An application of fractional derivatives to a thermo-convective viscous fluid with Dufour and Schmidt effects over a rotating disk

Abstract

The unsteady, free convective and incompressible fluid flow over a rotating porous disk with mass and heat transfer under the radiation effect has been studied. The governing equations are transformed into a set of first-order differential equations by using suitable transformation. The first-order differential equations are then transform into a system of fractional order differential equations through Caputo derivatives. The numerical solution of the system of fractional order differential-equations are obtained by using the predictor-corrector method. The Runge Kutta method order 4 method has been used to obtained the classical solution. The effect of all physical parameter involved in the problem are presented graphically. It shows that both the concentration and temperature get decreased with Dufour effect. The radial, axial and azimuthal velocities decreases with suction effect in the boundary layer. These observation are obtained by using Predictor-corrector method and to point out the validity of the result, the well known numerical technique Runge Kutta order 4 method is used.

Keywords

Fractional derivatives predictor-corrector method similarity transformation RK4 schmidt effect

1. Introduction

One of convential problem which play a significant role in fluid dynamics is heat transfer with rotating disk flow. It has a remarkable application in varies field. Some acridity has been found in research from previous many years, like wind energy, solar, or extraction energy from hydro power. From last few years the field of fractional calculus got much attention. As a result most of the gas-turbine manufacturers from different countries are providing different products of research. Many researcher have been discussed solutions in various aspects for the flow by a rotating disk, The first researcher was [26], who discussed fluid flow on the rotating disk surface. [27] identify few limitation in Karman’s work, and with help of two series expansion he presented more accurate result, Later with some modification in the work of Cochran, [8] solved the problem for unsteady case. [10] solved the heat transfer problem in steady state for various prandtl numbers. At any Prandtl number and constant temperature [12] examined the heat transfer at steady on the rotating disk. [11] considered the uniform blowing effect on the flow induced due to rotating disk. In the presence of radiative heat flux and variable thickness [22] examined flow due to rotating disk. UI Rizwan et al. [9] studied water-based squeezing flow between two parallel disks in the presence of carbon nanotubes. Mansha et al. [15] find analytical solutions of the rotational, unsteady flow of fractional order non-newtonian fluids. [13] investigated the heat transfer phenomena of non flate surface of the disk. [10] developed Crank-Niclson method for the system of non-linear differential equations, obtained through heat transfer and steady flow.

In space technology the heat transfer and the effect of radiation has a remarkable role and very less work so far has been done on it, The heat and mass transfer effects are very important in the presence of high temperature and have wide application in industry and specially in engineering, like food dry-freezing phenomena, cans theraml sterlization, power generation, aerospace, rotating machinery and industrial equipment, chemical processing and automotive. Bergman et al. [23] explained several concentration and thermal applications of heat and mass transfer. Nield and Bejan [7] discussed some important applications of heat and mass transfer in science and modern technology in their book. The gray fluid thermal radiation was examined by [25]. Due to its great application in technology field, The study of a rotating disk on the flow problem has been gradually extended to hydromagnetics. It is used in hard disk manufacturing, which has very high capacity of storage. Some important result was pointed by [24] by using magnetic field on rotating disk. The [20] solved momentum equation on porous rotating disk of an unsteady flow with time dependent velocities. Keep in view its application unsteady squeeze flow in motion between two disc have great importance in engineering and scientific studies. [3] formulated the phenomena of mass and heat transfer due to non isothermal wavy rotating disk. Between two dimensional or circular plates the [5] formulated the squeezing of fluid.

The importance of fractional calculus cannot be avoided in the field of engineering and advanced technology, because where the classical calculus fails at the unreachable points, such kind of function can be characterized by fractional derivatives at those points. Fractional calculus has become the great area of research, since the last half of twentieth century. It is actually the generalization to an arbitrary order of integration and differentiation. This field have many applications in biomathematics, quantum mechanics, control theory and in fluid mechanics. The origin of this field was in 1695 by raising a letter written to L’Hospital by Leibnitz. There are several definitions, presented by many mathematicians, of fractional derivatives. But among those the first well known definition was presented by Riemann Liouville. The only drawbacks in this definition was non zero result of constant terms in the functions. Caputo proposed another definition of fractional derivative, which over come on the limitations of Riemann Liouville’s definition. But still the kernel singularity problem were in both derivatives. Numerous fundamental and important applications has been recently studied by [6, 5, 17], Caputo’s concept was further modified by [2, 1, 4] by showing new fractional derivatives and its application. Later on, a new definition of fractional order derivative based on the exponential functions, with non singular kernel was introduced by Caputo and Fabrizio. Which was more reliable and suitable for most of the physical problems. The kernel was non-singular, but non local. Finally Atangana and Baleanu introduced a more generlized definition of fractional derivative with non local and non singular kernel. The analytical solution, using Caputo-Fabrizio derivative, of Maxwell fluid with slip effect has solved by Hammouch et al. [18]. [19] examined the unsteady viscous fluid behaviors with considering Dufour and Soret effects along with mass and heat transfer, using BVP4C package. But they do not consider the unsteadiness effects in both temperature and concentration.

The present work is basically the extension of [19], in which we consider the unsteadiness of temperature and concentration. The fractional behavior of free convective, unsteady and incompressible fluid flow over a rotating disk with mass and heat transfer under the radiation effects is formulated and discussed. For this purpose the modeled equations are investigated numerically by using two different approaches, the classical one RK4 and fractional Predictor-corrector method. For validity both results are graphically compared, which shows best agreement with each other, in the next portion, the problem will be formulated, analyzed and discussed.

Figure 1.

Geometry of the flow over a stretching rotating disk.

2. Mathematical formulation of the problem

In the z-direction the fluid is considered to be infinite. On the surface of rotating disk the incompressible, Axial symmetric and unsteady flow of viscous fluid with mass transfer and heat has been assumed. At the center polar coordinate $(r,\theta,z)$ has taken of the disk. In the direction of $(r,\theta,z)$ the velocity component $(u,v,w)$ is respectively. The disk surface is kept at uniform concentration $C_{0}$ and Temperature $T_{0}$ . Suppose with angular velocity $\Omega$ disk is rotating. At fixed concentration $C_{\infty}$ fixed temperature $T_{\infty}$ and fixed pressure $P_{0}$ away from the disk the free stream is maintain. But the disk surface is kept at uniform concentration $C_{0}$ and uniform temperature. According to above supposition the governing equation can be stated as [16, 21]:

$\displaystyle\frac{\partial u}{\partial r}+\frac{\partial w}{\partial z}+\frac% {u}{r}=0,$ (1) $\displaystyle\rho\left(\frac{\partial u}{\partial t}+u\frac{\partial u}{% \partial r}+w\frac{\partial u}{\partial z}-\frac{v^{2}}{r}\right)=-\frac{% \partial p}{\partial r}+\mu\left(\frac{\partial^{2}u}{\partial r^{2}}+\frac{% \partial^{2}u}{\partial z^{2}}+\frac{1}{r}\frac{\partial u}{\partial r}-\frac{% u}{r^{2}}\right),$ (2) $\displaystyle\rho\left(\frac{\partial v}{\partial t}+u\frac{\partial v}{% \partial r}+w\frac{\partial v}{\partial z}+\frac{uv}{r}\right)=\mu\left(\frac{% \partial^{2}v}{\partial r^{2}}+\frac{\partial^{2}v}{\partial z^{2}}+\frac{1}{r% }\frac{\partial v}{\partial r}-\frac{v}{r^{2}}\right),$ (3) $\displaystyle\rho\left(\frac{\partial w}{\partial t}+u\frac{\partial w}{% \partial r}+w\frac{\partial w}{\partial z}\right)=-\frac{\partial p}{\partial z% }+\mu\left(\frac{\partial^{2}w}{\partial r^{2}}+\frac{\partial^{2}w}{\partial z% ^{2}}+\frac{1}{r}\frac{\partial w}{\partial r}\right),$ (4) $\displaystyle\rho\left(\frac{\partial T^{*}}{\partial t}+u\frac{\partial T^{*}% }{\partial r}+w\frac{\partial T^{*}}{\partial z}\right)=\frac{k}{C_{P}}\left(% \frac{\partial^{2}T^{*}}{\partial r^{2}}+\frac{1}{r}\frac{\partial T^{*}}{% \partial r}+\frac{\partial^{2}T^{*}}{\partial z^{2}}\right)$ (5) $\displaystyle\quad-\frac{1}{C_{P}}\frac{\partial q_{r}}{\partial z}+\frac{DK_{% T^{*}}}{C_{s}C_{p}}\left(\frac{\partial^{2}C^{*}}{\partial r^{2}}+\frac{1}{r}% \frac{\partial C^{*}}{\partial r}+\frac{\partial^{2}C^{*}}{\partial z^{2}}% \right),$ $\displaystyle\rho\left(\frac{\partial C^{*}}{\partial t}+u\frac{\partial C^{*}% }{\partial r}+w\frac{\partial C^{*}}{\partial z}\right)=D\left(\frac{\partial^% {2}C^{*}}{\partial r^{2}}+\frac{1}{r}\frac{\partial C^{*}}{\partial r}+\frac{% \partial^{2}C^{*}}{\partial z^{2}}\right)$ (6) $\displaystyle\quad+\frac{DK_{T}}{T_{m}}\left(\frac{\partial^{2}T^{*}}{\partial r% ^{2}}+\frac{1}{r}\frac{\partial T^{*}}{\partial r}+\frac{\partial^{2}T^{*}}{% \partial z^{2}}\right),$

The boundary conditions for the fluid flow induced by an infinite rotating disk are given as

$\displaystyle u=0,v=\frac{\Omega_{0}r}{1-\alpha t},w=\omega_{0}(t),p=p_{0},T^{% *}=T_{0},C^{*}=C_{0},\ \text{at}\ z=0$ (7) $\displaystyle\text{and}\ u\longrightarrow 0,v\longrightarrow 0,T^{*}% \longrightarrow T_{\infty},C^{*}\longrightarrow C_{\infty}\ \text{as}\ z% \longrightarrow\infty.$

Here the thermal radiation term defined as [14]

$\displaystyle q_{r}=\frac{4\sigma^{*}}{3K^{*}}\frac{\partial T^{4}}{\partial z% }\text{,}$ (8)

where $K^{*}$ and $\sigma^{*}$ is the mean absorption coefficient and Stefan-Boltzmann constant respectively.

Figure 2.

The effect of parameter $S$ on the axial velocity profile (a) RK4 results, (b) classical order $\alpha=$ 1.0, (c) fractional order $\alpha=$ 0.9, (d) fractional order $\alpha=$ 0.8. When $Pr=$ 6.7, $Sc=$ 0.6, $Rd=$ 0.4, $Du=$ 0.4, $A=B=1$ .

Figure 3.

The effect of suction parameter $A$ on the axial velocity profile (a) RK4 results, (b) classical order $\alpha=$ 1.0, (c) fractional order $\alpha=$ 0.9, (d) fractional order $\alpha=$ 0.8. When $S=$ 0.2, $Pr=$ 6.7, $Sc=$ 0.6, $Rd=$ 0.4, $Du=$ 0.4, $B=$ 1.

To linearized $T^{4}$ at $T_{\infty}$ , let small temperature difference $T$ and $T_{\infty}$

$\displaystyle\left.\begin{array}[]{c}f(x)=T^{4}\ \text{about}\ x=x_{0}% \Rightarrow f(x)=T_{\infty},\\ f^{\prime}(x)=4T^{3}\hskip 62.596063pt\Rightarrow f^{\prime}(x)=4T_{\infty}^{3% },\\ f^{\prime\prime}(x)=12T^{2}\hskip 62.596063pt\Rightarrow f^{\prime\prime}(x)=1% 2T_{\infty}^{2},\\ \cdot\hskip 142.26378pt\cdot\\ \cdot\hskip 142.26378pt\cdot\\ \cdot\hskip 142.26378pt\cdot\\ f^{n}(x)=T^{4}=\Sigma_{n=0}^{\infty}\frac{f^{n}(T_{\infty})}{n!}(T-T_{\infty})% ^{n}\end{array}\right\}$ (9)

We get the following form by above summation

$\displaystyle T^{4}=f(T_{\infty})+f^{\prime}(T_{\infty})(T-T_{\infty})+\frac{f% ^{\prime\prime}(T_{\infty})(T-T_{\infty})^{2}}{2!}+\ldots$ (10)

Neglecting the second and higher terms and considered the term $T_{4}$ in a Taylor series about $T_{\infty}$ , we get

$\displaystyle T^{4}\simeq 4T_{\infty}^{3}T^{*}-3T^{4}_{\infty}.$ (11)

by using Eqs (8) and (11), Eq. (2) becomes

$\displaystyle\frac{\partial T^{*}}{\partial t}+u\frac{\partial T^{*}}{\partial r% }+w\frac{\partial T^{*}}{\partial z}=\frac{k}{\rho C_{P}}\left(\frac{\partial^% {2}T^{*}}{\partial r^{2}}+\frac{1}{r}\frac{\partial T^{*}}{\partial r}+\frac{% \partial^{2}T^{*}}{\partial z^{2}}\right)+\frac{16\sigma^{*}T^{*}3_{\infty}}{3% \rho K^{*}C_{P}}\frac{\partial^{2}T^{*}}{\partial z^{2}}+\frac{DK_{T}}{C_{s}C_% {p}}\left(\frac{\partial^{2}C^{*}}{\partial r^{2}}+\frac{1}{r}\frac{\partial C% ^{*}}{\partial r}+\frac{\partial^{2}C^{*}}{\partial z^{2}}\right).$ (12)

Figure 4.

The effect of suction parameter $A$ on the radial velocity profile (a) RK4 results, (b) classical order, $\alpha=$ 1.0, (c) fractional order $\alpha=$ 0.9, (d) fractional order $\alpha=$ 0.8. When $S=$ 0.2, $Pr=$ 6.7, $Sc=$ 0.6, $Rd=$ 0.4, $Du=$ 0.4, $B=$ 1.

Figure 5.

The effect of parameter $S$ on the radial velocity profile (a) RK4 results, (b) classical order $\alpha=$ 1.0, (c) fractional order $\alpha=$ 0.9, (d) fractional order $\alpha=$ 0.8. When $Pr=$ 6.7, $Sc=$ 0.6, $Rd=$ 0.4, $Du=$ 0.4, $A=B=1$ .

In view of [14], we can use the below transformations

$\displaystyle u=\frac{\Omega_{0}r}{(1-\alpha t)}F^{\prime}(\eta),v=\frac{% \Omega_{0}r}{(1-\alpha t)}G(\eta),w=-2\frac{(\nu\Omega_{0})^{\frac{1}{2}}}{(1-% \alpha t)^{\frac{1}{2}}}f(\eta),p=\frac{\rho\nu\Omega_{0}r}{(1-\alpha t)}p(% \eta),$ (13) $\displaystyle\Omega(t)=\frac{\Omega_{0}}{(1-\alpha t)}\text{,}\hskip 5.690551% pt\theta(\eta)=\frac{T^{*}-T_{\infty}}{T_{0}-T_{\infty}}\text{,}\hskip 5.69055% 1pt\phi(\eta)=\frac{C^{*}-C_{\infty}}{C_{0}-C_{\infty}}\hskip 5.690551pt\text{% and}\hskip 5.690551pt\eta=\frac{z}{\sqrt{1-\alpha t}}\sqrt{\frac{\Omega_{0}}{% \nu}}.$

By using Eq. (2) in Eqs (1)–(4), Eqs (2) and (12) reduced to nonlinear ODEs

$\displaystyle F^{\prime\prime\prime}+2FF^{\prime\prime}+G^{2}-F^{\prime 2}=S(% \frac{1}{2}\eta F^{\prime\prime}+F^{\prime}),$ (14) $\displaystyle G^{\prime\prime}-2F^{\prime}G+2FG^{\prime}=S(\frac{1}{2}\eta G^{% \prime}+G),$ (15) $\displaystyle p^{\prime}-2F^{\prime\prime}-4FF^{\prime}=-S(\eta F^{\prime}+F),$ (16) $\displaystyle\frac{\theta^{\prime\prime}}{Pr}\left(1+\frac{4}{3Rd}\right)+% \left(\frac{1}{2}S\eta+2F\right)\theta^{\prime}+Du\phi^{\prime\prime}=0,$ (17) $\displaystyle\frac{1}{Sc}\phi^{\prime\prime}+\left(\frac{1}{2}S\eta+2F\right)% \phi^{\prime}+\textit{So}\ \theta^{\prime\prime}=0.$ (18)

The transform boundary conditions are given as

$\displaystyle F(0)=\frac{A}{2},F^{\prime}(0)=0,G(0)=1,p(0)=B,\theta(0)=-1,\phi% (0)=-1,\ \text{at}\ \eta=0,$ (19) $\displaystyle\text{and}\ F^{\prime}(\eta)\longrightarrow 0,G(\eta)% \longrightarrow 0,\theta(\eta)\longrightarrow 0,\phi(\eta)\longrightarrow 0\ % \text{as}\ \eta\longrightarrow\infty,$

with physical parameters of suction and unsteadiness A and B given in nomenclature respectively.

Figure 6.

The effect of parameter $S$ on the azimuthal velocity profile (a) RK4 results, (b) classical order $\alpha=$ 1.0, (c) fractional order $\alpha=$ 0.9, (d) fractional order $\alpha=$ 0.8. When $Pr=$ 6.7, $Sc=$ 0.6, $Rd=$ 0.4, $Du=$ 0.4, $A=B=1$ .

Figure 7.

The effect of parameter $A$ on the azimuthal velocity profile (a) RK4 results, (b) classical order $\alpha=$ 1.0, (c) fractional order $\alpha=$ 0.9, (d) fractional order $\alpha=$ 0.8. When $S=$ 0.2, $Pr=$ 6.7, $Sc=$ 0.6, $Rd=$ 0.4, $Du=$ 0.4, $B=1$ .

3. Preliminaries on the Caputo fractional derivatives

The basic definition of Caputo and their relevant properties are presented below.

3.1 Caputo fractional derivatives

.

Let $a>0,t>a;a,t\in\Re.$ The Caputo fractional derivative of order Î± of function $f\in C^{n}$ is define as

$\displaystyle D_{t}^{\alpha}f(t)=\frac{1}{\Gamma(n-\alpha)}\int_{\alpha}^{t}% \frac{f^{n}(\xi)}{(t-\xi)^{a+1-n}}d\xi n-1<\alpha<n\in N$ (20)

Figure 8.

The effect of parameter $P r$ on the Temperature distribution profile (a) RK4 results, (b) classical order $\alpha=$ 1.0, (c) fractional order $\alpha$ = 0.9, (d) fractional order $\alpha=$ 0.8. When $S=$ 0.2, $Sc=$ 0.6, $Rd=$ 0.4, $Du=$ 0.4, $A=B=1$ .

Figure 9.

The effect of parameter $R d$ on the Temperature distribution profile (a) RK4 results, (b) classical order $\alpha=$ 1.0, (c) fractional order $\alpha=$ 0.9, (d) fractional order $\alpha=$ 0.8. When $S=$ 0.2, $Pr=$ 6.7, $Sc=$ 0.6, $Du=$ 0.4, $A=B=1$ .

3.1.1 Property

Let $f(t),g(t)$ : $[a,b]\longrightarrow\Re$ be such that $D_{t}^{\alpha}f(t)$ and $D_{t}^{\alpha}g(t)$ exist almost everywhere and Let $c_{1},c_{2}\in\Re.$ Then $D_{t}^{\alpha}\{c_{1}f(t)+c_{2}g(t)\}$ exist almost everywhere and

$\displaystyle D_{t}^{\alpha}\{c_{1}f(t)+c_{2}g(t)\}=c_{1}D_{t}^{\alpha}f(t)+c_% {2}D_{t}^{\alpha}g(t).$ (21)

3.1.2 Property

The constant function fractional derivative $f(t)\equiv c$ is zero:

$\displaystyle D_{t}^{\alpha}\equiv 0,$

Now considering the general fractional differential equation by using the Caputo derivative:

$\displaystyle D_{t}^{\alpha}x(t)=f(t,x(t)),\alpha\in(0,1)$ (22)

with initial conditions $x_{0}=x(t_{0})$

4. Solution methodology

The modeled Eqs (14)–(18) are transformed into a system of first-order differential equations by using the following variables as

$\displaystyle\left.\begin{array}[]{r}\text{Let}\ F=y_{1},F^{{}^{\prime}}=y_{2}% ,F^{{}^{\prime\prime}}=y_{3},G=y_{4},G^{{}^{\prime}}=y_{5},p=y_{6},\theta=y_{7% },\\ \theta^{{}^{\prime}}=y_{8},\Phi=y_{9}\ \text{and}\ \Phi^{{}^{\prime}}=y_{10}.% \end{array}\right\}$ (23) $\displaystyle y_{1}^{\prime}=y_{2},$ (24) $\displaystyle y_{2}^{\prime}=y_{3},$ (25) $\displaystyle y_{3}^{\prime}=S\left(\frac{1}{2}\eta y_{3}+y_{2}\right)+y_{2}^{% 2}-y_{4}^{2}-2y_{1}y_{3},$ (26)

$\displaystyle y_{4}^{\prime}=y_{5},$ (27) $\displaystyle y_{5}^{\prime}=S(\frac{1}{2}\eta y_{5}+y_{4})-2y_{1}y_{5}+2y_{2}% y_{4},$ (28) $\displaystyle y_{6}^{\prime}=-S(\eta y_{2}+y_{1})+4y_{1}y_{2}+2y_{3},$ (29) $\displaystyle y_{7}^{\prime}=y_{8},$ (30) $\displaystyle y_{8}^{\prime}=\frac{3P_{r}R_{d}(D_{u}S_{c}Sy_{10}+4y_{1}y_{8}-% \eta Sy_{8}-4S_{c}D_{u}y_{1}y_{8})}{-8+6R_{d}(-1+D_{u}P_{r}S_{0}S_{c})}$ (31) $\displaystyle y_{9}^{\prime}=y_{10},$ (32) $\displaystyle y_{10}^{\prime}=\frac{S_{c}(-(4+3R_{d})S\eta y_{10}+(3P_{r}SR_{d% }S_{0}-4y_{1}(-4-3R_{d}S_{0}P_{r}))y_{8}}{-8+6R_{d}(-1+D_{u}P_{r}S_{0}S_{c})},$ (33)

Figure 10.

The effect of parameter $D u$ on the Temperature distribution profile (a) RK4 results, (b) classical order $\alpha=$ 1.0, (c) fractional order $\alpha=$ 0.9, (d) fractional order $\alpha=$ 0.8. When $S=$ 0.2, $Pr=$ 6.7, $Sc=$ 0.6, $Rd=$ 0.4, $A=B=1$ .

Figure 11.

The effect of parameter $S c$ on the Temperature distribution profile (a) RK4 results, (b) classical order $\alpha=$ 1.0, (c) fractional order $\alpha=$ 0.9, (d) fractional order $\alpha=$ 0.8. When $S=$ 0.2, $Pr=$ 6.7, $Rd=$ 0.4, $Du=$ 0.4, $A=B=1$ .

By applying the Caputo fractional-order derivative to a system of first-order ordinary differential equations with given variables in Eq. (23) to obtain the system of fractional-order equation as

$\displaystyle D_{\eta}^{\alpha}y_{1}=y_{2},$ (34) $\displaystyle D_{\eta}^{\alpha}y_{2}=y_{3},$ (35) $\displaystyle D_{\eta}^{\alpha}y_{3}=S\left(\frac{1}{2}\eta y_{3}+y_{2}\right)% +y_{2}^{2}-y_{4}^{2}-2y_{1}y_{3},$ (36)

$\displaystyle D_{\eta}^{\alpha}y_{4}=y_{5},$ (37) $\displaystyle D_{\eta}^{\alpha}y_{5}=S\left(\frac{1}{2}\eta y_{5}+y_{4}\right)% -2y_{1}y_{5}+2y_{2}y_{4},$ (38) $\displaystyle D_{\eta}^{\alpha}y_{6}=-S(\eta y_{2}+y_{1})+4y_{1}y_{2}+2y_{3},$ (39) $\displaystyle D_{\eta}^{\alpha}y_{7}=y_{8},$ (40) $\displaystyle D_{\eta}^{\alpha}y_{8}=\frac{3P_{r}R_{d}(D_{u}S_{c}Sy_{10}+4y_{1% }y_{8}-\eta Sy_{8}-4S_{c}D_{u}y_{1}y_{8})}{-8+6R_{d}(-1+D_{u}P_{r}S_{0}S_{c})},$ (41) $\displaystyle D_{\eta}^{\alpha}y_{9}=y_{10},$ (42) $\displaystyle D_{\eta}^{\alpha}y_{10}=\frac{S_{c}(-(4+3R_{d})S\eta y_{10}+(3P_% {r}SR_{d}S_{0}-4y_{1}(-4-3R_{d}S_{0}P_{r}))y_{8}}{-8+6R_{d}(-1+D_{u}P_{r}S_{0}% S_{c})}.$ (43)

Figure 12.

The effect of parameter $P r$ on the concentration distribution (a) RK4 results, (b) Classical order $\alpha=$ 1.0, (c) fractional order $\alpha=$ 0.9, (d) fractional order $\alpha=$ 0.8. When $S=$ 0.2, $Sc=$ 0.6, $Rd=$ 0.4, $Du=$ 0.4, $A=B=1$ .

Figure 13.

The effect of parameter $R d$ on the concentration distribution (a) RK4 results, (b) classical order $\alpha=$ 1.0, (c) fractional order $\alpha=$ 0.9, (d) fractional order $\alpha=$ 0.8. When $S=$ 0.2, $Pr=$ 6.7, $Sc=$ 0.6, $Du=$ 0.4, $A=B=1$ .

The transformed boundary conditions for the first-order differential equations are as

$\displaystyle\left.\begin{array}[]{r}y_{1}(0)=\frac{A}{2},\hskip 5.690551pty_{% 2}(0)=0,y_{4}(0)=1,y_{6}(0)=B,y_{7}(0)=-1,y_{9}(0)=-1,\\ y_{2}(\infty)\longrightarrow 0,y_{2}(\infty)=0,y_{7}(\infty)=0\ \text{and}\ y_% {9}(\infty)=0.\end{array}\right\}$ (44)

5. Results and discussion

The non linear system of Eqs (12)–(16) along with boundary condition Eq. (2) are analyzed by two different method Predictor-corrector method and RK4 method for different interest physical parameters, like $\alpha$ , $R d$ , $B$ , $S$ , $D u$ , $P r$ , $S c$ , $S o$ and $A$ .

The physical interest parameters involved in this problem are discussed in detail. The two different technique show a great agreement with each other upto large extend. For this purpose the behavior of different function like pressure, temperature, velocity and mass transfer are shown graphically in Figs 2–15. Figure 1 shows the geometry of the problem. The left side Figs 2–15a represents the numerical method RK4 results and the rest figures represents fractional method results for different values of $\alpha$ .

Figure 14.

The effect of parameter $D u$ on the concentration distribution (a) RK4 results, (b) classical order $\alpha=$ 1.0, (c) fractional order $\alpha=$ 0.9, (d) fractional order $\alpha=$ 0.8. When $S=$ 0.2, $Pr=$ 6.7, $Sc=$ 0.6, $Rd=$ 0.4, $A=B=1$ .

Figure 15.

The effect of parameter $S c$ on the concentration distribution (a) RK4 results, (b) classical order $\alpha=$ 1.0, (c) fractional order $\alpha=$ 0.9, (d) fractional order $\alpha=$ 0.8. When $S=$ 0.2, $Pr=$ 6.7, $Rd=$ 0.4, $Du=$ 0.4, $A=B=1$ .

The effect of unsteadiness parameter $S$ on the axial velocity profile have been shown in Fig. 2a–d respectively. From Fig. 2a–d, It can be seen that for large values of unsteadiness parameter $S$ the velocity increase in both classical case as well as fractional case, Because the thickness of the momentum boundary layer increases for greater values of $S$ , As expected from Fig. 3a–d the axial velocity decline as with increase in suction parameter $A$ but here the classical case show greater axial velocity than fractional case.

Figure 4a–d describe the effect of suction parameter $A$ on the radial velocity, It can be observed from Fig. 4a–d that with increase in suction parameter $A$ radial velocity decrease. This is only due to the porousity of the rotating disk, The more and more incoming fluid directly passes through it, Therefore when suction parameter increase radial velocity decrease.

Figure 5a–d describe the effect of unsteadiness parameter $S$ on the radial velocity, It can be seen from Fig. 5a–d that with increase in unsteadiness parameter $S$ radial velocity increase.

In Fig. 6a–d we have examined the azimuthal component of velocity $g$ , It can be clearly observed that in these figures classical case and fractional case both show best agreement. The effect of unsteadiness parameter $S$ is shown in Fig. 6a–d respectively. It can be seen azimuthal velocity show its maximum at very slight increase in parameter $S$ while Fig. 7a–d depicts that azimuthal velocity decreases with increases in parameter $A$ , but Fig. 7c–d show some interesting result when we reduce $\alpha$ from 1.0 to 0.9 and 0.8, the azimuthal velocity show reverse behavior.

Figures 8 and 9a–d illustrated the effect of Prandtl number $P r$ and thermal radiation parameter $R d$ on the temperature profile $\theta$ . It can be observed that decreases in Prandtl $P r$ the thermal boundary layer and temperature increases shown in Fig. 8a–d, Physically higher thermal diffusivity is posses in smaller Prandtl fluid and lower thermal diffuivity is posses in large Prandtl fluids. The temperature profile $\theta$ decreases with increases in radiation parameter $R d$ illustrated in Fig. 9a–d, $Rd=\frac{K*K}{4\sigma*T^{3}_{\infty}}$ , The temperature show inverse relation with thermal radiation, mean with increases in radiation the fluid will start cooling and as result temperature will come down.

Figure 10a–d show the variation of temperature profile $\theta$ with different flow parameters. The Dufour term $D u$ effects is graphically displayed in Fig. 10a–d, throughout the boundary layer the temperature $\theta$ decreases due to rise in $D u$ , From Fig. 11a–d it can be observed that the decreases in temperature is due to increases in Schmidt number $S c$ , The boundary layer thickness decreases with enhancement of Schmidt number $S c$ which cause temperature decline.

The impact of physical parameters like Prandtl number $P r$ is shown in Fig. 12a–d. The square root of Prandtl number $P r$ is inversely proportional to the thickness of thermal boundary layer, due to variation in Prandtl number $P r$ the decreases in the concentration boundary layer occurred, Which reduce concentration distribution shown in Fig. 12a–d. Figure 13a–d shows that increases in parameter $R d$ rises the concentration distribution. Greater values of thermal radiation parameter $R d$ enhanced the thermal boundary layer and concentration distribution.

Figures 14a–d elucidates that the enhancement in boundary layer thickness and concentration distribution correspond to Dufour number $D u$ and similarly their decrease also reduce concentration distribution. Figure 15a–d illustrate the variation of Schmidt number $S c$ on concentration profile, That variation in Schmidt number $S c$ rises concentration distribution. In Fig. 8a–d the classical case and fractional case completely collapse with each other and gives exactly same result.

6. Conclusion

In this work, The free convective, unsteady flow with the influence of radiation effect on combined mass and heat transfer due to a rotating disk and also with Dufour and Schmidt effect has been investigated and presented. On the basis of this computation the following result has been drawn:

•

The fluid temperature reduced with radiation effect and rises the concentration distribution.

•

The decrease of axial, radial and azimuthal velocities with suction effect at the boundary layer .

•

The enhancement of Schimdt number result in rising the concentration distribution and lowering the fluid temperature .

•

With Dufour number $D u$ effect the temperature of fluid and concentration both get reduced.

•

Prandtl number $P r$ show direct relation with concentration and inverse relation with the fluid temperature.

•

The increase of unsteadiness parameter $S$ results in increase of radial, axial and azimuthal velocities.

Footnotes

Nomenclature

References

Atangana

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