Numerical Solution of Non-Newtonian Fluid Flow Due to Rotatory Rigid Disk

Abstract

In this article, the non-Newtonian fluid model named Casson fluid is considered. The semi-infinite domain of disk is fitted out with magnetized Casson liquid. The role of both thermophoresis and Brownian motion is inspected by considering nanosized particles in a Casson liquid spaced above the rotating disk. The magnetized flow field is framed with Navier’s slip assumption. The Von Karman scheme is adopted to transform flow narrating equations in terms of reduced system. For better depiction a self-coded computational algorithm is executed rather than to move-on with build-in array. Numerical observations via magnetic, Lewis numbers, Casson, slip, Brownian motion, and thermophoresis parameters subject to radial, tangential velocities, temperature, and nanoparticles concentration are reported. The validation of numerical method being used is given through comparison with existing work. Comparative values of local Nusselt number and local Sherwood number are provided for involved flow controlling parameters.

1. Introduction

The examination of non-Newtonian fluids has received remarkable attention from researchers and scientists because of their extensive use in industrial and technological areas. For instance, paints, synthetic lubricants, sugar solutions, certain oils, clay coating, drilling muds, and blood as a biological fluid are common examples of non-Newtonian fluids, just to mention a few. The fundamental mathematical equations given by Navier–Stokes cannot briefly delineate the flow field characteristics of non-Newtonian fluids because of the complex mathematical expression involved in the formulation of flow problem. In addition, the relation between strain rate and shear stress is non-linear so the single constitutive expressions are fruitless to report complete description of flows subject to non-Newtonian fluids. Numerous non-Newtonian fluid models are exposed to explore rheological characteristics, namely Bingham Herschel–Bulkley fluid model, Seely, Carreau Carreau–Yasuda, Sisko, Eyring, Cross, Ellis, Williamson, tangent hyperbolic, Generalized Burgers, Burgers, Oldroyd-8 constants, Oldroyd-A, Oldroyd-B fluid model, Maxwell, Jeffrey, Casson fluid model, etc. Researchers discussed flow characteristics of non-Newtonian fluid models via stretching surfaces by incorporating pertinent physical effects. Among these, Casson fluid model has many advantages as compared to rest of fluid models. This model can be used to approximate the properties of blood and daily life suspensions. One can assessed recent developments in this direction in References [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15].

The centrifugal filtration, gas turbine rotors, rotating air cleaning machines, food processing, medical equipment, system of electric-power generation, crystal growth processes, and many others are the practical applications of rotational fluids flow. Therefore, analysis of flows due to rotation of solid surfaces is widely recognized by scientists, and researchers like Karman [16] firstly report viscous fluid flow induced by rotating solid disk. A special transformation named as Karman transformation given by him for the first time in this attempt. These transformations are utilized for conversion of fundamental equations termed as Naviers–Stokes equations in terms of ordinary differential system. Later on, a number of studies were given by researchers to depict the flow characteristics of both Newtonian and non-Newtonian fluids model over a rotating disk. Preceding these analyses in 2013, the extension of Karman problem was given by Turkyilmazoglu and Senel [17]. In this attempt they discussed numerical results for heat transfer properties of rotating partial slip fluid flow. In 2014, the magnetized slip flow via porous disk was reported by Rashidi et al. [18]. In addition, they discussed entropy measurements for this case. The flow properties in the presence of nano-size particles were discussed by Turkyilmazoglu [19]. He used numerical algorithm for solution purpose. In fact, he dealt comparative execution to report the impact of various nanoparticles suspended in fluid flow regime. Afterwards, tremendous attempts are given in this direction by way of both analytical and numerical approach. One can find the concern developments on rotating flows in References [20,21,22,23,24,25,26,27,28,29,30,31].

The present article contains analysis of Casson liquid towards rotating rigid disk. The Casson flow field is magnetized and has nanoparticles. Further, slip effects are also taken into account. The physical model is translated in terms of mathematical model. For solution purposes, the van Karman way of study is adopted. A computational algorithm is applied and the obtained results of involved parameters of concerned quantities are discussed via graphs and tables. Further, the current attempt is compared with existing literature and we found a good agreement which leads to the surety of the present work.

2. Problem Formulation

The Casson liquid is quipped above the disk for $\overline{z}>0$. The constant frequency ($\overline{\mathsf{\Omega }}$) is constant. The semi bounded magnetized flow regime contains suspended nanoparticles. The surface is taken with velocity slip condition. The quantities ($\overline{u}$,$\overline{v}$,$\overline{w}$) are in ($\overline{r}$,$\overline{\varphi }$,$\overline{z}$) directions. The ultimate differential system of said problem is:

$$\frac{\partial \overline{w}}{\partial \overline{z}}+\frac{\partial \overline{u}}{\partial \overline{r}}+\frac{\overline{u}}{\overline{r}}=0,$$

(1)

$$\overline{w}\frac{\partial \overline{u}}{\partial \overline{z}}+\overline{u}\frac{\partial \overline{u}}{\partial \overline{r}}-\frac{{\overline{v}}^2}{\overline{r}}=\nu \left(1+\frac{1}{\lambda }\right)\left(\frac{{\partial }^2\overline{u}}{\partial {\overline{z}}^2}+\frac{1}{\overline{r}}\frac{\partial \overline{u}}{\partial \overline{r}}+\frac{{\partial }^2\overline{u}}{\partial {\overline{r}}^2}-\frac{\overline{u}}{{\overline{r}}^2}\right)-\frac{\sigma B_0{}^2}{{\rho }_f}\overline{u},$$

(2)

$$\overline{w}\frac{\partial \overline{v}}{\partial \overline{z}}+\overline{u}\frac{\partial \overline{v}}{\partial \overline{r}}+\frac{\overline{u}\overline{v}}{\overline{r}}=\nu \left(1+\frac{1}{\lambda }\right)\left(\frac{{\partial }^2\overline{v}}{\partial {\overline{z}}^2}+\frac{{\partial }^2\overline{v}}{\partial {\overline{r}}^2}-\frac{\overline{v}}{{\overline{r}}^2}+\frac{1}{\overline{r}}\frac{\partial \overline{v}}{\partial \overline{r}}\right)-\frac{\sigma B_0{}^2}{{\rho }_f}\overline{v},$$

(3)

$$\overline{w}\frac{\partial \overline{w}}{\partial \overline{z}}+\overline{u}\frac{\partial \overline{w}}{\partial \overline{r}}=\nu \left(1+\frac{1}{\lambda }\right)\left(\frac{{\partial }^2\overline{w}}{\partial {\overline{z}}^2}+\frac{1}{\overline{r}}\frac{\partial \overline{w}}{\partial \overline{r}}+\frac{{\partial }^2\overline{w}}{\partial {\overline{r}}^2}\right),$$

(4)

$$\begin{array}{l}\overline{w}\frac{\partial \overline{T}}{\partial \overline{z}}+\overline{u}\frac{\partial \overline{T}}{\partial \overline{r}}=\alpha \left(\frac{{\partial }^2\overline{T}}{\partial {\overline{z}}^2}+\frac{1}{\overline{r}}\frac{\partial \overline{T}}{\partial \overline{r}}+\frac{{\partial }^2\overline{T}}{\partial {\overline{r}}^2}\right)+\frac{{\left(\rho c\right)}_p}{{\left(\rho c\right)}_f}\left[D_B\left(\frac{\partial \overline{T}}{\partial \overline{z}}\frac{\partial \overline{C}}{\partial \overline{z}}+\frac{\partial \overline{T}}{\partial \overline{r}}\frac{\partial \overline{C}}{\partial \overline{r}}\right)\right]\\ +\frac{{\left(\rho c\right)}_p}{{\left(\rho c\right)}_f}\left[\frac{D_T}{{\overline{T}}_{\infty }}\left({\left(\frac{\partial \overline{T}}{\partial \overline{z}}\right)}^2+{\left(\frac{\partial \overline{T}}{\partial \overline{r}}\right)}^2\right)\right],\end{array}$$

(5)

$$\overline{w}\frac{\partial \overline{C}}{\partial \overline{z}}+\overline{u}\frac{\partial \overline{C}}{\partial \overline{r}}=D_B\left(\frac{{\partial }^2\overline{C}}{\partial {\overline{z}}^2}+\frac{1}{\overline{r}}\frac{\partial \overline{C}}{\partial \overline{r}}+\frac{{\partial }^2\overline{C}}{\partial {\overline{r}}^2}\right)+\frac{D_T}{{\overline{T}}_{\infty }}\left[\frac{{\partial }^2\overline{T}}{\partial {\overline{z}}^2}+\frac{1}{\overline{r}}\frac{\partial \overline{T}}{\partial \overline{r}}+\frac{{\partial }^2\overline{T}}{\partial {\overline{r}}^2}\right],$$

(6)

$$\overline{u}=L\frac{\partial \overline{u}}{\partial \overline{z}},\hspace{1em}\overline{v}=\overline{r}\overline{\mathsf{\Omega }}+L\frac{\partial \overline{v}}{\partial \overline{z}},\hspace{1em}\overline{w}=0,\hspace{1em}\overline{T}={\overline{T}}_w,\hspace{1em}\overline{C}={\overline{C}}_w\hspace{1em}\mathrm{at} \overline{z}=0,$$

(7)

$$\overline{u}\to 0,\hspace{1em}\overline{v}\to 0,\hspace{1em}\overline{T}\to {\overline{T}}_{\infty },\hspace{1em}\overline{C}\to {\overline{C}}_{\infty }\hspace{1em}\mathrm{as}\hspace{1em}\overline{z}\to \infty ,$$

(8)

for order reduction one can use the variables [16],

$$\begin{array}{l}\overline{u}=\overline{r}\overline{\mathsf{\Omega }}\frac{dF(\xi )}{d\xi },\hspace{1em}\overline{v}=G(\xi )\overline{r}\overline{\mathsf{\Omega }},\hspace{1em}\overline{w}=-F(\xi )\sqrt{2\overline{\mathsf{\Omega }}\nu },\\ C(\xi )=\frac{\overline{C}-{\overline{C}}_{\infty }}{{\overline{C}}_w-{\overline{C}}_{\infty }},\hspace{1em}T(\xi )=\frac{\overline{T}-{\overline{T}}_{\infty }}{{\overline{T}}_w-{\overline{T}}_{\infty }},\hspace{1em}\xi =\sqrt{\frac{2\overline{\mathsf{\Omega }}}{\nu }}\overline{z}.\end{array}$$

(9)

We get:

$$2\frac{d^{3}F(\xi )}{d{\xi }^3}\left(1+\frac{1}{\lambda }\right)+2F(\xi )\frac{d^{2}F(\xi )}{d{\xi }^2}-{\left(\frac{dF(\xi )}{d\xi }\right)}^2+{\left(G(\xi )\right)}^2-\gamma \frac{dF(\xi )}{d\xi }=0,$$

(10)

$$2\frac{d^{2}G(\xi )}{d{\xi }^2}\left(1+\frac{1}{\lambda }\right)+2F(\xi )\frac{dG(\xi )}{d\xi }-2G(\xi )\frac{dF(\xi )}{d\xi }-\gamma G(\xi )=0,$$

(11)

$$\frac{d^{2}T(\xi )}{d{\xi }^2}+\mathrm{Pr}\left(F(\xi )\frac{dT(\xi )}{d\xi }+N_B\frac{dT(\xi )}{d\xi }\frac{dC(\xi )}{d\xi }+N_T{\left(\frac{dT(\xi )}{d\xi }\right)}^2\right)=0,$$

(12)

$$\frac{d^{2}C(\xi )}{d{\xi }^2}+Le\mathrm{Pr}F(\xi )\frac{dC(\xi )}{d\xi }+\frac{N_T}{N_B}\frac{d^{2}T(\xi )}{d{\xi }^2}=0,$$

(13)

$$\begin{array}{l}F(\xi )=0,\frac{dF(\xi )}{d\xi }=\beta \frac{d^{2}F(\xi )}{d{\xi }^2},G(\xi )=1+\beta \frac{dG(\xi )}{d\xi },T(\xi )=1,C(\xi )=1,\hspace{1em}\mathrm{at}\xi =0,\\ \frac{dF(\xi )}{d\xi }\to 0,\hspace{1em}G(\xi )\to 0,\hspace{1em}T(\xi )\to 0,\hspace{1em}C(\xi )\to 0,\hspace{1em}\mathrm{as}\hspace{1em}\xi \to \infty .\end{array}$$

(14)

and:

$$\begin{array}{l}\gamma =\sqrt{\frac{\sigma B_0{}^2}{{\rho }_f\overline{\mathsf{\Omega }}}},\beta =L\sqrt{\frac{2\overline{\mathsf{\Omega }}}{\nu }},N_B=\frac{{\left(\rho c\right)}_p}{{\left(\rho c\right)}_f}\frac{({\overline{T}}_w-{\overline{T}}_{\infty })D_T}{{\overline{T}}_{\infty }\nu },\\ \\ Le=\frac{\alpha }{D_B},\hspace{1em}\mathrm{Pr}=\frac{\nu }{\alpha },\hspace{1em}{N}_T=\frac{{\left(\rho c\right)}_p}{{\left(\rho c\right)}_f}\frac{({\overline{C}}_w-{\overline{C}}_{\infty })D_B}{\nu },\end{array}$$

(15)

the surface quantities are defined as:

$$\begin{array}{l}\sqrt{{\mathrm{Re}}_{\overline{r}}}{C}_F=\left(1+\frac{1}{\lambda }\right)\frac{d^{2}F(0)}{d{\xi }^2},\hspace{1em}\sqrt{{\mathrm{Re}}_{\overline{r}}}{C}_G=\left(1+\frac{1}{\lambda }\right)\frac{dG(0)}{d\xi },\\ \frac{Nu}{\sqrt{{\mathrm{Re}}_{\overline{r}}}}=-\frac{dT(0)}{d\xi },\hspace{1em}\frac{Sh}{\sqrt{{\mathrm{Re}}_{\overline{r}}}}=-\frac{dC(0)}{d\xi },\end{array}$$

(16)

3. Computational Outline

To transform the system of Equations (10)–(13) into an initial value problem one can use the dummy substitutions:

$Y_2=F^{\prime }(\xi ),Y_3={F^{\prime }}_2=F^{″}(\xi ),Y_5=G^{\prime }(\xi ),Y_7=T^{\prime }(\xi ),Y_9=C^{\prime }(\xi ),$ so we have

$$\left[\begin{array}{l}{Y^{\prime }}_1\\ {Y^{\prime }}_2\\ {Y^{\prime }}_3\\ {Y^{\prime }}_4\\ {Y^{\prime }}_5\\ {Y^{\prime }}_6\\ {Y^{\prime }}_7\\ {Y^{\prime }}_8\\ {Y^{\prime }}_9\end{array}\right]=\left[\begin{array}{l}{Y}_2\\ Y_3\\ \frac{\gamma Y_2+{(Y_2)}^2-2Y_1{Y}_3-{(Y_4)}^2}{2\left(1+\frac{1}{\lambda }\right)}\\ Y_5\\ \frac{2Y_2{Y}_4+\gamma Y_4-2Y_1{Y}_5}{2\left(1+\frac{1}{\lambda }\right)}\\ Y_7\\ -\mathrm{Pr}\left[Y_1{Y}_7+N_B{Y}_7{Y}_9+N_T{Y}_7{}^2\right]\\ Y_9\\ -Le\mathrm{Pr}{Y}_9+\frac{N_T}{N_B}{Y^{\prime }}_7\end{array}\right]$$

(17)

$$\begin{array}{l}{Y}_1\left(\xi \right)=0,Y_2\left(\xi \right)=\beta F^{″}\left(\xi \right)=\beta {\alpha }_1,Y_3\left(\xi \right)=F^{″}\left(\xi \right),Y_4\left(\xi \right)=1+\beta G^{\prime }\left(\xi \right)=1+\beta {\alpha }_2,\\ Y_5\left(\xi \right)=G^{\prime }\left(\xi \right),Y_6\left(\xi \right)=1,Y_7\left(\xi \right)={\alpha }_3,Y_8\left(\xi \right)=1,Y_9\left(\xi \right)={\alpha }_4,\hspace{1em}\mathrm{when}\xi \to 0,\end{array}$$

(18)

with

$$Y_2\left(\xi \right)=0,Y_4\left(\xi \right)=0,Y_6\left(\xi \right)=0,Y_8\left(\xi \right)=0,\mathrm{when}\xi \to \infty ,$$

(19)

here,${\alpha }_1,{\alpha }_2,{\alpha }_3\mathrm{and}{\alpha }_4$ are initial guess values.

4. Analysis

The Casson fluid (CF) flow is considered on a rigid disk. The flow field is magnetized with suspended nanoparticles. The said problem is controlled mathematically and a numerical solution is offered through the shooting method. In detail, Figure 1, Figure 2, Figure 3, Figure 4, Figure 5 and Figure 6 are used to highlight the variations of both CF velocities ($F^{\prime }(\xi )\mathrm{and}G(\xi )$) via physical parameters, namely $\lambda$, $\gamma$, and $\beta .$ Figure 1 and Figure 2 are plotted to examine the impact of $\lambda$ on CF velocity. It is clear from Figure 1 and Figure 2 that the CF velocity decreases against $\lambda$.

The impact of $\gamma$ on CF velocity is examined and provided via Figure 3. The CF velocity decreases for higher values of $\gamma$. This is due to activation of Lorentz force via increasing $\gamma$. Similarly, the effect of $\gamma$ on tangential velocity $G(\xi )$ is examined and given by means of Figure 4. It is important to note that the tangential velocity decreases for $\gamma$ like radial one. The effect of $\beta$ on radial velocity is offer in Figure 5. It is noticed that the radial velocity reflects a diminishing nature for positive values of $\beta$ and the corresponding momentum boundary layer is also effected and admits decline values. Figure 6 gives the effect of $\beta$ on tangential velocity of Casson fluid parameter. It is observed that the tangential velocity decreases for slip parameter. The Casson fluid temperature is examined and provided via Figure 7, Figure 8 and Figure 9. Particularly, Figure 7 is plotted against $N_T$ while Figure 8 is used to identify the influence of Pr on $T(\xi )$. Figure 9 reports influence of $N_B$ on $T(\xi )$. From these figures we observed that Casson fluid temperature increases towards $N_T$, $N_B$ but opposite trend is testified for Pr. Figure 10, Figure 11 and Figure 12 reports the impact of $Le,N_B\mathrm{and}{N}_t$ on $C(\xi )$. In detail, Figure 10 paints the effect of $Le$ on $C(\xi )$. The Casson concertation decreases for positive variations in Le. The $C(\xi )$ effected significantly towards $N_B$. Figure 11 is evident that the $N_B$ results decline values in $C(\xi )$ for both zero and non-zero values of $\beta$. Such decreasing trend is due to higher values of Brownian force. The change in $C(\xi )$ is observed towards $N_t$ and offer in Figure 12. The higher values of $N_B$ corresponds increasing trends in $C(\xi )$ and related momentum boundary layer. In this attempt the MHD Casson nanofluid flow brought by rotating solid disk in the presence of slip conditions is examined. For comparison purpose, when Casson fluid parameter approaches to infinity our problem absolutely match with Hayat et al. [32]. In this work they studied nanoparticle aspects on viscous fluid flow due to rotating disk along with slip effects numerically. We have compared the variation of both Nusselt and Sherwood numbers with their findings as shown in

Table 1. Local Nusselt number comparison with Hayat et al. [32].

						$\frac{Nu}{\sqrt{{\mathrm{Re}}_{\overline{r}}}}=-\frac{dT(0)}{d\xi }$
$\beta$	$\gamma$	$N_T$	$Le$	$\mathrm{Pr}$	$N_B$	Hayat et al. [32]	Present values
0.2	-	-	-	-	-	0.32655	0.326600
0.5	-	-	-	-	-	0.30360	0.30363
0.8	-	-	-	-	-	0.28715	0.28724
-	0.0	-	-	-	-	0.30494	0.30502
-	0.7	-	-	-	-	0.24421	0.24434
-	1.4	-	-	-	-	0.17566	0.17575
-	-	0.5	-	-	-	0.25913	0.25916
-	-	0.7	-	-	-	0.23865	0.23879
-	-	1.0	-	-	-	0.21010	0.21025
-	-		0.5	-	-	0.29633	0.29642
-	-		1.0	-	-	0.28954	0.28963
-	-	-	1.5	-	-	0.28395	0.28398
-	-	-	-	0.5	-	0.24989	0.24999
-	-	-	-	1.0	-	0.29211	0.29224
-	-	-	-	1.5	-	0.32286	0.32294
-	-	-	-	-	0.5	0.26341	0.26358
-	-	-	-	-	0.7	0.23677	0.23687
-	-	-	-	-	1.0	0.20056	0.20068

and

Table 2. Local Sherwood number comparison with Hayat et al. [32].

						$\frac{Sh}{\sqrt{{\mathrm{Re}}_{\overline{r}}}}=-\frac{dC(0)}{d\xi }$
$\beta$	$\gamma$	$N_T$	$Le$	$\mathrm{Pr}$	$N_B$	Hayat et al. [32]	Present values
0.2	-	-	-	-	-	0.27583	0.27593
0.5	-	-	-	-	-	0.26933	0.26945
0.8	-	-	-	-	-	0.26493	0.26498
-	0.0	-	-	-	-	0.27000	0.27012
-	0.7	-	-	-	-	0.25387	0.25394
-	1.4	-	-	-	-	0.23722	0.23735
-	-	0.5	-	-	-	0.22206	0.22215
-	-	0.7	-	-	-	0.22539	0.22564
-	-	1.0	-	-	-	0.22285	0.22288
-	-		0.5	-	-	0.21373	0.21380
-	-		1.0	-	-	0.30132	0.30145
-	-	-	1.5	-	-	0.38690	0.38696
-	-	-	-	0.5	-	0.22934	0.22944
-	-	-	-	1.0	-	0.26624	0.26636
-	-	-	-	1.5	-	0.31262	0.31276
-	-	-	-	-	0.5	0.30338	0.30342
-	-	-	-	-	0.7	0..31875	0..31887
-	-	-	-	-	1.0	0.32959	0.32978

. One can see from these tables our finding match with existing values in a limiting sense. The trifling difference is due to choice of numerical method used in both attempts. Their values are obtained by build in command in Mathematica while we have used self-coded algorithm (shooting method with R-K scheme) subject to Casson nanofluid flow induced by solid rotating disk. Beside this one can extend idea to computational fluid dynamics in context of industrial and standpoints, see References [32,33,34,35,36,37,38,39,40,41,42].

5. Closing Remarks

A Casson fluid (CF) flow yield by rotating rigid disk is considered. Both the Brownian and thermophoresis aspects are entertained by incorporating nanoparticles. The flow characteristics are reported numerically with the support of computational algorithm. The summary is as follows:

• CF velocities which includes $[G(\xi ),F^{\prime }(\xi )]$ reflects decline trend towards $\beta$.
• CF velocities are decreasing function of $\lambda$ and $\gamma$.
• CFT $[T(\xi )]$ admits inciting nature towards both $N_T$ and $N_B$ but opposite trend is observed for Pr.
• CFC $[C(\xi )]$ shows decline values for both Le, and $N_B$.
• CFC $[C(\xi )]$ reflect inciting trend for $N_T$.
• Comparative values of HTR and MTR are provided for involved flow controlling parameters.