#
Numerical Computation of Ag/Al_{2}O_{3} Nanofluid over a Riga Plate with Heat Sink/Source and Non-Fourier Heat Flux Model

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## Abstract

**:**

## 1. Introduction

- Modify the current mathematical model to include nanofluids based on $\mathrm{Ag}/{\mathrm{Al}}_{2}{\mathrm{O}}_{3}$-water, Cattaneo–Christov heat flux, non-linear thermal radiation, and heat source/sink.
- In what ways does it affect Darcy–Forchheimer flow on a Riga plate?
- Exactly how do the Cattaneo–Christov heat flux phenomenon and non-linear thermal radiation influence heat transfer?
- When convective heating conditions are applied, how does the heat transfer gradient respond?

## 2. Mathematical Formulation

## 3. Numerical Solution

## 4. Results and Discussion

## 5. Conclusions

- The nanofluid velocity profile reduces for higher values of porosity, the Forchheimer number, the suction/injection parameter, and the slip parameter.
- The greater the thermal radiation, nanoparticle volume fraction, space and temperature dependent heat source parameter, the greater the nanofluid temperature profile.
- The nanofluid temperature declines for larger values of convection cooling, injection/suction and the thermal relaxation time parameter.
- The skin friction coefficient declines for increasing values of the Forchheimer number and suction/injection parameter, and increases when the modified Hartmann number increases.
- The heat transfer gradient increases with increasing values for the Hartmann number, radiation, suction/injection and the thermal relaxation time parameter, whereas it declines when the space and temperature dependent heat source parameter is increased.
- In future, we will expand this flow model by including hybrid and ternary hybrid nanofluids with different shape factors.

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

$\mathbf{Symbols}$ | $\mathbf{Description}$ |

${a}_{1}$ | $\mathrm{positive}\mathrm{constants}$ |

${A}^{*}$ | $\mathrm{space}\text{-}\mathrm{dependent}\mathrm{heat}\mathrm{source}\mathrm{parameter}$ |

${B}^{*}$ | $\mathrm{temperature}\text{-}\mathrm{dependent}\mathrm{heat}\mathrm{source}\mathrm{parameter}$ |

$Bi\left(=\frac{{h}_{c}}{{k}_{f}}\sqrt{\frac{{v}_{f}}{a}}\right)$ | $\mathrm{Biot}\mathrm{number}$ |

${C}_{p}$ | $\mathrm{specific}\mathrm{heat}\mathrm{capacity}$ |

${c}_{b}$ | $\mathrm{drag}\mathrm{coefficient}$ |

${C}_{f}$ | $\mathrm{skin}\mathrm{friction}\mathrm{coefficient}$ |

f | $\mathrm{subscript}\mathrm{represent}\mathrm{base}\mathrm{fluid}$ |

${f}_{w}\left(=\frac{{V}_{w}}{\sqrt{a{\left(\nu \right)}_{f}}}\right)$ | $\mathrm{suction}/\mathrm{injection}\mathrm{parameter}$ |

$Fr\left(=\frac{{c}_{b}}{\sqrt{{k}_{1}^{*}}}\right)$ | $\mathrm{Forchheimer}\mathrm{number}$ |

${h}_{c}$ | $\mathrm{heat}\mathrm{transfer}\mathrm{coefficient}$ |

$Ha\left(=\frac{\pi {J}_{0}Mx}{8{\left(\rho \right)}_{f}{a}^{2}}\right)$ | $\mathrm{modified}\mathrm{Hartmann}\mathrm{number}$ |

${J}_{0}$ | $\mathrm{current}\mathrm{density}\mathrm{applied}\mathrm{to}\mathrm{the}\mathrm{electrodes}s$ |

${k}_{1}^{*}$ | $\mathrm{permeability}\mathrm{of}\mathrm{porous}\mathrm{medium}$ |

${k}^{*}$ | $\mathrm{Rosseland}\mathrm{absorption}\mathrm{coefficient}$ |

M | $\mathrm{magnetic}\mathrm{field}$ |

$\mathrm{nf}$ | $\mathrm{subscript}\mathrm{represent}\mathrm{nanoliquid}$ |

$Nu$ | $\mathrm{Nusselt}\mathrm{number}$ |

$Pr\left(=\frac{{\alpha}_{f}}{{\nu}_{f}}\right)$ | $\mathrm{Prandtl}\mathrm{number}$ |

$Rd\left(=\frac{4{\sigma}^{*}{T}_{\infty}^{3}}{{k}^{*}{\left(k\right)}_{f}}\right)$ | $\mathrm{radiation}\mathrm{parameter}$ |

$Re\left(=\frac{a{x}^{2}}{{\nu}_{f}}\right)$ | $\mathrm{local}\mathrm{Reynolds}\mathrm{number}$ |

T | $\mathrm{fluid}\mathrm{temperature}$ |

${T}_{f}$ | $\mathrm{temperature}\mathrm{of}\mathrm{the}\mathrm{hot}\mathrm{fluid}$ |

${T}_{\infty}$ | $\mathrm{ambient}\mathrm{temperature}$ |

$u,v$ | $\mathrm{velocity}\mathrm{components}$ |

$x,y$ | $\mathrm{Cartesian}\mathrm{coordinates}$ |

${U}_{w},{V}_{w}$ | $\mathrm{surface}\mathrm{stretching}\mathrm{velocities}$ |

$\mathbf{Greek}\mathbf{Symbols}$ | |

$\rho $ | $\mathrm{density}$ |

$\mu $ | $\mathrm{dynamic}\mathrm{viscosity}$ |

$\varsigma $ | $\mathrm{dimensionless}\mathrm{variable}$ |

$\theta $ | $\mathrm{dimensionless}\mathrm{temperature}$ |

${\beta}_{R}\left(=\frac{\pi}{{a}_{1}}\sqrt{\frac{{\nu}_{f}}{a}}\right)$ | $\mathrm{dimensionless}\mathrm{parameter}$ |

${\theta}_{w}\left(=\frac{{T}_{f}}{{T}_{\infty}}\right)$ | $\mathrm{heating}\mathrm{variable}$ |

$\nu $ | $\mathrm{kinematic}\mathrm{viscosity}$ |

$\lambda \left(=\frac{{\nu}_{f}}{{k}_{1}^{*}a}\right)$ | $\mathrm{local}\mathrm{porosity}\mathrm{parameter}$ |

$\varphi $ | $\mathrm{nanoparticle}\mathrm{volume}\mathrm{fraction}$ |

$\psi $ | $\mathrm{stream}\mathrm{function}$ |

${\sigma}^{*}$ | $\mathrm{Stefen}-\mathrm{Boltzmann}\mathrm{constant}$ |

$\mathsf{\Lambda}$ | $\mathrm{slip}\mathrm{parameter}$ |

$\alpha $ | $\mathrm{thermal}\mathrm{diffusivity}$ |

${\mathsf{\Gamma}}_{1}\left(=\lambda a\right)$ | $\mathrm{thermal}\mathrm{relaxation}\mathrm{time}\mathrm{parameter}$ |

$\mathbf{Abbreviations}$ | |

$\mathrm{LNN}$ | $\mathrm{local}\mathrm{Nusselt}\mathrm{number}$ |

$\mathrm{MHD}$ | $\mathrm{magnetohydrodynamics}$ |

$\mathrm{ODEs}$ | $\mathrm{ordinary}\mathrm{differential}\mathrm{equations}$ |

$\mathrm{PDEs}$ | $\mathrm{partial}\mathrm{differential}\mathrm{equations}$ |

$\mathrm{SFC}$ | $\mathrm{skin}\mathrm{friction}\mathrm{coefficient}$ |

$\mathrm{SS}$ | $\mathrm{stretching}\mathrm{sheet}$ |

$\mathrm{TBL}$ | $\mathrm{thermal}\mathrm{boundary}\mathrm{layer}$ |

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**Figure 2.**The variation of ${f}^{\prime}\left(\varsigma \right)$ in relation to (

**a**) $\lambda $, (

**b**) $Fr$, (

**c**) $Ha$ and (

**d**) $fw$.

**Figure 3.**The variation of ${f}^{\prime}\left(\varsigma \right)$ in relation to (

**a**) $\mathsf{\Lambda}$ and (

**b**) $\varphi $.

**Figure 4.**The variation of $\theta \left(\varsigma \right)$ in relation to (

**a**) ${A}^{*}$, (

**b**) ${B}^{*}$, (

**c**) $Rd$ and (

**d**) $\varphi $.

**Figure 5.**The variation of $\theta \left(\varsigma \right)$ in relation to (

**a**) $fw$, (

**b**) ${\mathsf{\Gamma}}_{1}$, (

**c**) $Bi$ (convective heating) and (

**d**) $Bi$ (convective cooling).

**Figure 6.**SFC variation for diverging values of (

**a**) $Fr$ & $\lambda $, (

**b**) $Fr$ & $fw$, (

**c**) $Fr$ & $\mathsf{\Lambda}$ and (

**d**) $fw$ & $\mathsf{\Lambda}$.

**Figure 7.**SFC variation for diverging values of (

**a**) $fw$, (

**b**) $\lambda $, (

**c**) $\mathsf{\Lambda}$ and (

**d**) $Fr$.

**Figure 8.**LNN variation for diverging values of (

**a**) $Ha$, (

**b**) $Rd$, (

**c**) $fw$ and (

**d**) ${\mathsf{\Gamma}}_{1}$.

**Figure 10.**The increasing/declining percentage of SFC on (

**a**) $\lambda $, (

**b**) $Fr$, (

**c**) $Ha$ and (

**d**) $fw$.

**Figure 12.**The increasing/declining percentage of LNN on (

**a**) $\lambda $, (

**b**) $Fr$, (

**c**) $Ha$ and (

**d**) $Rd$.

**Figure 13.**The increasing/declining percentage of LNN on (

**a**) ${A}^{*}$, (

**b**) ${B}^{*}$, (

**c**) $Bi$ and (

**d**) $\mathsf{\Lambda}$.

**Table 1.**The thermo-physical properties of the nanomaterials and water, see Roja and Gireesha [56].

Physical Properties | Silver (Ag) | Aluminium Oxide (Al${}_{2}$O${}_{3}$) | Water (H${}_{2}$O) |
---|---|---|---|

$\rho /(\mathrm{kg}/{\mathrm{m}}^{-3})$ | $10,500$ | 3970 | $997.1$ |

${C}_{p}/(\mathrm{J}.{\mathrm{kg}}^{-1}.{\mathrm{K}}^{-1})$ | 235 | 765 | 4179 |

$\sigma /{(\mathsf{\Omega}.\mathrm{m})}^{-1}$ | $6.3\times {10}^{7}$ | $3.5\times {10}^{7}$ | $5.5\times {10}^{-6}$ |

$\mathrm{k}/(\mathrm{W}.{\mathrm{m}}^{-1}.{\mathrm{K}}^{-1})$ | 429 | 40 | $0.613$ |

**Table 2.**Physical characteristics, see Sharma [26].

$\mathbf{Properties}$ | $\mathbf{Nanofluid}$ |
---|---|

$\mathrm{Viscosity}\phantom{\rule{3.33333pt}{0ex}}\left(\mu \right)$ | ${A}_{1}=\frac{{\mu}_{f}}{{\mu}_{nf}}={\left(1-\varphi \right)}^{2.5}$ |

$\mathrm{Density}\phantom{\rule{3.33333pt}{0ex}}\left(\rho \right)$ | ${A}_{2}=\frac{{\rho}_{nf}}{{\rho}_{f}}=\left(1-\varphi +\varphi \frac{{\rho}_{s}}{{\rho}_{f}}\right)$ |

$\mathrm{Heat}\mathrm{capacity}\phantom{\rule{3.33333pt}{0ex}}\left(\rho Cp\right)$ | ${A}_{3}=\frac{{\left(\rho Cp\right)}_{nf}}{{\left(\rho Cp\right)}_{f}}=\left(1-\varphi +\varphi \frac{{\left(\rho Cp\right)}_{s}}{{\left(\rho Cp\right)}_{f}}\right)$ |

$\mathrm{Electrical}\mathrm{conductivity}\phantom{\rule{3.33333pt}{0ex}}\left(\sigma \right)$ | ${A}_{4}=\frac{{\sigma}_{nf}}{{\sigma}_{f}}=1+\frac{3\left(\frac{{\sigma}_{s}}{{\sigma}_{f}}-1\right)\varphi}{\left(\frac{{\sigma}_{s}}{{\sigma}_{f}}+2\right)-\left(\frac{{\sigma}_{s}}{{\sigma}_{f}}-1\right)\varphi}$ |

$\mathrm{Thermal}\mathrm{conductivity}\phantom{\rule{3.33333pt}{0ex}}\left(k\right)$ | ${A}_{5}=\frac{{k}_{nf}}{{k}_{f}}=\frac{{k}_{s}+(m-1){k}_{f}-(m-1)\varphi ({k}_{f}-{k}_{s})}{{k}_{s}+(m-1){k}_{f}+\varphi ({k}_{f}-{k}_{s})}$ |

$\mathbf{fw}$ | $\mathbf{Present}\mathbf{Study}$ | Ref. [54] | Ref. [55] |
---|---|---|---|

0 | $1.000001$ | $1.000000$ | $1.0000$ |

$0.5$ | $1.280776$ | $1.280776$ | $1.2808$ |

**Table 4.**SFC & LNN comparison for diverse combo of $\lambda $, $Fr$, $Ha$, $fw$, $\mathsf{\Lambda}$, $\varphi $.

$\mathit{\lambda}$ | $\mathit{Fr}$ | $\mathit{Ha}$ | $\mathit{fw}$ | $\mathbf{\Lambda}$ | $\mathit{\varphi}$ | Ag | |
---|---|---|---|---|---|---|---|

${\mathbf{C}}_{\mathbf{f}}$ | $\mathbf{Nu}$ | ||||||

$0.2$ | $0.4$ | $0.3$ | $0.5$ | 1 | $0.05$ | $-0.540406$ | $0.722099$ |

$0.3$ | $-0.549413$ | $0.721707$ | |||||

$0.4$ | $-0.557932$ | $0.721327$ | |||||

$0.5$ | $-0.566000$ | $0.720961$ | |||||

$0.6$ | $-0.573647$ | $0.720606$ | |||||

$0.2$ | $0.4$ | $0.3$ | $0.5$ | 1 | $0.05$ | $-0.540406$ | $0.722099$ |

$0.8$ | $-0.553538$ | $0.721507$ | |||||

$1.2$ | $-0.565041$ | $0.720977$ | |||||

$1.6$ | $-0.575249$ | $0.720498$ | |||||

2 | $-0.584406$ | $0.720062$ | |||||

$0.2$ | $0.4$ | 0 | $0.5$ | 1 | $0.05$ | $-0.598291$ | $0.719486$ |

$0.1$ | $-0.578264$ | $0.720432$ | |||||

$0.2$ | $-0.559002$ | $0.721299$ | |||||

$0.3$ | $-0.540406$ | $0.722099$ | |||||

$0.4$ | $-0.522401$ | $0.722842$ | |||||

$0.2$ | $0.4$ | $0.3$ | $-0.6$ | 1 | $0.05$ | $-0.409777$ | $0.139884$ |

$-0.2$ | $-0.453539$ | $0.455283$ | |||||

0 | $-0.477364$ | $0.571358$ | |||||

$0.2$ | $-0.502169$ | $0.649742$ | |||||

$0.6$ | $-0.553210$ | $0.738959$ | |||||

$0.2$ | $0.4$ | $0.3$ | $0.5$ | $0.2$ | $0.05$ | $-1.198806$ | $0.730271$ |

$0.4$ | $-0.911637$ | $0.727170$ | |||||

$0.6$ | $-0.739594$ | $0.725004$ | |||||

$0.8$ | $-0.623925$ | $0.723378$ | |||||

1 | $-0.540406$ | $0.722099$ | |||||

$0.2$ | $0.4$ | $0.3$ | $0.5$ | 1 | 0 | $-0.474647$ | $0.772332$ |

$0.05$ | $-0.540406$ | $0.722099$ | |||||

$0.1$ | $-0.590914$ | $0.680030$ | |||||

$0.15$ | $-0.632065$ | $0.644168$ | |||||

$0.2$ | $-0.666931$ | $0.613247$ |

**Table 5.**SFC & LNN comparison for a diverse combination of $\lambda $, $Fr$, $Ha$, $fw$, $\mathsf{\Lambda}$, $\varphi $.

$\mathit{\lambda}$ | $\mathit{Fr}$ | $\mathit{Ha}$ | $\mathit{fw}$ | $\mathbf{\Lambda}$ | $\mathit{\varphi}$ | Al_{2}O_{3} | |
---|---|---|---|---|---|---|---|

${\mathbf{C}}_{\mathbf{f}}$ | $\mathbf{Nu}$ | ||||||

$0.2$ | $0.4$ | $0.3$ | $0.5$ | 1 | $0.05$ | $-0.507314$ | $0.728057$ |

$0.3$ | $-0.518302$ | $0.727571$ | |||||

$0.4$ | $-0.528618$ | $0.727103$ | |||||

$0.5$ | $-0.538316$ | $0.726652$ | |||||

$0.6$ | $-0.547446$ | $0.726219$ | |||||

$0.2$ | $0.4$ | $0.3$ | $0.5$ | 1 | $0.05$ | $-0.507314$ | $0.728057$ |

$0.8$ | $-0.521096$ | $0.727438$ | |||||

$1.2$ | $-0.533185$ | $0.726883$ | |||||

$1.6$ | $-0.543925$ | $0.726380$ | |||||

2 | $-0.553570$ | $0.725921$ | |||||

$0.2$ | $0.4$ | 0 | $0.5$ | 1 | $0.05$ | $-0.569436$ | $0.725170$ |

$0.1$ | $-0.548014$ | $0.726214$ | |||||

$0.2$ | $-0.527336$ | $0.727172$ | |||||

$0.3$ | $-0.507314$ | $0.728057$ | |||||

$0.4$ | $-0.487879$ | $0.728880$ | |||||

$0.2$ | $0.4$ | $0.3$ | $-0.6$ | 1 | $0.05$ | $-0.393978$ | $0.175973$ |

$-0.2$ | $-0.432415$ | $0.479912$ | |||||

0 | $-0.453043$ | $0.587601$ | |||||

$0.2$ | $-0.474401$ | $0.660208$ | |||||

$0.6$ | $-0.518389$ | $0.744083$ | |||||

$0.2$ | $0.4$ | $0.3$ | $0.5$ | $0.2$ | $0.05$ | $-1.074296$ | $0.736057$ |

$0.4$ | $-0.833493$ | $0.733090$ | |||||

$0.6$ | $-0.684442$ | $0.730967$ | |||||

$0.8$ | $-0.582187$ | $0.729347$ | |||||

1 | $-0.507314$ | $0.728057$ | |||||

$0.2$ | $0.4$ | $0.3$ | $0.5$ | 1 | 0 | $-0.474647$ | $0.772332$ |

$0.05$ | $-0.507314$ | $0.728057$ | |||||

$0.1$ | $-0.538018$ | $0.690530$ | |||||

$0.15$ | $-0.567324$ | $0.658335$ | |||||

$0.2$ | $-0.595710$ | $0.630453$ |

**Table 6.**Variations of LNN for a diverse combination of ${A}^{*}$, ${B}^{*}$, ${\mathsf{\Gamma}}_{1}$, $Rd$, $Bi$.

${\mathit{A}}^{*}$ | ${\mathit{B}}^{*}$ | ${\mathbf{\Gamma}}_{1}$ | $\mathit{Rd}$ | $\mathit{Bi}$ | $\mathit{Ag}$ | ${\mathit{Al}}_{2}{\mathit{O}}_{3}$ |
---|---|---|---|---|---|---|

0 | $0.1$ | $0.1$ | $0.6$ | $0.5$ | $0.730125$ | $0.734846$ |

$0.2$ | $0.714032$ | $0.721239$ | ||||

$0.4$ | $0.697778$ | $0.707516$ | ||||

$0.6$ | $0.681362$ | $0.693676$ | ||||

$0.8$ | $0.664781$ | $0.679719$ | ||||

$0.1$ | 0 | $0.1$ | $0.6$ | $0.5$ | $0.723775$ | $0.729211$ |

$0.2$ | $0.720341$ | $0.726864$ | ||||

$0.4$ | $0.716550$ | $0.724352$ | ||||

$0.6$ | $0.712333$ | $0.721652$ | ||||

$0.8$ | $0.707854$ | $0.718741$ | ||||

$0.1$ | $0.1$ | 0 | $0.6$ | $0.5$ | $0.712420$ | $0.718396$ |

$0.1$ | $0.722099$ | $0.728057$ | ||||

$0.2$ | $0.731930$ | $0.737842$ | ||||

$0.3$ | $0.741850$ | $0.747699$ | ||||

$0.4$ | $0.759113$ | $0.757502$ | ||||

$0.1$ | $0.1$ | $0.1$ | 0 | $0.5$ | $0.440869$ | $0.442943$ |

1 | $0.898688$ | $0.907764$ | ||||

2 | $1.305605$ | $1.323422$ | ||||

3 | $1.675997$ | $1.701921$ | ||||

4 | $2.025144$ | $2.057481$ | ||||

$0.1$ | $0.1$ | $0.1$ | $0.6$ | $0.1$ | $0.162427$ | $0.163222$ |

$0.3$ | $0.459119$ | $0.462158$ | ||||

$0.5$ | $0.722099$ | $0.728057$ | ||||

$0.7$ | $0.955640$ | $0.964981$ | ||||

1 | $1.259007$ | $1.273932$ |

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## Share and Cite

**MDPI and ACS Style**

Divya, S.; Eswaramoorthi, S.; Loganathan, K.
Numerical Computation of Ag/Al_{2}O_{3} Nanofluid over a Riga Plate with Heat Sink/Source and Non-Fourier Heat Flux Model. *Math. Comput. Appl.* **2023**, *28*, 20.
https://doi.org/10.3390/mca28010020

**AMA Style**

Divya S, Eswaramoorthi S, Loganathan K.
Numerical Computation of Ag/Al_{2}O_{3} Nanofluid over a Riga Plate with Heat Sink/Source and Non-Fourier Heat Flux Model. *Mathematical and Computational Applications*. 2023; 28(1):20.
https://doi.org/10.3390/mca28010020

**Chicago/Turabian Style**

Divya, S., S. Eswaramoorthi, and Karuppusamy Loganathan.
2023. "Numerical Computation of Ag/Al_{2}O_{3} Nanofluid over a Riga Plate with Heat Sink/Source and Non-Fourier Heat Flux Model" *Mathematical and Computational Applications* 28, no. 1: 20.
https://doi.org/10.3390/mca28010020