expanding Riga Wedge

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## Abstract

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## 1. Introduction

_{2}O

_{3}particles through a vertical stretched surface with mixed convection and viscous dissipation. The features of heat transport and entropy impact induced by a non-Newtonian nanofluid past an expandable linearly irregular medium are investigated by Jamshed et al. [8]. Jamshed et al. [9] inspected the thermal characteristics of Casson fluid containing nanoparticles through a parabolic solar trough collector. They scrutinized that the thermal efficiency enhances up to 18.5% in the presence of nanofluid. Some recent developments on the performance of nanofluid in the heat transfer enhancement are summarized in refs. [10,11,12].

_{2}O

_{3}/MWCNTs hybrid nanofluid. The different features and techniques containing hybrid nanofluid were examined by Nine et al. [14]. Huang et al. [15] scrutinized the heat transport features and nanofluids’ pressure drop involving Al

_{2}O

_{3}/MWCNTs hybrid nanofluid. The impact of nonlinear radiation on the 3D rotated flow of Cu-Al

_{2}O

_{3}hybrid nanocomposites past a stretched surface with unsteady thermal conductivity was scrutinized by Usman et al. [16]. Waini et al. [17] surveyed the time-dependent flow with characteristics of heat transport over a continuous porous shrinkable/stretchable curved sheet induced by a hybrid nanofluid and multiple solutions were found by using the bvp4c technique.

## 2. Mathematical Background of the Problem

#### 2.1. Physical Boundary Conditions

#### 2.2. Model Relative Expression and Thermo-Physical Data of the Hybrid Nanofluid

**Table 1.**Thermophysical data of tiny nanoparticles and base fluid [44].

Physical Properties | Water | Ag | MgO |
---|---|---|---|

${c}_{p}$ (J/Kg K) | 4179 | 235 | 955 |

$\rho $ (Kg/m^{3}) | 997.1 | 10,500 | 3560 |

${\beta}_{T}\times {10}^{-5}$ (1/K) | 21 | 1.89 | 1.05 |

$k$ (W/mK) | 0.613 | 429 | 45 |

Pr | 6.2 | - | - |

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

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

Thermal expansion | ${\left({\beta}_{T}\rho \right)}_{hnf}=\left[\left(1-{\varphi}_{1}\right){\left({\beta}_{T}\rho \right)}_{f}+{\varphi}_{1}{\left({\beta}_{T}\rho \right)}_{{s}_{1}}\right]\left(1-{\varphi}_{2}\right)+{\varphi}_{2}{\left({\beta}_{T}\rho \right)}_{{s}_{2}}$ |

Thermal conductivity | $\begin{array}{l}{k}_{hnf}=\frac{\left({k}_{{s}_{2}}+2{k}_{nf}\right)-2{\varphi}_{1}\left({k}_{nf}-{k}_{{s}_{2}}\right)}{\left({k}_{{s}_{2}}+2{k}_{nf}\right)+{\varphi}_{2}\left({k}_{nf}-{k}_{{s}_{2}}\right)}\\ \mathrm{with}\text{}{k}_{nf}=\frac{\left(2{k}_{f}+{k}_{{s}_{1}}\right)-2{\varphi}_{1}\left({k}_{f}-{k}_{{s}_{1}}\right)}{\left(2{k}_{f}+{k}_{{s}_{1}}\right)+{\varphi}_{2}\left({k}_{f}-{k}_{{s}_{1}}\right)}\times {k}_{f}\end{array}$ |

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

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

#### 2.3. Similarity Transformations

#### 2.4. Engineering Physical Quantities of Interest

## 3. Methodology of the Considered Numerical Solution

#### 3.1. Single Solution through bvp4c Technique

^{−6}level error tolerance in all cases.

#### 3.2. Confirmation of the Existing Numerical Code

## 4. Results and Discussion

#### 4.1. Effect of the Modified Hartmann Number on Dimensionless Velocity and Temperature Distribution Profiles

_{2}O-based Ag-MgO hybrid nanofluid are shown in Figure 2a,b, respectively. It is clear that when the flow along with the flat plate, wedge flow, and stagnation point flow is considered the velocity is higher and the temperature is lower. A similar tendency of the velocity and temperature is detected with the upsurge of the modified Hartmann number. In this regard, the higher value of ${M}_{HA}$ diminishes the momentum boundary layer and thickens the thermal boundary layer. Physically, the external electric field is increased due to the higher values of M

_{HA}, which creates the wall Lorentz or drag force parallel to the Riga stagnation point, flat plate, and the wedge. As a consequence, the stronger magnetic field causes an improvement in flow dimensionless velocity, and thus the temperature distribution of the fluid decreases.

**Figure 2.**(

**a**) Influence of ${M}_{HA}$ on dimensionless velocity (${f}^{\prime}\left(\eta \right)$) for the three different cases of geometry such as flat plate (${\beta}_{1}=0.0$), wedge (${\beta}_{1}=0.5$) and stagnation point (${\beta}_{1}=1.0$) when ${\lambda}_{T}=-5.5$, ${\lambda}_{MC}=0.5$, ${\alpha}_{HA}=2.0$, $B=0.5$, $\mathrm{Pr}=6.2$, ${\varphi}_{1}=0.025$, $A=2.0$, $E=1/6$, ${f}_{w}=1.0$, and ${\varphi}_{2}=0.025$. (

**b**) Influence of ${M}_{HA}$ on dimensionless temperature distribution ($\theta \left(\eta \right)$) for the three different cases of geometry such as flat plate (${\beta}_{1}=0.0$), wedge (${\beta}_{1}=0.5$) and stagnation point (${\beta}_{1}=1.0$) when ${\lambda}_{T}=-5.5$, ${\lambda}_{MC}=0.5$, ${\alpha}_{HA}=2.0$, $B=0.5$, $\mathrm{Pr}=6.2$, ${\varphi}_{1}=0.025$, $A=2.0$, $E=1/6$, ${f}_{w}=1.0$, and ${\varphi}_{2}=0.025$.

#### 4.2. Effect of the Solid Nanoparticle Volume Fractions on Dimensionless Velocity and Temperature Distribution Profiles

_{2}O-based Ag-MgO hybrid nanofluid due to varying the volume fraction of nanoparticles are presented in Figure 3a,b, respectively. It is found that the velocity decreases and the temperature increase for higher values of ${\varphi}_{1}$ and ${\varphi}_{2}$. Conversely, the dimensionless velocity increases, and profiles of temperature decrease according to the flat plate, wedge flow, and stagnation point flow irrespective of ${\varphi}_{1}$ and ${\varphi}_{2}$. Since the thermophysical features of the hybrid nanofluid are altered with the inclusion of nanoparticles and it is heavier, hence the velocity of the fluid diminishes.

**Figure 3.**(

**a**) Influence of ${\varphi}_{1}$ and ${\varphi}_{2}$ on dimensionless velocity (${f}^{\prime}\left(\eta \right)$) for the three different cases of geometry such as flat plate (${\beta}_{1}=0.0$), wedge (${\beta}_{1}=0.5$) and stagnation point (${\beta}_{1}=1.0$) when ${\lambda}_{T}=-5.5$, ${\lambda}_{MC}=0.5$, ${\alpha}_{HA}=2.0$, $B=0.5$, $\mathrm{Pr}=6.2$, $A=2.0$, $E=1/6$, ${f}_{w}=1.0$, and ${M}_{HA}=0.2$. (

**b**) Influence of ${\varphi}_{1}$ and ${\varphi}_{2}$ on dimensionless temperature distribution ($\theta \left(\eta \right)$) for the three different cases of geometry such as flat plate (${\beta}_{1}=0.0$), wedge (${\beta}_{1}=0.5$) and stagnation point (${\beta}_{1}=1.0$) when ${\lambda}_{T}=-5.5$, ${\lambda}_{MC}=0.5$, ${\alpha}_{HA}=2.0$, $B=0.5$, $\mathrm{Pr}=6.2$, $A=2.0$, $E=1/6$, ${f}_{w}=1.0$, and ${M}_{HA}=0.2$.

#### 4.3. Effect of the Suction Parameter on Dimensionless Velocity and Dimensionless Temperature Distribution Profiles

_{2}O-based Ag-MgO hybrid nanofluid are illustrated, respectively. With the increase of ${f}_{w}$, the profiles of dimensionless velocity for the hybrid nanofluid increase but its temperature decreases. Whatever the choice of the suction parameter is considered the highest velocity is for a stagnation point and the lowest velocity is for a flat plate. The contrary trend is observed in the phenomenon of temperature. As the suction of fluid through the surface accelerates the flow velocity hence the cold fluid particles come close to the surface. For this reason, the aforesaid characteristics are identified.

**Figure 4.**(

**a**) Influence of ${f}_{w}$ on dimensionless velocity (${f}^{\prime}\left(\eta \right)$) for the three different cases of geometry such as flat plate (${\beta}_{1}=0.0$), wedge (${\beta}_{1}=0.5$) and stagnation point (${\beta}_{1}=1.0$) when ${\lambda}_{T}=-5.5$, ${\lambda}_{MC}=0.5$, ${\alpha}_{HA}=2.0$, $B=0.5$, $\mathrm{Pr}=6.2$, ${\varphi}_{1}=0.025$, $A=2.0$, $E=1/6$, ${M}_{HA}=0.2$, and ${\varphi}_{2}=0.025$. (

**b**) Influence of ${f}_{w}$ on dimensionless temperature distribution ($\theta \left(\eta \right)$) for the three different cases of geometry such as flat plate (${\beta}_{1}=0.0$), wedge (${\beta}_{1}=0.5$) and stagnation point (${\beta}_{1}=1.0$) when ${\lambda}_{T}=-5.5$, ${\lambda}_{MC}=0.5$, ${\alpha}_{HA}=2.0$, $B=0.5$, $\mathrm{Pr}=6.2$, ${\varphi}_{1}=0.025$, $A=2.0$, $E=1/6$, ${M}_{HA}=0.2$, and ${\varphi}_{2}=0.025$.

#### 4.4. Influence of the Solid Nanoparticle Volume Fractions on Shear Stress and Heat Transfer

_{2}O-based Ag-MgO hybrid nanofluid against the expanding/contracting parameter are exhibited in Figure 5a,b respectively. It is more clearly observable from these figures that the local skin friction coefficient augments for ${\mathsf{\lambda}}_{T}\le 1$ and shrinkages for ${\mathsf{\lambda}}_{T}\ge 1$. Due to the increase of the values of ${\varphi}_{1}$ and ${\varphi}_{2}$ it also increases in the range ${\mathsf{\lambda}}_{T}\le 1$, but a decreasing behavior is seen in the range ${\mathsf{\lambda}}_{T}\ge 1$. In contrast, the heat transfer gradually increases by diminishing the contracting parameter and increasing the expanding parameter. Moreover, for larger values of ${\varphi}_{1}$ and ${\varphi}_{2}$ the heat transfer increases which eventually uplifts the thermal conductvity.

**Figure 5.**(

**a**) Deviation of local skin friction coefficient for several values of ${\varphi}_{1}$ and ${\varphi}_{2}$ against ${\mathsf{\lambda}}_{T}$ for the three different cases of geometry such as flat plate (${\beta}_{1}=0.0$), wedge (${\beta}_{1}=0.5$) and stagnation point (${\beta}_{1}=1.0$) when ${\lambda}_{MC}=0.5$, ${\alpha}_{HA}=2.0$, $B=0.5$, $\mathrm{Pr}=6.2$, $A=2.0$, $E=1/6$, ${f}_{w}=1.0$, and ${M}_{HA}=0.2$. (

**b**) Deviation of the heat transfer for several values of ${\varphi}_{1}$ and ${\varphi}_{2}$ against ${\mathsf{\lambda}}_{T}$ for the three different cases of geometry such as flat plate (${\beta}_{1}=0.0$), wedge (${\beta}_{1}=0.5$) and stagnation point (${\beta}_{1}=1.0$) when ${\lambda}_{MC}=0.5$, ${\alpha}_{HA}=2.0$, $B=0.5$, $\mathrm{Pr}=6.2$, $A=2.0$, $E=1/6$, ${f}_{w}=1.0$, and ${M}_{HA}=0.2$.

#### 4.5. Effect of the Temperature and Velocity Slip Constraint on Friction Factor and Nusselt Number

**Figure 6.**(

**a**) Deviation of skin friction coefficient for several values of $B$ against ${\mathsf{\lambda}}_{T}$ for the three different cases of geometry such as flat plate (${\beta}_{1}=0.0$), wedge (${\beta}_{1}=0.5$) and stagnation point (${\beta}_{1}=1.0$) when ${\lambda}_{MC}=0.5$, ${\alpha}_{HA}=2.0$, $\mathrm{Pr}=6.2$, ${\varphi}_{1}=0.025$, $A=2.0$, $E=1/6$, ${M}_{HA}=0.2$, ${f}_{w}=1.0$, and ${\varphi}_{2}=0.025$. (

**b**) Deviation of Nusselt number for several values of $B$ against ${\mathsf{\lambda}}_{T}$ for the three different cases of geometry such as flat plate (${\beta}_{1}=0.0$), wedge (${\beta}_{1}=0.5$) and stagnation point (${\beta}_{1}=1.0$) when ${\lambda}_{MC}=0.5$, ${\alpha}_{HA}=2.0$, $\mathrm{Pr}=6.2$, ${\varphi}_{1}=0.025$, $A=2.0$, $E=1/6$, ${M}_{HA}=0.2$, ${f}_{w}=1.0$, and ${\varphi}_{2}=0.025$.

**Figure 7.**(

**a**) Deviation of skin friction coefficient for several values of $A$ against ${\mathsf{\lambda}}_{T}$ for the three different cases of geometry such as flat plate (${\beta}_{1}=0.0$), wedge (${\beta}_{1}=0.5$) and stagnation point (${\beta}_{1}=1.0$) when ${\lambda}_{MC}=0.5$, ${\alpha}_{HA}=2.0$, $\mathrm{Pr}=6.2$, ${\varphi}_{1}=0.025$, $B=0.5$, $E=1/6$, ${M}_{HA}=0.2$, ${f}_{w}=1.0$, and ${\varphi}_{2}=0.025$. (

**b**) Deviation of heat transfer for several values of $A$ against ${\mathsf{\lambda}}_{T}$ for the three different cases of geometry such as flat plate (${\beta}_{1}=0.0$), wedge (${\beta}_{1}=0.5$) and stagnation point (${\beta}_{1}=1.0$) when ${\lambda}_{MC}=0.5$, ${\alpha}_{HA}=2.0$, $\mathrm{Pr}=6.2$, ${\varphi}_{1}=0.025$, $B=0.5$, $E=1/6$, ${M}_{HA}=0.2$, ${f}_{w}=1.0$, and ${\varphi}_{2}=0.025$.

#### 4.6. The Computational Values of the Shear Stress and Nusselt Number for the Flow Geometries of the Wedge, Flat Plate, and Stagnation Point Due to the Effect of Various Varying Parameters

#### 4.7. Grid Sensitivity Analysis

## 5. Conclusions

_{2}O-based Ag-MgO hybrid nanofluid on the combined effect of free and forced convection near a stagnation point in a contracting/expanding Riga wedge along with the significant impacts of slip. The key findings of the present flow symmetry investigation are listed as follows.

- The dimensionless velocity profile and momentum boundary layer width is enhancing due to greater values of ${M}_{HA}$ and ${f}_{w}$ for the dynamics of flow cases such as stagnation point, flat plate, and wedge, while the temperature shows the opposite behavior.
- It is established that the dimensionless velocity profiles declines and the temperature uplifts forthe higher volume fraction of nanoparticles.
- The stagnation point case has a larger velocity profile as compared to the flat plate and wedge cases, while the temperature profile is larger for the case of the flat plate than for the stagnation point and wedge.
- The outcomes indicate that the local SFC upsurges for the range of ${\mathsf{\lambda}}_{T}\le 1$ and decreases for the range of ${\mathsf{\lambda}}_{T}\ge 1$ owing to the higher nanoparticles volume fraction and the temperature slip parameter. Instead, the heat transfer rate is increasing for the larger contracting parameter and decreasesfor the expanding parameter owed to the advanced values of the hybrid nanoparticles while the change tendency is detected for the temperature slip parameter.
- With the higher values of the velocity slip parameter, a decrease in the skin friction coefficient is seen in the range ${\mathsf{\lambda}}_{T}\le 1$ and it is increased for ${\mathsf{\lambda}}_{T}\ge 1$ while the heat transfer rate is increased.
- The skin friction coefficient is increased with the percentage of 0.540%, 0.766%, 2.812%, 0.063%, 0.023%, and 0.030% for the values of the mixed convection parameter, and the values of the dimensionless exponent parameter, respectively. In addition, the shear stress is reduced due to the higher impact of suction and velocity slip parameter with the percentage of 0.738%, 0.463%, and 7.3835 and 37.531%, 37.337%, and 33.247%, respectively.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

$\left(u,v\right)$ | Velocity componentsalong the $x$- and $y$-axes directions (m/s) |

$\left(x,y\right)$ | Cartesian coordiante system (m) |

${d}_{1}$ | Width of magnet and electrodes |

${U}_{\infty}(x)$ | Variable fress stream velocity (m/s) |

${M}_{1}$ | Magnetization of the permanent magnets |

${J}_{1}$ | Current density in electrodes |

$g$ | Acceleration due to gravity (m/s^{2}) |

$T$ | Temperature (K) |

${T}_{\infty}$ | Fress stream constant temperature (K) |

$\mathsf{\Omega}$ | Entire angle of the wedge surface of the Riga plate |

${c}_{p}$ | Specific heat at constant pressure (J/KgK) |

${V}_{w}\left(x\right)$ | Mass flux velocity (m/s) |

${U}_{w}(x)$ | Uniform variable velocity (m/s) |

$b$ | Expanding/Contracting parameter |

$m$ | Hartree pressure gradient |

$A,B$ | Arbitrary positive slips parameter |

$\left(c,{T}_{0}\right)$ | Positive arbitrary constants |

$\mathrm{Pr}$ | Prandtl number |

$f$ | Dimensionless velocity |

${M}_{HA}$ | Modified Hartmann number |

${M}_{1}^{*}$ | Characteristic magnitization of the permanent magnet |

${d}_{1}^{*}$ | Characteristic width of magnet and electrodes |

${f}_{w}$ | Mass suction parameter |

${C}_{f}$ | Coefficient of skin friction |

$k$ | Thermal conductivity |

$N{u}_{x}$ | Local Nusselt number |

${\mathrm{Re}}_{x}$ | Local Reynolds number |

${q}_{w}$ | Wall heat flux |

Greek symbols | |

${\alpha}^{*}$ | Thermal diffusivity |

${\alpha}_{HA}$ | Exponent parameter |

${\lambda}_{MC}$ | Mixed convection parameter |

${\beta}_{1}$ | Hartree pressure gradient parameter |

${\lambda}_{0}$ | Arbitrary constant parameter |

${\gamma}_{1}\left(x\right)$ | Variable slip velocity parameter |

${\gamma}_{2}\left(x\right)$ | Variable temperature slip parameter |

${\beta}_{T}$ | Thermal expansion coefficient |

$\rho $ | Density |

$\mu $ | Absolute viscosity |

${\nu}_{f}$ | Kinematic viscosity |

$\eta $ | similarity variable |

$\psi $ | Stream function |

${\tau}_{w}$ | Wall shear stress |

${\beta}_{T}^{*}$ | Constant thermal expansion coefficient |

${\lambda}_{T}$ | Expanding/Contracting parameter |

$\theta $ | Dimensionless temperature |

Acronyms | |

PDEs | Partial differential equations |

BL | Boundary layer |

HPG | Hartree pressure gradient |

3D | Three-dimensional |

MHD | Magnetohydrodynamics |

ODEs | Ordinary differential equations |

EMHD | Electro magnetohydrodynamics |

bvp4c | Boundary value problem of fourth-order |

Subscripts | |

$w$ | Condition at surface |

$nf$ | Nanofluid |

$hnf$ | Hybrid nanofluid |

$f$ | Base fluid |

$\infty $ | Free-stream condition |

Superscripts | |

$\prime $ | Differentiation with respect to $\eta $. |

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**Figure 1.**(

**a**) Physical configuration of the flow Riga surface. (

**b**) Flow geometry and Cartesian coordinate system.

**Table 3.**Comparison of the heat transfer numerical outcome values for the limiting case of $T\left(x,y=0\right)={T}_{w}$ and ${\lambda}_{T}=1$ in the absence of ${\varphi}_{1}={\varphi}_{2}={f}_{w}=A=B={M}_{HA}=0$. (Error % is calculated with reference to [43]).

$\mathbf{Pr}$ | Stagnation Point | Wedge | ||||||
---|---|---|---|---|---|---|---|---|

Ref. [43] | Ref. [44] | Current Solution | Error % | Ref. [43] | Ref. [44] | Current Solution | Error % | |

0.01 | 0.76098 | 0.76098 | 0.76099567 | 0.002 | 0.61437 | 0.61440 | 0.61439867 | 0.004 |

0.1 | 0.70524 | 0.70524 | 0.70523953 | 0.000 | 0.55922 | 0.55926 | 0.55925644 | 0.006 |

1.0 | 0.64032 | 0.64032 | 0.64031945 | 0.000 | 0.49396 | 0.49401 | 0.49421435 | 0.051 |

10 | 0.63136 | 0.63192 | 0.63191758 | 0.088 | 0.47703 | 0.47824 | 0.47819435 | 0.243 |

**Table 4.**Numerical values of the Shear Stress for the three different geometries varying the mixed convection parameter, velocity slip parameter, mass suction parameter, and dimensionless exponent parameter while the rest are fixed.

${\mathit{\lambda}}_{\mathit{M}\mathit{C}}$ | ${\mathit{f}}_{\mathit{w}}$ | $\mathit{A}$ | ${\mathit{\alpha}}_{\mathit{H}\mathit{A}}$ | $\left(1/2\right){\mathbf{Re}}_{\mathit{x}}^{1/2}{\mathit{\delta}}^{-1}{\mathit{C}}_{\mathit{f}}$ | ||
---|---|---|---|---|---|---|

${\mathit{\beta}}_{1}=0.0$ | ${\mathit{\beta}}_{1}=0.5$ | ${\mathit{\beta}}_{1}=1.0$ | ||||

0.5 | 1.0 | 2.0 | 1.0 | 1.2585 | 1.2525 | 1.3191 |

0.7 | - | - | - | 1.2653 | 1.2621 | 1.3562 |

1.0 | - | - | - | 1.2751 | 1.2758 | 1.3935 |

0.5 | 1.0 | 2.0 | 1.0 | 1.2585 | 1.2525 | 1.3191 |

- | 2.0 | - | 1.2592 | 1.2467 | 1.5943 | |

- | 3.0 | - | - | 1.2428 | 1.2462 | 1.7214 |

0.5 | 1.0 | 1.0 | 1.0 | 2.0146 | 1.9988 | 1.9761 |

- | - | 2.0 | - | 1.2585 | 1.2525 | 1.3191 |

- | - | 3.0 | - | 0.9277 | 0.9405 | 1.2118 |

0.5 | 1.0 | 2.0 | 1.0 | 1.2585 | 1.2525 | 1.3191 |

- | - | - | 1.5 | 1.2593 | 1.2528 | 1.3047 |

- | - | - | 2.0 | 1.2595 | 1.2527 | 1.2970 |

**Table 5.**Numerical values of the heat transfer for the three different geometries varying the mixed convection parameter, temperature slip parameter, mass suction parameter, and dimensionless exponent parameter while the rest are fixed.

${\mathit{\lambda}}_{\mathit{M}\mathit{C}}$ | ${\mathit{f}}_{\mathit{w}}$ | $\mathit{B}$ | ${\mathit{\alpha}}_{\mathit{H}\mathit{A}}$ | ${\mathbf{Re}}_{\mathit{x}}^{-\frac{1}{2}}{\mathit{\delta}}^{-1}\mathit{N}{\mathit{u}}_{\mathit{x}}$ | ||
---|---|---|---|---|---|---|

${\mathit{\beta}}_{1}=0.0$ | ${\mathit{\beta}}_{1}=0.5$ | ${\mathit{\beta}}_{1}=1.0$ | ||||

0.5 | 1.0 | 0.5 | 1.0 | −13.2431 | −9.2025 | −1.2299 |

0.7 | - | - | - | −9.3317 | −6.3996 | −0.5940 |

1.0 | - | - | - | −6.3905 | −4.2928 | −0.2116 |

0.5 | 1.0 | 0.5 | 1.0 | −13.2431 | −9.2025 | −1.2299 |

- | 2.0 | - | - | −12.0503 | −5.8362 | 0.9478 |

- | 3.0 | - | - | −9.9226 | −0.7498 | 1.2777 |

0.5 | 1.0 | 0.1 | 1.0 | −14.3720 | −11.2200 | −4.9021 |

- | - | 0.3 | - | −14.1125 | −10.2073 | −1.6997 |

- | - | 0.5 | - | −13.2431 | −9.2025 | −1.2299 |

0.5 | 1.0 | 0.5 | 1.0 | −13.2431 | −9.2025 | −1.2299 |

- | - | - | 1.5 | −13.5935 | −9.5456 | −1.6407 |

- | - | - | 2.0 | −13.8065 | −9.7651 | −1.9222 |

**Table 6.**Grid sensitivity analysis for the case of flat plate (${\beta}_{1}=0.0$) when ${\lambda}_{T}=-3.5$, ${\lambda}_{MC}=0.5$, ${\alpha}_{HA}=2.0$, ${M}_{HA}=0.2$, $B=0.5$, $\mathrm{Pr}=6.2$, ${\varphi}_{1}=0.025$, $A=2.0$, $E=1/6$, ${f}_{w}=1.0$, and ${\varphi}_{2}=0.025$.

Function | $\mathit{h}$ | $\mathit{\eta}$ | |||||
---|---|---|---|---|---|---|---|

0.5 | 2.0 | 4.0 | 6.0 | 8.0 | 10.0 | ||

${f}^{\prime}\left(\eta \right)$ | 50.0 | −1.4733 | 0.2943 | 1.2159 | 1.6600 | 1.8729 | 1.9310 |

75.0 | −1.4923 | 0.2982 | 1.2124 | 1.6595 | 1.8728 | 1.9310 | |

100.0 | −1.4920 | 0.2918 | 1.2123 | 1.6593 | 1.8728 | 1.9310 | |

$\theta \left(\eta \right)$ | 50.0 | 10.7621 | 7.0693 | 0.8839 | 0.0192 | 0.0000 | 0.0000 |

75.0 | 10.7265 | 7.0450 | 0.8993 | 0.0194 | 0.0000 | 0.0000 | |

100.0 | 10.7270 | 7.0854 | 0.8998 | 0.0194 | 0.0000 | 0.0000 |

**Table 7.**Grid sensitivity analysis for the case of wedge (${\beta}_{1}=0.5$) when ${\lambda}_{T}=-3.5$, ${\lambda}_{MC}=0.5$, ${\alpha}_{HA}=2.0$, ${M}_{HA}=0.2$, $B=0.5$, $\mathrm{Pr}=6.2$, ${\varphi}_{1}=0.025$, $A=2.0$, $E=1/6$, ${f}_{w}=1.0$, and ${\varphi}_{2}=0.025$.

Function | $\mathit{h}$ | $\mathit{\eta}$ | |||||
---|---|---|---|---|---|---|---|

0.5 | 2.0 | 4.0 | 6.0 | 8.0 | 10.0 | ||

${f}^{\prime}\left(\eta \right)$ | 50.0 | −1.3034 | 0.5428 | 1.6202 | 1.8929 | 1.9287 | 1.9310 |

75.0 | −1.4938 | 0.5613 | 1.6189 | 1.8921 | 1.9287 | 1.9310 | |

100.0 | −1.4506 | 0.5551 | 1.6183 | 1.8916 | 1.9287 | 1.9310 | |

$\theta \left(\eta \right)$ | 50.0 | 6.4723 | 3.5353 | 0.1865 | 0.0003 | 0.0000 | 0.0000 |

75.0 | 6.3718 | 3.4750 | 0.1884 | 0.0003 | 0.0000 | 0.0000 | |

100.0 | 6.4017 | 3.4954 | 0.1893 | 0.0003 | 0.0000 | 0.0000 |

**Table 8.**Grid sensitivity analysis for the case of stagnation point (${\beta}_{1}=1.0$) when ${\lambda}_{T}=-3.5$, ${\lambda}_{MC}=0.5$, ${\alpha}_{HA}=2.0$, ${M}_{HA}=0.2$, $B=0.5$, $\mathrm{Pr}=6.2$, ${\varphi}_{1}=0.025$, $A=2.0$, $E=1/6$, ${f}_{w}=1.0$, and ${\varphi}_{2}=0.025$.

Function | $\mathit{h}$ | $\mathit{\eta}$ | |||||
---|---|---|---|---|---|---|---|

0.5 | 2.0 | 4.0 | 6.0 | 8.0 | 10.0 | ||

${f}^{\prime}\left(\eta \right)$ | 50.0 | −1.2187 | 0.8625 | 1.8086 | 1.9250 | 1.9309 | 1.9310 |

75.0 | −1.2258 | 0.8811 | 1.8024 | 1.9250 | 1.9309 | 1.9310 | |

100.0 | −1.2449 | 0.8748 | 1.8020 | 1.9249 | 1.9309 | 1.9310 | |

$\theta \left(\eta \right)$ | 50.0 | 2.1522 | 0.7952 | 0.0091 | 0.0000 | 0.0000 | 0.0000 |

75.0 | 2.1530 | 0.7758 | 0.0103 | 0.0000 | 0.0000 | 0.0000 | |

100.0 | 2.1594 | 0.7823 | 0.0104 | 0.0000 | 0.0000 | 0.0000 |

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**MDPI and ACS Style**

Khan, U.; Zaib, A.; Ishak, A.; Waini, I.; Madhukesh, J.K.; Raizah, Z.; Galal, A.M.
Impact of Buoyancy and Stagnation-Point Flow of Water Conveying Ag-MgO Hybrid Nanoparticles in a Vertical Contracting/Expanding Riga Wedge. *Symmetry* **2022**, *14*, 1312.
https://doi.org/10.3390/sym14071312

**AMA Style**

Khan U, Zaib A, Ishak A, Waini I, Madhukesh JK, Raizah Z, Galal AM.
Impact of Buoyancy and Stagnation-Point Flow of Water Conveying Ag-MgO Hybrid Nanoparticles in a Vertical Contracting/Expanding Riga Wedge. *Symmetry*. 2022; 14(7):1312.
https://doi.org/10.3390/sym14071312

**Chicago/Turabian Style**

Khan, Umair, Aurang Zaib, Anuar Ishak, Iskandar Waini, Javali K. Madhukesh, Zehba Raizah, and Ahmed M. Galal.
2022. "Impact of Buoyancy and Stagnation-Point Flow of Water Conveying Ag-MgO Hybrid Nanoparticles in a Vertical Contracting/Expanding Riga Wedge" *Symmetry* 14, no. 7: 1312.
https://doi.org/10.3390/sym14071312