# Fractional Modeling of Viscous Fluid over a Moveable Inclined Plate Subject to Exponential Heating with Singular and Non-Singular Kernels

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## Abstract

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## 1. Introduction

## 2. Mathematical Model

## 3. Mathematical Preliminaries

#### Special Functions

**Mittag-Leffler function.**The Mittag-Leffler function is the generalization of the exponential function and is defined as [33]$${E}_{\wr}\left(t\right)=\sum _{\wp =0}^{\infty}\frac{{t}^{\wp}}{\Gamma \left(\wp \wr +1\right)}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}};\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\wr >0.$$The exponential function is a special case of this function; for $\wr =1$, we get$${E}_{1}\left(t\right)=\sum _{\wp =0}^{\infty}\frac{{t}^{\wp}}{\Gamma \left(\wp +1\right)}={e}^{t}.$$Moreover,$$\mathcal{L}\left\{{E}_{\wr}\left(-a{t}^{\wr}\right)\right\}=L\left\{\sum _{\wp =0}^{\infty}\frac{{\left(-a\right)}^{\wp}{t}^{\wp \wr}}{\Gamma \left(\wp \wr +1\right)}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\right\}=\frac{{q}^{\wr}}{q\left({q}^{\wr}+a\right)};\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\wr >0.$$**Erdelyi’s function.**This function is the generalization of the Mittag-Leffler function and is described as [36]$${E}_{\wr ,\beta}\left(t\right)=\sum _{\wp =0}^{\infty}\frac{{t}^{\wp}}{\Gamma \left(\wp \wr +\beta \right)}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}};\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\wr ,\beta >0.$$Setting $\beta =1$, we have$${E}_{\wr ,1}\left(t\right)=\sum _{\wp =0}^{\infty}\frac{{t}^{\wp}}{\Gamma \left(\wp \wr +1\right)}={E}_{\wr}\left(t\right).$$For $\wr =1$ and $\beta =2$, we have$${E}_{1,2}\left(t\right)=\frac{{e}^{t}-1}{t}.$$Similarly, for $\wr =\frac{1}{2}$ and $\beta =1$, we get$${E}_{\frac{1}{2},1}\left(t\right)={e}^{{t}^{2}}erfc\left(-t\right)$$When $\wr =2$ and $\beta =2$, we have$${E}_{2,2}\left({t}^{2}\right)=\frac{sinh\left(t\right)}{t},$$Further,$$\mathcal{L}\left\{{E}_{\wr ,\beta}\left(t\right)\right\}=\sum _{\wp =0}^{\infty}\frac{\Gamma \left(\wp +1\right)}{\Gamma \left(\wp \wr +\beta \right)}\phantom{\rule{0.166667em}{0ex}}\frac{1}{{q}^{\wp +1}};\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\wr ,\beta >0.$$**Robotnov and Hartley function.**This was presented by Hartley and Lorenzo [34] and later on studied by Robotnov for utilization in solid mechanics as well. It is confined as$${F}_{\wr}\left(-at\right)={t}^{\wr -1}\sum _{\wp =0}^{\infty}\frac{{\left(-a\right)}^{\wp}{t}^{\wp \wr}}{\Gamma \left(\wp \wr +1\right)}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}};\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\wr >0.$$Here,$${E}_{\wr}\left(-a{t}^{\wr}\right)=\sum _{\wp =0}^{\infty}\frac{{\left(-a\right)}^{\wp}{t}^{\wp \wr}}{\Gamma \left(\wp \wr +1\right)},F\wr \left(-at\right)={t}^{\wr -1}{E}_{\wr}\left(-a{t}^{\wr}\right),$$$$\mathcal{L}\left\{{F}_{\wr}\left(-at\right)\right\}=\frac{1}{{q}^{\wr}+a};\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\wr >0.$$**Miller and Ross’ function.**It was proposed by Miller, and Ross [37]. This function is stated as:$${E}_{t}\left(\wr ,a\right)={t}^{\wr}\sum _{\wp =0}^{\infty}\frac{{\left(at\right)}^{\wp}}{\Gamma \left(\wr +\wp +1\right)}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}};\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}Re\left(\wr \right)>1,$$$$\mathcal{L}\left\{{E}_{t}\left(\wr ,a\right)\right\}=\frac{{q}^{-\wr}}{q-a};\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}Re\left(\wr \right)>1.$$**Generalized R-function.**Lorenzo and Hartley [35] developed this function; it is written as:$${R}_{\wr ,\beta}\left(a,t\right)=\sum _{\wp =0}^{\infty}\frac{{a}^{k}{t}^{\left(\wp +1\right)\wr -\beta -1}}{\Gamma \left(\left(\wp +1\right)\wr -\beta \right)}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}};\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}Re\left(\wr -\beta \right)>0.$$It is easy to see that ${R}_{1,0}\left(a,t\right)={e}^{at}$, $a{R}_{2,0}\left(-{a}^{2},t\right)=sin\left(at\right)$ and ${R}_{2,1}\left(-{a}^{2},t\right)=cos\left(at\right)$.When $a=1,\beta =\wr -1$, we get$${R}_{\wr ,\wr -1}\left(1,t\right)=\sum _{\wp =0}^{\infty}\frac{{\left({t}^{\wr}\right)}^{\wp}}{\Gamma \left(\wp \wr +1\right)}={E}_{\wr}\left({t}^{\wr}\right).$$Similarly, for $a=1,\beta =\wr -\nu $, yields$${R}_{\wr ,\wr -\nu}\left(1,t\right)={t}^{\nu -1}\sum _{\wp =0}^{\infty}\frac{{\left({t}^{\wr}\right)}^{\wp}}{\Gamma \left(\wp \wr +\nu \right)}={t}^{\nu -1}{E}_{\wr ,\nu}\left({t}^{\wr}\right).$$Moreover,$$\mathcal{L}\left\{{R}_{\wr ,\beta}\left(a,t\right)\right\}=\frac{{q}^{\beta}}{{q}^{\wr}-a};\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}Re\left(\wr -\beta \right)>0,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}Re\left(q\right)>0.$$**Generalized G-function.**Lorenzo and Hartley [35] also introduced this function which is the generalization of R-function and is specified as:$${G}_{\wr ,b,j}\left(a,t\right)=\sum _{\wp =0}^{\infty}\frac{{a}^{\wp}\Gamma \left(j+\wp \right)}{\Gamma \left(j\right)\Gamma \left(\wp +1\right)}\frac{{t}^{\left(j+\wp \right)\wr -b-1}}{\Gamma \left(\left(\wp +1\right)\wr -b\right)}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}};\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}Re\left(\wr j-b\right)>0.$$For $j=1$, we have$${G}_{\wr ,b,1}\left(a,t\right)=\sum _{\wp =0}^{\infty}\frac{{a}^{\wp}{t}^{\left(1+\wp \right)\wr -b-1}}{\Gamma \left(\left(\wp +1\right)\wr -b\right)}={R}_{\wr ,b}\left(a,t\right).$$Moreover,$$\begin{array}{c}{\int}_{0}^{s}{G}_{\wr ,b,j}\left(a,t\right)dt={\displaystyle \sum _{\wp =0}^{\infty}}\frac{{a}^{\wp}\Gamma \left(j+\wp \right)}{\Gamma \left(j\right)\Gamma \left(\wp +1\right)}\frac{{s}^{\left(j+\wp \right)\wr -b}}{\left({}^{\left(j+\wp \right)\wr -b}\right)\Gamma \left(\left(\wp +1\right)\wr -b\right)}\\ ={\displaystyle \sum _{\wp =0}^{\infty}}\frac{{a}^{\wp}\Gamma \left(j+\wp \right)}{\Gamma \left(j\right)\Gamma \left(\wp +1\right)}\frac{{s}^{\left(j+\wp \right)\wr -b}}{\Gamma \left(\left(\wp +1\right)\wr -b+1\right)}={G}_{\wr ,b-1,j}\left(a,s\right).\end{array}$$Moreover,$$\mathcal{L}\left\{{G}_{\wr ,b,j}\left(a,t\right)\right\}=\frac{{q}^{b}}{{\left({q}^{\wr}-a\right)}^{j}};\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}Re\left(\wr j-b\right)>0,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}Re\left(q\right)>0,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\left|\frac{a}{{q}^{\wr}}\right|<1.$$Next, we define Caputo, CF and ABC fractional operators used in this paper to fractionalize the proposed problem.

- Caputo fractional operator having power law kernel is described as:$${}^{C}{D}_{\eta}^{\wp}f\left(z,\eta \right)=\frac{1}{\Gamma (1-\wp )}{\int}_{0}^{\eta}\frac{1}{{(\eta -\tau )}^{\wp}}\frac{\partial f(z,\tau )}{\partial \tau}d\tau ,\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}0<\wp <1.$$$$\mathcal{L}\left({}^{C}{D}_{\eta}^{\wp}f\left(z,\eta \right)\right)={s}^{\wp}\mathcal{L}\left(f(z,\eta )\right)-{s}^{\wp -1}f(z,0).$$
- CF fractional operator with a non-singularized and local kernel is described as:$${}^{CF}{D}_{\eta}^{\wp}f\left(z,\eta \right)=\frac{1}{1-\wp}{\int}_{0}^{\eta}exp\left(-\frac{\wp (\eta -\tau )}{1-\wp}\right)\frac{\partial f(z,\tau )}{\partial \tau}d\tau ,\phantom{\rule{1.em}{0ex}}0<\wp <1.$$Its Laplace transformation is obtained as:$$\mathcal{L}\left({}^{CF}{D}_{\eta}^{\wp}f\left(z,\eta \right)\right)=\frac{s\mathcal{L}\left(f(z,\eta )\right)-f(z,0)}{(1-\wp )s+\wp}.$$
- The Atangana–Baleanu fractional operator in a Caputo sense (ABC) with non-singularized and non-local kernel is defined in the following way:$${}^{ABC}{D}_{\eta}^{\wp}f\left(z,\eta \right)=\frac{1}{1-\wp}{\int}_{0}^{\eta}{E}_{\wp}\left(-\frac{\wp {(\eta -\tau )}^{\wp}}{1-\wp}\right)\frac{\partial f(z,\tau )}{\partial \tau}d\tau ,\phantom{\rule{1.em}{0ex}}0<\wp <1.$$Its Laplace transformation is obtained as:$$\mathcal{L}\left({}^{ABC}{D}_{\eta}^{\wp}f\left(z,\eta \right)\right)=\frac{{s}^{\wp}\mathcal{L}\left(f(z,\eta )\right)-{s}^{\wp -1}f(z,0)}{(1-\wp ){s}^{\wp}+\wp}.$$

## 4. Solution of the Problem

#### 4.1. Exact Solution of Heat Profile with CF Time Fractional Derivative

#### 4.2. Exact Solution of Heat Profile with ABC Time Fractional Derivative

#### 4.3. Exact Solution of Mass Profile with CF Time Fractional Derivative

#### 4.4. Exact Solution of Mass Profile with ABC Time Fractional Derivative

#### 4.5. Exact Solution of Velocity Profile with CF Time Fractional Derivative

#### 4.6. Exact Solution of Velocity Profile with ABC Time Fractional Derivative

## 5. Various Cases Concerning the Motion of the Plate

#### 5.1. Case-I: $f\left(\eta \right)=H\left(\eta \right){\eta}^{\gamma}$ (for Variable Accelerating Plate)

#### 5.2. Case-II: $f\left(\eta \right)=cos\left(\omega \eta \right)\phantom{\rule{0.166667em}{0ex}}H\left(\eta \right)\phantom{\rule{0.166667em}{0ex}}or\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}sin\left(\omega \eta \right)H\left(\eta \right)$ (for Oscillating Plate)

## 6. Results Validation

## 7. Results and Discussion

## 8. Conclusions

- It is detected that the velocity field declined with the larger values of ${R}_{c}$. Moreover, reduction in the velocity and concentration profile are observed for growing values of $Sc$ for varying values of $\alpha $.
- It is found that the fluid velocity intensifies for $N>0$, but the opposite trend is observed for $N<0$.
- The increasing values of the time $\eta $ stimulate the velocity distribution.
- The accumulative values of the parameter $P{r}_{eff}$ decline in the temperature profile are noticed.
- Involvement of concentration factor of fluid velocity in the fluid movement is significant and cannot be overlooked.
- It is depicted that for both non-integer operators CF and ABC, velocity field, concentration and temperature profile represent the same behavior for parametric analysis of the proposed problem.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 13.**Comparison of dimensionless velocity profil for dissimilar values of $\eta $ between CF and ABC.

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**MDPI and ACS Style**

Rehman, A.U.; Riaz, M.B.; Rehman, W.; Awrejcewicz, J.; Baleanu, D.
Fractional Modeling of Viscous Fluid over a Moveable Inclined Plate Subject to Exponential Heating with Singular and Non-Singular Kernels. *Math. Comput. Appl.* **2022**, *27*, 8.
https://doi.org/10.3390/mca27010008

**AMA Style**

Rehman AU, Riaz MB, Rehman W, Awrejcewicz J, Baleanu D.
Fractional Modeling of Viscous Fluid over a Moveable Inclined Plate Subject to Exponential Heating with Singular and Non-Singular Kernels. *Mathematical and Computational Applications*. 2022; 27(1):8.
https://doi.org/10.3390/mca27010008

**Chicago/Turabian Style**

Rehman, Aziz Ur, Muhammad Bilal Riaz, Wajeeha Rehman, Jan Awrejcewicz, and Dumitru Baleanu.
2022. "Fractional Modeling of Viscous Fluid over a Moveable Inclined Plate Subject to Exponential Heating with Singular and Non-Singular Kernels" *Mathematical and Computational Applications* 27, no. 1: 8.
https://doi.org/10.3390/mca27010008