# A Model Development for Thermal and Solutal Transport Analysis of Non-Newtonian Nanofluid Flow over a Riga Surface Driven by a Waste Discharge Concentration

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## Abstract

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## 1. Introduction

- How does the modified Hartmann number impact the velocity profile in the presence of second-grade fluid and Walter’s liquid B fluid?
- What are the behavioral changes observed in the concentration profile when external pollutant source variation parameter are varied?
- How will the local pollutant external source parameter and solid volume fractions influence the mass transfer rate?

## 2. Mathematical Formulation

## 3. Numerical Scheme

^{−6}set for tolerance of error. Further, our present numerical scheme is validated by the works of [40,41,42] for some limiting parameters and obtained good agreement (see Table 4).

## 4. Results and Discussion

## 5. Final Remarks

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

Symbols | |

$x\mathrm{and}y$ | Directions |

$\overline{u}\mathrm{and}\overline{v}$ | Velocity components |

${M}_{0}$ | Magnetization of permanent magnets |

${T}^{*}$ | Temperature |

${j}_{0}$ | Applied current density in electrodes |

$C$ | Concentration |

${D}_{f}$ | Diffusivity |

${C}_{\infty}$ | Ambient concentration |

${C}_{w}$ | Surface concentration |

${U}_{w}$ | Uniform velocity |

${K}^{*}$ | Permeability of the porous medium |

${K}_{1}^{*}$ | Viscoelastic constraint |

${Q}_{1}^{*}$ | Modified Hartmann number |

$R{d}_{1}^{*}$ | Radiation constant |

$S{c}^{*}$ | Schmidt number |

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

$Cf$ | Skin friction |

$Nu$ | Nusselt number |

$Sh$ | Sherwood number |

${F}_{m}^{*}$ | Electromagnetic force |

${T}_{\infty}$ | Ambient temperature |

${C}_{p}$ | Heat capacitance |

$k$ | Thermal conductivity |

${q}_{r}$ | Radiation heat flux |

${Q}^{*}$ | Pollutant external source variation parameter |

$c$ | Width of the electrodes |

${b}_{3}^{*}$ | Pollutant external source variation parameter |

${k}^{*}$ | Absorption coefficient |

Greek Letters | |

${\alpha}_{1}^{*}$ | Material constant |

${\sigma}^{*}$ | Stefan–Boltzmann coefficient |

${\beta}_{1}^{*}$ | Parameter related to width and magnitude of electrode |

${\lambda}_{1}^{*}$ | Porous constant |

${\delta}_{1}^{*}$ | Parameter related to local pollutant external source |

${\gamma}_{1}^{*}$ | Parameter related to external pollutant source variation |

${\varphi}^{*}$ | Solid volume fraction |

$\mu $ | Dynamic viscosity |

$\upsilon $ | Kinematic viscosity |

$\rho $ | Density |

$\alpha $ | Thermal diffusivity |

Subscripts | |

$f$ | Fluid |

$nf$ | Nanofluid |

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**Figure 11.**Streamline pattern for second-grade fluid and Walter’s liquid B fluid in the presence and absence of ${\lambda}_{1}^{*}$.

Sl. No | Parameter Definition | Parameter Name |
---|---|---|

01 | ${K}_{1}^{*}=\frac{{\alpha}_{1}^{*}a}{{\rho}_{f}{\nu}_{f}}$ | Viscoelastic constraint ${K}_{1}^{*}>0$ second-grade fluid ${K}_{1}^{*}<0$ Walter’s liquid B fluid |

02 | ${Q}_{1}^{*}=\frac{\pi {j}_{0}{M}_{0}}{8{\rho}_{f}a{U}_{w}}$ | Modified Hartmann number |

03 | ${\beta}_{1}^{*}=\sqrt{\frac{{\pi}^{2}{\nu}_{f}}{{c}^{2}a}}$ | Parameter related to width and magnitude of electrodes |

04 | ${\lambda}_{1}^{*}=\frac{{\nu}_{f}}{{K}^{*}a}$ | Porous constraint |

05 | $R{d}_{1}^{*}=\frac{4{\sigma}^{*}{T}_{\infty}^{3}}{{k}^{*}{k}_{f}}$ | Radiation constraint |

06 | $\mathrm{Pr}=\frac{{\nu}_{f}}{{\alpha}_{f}}$ | Prandtl number |

07 | ${\delta}_{1}^{*}=\frac{Q}{a({C}_{w}-{C}_{\infty})}$ | Parameter related to local pollutant external source |

08 | ${\gamma}_{1}^{*}={b}_{3}({C}_{w}-{C}_{\infty})$ | Parameter related to external pollutant source variation |

09 | $S{c}_{1}^{*}=\left(\frac{{\upsilon}_{f}}{{D}_{f}}\right)$ | Schmidt number |

10 | $\mathrm{Re}=\frac{{x}^{2}a}{{\nu}_{f}}$ | Local Reynolds number |

11 | ${A}_{1}^{*}={\left(1-{\varphi}^{*}\right)}^{2.5}$ | |

12 | ${A}_{2}^{*}=\left(1-{\varphi}^{*}+{\varphi}^{*}\frac{{\rho}_{s}}{{\rho}_{f}}\right)$ | |

13 | ${A}_{3}^{*}=\left(1-{\varphi}^{*}+{\varphi}^{*}\frac{{\left(\rho {C}_{p}\right)}_{s}}{{\left(\rho {C}_{p}\right)}_{f}}\right)$ |

Thermophysical Characteristics | Name |
---|---|

${\left(\rho {C}_{p}\right)}_{nf}=\left(\left(1-{\varphi}^{*}\right)+{\varphi}^{*}\frac{{\left(\rho {C}_{p}\right)}_{s}}{{\left(\rho {C}_{p}\right)}_{f}}\right){\left(\rho {C}_{p}\right)}_{f}$ | Specific heat capacity |

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

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

${\rho}_{nf}=\left(1-{\varphi}^{*}+{\varphi}^{*}\frac{{\rho}_{s}}{{\rho}_{f}}\right){\rho}_{f}$ | Density |

Properties | $\mathbf{Pr}$ | ${\mathit{C}}_{\mathit{p}}\mathbf{\left(}\mathbf{J}\mathbf{k}{\mathbf{g}}^{\mathbf{-}\mathbf{1}}{\mathbf{K}}^{\mathbf{-}\mathbf{1}}\mathbf{\right)}$ | $\mathit{k}\mathbf{\left(}\mathbf{k}\mathbf{g}\mathbf{m}{\mathbf{s}}^{\mathbf{-}\mathbf{3}}{\mathbf{K}}^{\mathbf{-}\mathbf{1}}\mathbf{\right)}$ | $\mathit{\rho}\mathbf{\left(}\mathbf{k}\mathbf{g}{\mathbf{m}}^{\mathbf{-}\mathbf{3}}\mathbf{\right)}$ |
---|---|---|---|---|

$SA\left({\mathrm{C}}_{6}{\mathrm{H}}_{9}\mathrm{N}\mathrm{a}{\mathrm{O}}_{7}\right)$ | $6.45$ | $4175$ | $0.6376$ | $989$ |

$\mathrm{T}\mathrm{i}{\mathrm{O}}_{2}$ | - | $686.2$ | $8.9528$ | $4250$ |

**Table 4.**Assessment of numerical values $-{\theta}^{\prime}{\left(\eta \right)}_{\eta =0}$ for variation in $\mathrm{Pr}$ with the absence of ${\lambda}_{1}^{*}$, ${K}_{1}^{*}$, $R{d}_{1}^{*}$, ${\varphi}^{*}$, and ${Q}_{1}^{*}$.

Pr | Ishak et al. [42] | Abolbashari et al. [43] | Das et al. [44] | Present Numerical Outcome |
---|---|---|---|---|

0.72 | 0.8086 | 0.80863135 | 0.80876122 | 0.80876153 |

1.0 | 1.0000 | 1.00000000 | 1.00000000 | 1.00000000 |

3.0 | 1.9237 | 1.92368259 | 1.92357431 | 1.92357446 |

7.0 | 3.0723 | 3.07225021 | 3.07314679 | 3.07314636 |

10 | 3.7207 | 3.72067390 | 3.72067390 | 3.72067335 |

**Table 5.**Comparison table of rate of mass transfer percentage for second-grade fluid and Walter’s liquid B fluid in the presence of ${\delta}_{1}^{*}$ and ${\gamma}_{1}^{*}$.

Parameter | Values | Second-Grade Fluid | Walter’s Liquid B Fluid |
---|---|---|---|

$\left|\frac{\mathit{S}{\mathit{h}}_{{\mathit{\varphi}}^{*}=0.01}-\mathit{S}{\mathit{h}}_{{\mathit{\varphi}}^{*}=0}}{\mathit{S}{\mathit{h}}_{{\mathit{\varphi}}^{*}=0}}\right|\times 100$ | $\left|\frac{\mathit{S}{\mathit{h}}_{{\mathit{\varphi}}^{*}=0.01}-\mathit{S}{\mathit{h}}_{{\mathit{\varphi}}^{*}=0}}{\mathit{S}{\mathit{h}}_{{\mathit{\varphi}}^{*}=0}}\right|\times 100$ | ||

${\delta}_{1}^{*}$ | 0.01 | 0.140264% | 0.560122% |

0.02 | 0.140338% | 0.570178% | |

0.03 | 0.161919% | 0.601998% | |

${\gamma}_{1}^{*}$ | 0.1 | 0.140853% | 0.650966% |

0.2 | 0.122659% | 0.631004% | |

0.3 | 0.123269% | 0.648976% |

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## Share and Cite

**MDPI and ACS Style**

Madhukesh, J.K.; Kalleshachar, V.; Kumar, C.; Khan, U.; Nagaraja, K.V.; Sarris, I.E.; Sherif, E.-S.M.; Hassan, A.M.; Chohan, J.S.
A Model Development for Thermal and Solutal Transport Analysis of Non-Newtonian Nanofluid Flow over a Riga Surface Driven by a Waste Discharge Concentration. *Water* **2023**, *15*, 2879.
https://doi.org/10.3390/w15162879

**AMA Style**

Madhukesh JK, Kalleshachar V, Kumar C, Khan U, Nagaraja KV, Sarris IE, Sherif E-SM, Hassan AM, Chohan JS.
A Model Development for Thermal and Solutal Transport Analysis of Non-Newtonian Nanofluid Flow over a Riga Surface Driven by a Waste Discharge Concentration. *Water*. 2023; 15(16):2879.
https://doi.org/10.3390/w15162879

**Chicago/Turabian Style**

Madhukesh, Javali Kotresh, Vinutha Kalleshachar, Chandan Kumar, Umair Khan, Kallur Venkat Nagaraja, Ioannis E. Sarris, El-Sayed M. Sherif, Ahmed M. Hassan, and Jasgurpreet Singh Chohan.
2023. "A Model Development for Thermal and Solutal Transport Analysis of Non-Newtonian Nanofluid Flow over a Riga Surface Driven by a Waste Discharge Concentration" *Water* 15, no. 16: 2879.
https://doi.org/10.3390/w15162879