# Statistical Modelling for the Darcy–Forchheimer Flow of Casson Cobalt Ferrite-Water/Ethylene Glycol Nanofluid under Nonlinear Radiation

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

_{2}O

_{3}in ethylene glycol, Al

_{2}O

_{3}in water, CuO in ethylene glycol, and CuO in water) and found that the ethylene glycol–CuO nanoliquid increased thermal conductivity by more than 20%. However, a 40% increase in thermal conductivity in ethylene glycol–Cu nanoliquid documented by Eastman et al. [6] resulted in an improvement. Xie et al. [7] explored the role of base liquids on the thermal conductivity of nanofluids using a variety of base fluids. With the increased thermal conductivity of the base fluid, the thermal conductivity ratio decreased. As a result, nanofluids could be an intriguing choice for enhanced heat-transport applications in the future, specifically those in the micro-scale. Furthermore, Dogonchi et al. [8] studied the significance of natural convective magnetic nanoliquids in an enclosure with a porous medium by taking Brownian motion phenomena. Shafiq et al. [9] analyzed the single as well as multiple wall carbon nanotubes on magnetohydrodynamic stagnant-point nanoliquid flow towards variable thick plates towards concave and convex phenomena. Hayat et al. [10] examined the Stratification impact on MHD Tangent hyperbolic nanofluid flow, which is induced by inclined surfaces. Shafiq et al. [11] investigated the Casson magnetohydrodynamic axisymmetric Marangoni forced-convection nanofluid flow towards a flat surface. Rehman et al. [12] analyzed the heat transport with nano-sized materials suspended in a magnetized rotatory flow field. Marangoni convective boundary-layer carbon-nanotube flow over a Riga surface was studied by Shafiq et al. [13]. Shafiq et al. [14] developed an artificial neural network (ANN) model to predict the flow of a single-walled carbon nanotube liquid over three various non-linear thin isothermal needles of cone, paraboloid, and cylinder shapes under convective conditions. The temperature profile increased according to the Biot number. Moreover, Shafiq et al. [15] studied hydromagnetic unsteady Williamson nanoliquid flows towards a radiative plate via numerical as well as artificial neural network modeling. They found good agreements according to the numerical results with ANN results. Noteworthy developments concerning nanoliquids can be found in refs. [16,17,18,19,20].

## 2. Mathematical Modelling

## 3. Methodology

## 4. Model Selection and Discussion

_{2}O-CoFe

_{2}O

_{4}), and ethylene glycol-cobalt ferrite (C

_{2}H

_{6}O

_{2}-CoFe

_{2}O

_{4}). Moreover, we generated another three datasets for LNN when ${R}_{i}$ = 0.1, 0.3, 0.6 and ${N}_{i}$ = 0.1, 0.2, 0.3 under water-cobalt ferrite (H

_{2}O-CoFe

_{2}O

_{4}), and ethylene glycol-cobalt ferrite (C

_{2}H

_{6}O

_{2}-CoFe

_{2}O

_{4}). For each data set, we estimate the model’s parameters by using maximum likelihood estimation from the above fitted models and listed them in Table 3, Table 4, Table 5 and Table 6. The following excellent statistical benchmarks have been used to compare these models: Anderson–Darling (A *), Cramer–von Mises (W *), and Kolmogorov–Smirnov (K-S) with p-values. The best model for the real data set might be the one with the lowest values of the above-mentioned goodness-of-fit (GoF) measures. These tests are used to determine the model’s goodness of fit and to identify which model best fits the data. The statistics A * and W * for each model are calculated using the algorithm in the R package [35], while the K-S is calculated using the algorithm in the R package GLDEX [36]. The statistics A * and W * were provided in detail by Chen and Balakrishnan [37]. The data are better represented by the model with the minimum value of these metrics and a large p-value (PV) than the other models. The values of goodness-of-fit (GOF) measures for the fitted models can be found in Table 7, Table 8, Table 9 and Table 10. Table 7 provides the values of GoF Statistics for the LSFC when ${\beta}_{1}$= 0.1, 0.5, 0.9 under water-cobalt ferrite (H

_{2}O-CoFe

_{2}O

_{4}) and ethylene glycol-cobalt ferrite (C

_{2}H

_{6}O

_{2}-CoFe

_{2}O

_{4}). It is revealed from Table 7 that for the LSFC under water-cobalt ferrite (H

_{2}O-CoFe

_{2}O

_{4}), the EW model could be chosen as a best model among the other fitted models since it has the lowest values of the A * and W * and K-S statistics. It also has the largest PV of the KS test. For the LSFC of C

_{2}H

_{6}O

_{2}-CoFe

_{2}O

_{4}, three models could be chosen as the best model among the fitted models. As it can be seen, the results indicate that the Frechet model has the smallest values of Gof among the fitted models at ${\beta}_{1}$ = 0.1; therefore, it can be considered as the best model. It is noted that, the outcomes at ${\beta}_{1}$ = 0.5 indicates that the Weibull model has the smallest values of Gof among the fitted models. As a consequence, it may be called the best model. On the other hand, for the case of ${\beta}_{1}$ = 0.9, the EExF model has the smallest values of Gof among the fitted models; therefore, it can be considered as the best model.

_{2}O-CoFe

_{2}O

_{4}) at ${F}_{r}$ = 1.7, 2.1 and 2.5. The PV of the KS test is likewise the highest. Three different models could perhaps be considered as the best among the fitted models for the LSFC of C

_{2}H

_{6}O

_{2}-CoFe

_{2}O

_{4}, at ${F}_{r}$ = 1.7, 2.1 and 2.5 because the goodness-of-fit criterion is met by three models, namely, GHL, Weibull, and EW, at ${F}_{r}$ = 1.7, 2.1 and 2.5 respectively.

_{2}O-CoFe

_{2}O

_{4}) and ethylene glycol-cobalt ferrite (C

_{2}H

_{6}O

_{2}-CoFe

_{2}O

_{4}) for ${R}_{i}$ = 0, 0.3 and 0.6. Whereas for ${R}_{i}$ = 0.3, the best model among the fitted models for the LNN of water-cobalt ferrite (H

_{2}O-CoFe

_{2}O

_{4}) is the Weibull model with the lowest values of the above-mentioned goodness-of fit (GoF) measures.

_{2}O-CoFe

_{2}O

_{4}), two models (Weibull and EW) could be chosen as the best model among the fitted models. At ${N}_{i}$ = 0.1, the Weibull model is the best since it has the lowest values of the above-mentioned goodness-of fit (GoF) measures; on the other hand, for ${N}_{i}$ = 0.2 and 0.3, Ew showed its suitability because of having the minimum values of GoF metrics. On the other hand, among the fitted models for the LNN of ethylene glycol-cobalt ferrite, the two models EW and EExF may be considered as the most appropriate ethylene glycol-cobalt ferrite (C

_{2}H

_{6}O

_{2}-CoFe

_{2}O

_{4}) for ${N}_{i}$ = 0.1 and ${N}_{i}$ = 0.2, 0.3, respectively, because the lowest values of GoF metrics for Ew and EExF demonstrated suitable suitability. Figure 2, Figure 3, Figure 4 and Figure 5 show the fitted models that support the findings of Table 7, Table 8, Table 9 and Table 10.

_{2}O-CoFe

_{2}O

_{4}), and ethylene glycol-cobalt ferrite (C

_{2}H

_{6}O

_{2}-CoFe

_{2}O

_{4}) at various levels of ${\beta}_{1}$. From the upper panel of Figure 2, it is noted that the EW model is a suitable candidate for the data sets generated for water-cobalt ferrite (H

_{2}O-CoFe

_{2}O

_{4}) at all considered levels of ${\beta}_{1}$. On the other hand, from the lower panel of Figure 2, the Frechet model showed good suitability with the data behavior generated for ethylene glycol-cobalt ferrite (C

_{2}H

_{6}O

_{2}-CoFe

_{2}O

_{4}) at ${\beta}_{1}$ = 0.1, whereas Weibull and EExF models provide the correct fit to the data generated at ${\beta}_{1}$ = 0.5, 0.9 compared to the other models. The behavior of estimated models under water-cobalt ferrite (H

_{2}O-CoFe

_{2}O

_{4}) and ethylene glycol-cobalt ferrite (C

_{2}H

_{6}O

_{2}-CoFe

_{2}O

_{4}) is plotted in Figure 3 at various levels of ${F}_{r}$. It can be seen in the upper panel of Figure 3 that the EW model is a good fit for the data obtained for water-cobalt ferrite (H

_{2}O-CoFe

_{2}O

_{4}) at all values of ${F}_{r}$. The GHL model, on the other hand, exhibited good appropriateness with the data behavior obtained for ethylene glycol-cobalt ferrite (C

_{2}H

_{6}O

_{2}-CoFe

_{2}O

_{4}) at ${F}_{r}$ = 1.7, besides the Weibull and EW models, when compared to the other models and provides the correct match to the data generated at ${F}_{r}$ = 2.1, 2.5. Figure 4 presents the characteristics of the estimated models for water-cobalt ferrite (H

_{2}O-CoFe

_{2}O

_{4}) and ethylene glycol-cobalt ferrite (C

_{2}H

_{6}O

_{2}-CoFe

_{2}O

_{4}) at different levels of ${R}_{i}$. This is visible in the top panel of Figure 4 in which the EW model fits the data obtained for the water-cobalt ferrite (H

_{2}O-CoFe

_{2}O

_{4}) at ${R}_{i}$ = 0.1 and 0.6 values. The Weibull model, on the other hand, exhibited good appropriateness with the data at ${R}_{i}$ = 0.3. On the other hand, the EW model when compared to the other models, provided the best match for the data generated for ethylene glycol-cobalt ferrite (C

_{2}H

_{6}O

_{2}-CoFe

_{2}O

_{4}) at all values of ${R}_{i}$. At varying quantities of ${N}_{i}$, Figure 5 shows the pattern of estimated models for water-cobalt ferrite (H

_{2}O-CoFe

_{2}O

_{4}) (top panel) and ethylene glycol-cobalt ferrite (C

_{2}H

_{6}O

_{2}-CoFe

_{2}O

_{4}) (bottom panel). The EW model fits the data observed for the water-cobalt ferrite (H

_{2}O-CoFe

_{2}O

_{4}) for ${N}_{i}$ = 0.2 and 0.3, as shown in the top panel of Figure 5. At ${N}_{i}$ = 0.1, the Weibull model, on the other hand, showed good adequacy with the data. When compared to other models, the EW model provided the best fit to the data generated for the ethylene glycol-cobalt ferrite (C

_{2}H

_{6}O

_{2}-CoFe

_{2}O

_{4}at ${N}_{i}$ = 0.1 and EExF model at ${N}_{i}$ = 0.2 and 0.3, respectively.

## 5. Concluding Remarks

- For the LSFC under water-cobalt ferrite (H
_{2}O-CoFe_{2}O_{4}), the EW model can be chosen as the best model among the other fitted models at different levels of ${\beta}_{1}$. - For the LSFC of ethylene glycol-cobalt ferrite (C
_{2}H_{6}O_{2}-CoFe_{2}O_{4}), three models, such as the Frechet, the Weibull, and EExF, can be chosen as the best models at ${\beta}_{1}$ = 0.1, 0.5, and 0.9, respectively. - The EW model, which has the lowest values of the A * and W * and K-S statistics, is the best model among the fitted models for the LSFC of water-cobalt ferrite (H
_{2}O-CoFe_{2}O_{4}) at ${F}_{r}$ = 1.7, 2.1 and 2.5. - For the LSFC under water-cobalt ferrite (H
_{2}O-CoFe_{2}O_{4}), the EW model can be chosen as a best model among the other fitted models since it has the lowest values of the A * and W * and K-S statistics and has the largest PV of the KS test. - For LNN, the EW model provided the best fit to the data generated for the ethylene glycol-cobalt ferrite (C
_{2}H_{6}O_{2}-CoFe_{2}O_{4}) at ${N}_{i}$ = 0.1 and EExF model at ${N}_{i}$ = 0.2 and 0.3, respectively

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

$\theta $ | dimensionless temperature | ${U}_{w}$ | stretching velocity |

${h}_{s}$ | heat transfer coefficient W·m^{−2}·K^{−1} | f | dimensionless velocity |

${\stackrel{\u02c7}{\mu}}_{{n}_{f}}$ | nanofluid’s dynamic viscosity kg·m^{−1}·s^{−1} | ${\stackrel{\u02c7}{\rho}}_{{n}_{f}}$ | nanofluid’s density
kg·m^{−3} |

${\stackrel{\u02c7}{\mu}}_{{n}_{f}}$ | nanofluid’s dynamic viscosity kg·m^{−1}·s^{−1} | ${c}_{p}$ | specific heat |

$\alpha $ | nanomaterials solid volume fraction | ${\rho}_{p}$ | density of nanomaterials kg·m^{−3} |

${k}_{{}_{s}}$ | nanomaterials thermal conductivity W·m^{−1}·K^{−1} | ${\delta}_{1}$ | stretching-strength parameter |

${k}_{{n}_{f}}$ | nanofluid’s thermal conductivity W·m^{−1}·K^{−1} | ${\beta}_{1}$ | Casson fluid parameter |

${k}_{{}_{f}}$ | base fluid’s thermal conductivity W·m^{−1}·K^{−1} | ${F}_{r}$ | Inertia coefficient |

${\left(\rho {c}_{p}\right)}_{{}_{s}}$ | nanoparticles heat | ${S}_{1}$ | suction parameter |

capacitance J·kg^{2} m^{3}·K^{−1} | $Ec$ | Eckert number | |

$\sigma $ | electrical conductivity of nanoliquid | Pr | Prandtl number |

$\tau $ | the ratio of heat capacity of nanoparticles | ${R}_{1}$ | radiation parameter |

by heat capacity of nanofluid | ${N}_{i}$ | Conjugate parameter | |

$\stackrel{\u02c7}{T}$ | temperature of liquid K | $\Omega $ | constant angular velocity |

${\stackrel{\u02c7}{T}}_{\infty}$ | ambient temperature K | ${h}_{f}$ | coefficient of heat transfer |

${K}^{*}$ | permeability of porous medium | ${F}^{*}=\frac{{c}_{b}}{x\sqrt{{K}^{*}}}$ non-uniform inertia | |

coefficient of porous medium | |||

${c}_{b}$ | drag coefficient | $\lambda $ | mixed convective number |

$\left(\stackrel{\u02c7}{u},\stackrel{\u02c7}{v},\stackrel{\u02c7}{w}\right)$ | velocity components in $\left(r,\phi ,z\right)$ directions | ${k}_{1}$ | permeability parameter |

respectively ms^{−1} |

## Abbreviations

LNN | Local Nusselt Number | EG | ethylene glycol |

LSFC | Local Skin friction Coefficient | IEx | Inverse Exponential |

GHL | Generalized Half Logistic | DF | Darcy–Forchheimer |

EExF | Exponentiated Exponential Family | Ex | Exponential |

EW | Exponentiated Weibull |

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Base Fluid Physical Properties | ||||
---|---|---|---|---|

$\rho $/$\left(\mathrm{kg}\xb7{\mathrm{m}}^{-3}\right)$ | ${c}_{p}$/$(\mathrm{J}\xb7{\mathrm{kg}}^{-1}\xb7{\mathrm{K}}^{-1})$ | k/$(\mathrm{W}\xb7{\mathrm{m}}^{-1}\xb7{\mathrm{K}}^{-1})$ | Pr | |

(C_{2}H_{6}O_{2}) Ethylene glycol (EG) | 1115 | 2430 | 0.253 | 203.63 |

H_{2}O (Water) | 997.1 | 4179 | 0.613 | 6.2 |

Nanomaterial physical properties | ||||

CoFe_{2}O_{4} (Cobalt ferrite) | 4907 | 700 | 3.7 | - |

Distribution | Probability Distribution Function $\mathit{g}(\mathit{x})$ |
---|---|

IEx | $g(x;\stackrel{\u02d8}{\theta})=\frac{\stackrel{\u02d8}{\theta}}{{x}^{2}}exp(-\frac{\stackrel{\u02d8}{\theta}}{x})$$x>0,$$\stackrel{\u02d8}{\theta}>0,$ |

where $\stackrel{\u02d8}{\theta}$ is regarded as the scale parameter | |

EExF | $g(x;\widehat{\eta},\widehat{\delta})=\widehat{\eta}\widehat{\delta}{(1-exp(-\widehat{\delta}x))}^{\widehat{\eta}-1}exp(-\widehat{\delta}x)$$x>0,$$\widehat{\eta},\widehat{\delta}>0,$ |

here $\widehat{\eta}$ is the shape parameter and $\widehat{\delta}$ is the scale parameter. | |

GHL | $g(x;\stackrel{\u02d8}{\zeta},\widehat{\delta})=\frac{2\stackrel{\u02d8}{\zeta}\widehat{\delta}exp(-\widehat{\delta}x){(1-exp(-\widehat{\delta}x))}^{\stackrel{\u02d8}{\zeta}-1}}{{(1+exp(-\widehat{\delta}x))}^{\stackrel{\u02d8}{\zeta}+1}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}x,$$\stackrel{\u02d8}{\zeta},\widehat{\delta}>0.$ |

Where $\stackrel{\u02d8}{\zeta}$ is the shape parameter and $\widehat{\delta}$ is the scale parameter. | |

Weibull | $g(x;\stackrel{\u02d8}{\gamma},\widehat{\beta})=\frac{\widehat{\beta}}{\stackrel{\u02d8}{\gamma}}{(\frac{x}{\stackrel{\u02d8}{\gamma}})}^{\widehat{\beta}-1}{e}^{-{(\frac{x}{\stackrel{\u02d8}{\gamma}})}^{\widehat{\beta}}}$$x>0,$$\stackrel{\u02d8}{\gamma},\widehat{\beta}>0,$ |

where $\widehat{\beta}>0$ is the shape parameter and $\stackrel{\u02d8}{\gamma}>0$ is the scale parameter. | |

EW | $g(x;\stackrel{\u02d8}{\alpha},\stackrel{\u02d8}{\beta},\stackrel{\u02d8}{\vartheta})=\stackrel{\u02d8}{\alpha}\stackrel{\u02d8}{\beta}{\stackrel{\u02d8}{\vartheta}}^{\stackrel{\u02d8}{\beta}}{x}^{\stackrel{\u02d8}{\beta}-1}{(1-exp(-{(\stackrel{\u02d8}{\vartheta}x)}^{\stackrel{\u02d8}{\beta}}))}^{\stackrel{\u02d8}{\alpha}-1}exp(-{(\stackrel{\u02d8}{\vartheta}x)}^{\stackrel{\u02d8}{\beta}})$$x,\stackrel{\u02d8}{\alpha},\stackrel{\u02d8}{\beta},\stackrel{\u02d8}{\vartheta}>0,$ |

where $\stackrel{\u02d8}{\alpha},\stackrel{\u02d8}{\beta}$ are shape parameters and $\stackrel{\u02d8}{\vartheta}$ is the scale parameter. | |

Ex | $g(x;\widehat{\sigma})=\widehat{\sigma}exp(-\widehat{\sigma}x),$$x>0,$$\widehat{\sigma}>0,$ |

where $\widehat{\sigma}$ is the scale parameter. | |

Fréchet | $g(x;\stackrel{\u02d8}{\omega},\stackrel{\u02d8}{\varrho})=\frac{\stackrel{\u02d8}{\omega}}{x}{(\frac{\stackrel{\u02d8}{\varrho}}{x})}^{\stackrel{\u02d8}{\omega}}{e}^{-{(\frac{\stackrel{\u02d8}{\varrho}}{x})}^{\stackrel{\u02d8}{\omega}}}$$x>0,$$\alpha ,\stackrel{\u02d8}{\omega}>0,$ |

here $\stackrel{\u02d8}{\omega}$ is the shape parameter and $\stackrel{\u02d8}{\varrho}$ is the scale parameter. |

$\left({\mathrm{Re}}_{r}\right)$${}^{1/2}{C}_{\mathit{fr}}$ | |||||||
---|---|---|---|---|---|---|---|

H_{2}O-CoFe_{2}O_{4} | C_{2}H_{6}O_{2}-CoFe_{2}O_{4} | ||||||

β_{1} = 0.1 | β_{1} = 0.5 | β_{1} = 0.9 | β_{1} = 0.1 | β_{1} = 0.5 | β_{1} = 0.9 | ||

IEx | $\stackrel{\u02d8}{\theta}$ | 0.00492572 | 0.0080338 | 0.01161163 | 12.54911 | 8.249853 | 8.687495 |

$\widehat{\eta}$ | 0.3777665 | 0.6825908 | 0.7607601 | 48.1333741 | 17.4752617 | 5.8528827 | |

EExF | $\widehat{\delta}$ | 0.8838523 | 2.2593711 | 2.9989145 | 0.3290053 | 0.3723293 | 0.2168392 |

$\stackrel{\u02d8}{\zeta}$ | 0.3628408 | 0.6191486 | 0.6781996 | 55.1142332 | 9.8025548 | 3.745755 | |

GHL | $\widehat{\delta}$ | 1.3514776 | 3.0932756 | 4.0255717 | 0.3973598 | 0.3815363 | 0.232938 |

$\stackrel{\u02d8}{\gamma}$ | 0.4971002 | 0.8588737 | 0.9237080 | 4.097241 | 3.695815 | 2.33285 | |

Weibull | $\widehat{\beta}$ | 0.3865549 | 0.3286506 | 0.2723273 | 14.436267 | 10.230407 | 12.60927 |

$\stackrel{\u02d8}{\alpha}$ | 0.1455637 | 0.1354167 | 0.1231075 | 58.0759309 | 38.3896169 | 2.2014263 | |

EW | $\stackrel{\u02d8}{\beta}$ | 2.5168136 | 4.5420460 | 4.7893046 | 1.2435705 | 0.8194937 | 1.5521847 |

$\stackrel{\u02d8}{\vartheta}$ | 0.6813138 | 1.3888280 | 1.4361275 | 0.2609584 | 0.6343706 | 0.1139442 | |

Ex | $\widehat{\sigma}$ | 0.5821886 | 0.3463785 | 3.573092 | 0.07594189 | 0.1085452 | 0.08977783 |

Fréchet | $\stackrel{\u02d8}{\omega}$ | 0.02167272 | 0.0463213 | 0.0462311 | 11.612875 | 7.344389 | 7.652290 |

$\stackrel{\u02d8}{\varrho}$ | 0.34232982 | 0.3942654 | 0.4377827 | 5.988154 | 2.935294 | 1.881057 |

$\left({\mathrm{Re}}_{r}\right)$${}^{1/2}{C}_{\mathit{fr}}$ | |||||||
---|---|---|---|---|---|---|---|

H_{2}O-CoFe_{2}O_{4} | C_{2}H_{6}O_{2}-CoFe_{2}O_{4} | ||||||

F_{r} = 1.7 | F_{r} = 2.1 | F_{r} = 2.5 | F_{r} = 1.7 | F_{r} = 2.1 | F_{r} = 2.5 | ||

IEx | $\stackrel{\u02d8}{\theta}$ | 0.054694 | 0.049982 | 0.034184 | 0.0471712 | 8.80131 | 6.61957 |

$\widehat{\eta}$ | 1.174048 | 1.159579 | 1.059286 | 21.7080495 | 16.37802 | 3.70824 | |

EExF | $\widehat{\delta}$ | 4.079907 | 4.163169 | 4.621208 | 0.3863725 | 0.33935 | 0.18816 |

$\stackrel{\u02d8}{\zeta}$ | 0.983694 | 0.980645 | 0.898366 | 12.053610 | 9.39377 | 2.75686 | |

GHL | $\widehat{\delta}$ | 5.123580 | 5.271947 | 5.882473 | 0.394837 | 0.34931 | 0.21529 |

$\stackrel{\u02d8}{\gamma}$ | 1.239291 | 1.254754 | 1.133189 | 4.133696 | 4.64735 | 3.31257 | |

Weibull | $\widehat{\beta}$ | 0.286571 | 0.278797 | 0.233099 | 10.374723 | 10.72168 | 11.40295 |

$\stackrel{\u02d8}{\alpha}$ | 0.122339 | 0.125279 | 0.310632 | 0.45824259 | 0.20764 | 0.17935 | |

EW | $\stackrel{\u02d8}{\beta}$ | 6.361322 | 6.459926 | 2.542951 | 6.82330884 | 14.37007 | 12.35625 |

$\stackrel{\u02d8}{\vartheta}$ | 1.660942 | 1.799889 | 2.294579 | 0.08442887 | 0.07778 | 0.07079 | |

Ex | $\widehat{\sigma}$ | 3.700369 | 3.805439 | 4.458932 | 0.106422 | 0.10216 | 0.09638 |

Fréchet | $\stackrel{\u02d8}{\omega}$ | 0.083546 | 0.079480 | 0.061572 | 7.631669 | 7.80095 | 6.22488 |

$\stackrel{\u02d8}{\varrho}$ | 0.654382 | 0.638110 | 0.597793 | 2.990102 | 2.48381 | 1.14402 |

${\left({\mathbf{Re}}_{\mathit{r}}\right)}^{-1/2}{\mathit{Nu}}_{r}$ | |||||||
---|---|---|---|---|---|---|---|

H_{2}O-CoFe_{2}O_{4} | C_{2}H_{6}O_{2}-CoFe_{2}O_{4} | ||||||

R_{i} = 0 | R_{i} = 0.3 | R_{i} = 0.6 | R_{i} = 0 | R_{i} = 0.3 | R_{i} = 0.6 | ||

IEx | $\stackrel{\u02d8}{\theta}$ | 4.79946 | 2.85449 | 0.13902 | 2.93598 | 132.68050 | 238.9850 |

$\widehat{\eta}$ | 2.94301 | 69.67955 | 64.30566 | 2.05791 | 1.51425 | 5.70860 | |

EExF | $\widehat{\delta}$ | 0.25067 | 1.57015 | 2.145285 | 0.28546 | 0.00375 | 0.00747 |

$\stackrel{\u02d8}{\zeta}$ | 1.98471 | 63.58090 | 85.70160 | 1.47689 | 1.24305 | 3.79916 | |

GHL | $\widehat{\delta}$ | 0.27438 | 1.78448 | 2.57725 | 0.32109 | 0.00455 | 0.00807 |

$\stackrel{\u02d8}{\gamma}$ | 1.64390 | 10.96689 | 19.28294 | 1.45369 | 0.65533 | 2.49353 | |

Weibull | $\widehat{\beta}$ | 8.20245 | 3.06479 | 2.14166 | 5.91572 | 110.95909 | 261.1334 |

$\stackrel{\u02d8}{\alpha}$ | 12.00428 | 0.17409 | 1.268812 | 2.83422 | 2.86082 | 0.87337 | |

EW | $\stackrel{\u02d8}{\beta}$ | 0.59147 | 38.55997 | 17.03825 | 0.85722 | 0.74147 | 4.78644 |

$\stackrel{\u02d8}{\vartheta}$ | 1.01602 | 0.30078 | 0.47230 | 0.38172 | 0.00669 | 0.00283 | |

Ex | $\widehat{\sigma}$ | 0.137270 | 0.34275 | 0.47978 | 0.18734 | 0.00292 | 0.00320 |

Fréchet | $\stackrel{\u02d8}{\omega}$ | 4.25925 | 2.65483 | 2.00511 | 2.69145 | 95.44180 | 162.58202 |

$\stackrel{\u02d8}{\varrho}$ | 1.68032 | 5.24668 | 12.40504 | 1.29471 | 0.94593 | 1.46931 |

**Table 6.**Estimates of the parameters of statistical distribution for LNN when ${\delta}_{1}={\delta}_{2}={\delta}_{3}=0.7$.

${\left({\mathbf{Re}}_{\mathit{r}}\right)}^{-1/2}{\mathbf{Nu}}_{\mathit{r}}$ | |||||||
---|---|---|---|---|---|---|---|

H_{2}O-CoFe_{2}O_{4} | C_{2}H_{6}O_{2}-CoFe_{2}O_{4} | ||||||

N_{i} = 0.1 | N_{i} = 0.2 | N_{i} = 0.3 | N_{i} = 0.1 | N_{i} = 0.2 | N_{i} = 0.3 | ||

IEx | $\stackrel{\u02d8}{\theta}$ | 7.04387 | 4.10631 | 3.24670 | 10.72483 | 23.29198 | 38.38786 |

$\widehat{\eta}$ | 65.53492 | 67.36743 | 88.57084 | 66.3001118 | 72.03932 | 44.40364 | |

EExF | $\widehat{\delta}$ | 0.62587 | 1.09531 | 1.45271 | 0.4163272 | 0.19580 | 0.10546 |

$\stackrel{\u02d8}{\zeta}$ | 66.40417 | 66.92848 | 71.37620 | 48.7389196 | 61.86919 | 43.62420 | |

GHL | $\widehat{\delta}$ | 0.72377 | 1.25615 | 1.60937 | 0.4534334 | 0.22058 | 0.12402 |

$\stackrel{\u02d8}{\gamma}$ | 13.04798 | 12.80603 | 7.31229 | 5.071224 | 4.77826 | 4.41743 | |

Weibull | $\widehat{\beta}$ | 7.38282 | 4.30113 | 3.47163 | 12.202111 | 26.74031 | 44.73431 |

$\stackrel{\u02d8}{\alpha}$ | 15.65368 | 12.10827 | 30.82324 | 5.383780 | 17.82266 | 17.29216 | |

EW | $\stackrel{\u02d8}{\beta}$ | 4.18284 | 4.53642 | 2.85223 | 124.6565821 | 1.53965 | 1.44738 |

$\stackrel{\u02d8}{\vartheta}$ | 0.18618 | 0.30658 | 0.49022 | 1.0499519 | 0.09088 | 0.05712 | |

Ex | $\widehat{\sigma}$ | 0.14091 | 0.24172 | 0.30423 | 0.4447017 | 0.04083 | 0.02451 |

Fréchet | $\stackrel{\u02d8}{\omega}$ | 6.774369 | 3.94931 | 3.10566 | 9.879902 | 21.38523 | 35.16524 |

$\stackrel{\u02d8}{\varrho}$ | 13.43649 | 12.76804 | 10.42570 | 5.383780 | 5.09862 | 4.75391 |

${\left({\mathrm{Re}}_{r}\right)}^{1/2}{C}_{\mathit{fr}}$ | |||||||
---|---|---|---|---|---|---|---|

H_{2}O-CoFe_{2}O_{4} | C_{2}H_{6}O_{2}-CoFe_{2}O_{4} | ||||||

β_{1} = 0.1 | β_{1} = 0.5 | β_{1} = 0.9 | β_{1} = 0.1 | β_{1} = 0.5 | β_{1} = 0.9 | ||

IEx | W * | 0.26059 | 0.30331 | 0.30421 | 0.03525 | 0.09233 | 0.05789 |

A * | 1.50475 | 1.74873 | 1.72260 | 0.27508 | 0.56991 | 0.40980 | |

K-S | 0.65411 | 0.69733 | 0.71474 | 0.45531 | 0.44686 | 0.34439 | |

PV | 0.00027 | 6.6 × 10^{−5} | 3.6 × 10^{−5} | 0.03181 | 0.03714 | 0.18630 | |

EExF | W * | 0.15731 | 0.09070 | 0.08664 | 0.03528 | 0.07932 | 0.04200 |

A * | 0.99865 | 0.57101 | 0.49475 | 0.27537 | 0.49740 | 0.27055 | |

K-S | 0.29435 | 0.26968 | 0.28921 | 0.16930 | 0.21140 | 0.16701 | |

PV | 0.34620 | 0.45210 | 0.36680 | 0.92240 | 0.74210 | 0.92940 | |

GHL | W * | 0.14085 | 0.07679 | 0.06855 | 0.03258 | 0.07646 | 0.04202 |

A * | 0.91146 | 0.48509 | 0.39169 | 0.25264 | 0.48188 | 0.27061 | |

K-S | 0.27790 | 0.26067 | 0.26298 | 0.15706 | 0.20658 | 0.17693 | |

PV | 0.41500 | 0.49460 | 0.48350 | 0.95530 | 0.76620 | 0.89670 | |

Weibull | W * | 0.18402 | 0.09656 | 0.09111 | 0.08753 | 0.05159 | 0.05599 |

A * | 1.13647 | 0.60744 | 0.52036 | 0.62745 | 0.33021 | 0.32478 | |

K-S | 0.30034 | 0.26270 | 0.26929 | 0.20070 | 0.18983 | 0.21735 | |

PV | 0.32310 | 0.48490 | 0.45390 | 0.79500 | 0.84500 | 0.71200 | |

EW | W * | 0.11664 | 0.04739 | 0.05658 | 0.03172 | 0.05175 | 0.04601 |

A * | 0.77064 | 0.30826 | 0.31238 | 0.24521 | 0.32975 | 0.28155 | |

K-S | 0.26159 | 0.25175 | 0.22763 | 0.16338 | 0.19548 | 0.19333 | |

PV | 0.49020 | 0.53820 | 0.66010 | 0.93970 | 0.81960 | 0.82940 | |

Ex | W * | 0.15649 | 0.09126 | 0.08717 | 0.04941 | 0.06418 | 0.04409 |

A * | 0.99511 | 0.57380 | 0.49782 | 0.38190 | 0.40972 | 0.27689 | |

K-S | 0.32987 | 0.25594 | 0.24899 | 0.52133 | 0.41542 | 0.35322 | |

PV | 0.22540 | 0.51760 | 0.55200 | 0.00841 | 0.06398 | 0.16510 | |

Fréchet | W * | 0.24565 | 0.22472 | 0.24243 | 0.02549 | 0.12714 | 0.08116 |

A * | 1.44050 | 1.36232 | 1.39132 | 0.17654 | 0.75849 | 0.56017 | |

K-S | 0.35622 | 0.28619 | 0.35255 | 0.14045 | 0.25032 | 0.20570 | |

PV | 0.15840 | 0.37930 | 0.16670 | 0.98320 | 0.54530 | 0.77060 |

${\left({\mathrm{Re}}_{r}\right)}^{1/2}{C}_{\mathit{fr}}$ | |||||||
---|---|---|---|---|---|---|---|

H_{2}O-CoFe_{2}O_{4} | C2_{H}_{6}O_{2}-CoFe_{2}O_{4} | ||||||

F_{r} = 1.7 | F_{r} = 2.1 | F_{r} = 2.5 | F_{r} = 1.7 | F_{r} = 2.1 | F_{r} = 2.5 | ||

IEx | W * | 0.22199 | 0.24126 | 0.24980 | 0.04717 | 0.14382 | 0.36806 |

A * | 1.31178 | 1.39296 | 1.44821 | 0.35947 | 0.90885 | 2.02909 | |

K-S | 0.46608 | 0.52772 | 0.57170 | 0.47812 | 0.48500 | 0.36469 | |

PV | 0.02599 | 0.00730 | 0.00260 | 0.02059 | 0.01797 | 0.14040 | |

EExF | W * | 0.06645 | 0.07853 | 0.06926 | 0.03968 | 0.11522 | 0.24185 |

A * | 0.39673 | 0.47170 | 0.40534 | 0.30953 | 0.74672 | 1.42151 | |

K-S | 0.24897 | 0.24079 | 0.25325 | 0.13993 | 0.27834 | 0.31886 | |

PV | 0.55200 | 0.59320 | 0.53080 | 0.98380 | 0.41300 | 0.25880 | |

GHL | W * | 0.05185 | 0.06022 | 0.05567 | 0.03788 | 0.10808 | 0.21581 |

A * | 0.30923 | 0.36917 | 0.32651 | 0.29714 | 0.70509 | 1.29001 | |

K-S | 0.22720 | 0.21320 | 0.22672 | 0.13503 | 0.27029 | 0.30256 | |

PV | 0.66220 | 0.73300 | 0.6647 | 0.98870 | 0.44930 | 0.31480 | |

Weibull | W * | 0.058079 | 0.06775 | 0.06475 | 0.02960 | 0.04983 | 0.13728 |

A * | 0.346025 | 0.41065 | 0.37861 | 0.21125 | 0.33471 | 0.86829 | |

K-S | 0.22648 | 0.20214 | 0.22800 | 0.15506 | 0.19436 | 0.25317 | |

PV | 0.66590 | 0.78800 | 0.65820 | 0.95970 | 0.82480 | 0.53120 | |

EW | W * | 0.03974 | 0.03917 | 0.05183 | 0.03665 | 0.05421 | 0.07977 |

A * | 0.23581 | 0.23391 | 0.30557 | 0.23505 | 0.32519 | 0.50882 | |

K-S | 0.16218 | 0.16801 | 0.19994 | 0.16418 | 0.16840 | 0.24120 | |

PV | 0.94280 | 0.92640 | 0.79860 | 0.93750 | 0.92520 | 0.59110 | |

Ex | W * | 0.06589 | 0.07793 | 0.06911 | 0.03042 | 0.08841 | 0.22554 |

A * | 0.39337 | 0.46826 | 0.40445 | 0.24313 | 0.58743 | 1.33893 | |

K-S | 0.26172 | 0.26144 | 0.26118 | 0.39793 | 0.43772 | 0.44141 | |

PV | 0.48960 | 0.49090 | 0.49220 | 0.08492 | 0.04371 | 0.04095 | |

Fréchet | W * | 0.18983 | 0.20977 | 0.20714 | 0.07827 | 0.19744 | 0.37788 |

A * | 1.13309 | 1.21657 | 1.21614 | 0.55284 | 1.196827 | 2.07480 | |

K-S | 0.27592 | 0.32265 | 0.33118 | 0.19054 | 0.30950 | 0.37796 | |

PV | 0.42380 | 0.24690 | 0.22170 | 0.84190 | 0.2900 | 0.11550 |

${\left({\mathrm{Re}}_{r}\right)}^{-1/2}{\mathit{Nu}}_{r}$ | |||||||
---|---|---|---|---|---|---|---|

H_{2}O-CoFe_{2}O_{4} | C_{2}H_{6}O_{2}-CoFe_{2}O_{4} | ||||||

R_{1} = 0 | R_{1} = 0.3 | R_{1} = 0.6 | R_{1} = 0 | R_{1} = 0.3 | R_{1} = 0.6 | ||

IEx | W * | 0.01515 | 0.1813 | 0.13756 | 0.02869 | 0.19041 | 0.38233 |

A * | 0.13126 | 1.10529 | 0.81700 | 0.22541 | 1.08513 | 2.09143 | |

K-S | 0.24318 | 0.57470 | 0.59488 | 0.19804 | 0.37353 | 0.44431 | |

PV | 0.58120 | 0.00241 | 0.00144 | 0.80760 | 0.12340 | 0.03888 | |

EExF | W * | 0.01979 | 0.18611 | 0.13902 | 0.01255 | 0.14209 | 0.29568 |

A * | 0.15219 | 1.13060 | 0.82542 | 0.10925 | 0.79318 | 1.68073 | |

K-S | 0.12397 | 0.30886 | 0.37364 | 0.09444 | 0.32821 | 0.36948 | |

PV | 0.99570 | 0.29220 | 0.12320 | 0.99980 | 0.2302 | 0.13100 | |

GHL | W * | 0.02457 | 0.19387 | 0.14370 | 0.01603 | 0.14841 | 0.27457 |

A * | 0.18181 | 1.17111 | 0.85214 | 0.12976 | 0.82751 | 1.57840 | |

K-S | 0.13603 | 0.28856 | 0.36917 | 0.10781 | 0.32741 | 0.34997 | |

PV | 0.98780 | 0.3695 | 0.13160 | 0.99950 | 0.23260 | 0.17270 | |

Weibull | W * | 0.02965 | 0.07688 | 0.09455 | 0.01677 | 0.14659 | 0.20054 |

A * | 0.21405 | 0.50810 | 0.55809 | 0.13420 | 0.81758 | 1.20425 | |

K-S | 0.13189 | 0.16984 | 0.23597 | 0.10663 | 0.56127 | 0.59595 | |

PV | 0.99120 | 0.92070 | 0.61770 | 0.99960 | 0.00336 | 0.00140 | |

EW | W * | 0.01483 | 0.08454 | 0.09349 | 0.01178 | 0.13291 | 0.16038 |

A * | 0.12260 | 0.50744 | 0.55295 | 0.10535 | 0.74479 | 0.98821 | |

K-S | 0.10006 | 0.20190 | 0.23430 | 0.09353 | 0.31061 | 0.23122 | |

PV | 0.99990 | 0.78920 | 0.61810 | 1.00000 | 0.28610 | 0.64190 | |

Ex | W * | 0.02206 | 0.14931 | 0.12721 | 0.01319 | 0.14064 | 0.27278 |

A * | 0.16630 | 0.93418 | 0.75714 | 0.11274 | 0.78537 | 1.56886 | |

K-S | 0.24402 | 0.50110 | 0.57218 | 0.16601 | 0.33868 | 0.48518 | |

PV | 0.57690 | 0.01294 | 0.00257 | 0.9323 | 0.20100 | 0.01790 | |

Fréchet | W * | 0.01946 | 0.26036 | 0.21663 | 0.0355 | 0.18727 | 0.40106 |

A * | 0.16466 | 1.506904 | 1.25046 | 0.2700 | 1.06500 | 2.17841 | |

K-S | 0.11154 | 0.33388 | 0.35455 | 0.12546 | 0.43081 | 0.53184 | |

PV | 0.99900 | 0.21400 | 0.16210 | 0.99500 | 0.04932 | 0.00666 |

${\left({\mathrm{Re}}_{r}\right)}^{1/2}{\mathit{Nu}}_{r}$ | |||||||
---|---|---|---|---|---|---|---|

H_{2}O-CoFe_{2}O_{4} | C_{2}H_{6}O_{2}-CoFe_{2}O_{4} | ||||||

N_{i} = 0.1 | N_{i} = 0.2 | N_{i} = 0.3 | N_{i} = 0.1 | N_{i} = 0.2 | N_{i} = 0.3 | ||

IEx | W * | 0.07553 | 0.03313 | 0.23306 | 0.02021 | 0.02021 | 0.02020 |

A * | 0.49281 | 0.21902 | 1.22388 | 0.16220 | 0.16217 | 0.16210 | |

K-S | 0.58587 | 0.57765 | 0.52521 | 0.49754 | 0.48898 | 0.47613 | |

PV | 0.00040 | 0.00052 | 0.00236 | 0.00486 | 0.00601 | 0.00820 | |

EExF | W * | 0.07542 | 0.03322 | 0.22973 | 0.02018 | 0.02013 | 0.02014 |

A * | 0.49272 | 0.21980 | 1.20477 | 0.16207 | 0.16152 | 0.16299 | |

K-S | 0.35769 | 0.30646 | 0.34323 | 0.09330 | 0.08986 | 0.08841 | |

PV | 0.09145 | 0.20530 | 0.11640 | 0.99980 | 0.99990 | 0.99990 | |

GHL | W * | 0.07516 | 0.03337 | 0.22780 | 0.02021 | 0.02013 | 0.02016 |

A * | 0.49228 | 0.22119 | 1.19368 | 0.16223 | 0.16169 | 0.16366 | |

K-S | 0.33566 | 0.28610 | 0.32823 | 0.09001 | 0.10488 | 0.10476 | |

PV | 0.13160 | 0.27290 | 0.14790 | 0.99990 | 0.99850 | 0.99850 | |

Weibull | W * | 0.07277 | 0.04138 | 0.31709 | 0.03747 | 0.03814 | 0.03928 |

A * | 0.49525 | 0.27503 | 1.71851 | 0.26762 | 0.27172 | 0.27863 | |

K-S | 0.19098 | 0.17817 | 0.32542 | 0.12878 | 0.12926 | 0.13003 | |

PV | 0.75080 | 0.8174 | 0.15450 | 0.98220 | 0.98150 | 0.98050 | |

EW | W * | 0.07609 | 0.03327 | 0.22931 | 0.02014 | 0.02096 | 0.02118 |

A * | 0.49548 | 0.21917 | 1.20294 | 0.16169 | 0.16691 | 0.16826 | |

K-S | 0.20552 | 0.14244 | 0.27982 | 0.10529 | 0.10367 | 0.10806 | |

PV | 0.6698 | 0.95450 | 0.29660 | 0.99840 | 0.99870 | 0.99770 | |

Ex | W * | 0.07657 | 0.03306 | 0.24789 | 0.022909 | 0.02331 | 0.02399 |

A * | 0.49533 | 0.21719 | 1.31113 | 0.178812 | 0.18127 | 0.18537 | |

K-S | 0.58928 | 0.57984 | 0.55696 | 0.50872 | 0.50172 | 0.49199 | |

PV | 0.00036 | 0.00049 | 0.00097 | 0.00365 | 0.00437 | 0.00558 | |

Fréchet | W * | 0.07212 | 0.04473 | 0.22513 | 0.02504 | 0.02464 | 0.02400 |

A * | 0.49332 | 0.29660 | 1.18478 | 0.19155 | 0.18912 | 0.18512 | |

K-S | 0.22469 | 0.14372 | 0.30281 | 0.11411 | 0.11385 | 0.11332 | |

PV | 0.56130 | 0.95310 | 0.21640 | 0.99530 | 0.99540 | 0.99570 |

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**MDPI and ACS Style**

Shafiq, A.; Lone, S.A.; Sindhu, T.N.; Nonlaopon, K.
Statistical Modelling for the Darcy–Forchheimer Flow of Casson Cobalt Ferrite-Water/Ethylene Glycol Nanofluid under Nonlinear Radiation. *Symmetry* **2022**, *14*, 1717.
https://doi.org/10.3390/sym14081717

**AMA Style**

Shafiq A, Lone SA, Sindhu TN, Nonlaopon K.
Statistical Modelling for the Darcy–Forchheimer Flow of Casson Cobalt Ferrite-Water/Ethylene Glycol Nanofluid under Nonlinear Radiation. *Symmetry*. 2022; 14(8):1717.
https://doi.org/10.3390/sym14081717

**Chicago/Turabian Style**

Shafiq, Anum, Showkat A. Lone, Tabassum Naz Sindhu, and Kamsing Nonlaopon.
2022. "Statistical Modelling for the Darcy–Forchheimer Flow of Casson Cobalt Ferrite-Water/Ethylene Glycol Nanofluid under Nonlinear Radiation" *Symmetry* 14, no. 8: 1717.
https://doi.org/10.3390/sym14081717