Variable thickness flow over a rotating disk under the influence of variable magnetic field: An application to parametric continuation method

Abstract

The present work explores the behavior of three-dimensional incompressible viscous fluid flow and heat transfer over the surface of a non-flat stretchable rotating disk. A variable thickness fluid is subjected under the influence of an external variable magnetic field and heat transfer. Navier–Stokes equation is coupled with Maxwell equations to examine the hydrothermal properties of fluid. The basic governing equations of motion are diminished to a system of nonlinear ordinary differential equations using appropriate similarity framework, which are further treated with numerical scheme known as parametric continuation method. The parametric continuation method has combined interesting characteristics of both shooting and implicit finite difference methods. For validity of the present numerical scheme, a comparison with the published work is performed and it is found that the results are in excellent agreement with each other. Numerical and graphical results for the velocity, temperature, and magnetic strength profiles as well as skin fractions and Nusselt number are presented and discussed in detail for various physical parameters. The heat transfer process is reduced with positive increment of no-flatness parameter $\zeta$ , while Prandtl number increases the heat transfer rate at the surface of the disk.

Introduction

The rotating disk flows are of much involvement in many hardheaded applications field engineering, for instant, in computers, the heat convection affects temperature of the adjacent electronic components, which results in the flow near a rotating hard disk. The importance of the rotating disk flows in industrial field is worked out by the production of centrifugal separators, automotive industries, aerospace industries, internal combustion engines, ship propellers, centrifugal pumps, marine industries, disk viscometers, flywheels, and methods creating crystals which are used in computer memories.

Von-Karman ¹ carried out the pioneering study of ordinary viscous fluid flow, due to an infinite rotating disk. To get some physical insight into the rotating disk flow structure, Von Karman used Navier–Stokes equations for the first time for a boundary layer flow. Cochran ² corrected and solved the Von Karman problem numerically by making some changes. He concluded that the rotating disk is just like a centrifugal fan which throws and draws the fluid radially. Bdewadt ³ also investigated the Von-Karman’s flow and assumed that the outer fluid is rotating like a solid body and considers a stationary disk. Batchelor ⁴ held open the same structure as that of Von-Karman and meditated the effect of both Von-Karman and Bdewadt.

Stretching boundary has so many applications in engineering and industries, for instant, extrusion of a polymer, process of liquid film condensation, coating emulsion on photographic films, melt-spinning, and hot rolling. A nonuniform velocity is created by a thin polymer sheet on continuously changing surface. It has been shown experimentally that the fluid velocity on the surface of a stretching boundary is proportional to the distance from the orifice. ⁵ Sakiadis^6,7 was the first to open up the study of boundary layer flow on continuously stretching surface with constant speed. Tsou et al. ⁸ experimentally verify Sakiadis work. Crane ⁹ investigated the two-dimensional boundary layer flow of a Newtonian liquid due to a stretching surface which moves in its own plane with linear velocity. Power-law velocity function was included in the fluid flow on a stretching surface. Liao^10,11 obtain a new branch solution for both permeable and impermeable stretching sheets and observed that under certain conditions more than one solutions are possible.

A very little attention has been given to a variable thickness fluid, although it has very strong applications in engineering and natural sciences as it is used in mechanical marine, aeronautical, civil systems, and so on. The variable thickness phenomena are used for decreasing the structural element weight and enriched the application of material only. ¹² Subhashini et al. ¹³ formulated and studied the behavior of two-dimensional viscous fluid with thermal diffusion over a variable thickness stretching surface. Bayat et al. ¹⁴ subjected the variable thickness rotating disk flow by magneto-thermo-mechanical of a functionally graded magneto-electric material (FGMM). The flow of Maxwell fluid in the presence of Cattaneo–Christov heat flux over a variable thickness surface was studied by Hayat et al. ¹⁵ A nanofluid flow over a variable thickness stretching surface under the influence of Brownian motion is investigated by Abdel-Wahed et al. ¹⁶ Ostwald-de Waele fluid and heat transfer by a rotating disk with variable thickness has been investigated by Xun et al. ¹⁷ Hayat et al. ¹⁸ investigated the flow of nanofluid by a rotating disk with homogeneous–heterogeneous reactions and variable thickness. Zhang et al. ¹⁹ and Bhatti et al. ²⁰ used a generalized structure of Reynolds number and modeled three-dimensional nanoliquid flow between circular rotating plates filled with gyrotactic microorganisms and nanosize particles.

One physical involvement in rotating flow is the use of an external magnetic field to shield a rotating body from excessive heating. Due to the effect of a magnetic field inside the boundary layer, the total pressure remains constant across the boundary layer over the disk that is the reason that the fluid pressure on the surface of the disk reduced by an amount of magnetic intensity of an applied magnetic field. Glauret ²¹ discussed the same situation of a boundary layer flow over a magnetized plate. Pao and Long ²² also concluded a very interesting result of an external magnetic field that intensifying magnetic intensity results in boundary layer thickness and weakness of the axial flow component. Bhatti et al. ²³ examined the interactions of hydrodynamics metachronal waves on liquid motion through an annulus. The numerical investigation of hall current and heat transfer on peristaltic propulsion of particle-liquid suspension was calculated by Bhatti et al. ²⁴ The impact of thermal radiation and chemical reaction on magnetohydrodynamic (MHD) micropolar flow with mass and heat transfer over a moving porous plate was discussed by Mohamed and colleagues^25,26 Osman et al. ²⁷ find the solution of chemical reaction and thermal radiation influence on MHD unsteady convection flow with heat source/sink through porous media. Mohamed et al. ²⁸ scrutinized MHD unsteady double-diffusive boundary layer flow over a hot vertical plate in porous media along with heat sink and chemical reaction.

In the process of modeling some engineering phenomena, the researchers are facing very nonlinear boundary value problems (BVPs), which could not be solved easily by a numerical method. Numerical schemes convergence is of great concern and sensitive to the first approaches. Ambarzsumian ²⁹ and Chandrasekhar ³⁰ were the pioneers to introduce parametric continuation method (PCM). They transformed the BVP to a Cauchy problem. Bellman and Kalaba generalized this problem. ³¹ This method is very useful in many fields of physics such as neutron transport theory, ³² random walk and scattering, ³³ wave propagation, ³⁴ rare field gas dynamics, ³⁵ Hamilton’s equations of motion, ³⁶ and the flow in chemical reactors. ³⁷ A large number of new studies are developed and applied to PCM method. Grigoluyk and Shalashilin ³⁸ applied the PCM to various solid mechanics problems. PCM method is very sensitive on to make a choice an appropriate continuation parameter. Vorovich and Zipalova ³⁹ and Ricks ⁴⁰ worked on how to choose a suitable direction for continuation parameter to get a well conditioning of the solution of the linearized system of $N\times N$ equations. The geometrical interpretation with applications to different problems of nonlinear mechanics is given in literature.^41,42 Valli et al. ⁴³ applied PCM on Navier–Stokes equations. They used the Reynolds number as a parameter and varying it along a continuation path. They pointed out that ill conditioning of the Jacobian matrix occurred during the iterations. To get the best, the arch-length continuation technique, described in Keller, ⁴⁴ is used.

Nonlinear ordinary differential equations (ODEs) or partial differential equations (PDEs) are used to model almost all the engineering problems and phenomena. Therefore, the researchers apply various methods for solving nonlinear differential equations to analyze some practical problems. There are a number of methods for solutions of nonlinear equations, which scope from purely analytic to purely numerical techniques. Besides all advantages of using analytic or closed form solutions, numerical techniques are more appealing because of its versatility to solve almost all scientific and engineering problems.

The application of variable stretching body to engineering problem may happen to a greater extent than a flat body. The speed of a variable stretching body, at different positions, depends on the stretching body material, the body thickness can change with the motion of the body. For instant, if the stretching speed is linearly proportional to the distance from a specified position, a linear decrease occurs in the body thickness with the distance. The stretching disk with variable thickness can be more relevant to the practical situation. Consequently, the purpose of the present article is to model and investigate the behavior of viscous boundary layer flow and heat transfer aspects, under influence of a variable magnetic field. The Navier–Stokes, magnetic force equations are coupled with energy equation. An appropriate transformation has been used to transform the basic governing equations to a system of ODEs and solved numerically using PCM. The flow pattern has been analyzed by the effects of natural parameters such as the force based on magnetic strength $R_3$ , magnetic Reynolds number $R{e}_m$ , shaped parameters $a$ and $\zeta$ , stretching parameter $A$ , and thickness coefficient $\alpha$ . According to our knowledge, no literature is available for dealing rotating disk flow by PCM method.

The rest of the article is organized in the following fashion. In section “Mathematical formulation of the problem,” the governing equations and Maxwell equations are presented in similarity reduced form. Section “Solution by PCM” develops the analysis for the numerical solution by PCM method. The results and discussion are conducted in section “Results and discussions.” Section “Conclusion” contains the conclusion drawn from the present study.

Mathematical formulation of the problem

Consider a steady, incompressible, axial symmetric, and thermo-magnetically conducting fluid flow due to a stretchable rotating disk with stretching rate $C$ . The disk rotates with an angular velocity $\mathrm{\Omega }$ and is placed at $z=a((r/R_0)+1{)}^{-\zeta }$ , where $a$ and $\zeta$ are shaped parameters. If $a=0$ , then for any value of $\zeta$ we have a smooth disk at $z=0$ . For $\zeta =0$ , we have again a smooth surface (for any $a$ ) at $z=a$ , and when $a$ and $\zeta$ are non-zero a flat disk. For large values of $\zeta$ , we have a smooth surface at $z=0$ . A fluid of variable thickness is subjected by a temperature as well as a variable magnetic field simultaneously. It is assumed that the fluid is infinite and extends in the positive z-direction. The components of velocity and magnetic field intensity are $U(u,v,w)$ and $B(0,0,B_z)$ . The surface of the disk is maintained at a uniform temperature $T_w$ , while the temperature of the free stream surface is $T_{\infty }$ . The geometry of the problem is presented in Figure 1. Under these assumptions and the proposed model of Andersson et al., ⁴⁵ the governing equations of motion are modified with Euler and induced equations for variable magnetic field as

Figure 1.

Geometry of the fluid flow.

Continuity equation

$$\overrightarrow{\nabla }·\overrightarrow{U}=0$$ (1)

Navier–Stokes equations

$$\rho (\overrightarrow{U}·\overrightarrow{\nabla })\overrightarrow{U}=-\overrightarrow{\nabla }\overrightarrow{P}+\mu {\overrightarrow{\nabla }}^2\overrightarrow{U}+\frac{1}{{\mu }_2}(\overrightarrow{\nabla }\overrightarrow{B}\times \overrightarrow{B})$$ (2)

Energy equation

$$(\rho C_p)(\overrightarrow{U}·\overrightarrow{\nabla })T=\overrightarrow{\nabla }·(\kappa \overrightarrow{\nabla })T-\overrightarrow{\nabla }{q}_r$$ (3)

Maxwell’s equations

$$\overrightarrow{\nabla }·\overrightarrow{B}=0$$ (4)

$$\overrightarrow{\nabla }\times (\overrightarrow{U}\times \overrightarrow{B})=\frac{\rho }{\sigma {\mu }_2}{\overrightarrow{\nabla }}^2\overrightarrow{B}$$ (5)

The appropriate boundary conditions for the flow induced by an infinite disk which rotates with constant angular velocity $\mathrm{\Omega }$ are given by

$$\begin{array}{c}u=\mathrm{Cr},v=r\mathrm{\Omega },w=0\hfill \\ \multicolumn{1}{c}{B_z=\frac{M_0{(1+r)}^{\zeta }}{\sqrt{R_0}}{\left(\frac{\mathrm{\Omega }{R}_0^2\rho }{\mu }\right)}^{\frac{1}{n+1}}}\\ \multicolumn{1}{c}{T=T_w,\mathrm{at}z=a{(\frac{r}{R_0}+1)}^{-\zeta }}\\ \multicolumn{1}{c}{u\to 0,v\to 0,B_z\to 0}\\ \multicolumn{1}{c}{T\to T_{\infty }\mathrm{as}z\to \infty }\end{array}\}$$ (6)

where $\overrightarrow{U}$ is the velocity of the fluid, $\overrightarrow{B}$ is the applied magnetic field, $T$ is fluid temperature, $\overrightarrow{P}$ is the pressure, $\sigma$ the fluid electric conductivity, and $(T-T_{\infty })$ is the mean fluid temperature. For liquid metals ${\mu }_2={\mu }_0$ , where ${\mu }_0$ is the permeability of the free surface, $q_r=(-4\sigma */3\kappa *)(\partial T^4/\partial z)$ is the radiation heat flux, $\sigma *$ is the Stefan–Boltzmann constant, $\kappa *$ is the mean absorption coefficient, $\mu$ is the dynamic viscosity of the fluid, $\rho$ is the density, $a$ is the small thickness of the disk, $R_0$ is the future radius, and $\zeta$ is the thickness index. The following transformations are used to convert the above system of PDE to a system of ODE

$$\begin{array}{c}u=r^{*}\mathrm{\Omega }F\prime (\eta ),v=r^{*}{R}_0\mathrm{\Omega }G(\eta )\hfill \\ \multicolumn{1}{c}{w=R_0\mathrm{\Omega }(1+r^{*}{)}^{-\zeta }{\left(\frac{\mathrm{\Omega }{R}_0^2\rho }{\mu }\right)}^{-\frac{1}{n+1}}H(\eta )}\\ \multicolumn{1}{c}{T=T_{\infty }+(T_{\omega }-T_{\infty })\vartheta (\eta )}\\ \multicolumn{1}{c}{B_z=\frac{M_0{(1+r)}^{\zeta }}{\sqrt{R_0}}{\left(\frac{\mathrm{\Omega }{R}_0^2\rho }{\mu }\right)}^{\frac{1}{n+1}}}\\ \multicolumn{1}{c}{\eta =\frac{z}{R_0}(1+r^{*}{)}^{\zeta }{\left(\frac{\mathrm{\Omega }{R}_0^2\rho }{\mu }\right)}^{\frac{1}{n+1}}}\end{array}\}$$ (7)

After applying the above transformation to equations (1)–(7), we find the following reduced equations

$$2F+H\prime +\eta \epsilon \zeta F\prime =0$$ (8)

$$\begin{array}{c}(\mathrm{Re}{)}^{\frac{1-n}{1+n}}(1+r^{*}{)}^{2\zeta }F″-F^2+G^2-\mathrm{HF}\prime -\eta \epsilon \zeta \mathrm{FF}\prime \hfill \\ -R_3^2(1+r^{*}{)}^{2\zeta }(\mathrm{Re}{)}^{\frac{1-n}{1+n}}F{M}^2=0\hfill \end{array}$$ (9)

$$\begin{array}{c}(\mathrm{Re}{)}^{\frac{1-n}{1+n}}(1+r^{*}{)}^{2\zeta }G″-2\mathrm{FG}-\mathrm{HG}\prime -\eta \epsilon \zeta \mathrm{FG}\prime \hfill \\ -R_3^2(1+r^{*}{)}^{2\zeta }(\mathrm{Re}{)}^{\frac{1-n}{1+n}}G{M}^2=0\hfill \end{array}$$ (10)

$$\begin{array}{c}\frac{1}{\mathrm{Pr}}(\mathrm{Re}{)}^{\frac{1-n}{1+n}}(1+R)(1+r^{*}{)}^{2\zeta }\vartheta ″-\eta \epsilon \zeta F\vartheta \prime \hfill \\ -H\vartheta \prime =0\hfill \end{array}$$ (11)

$$\begin{array}{c}(\mathrm{Re}{)}^{\frac{1-n}{1+n}}(1+r^{*}{)}^{2\zeta }M″-\mathrm{Rem}(2\mathrm{MF}+\zeta \epsilon \mathrm{FM}\hfill \\ +\eta \epsilon \zeta \mathrm{FM}\prime +\eta \epsilon \mathrm{MF}\prime )=0\hfill \end{array}$$ (12)

The transform boundary conditions are given as

$$\begin{array}{c}F(\alpha )=A,G(\alpha )=\vartheta (\alpha )=M(\alpha )=1\hfill \\ \multicolumn{1}{c}{H(\alpha )=0,F(\infty )\to 0,G(\infty )\to 0}\\ \multicolumn{1}{c}{\vartheta (\infty )\to 0,M(\infty )\to 0}\end{array}\}$$ (13)

where $F$ , $G$ , and $H$ are non-dimensional component of velocity in direction of $r\mathrm{-}$ , $\theta \mathrm{-}$ , and $z\mathrm{-}\mathrm{axes}$ , respectively. $M$ is non-dimensional components of magnetic velocities along $z\mathrm{-axis}$ , $R_3$ is the magnetic strength numbers, $R{e}_m$ is the magnetic Reynolds number, $\mathrm{Pr}$ is the Prandtl number, $\epsilon$ is a dimensionless constant, $A$ is the stretching parameter, $R$ is the radiation parameter, $\alpha$ is the thickness coefficient of disk, $r^{*}$ is the dimensionless radius, $\eta$ is the similarity variable and prime denotes derivative with respect to $\eta$ . These parameters are defined as

$$\begin{array}{c}{R}_3=\frac{M_0}{\sqrt{{\mu }_2\nu }},R{e}_m={\mu }_0\sigma R_0\nu ,\mathrm{Pr}=\frac{\rho C_p\nu }{K}\hfill \\ \multicolumn{1}{c}{\epsilon =\frac{r^{*}}{1+r^{*}},A=\frac{C}{\mathrm{\Omega }},R=\frac{16{\sigma }^{*}{T}_{\infty }^3}{3K{K}^{*}}}\\ \multicolumn{1}{c}{\alpha =\frac{a}{R_0^2}{\left(\frac{\mathrm{\Omega }{R}_0^2\rho }{\mu }\right)}^{\frac{-1}{n+1}},r^{*}=\frac{r}{R_0}}\end{array}\}$$ (14)

Furthermore, the following deformations are introduced

$$\begin{array}{c}F=f(\eta -\alpha )=f(\xi ),G=g(\eta -\alpha )=g(\xi )\hfill \\ \multicolumn{1}{c}{H=h(\eta -\alpha )=h(\xi ),\vartheta =\theta (\eta -\alpha )=\theta (\xi )}\\ \multicolumn{1}{c}{M=m(\eta -\alpha )=m(\xi )}\end{array}\}$$ (15)

Equations (8)–(13) takes the form as

$$2f+h\prime +\epsilon \zeta (\xi +\alpha )f\prime =0$$ (16)

$$\begin{array}{c}(\mathrm{Re}{)}^{\frac{1-n}{1+n}}(1+r^{*}{)}^{2\zeta }f″-f^2+g^2-\mathrm{hf}\prime -\epsilon \zeta (\xi +\alpha )\mathrm{ff}\prime \hfill \\ -R_3^2(1+r^{*}{)}^{2\zeta }(\mathrm{Re}{)}^{\frac{1-n}{1+n}}f{m}^2=0\hfill \end{array}$$ (17)

$$\begin{array}{c}(\mathrm{Re}{)}^{\frac{1-n}{1+n}}(1+r^{*}{)}^{2\zeta }g″-2\mathrm{fg}-\mathrm{hg}\prime -\epsilon \zeta (\xi +\alpha )\mathrm{fg}\prime \hfill \\ -R_3^2(1+r^{*}{)}^{2\zeta }(\mathrm{Re}{)}^{\frac{1-n}{1+n}}g{m}^2=0\hfill \end{array}$$ (18)

$$\begin{array}{c}\frac{1}{\mathrm{Pr}}(\mathrm{Re}{)}^{\frac{1-n}{1+n}}(1+R)(1+r^{*}{)}^{2\zeta }\theta ″\hfill \\ -\epsilon \zeta (\xi +\alpha )f\theta \prime -h\theta \prime =0\hfill \end{array}$$ (19)

$$\begin{array}{c}(\mathrm{Re}{)}^{\frac{1-n}{1+n}}(1+r^{*}{)}^{2\zeta }m″-\mathrm{Rem}(2\mathrm{mf}+\zeta \epsilon \mathrm{fm}\hfill \\ +\epsilon \zeta (\xi +\alpha )\mathrm{fm}\prime +\epsilon (\xi +\alpha )f\prime m)=0\hfill \end{array}$$ (20)

with boundary conditions

$$\begin{array}{c}f(0)=A,g(0)=\theta (0)=m(0)=1\hfill \\ \multicolumn{1}{c}{h(0)=0,f(\infty )\to 0,g(\infty )\to 0}\\ \multicolumn{1}{c}{\theta (\infty )\to 0,m(\infty )\to 0}\end{array}\}$$ (21)

Solution by PCM

The basic idea of application of PCM method, to the system of ODE (equations (16)–(20)) with the boundary condition (equation (21)), is presented with the following steps.

Step 1: converting the system of BVP as a system of first-order ODE and introducing the embedding parameter p

For this purpose, introducing the following functions

$$\begin{array}{c}\begin{array}{cc}{F}_1\hfill & =f,F_2=f\prime ,F_3=g\hfill \\ \multicolumn{1}{c}{F_4}& =g\prime ,F_5=h,F_6=\theta \hfill \\ \multicolumn{1}{c}{F_7}& =\theta \prime ,F_8=M,F_9=M\prime \hfill \end{array}\}\hfill \end{array}$$ (22)

Rewriting BVP (equations (16)–(20)) in the following form, to obtain a system of ODE in a p-parametric family

$$\begin{array}{c}\multicolumn{1}{c}{{F\prime }_2=(\mathrm{Re}{)}^{\frac{n-1}{n+1}}(1+r{)}^{-2\zeta }(F_1^2-F_3^2+((F_2-1)p+1)F_5}\\ +\epsilon \zeta (\xi +\alpha )F_1{F}_2+R_3^2{F}_1{F}_8^2)\hfill \end{array}$$ (23)

$$\begin{array}{c}\multicolumn{1}{c}{{F\prime }_4=(\mathrm{Re}{)}^{\frac{n-1}{n+1}}(1+r{)}^{-2\zeta }(2F_1{F}_3+((F_4-1)p+1)F_5}\\ +\epsilon \zeta (\xi +\alpha )F_1{F}_4+R_3^2{F}_3{F}_8^2)\hfill \end{array}$$ (24)

$$F{\prime }_5=-2F_1-\epsilon \zeta (\xi +\alpha )F_2$$ (25)

$$\begin{array}{c}\multicolumn{1}{c}{{F\prime }_7=\mathrm{Pr}(\mathrm{Re}{)}^{\frac{n-1}{n+1}}(1+r{)}^{-2\zeta }(1+R{)}^{-1}(F_5((F_7-1)p+1)}\\ +\epsilon \zeta (\xi +\alpha )F_1((F_7-1)p+1))\hfill \end{array}$$ (26)

$$\begin{array}{c}\multicolumn{1}{c}{{F\prime }_9=(\mathrm{Re}{)}^{\frac{n-1}{n+1}}(1+r{)}^{-2\zeta }R{e}_m(2F_1{F}_8^2+\epsilon \zeta F_1{F}_8}\\ +\epsilon \zeta (\xi +\alpha )F_1((F_9-1)p+1)\hfill \\ +\epsilon (\xi +\alpha )F_2{F}_8)\hfill \end{array}$$ (27)

with the corresponding boundary conditions

$$\begin{array}{c}{F}_1(\xi )=A,F_3(\xi )=1,F_5(\xi )=0,F_6(\xi )=1\hfill \\ \multicolumn{1}{c}{F_8(\xi )=1,\mathrm{at}\xi =0}\\ \multicolumn{1}{c}{F_1(\xi )\to 0,F_3(\xi )\to 0,F_6(\xi )\to 0}\\ \multicolumn{1}{c}{F_8(\xi )\to 0,\mathrm{as}\xi \to \infty }\end{array}\}$$ (28)

Step 2: differentiating by parameter “p”

After differentiating equations (23)–(27) with respect to parameter p, arrive at the following system with respect to sensitivities to the parameter p

$$V\prime =\mathrm{AV}+R$$ (29)

where A is the coefficient matrix, R is the remainder and

$$V=\frac{\partial F_i}{\partial \tau }$$ (30)

where $i=1,2,\dots ,9.$

Step 3: apply superposition principle and specify Cauchy problem for each component

$$V=\mathrm{aU}+W$$ (31)

where U and W are the unknown vector functions; a is the unknown blend coefficient.

Solve the following two Cauchy problems for each component

$$U\prime =\mathrm{AU}$$ (32)

$$W\prime =\mathrm{AW}+R$$ (33)

Putting the approximate solution equation (31) into the original equation (29), we obtain

$$(\mathrm{aU}+W)\prime =A(\mathrm{aU}+W)+R$$ (34)

Step 4: solving the Cauchy problems

Numerical implicit scheme is used in this work, which is presented as below.

From equation (32) and equation (33)

$$\frac{U^{i+1}-U^i}{\mathrm{\Delta }\eta }=A{U}^{i+1},\mathrm{or}(I-\mathrm{\Delta }\eta A)U^{i+1}=U^i$$ (35)

$$\begin{array}{c}\frac{W^{i+1}-W^i}{\mathrm{\Delta }\eta }=A{W}^{i+1}+R,\mathrm{or}\hfill \\ (I-\mathrm{\Delta }\eta A)W^{i+1}=W^i+\mathrm{\Delta }\eta R\hfill \end{array}$$ (36)

from where

$$U^{i+1}=(I-\mathrm{\Delta }\eta A{)}^{-1}{U}^i$$ (37)

$$W^{i+1}=(I-\mathrm{\Delta }\eta A{)}^{-1}(W^i+\mathrm{\Delta }\eta R)$$ (38)

Results and discussions

Skin effects

The numerical findings are briefly summed up in Tables 1 and 2 and Figures 2–5. In order to verify the validity of the present numerical scheme, the numerical findings are compared with the previous published results for a rotating disk flow. It is worth mentioning here that the present results are in an excellent agreement with the previous one Table 1. The numerical values of the physical interest quantities such as radial ( $f\prime (0)$ ), tangential ( $-g\prime (0)$ ) skin fractions, axial in flow ( $-h(\infty )$ ), heat transfer coefficient ( $-\theta \prime (0)$ ), and magnetic skin depth ( $-m(0)$ ) are tabulated in Table 2. Skin effect is the uniform distribution of time varying currents. The cause of the skin effect is electromagnetic induction. The skin effect is of considerable practical importance. A very thin layer of conductor carries most of the current, so any conductor can be coated with silver (a best available conductor). Keeping in view the importance of skin effect, the radial skin fraction ( $f\prime (0)$ ), tangential skin fraction ( $-g\prime (0)$ ), the axial inflow ( $-h\prime (\infty )$ ), heat transfer coefficient ( $-\theta \prime (0)$ ), and magnetic skin depth ( $-m\prime (0)$ ) are given in Table 2 for different values of physical parameters. Table 2 depicts the effect of the stretching parameter $A$ on various physical importance quantities. It can be observed from Table 2 that with the increasing values of $A$ , the axial inflow, heat transfer coefficient, and magnetic skin depth increase while the radial skin fraction decreases. The reason of this affect is that relatively large drag force is acted upon the radial skin fraction as compared to other skin fractions. The impact of disk thickness index $\zeta$ can be observed in Table 2. It shows that radial skin fraction and axial inflow increase while tangential skin fraction, heat transfer coefficient, and magnetic skin depth decrease with the increasing values of $\zeta$ . Only tangential skin fraction increases while all other quantities decrease for increasing values of force based on the magnetic strength $R_3$ , this effect can be observed in Table 2. And a very opposite result can be observed for magnetic Reynolds number $R{e}_m$ in Table 2. The power law exponent $n$ effect can be depicted in Table 2, which shows that increase in values of power law exponent results in a decrease in radial, tangential skin fractions, heat transfer coefficient, and magnetic skin depth while the axial inflow increases.

Table 1.

Comparison of present result with literature^17,45–47 where $r=0.2$ , $\varphi =0$ , ${\rho }_s={\rho }_f=(C_p{)}_s=(C_p{)}_f=K_{\mathrm{nf}}=K_f=1,A=\zeta =\xi =\epsilon =0$ .

Author	$f\prime (0)$	$-g\prime (0)$	$-h\prime (\infty )$	$-\theta (0)$
Present	0.5170	0.6221	0.8889	0.4003
Andersson et al. 45	0.510	0.616	0.883	–
Ming et al. 46	0.51021	0.61591	0.88230	0.39632
Xun et al. 17	0.510231	0.615921	–	0.396271
Hayat et al. 47	0.5109	0.61598	–	0.3959

Table 2.

Numerical values of radial and tangential skin frictions, axial inflow, Nusselt number, and magnetic skin depth for different values of the physical parameters, for $\zeta =0.5$ , $\xi =1$ , $R_{\mathrm{em}}=10$ , $n=1.5$ , $\mathrm{Pr}=6.7$ , $\mathrm{Re}=0.9$ , $\epsilon =0.3$ , $R=0.4$ , $R_3=0.7$ .

$n$	$f\prime (0)$	$-g\prime (0)$	$-h\prime (\infty )$	$-\theta \prime (0)$	$-m\prime (0)$
$A$
0.0	0.4448	0.6645	0.9612	0.7304	1.6101
0.5	−0.1153	1.0251	1.2959	1.0766	2.1755
1.0	−0.7976	1.2997	1.6234	1.2517	2.4390
$\zeta$
0.0	0.3922	0.7961	0.9403	0.9637	2.1024
0.5	0.3533	0.7369	1.0155	0.8117	1.7486
1.0	0.3167	0.6869	1.0949	0.6902	1.4841
$R_3$
0.0	0.3807	0.6412	1.0343	0.8252	1.7857
0.5	0.3667	0.6899	1.0248	0.8183	1.7669
1.0	0.3248	0.8375	0.9951	0.7971	1.7090
$\mathrm{Rem}$
0.0	0.2602	0.8634	0.6957	0.7138	0.1010
0.5	0.2833	0.8352	0.8066	0.7433	0.3071
1.0	0.3005	0.8148	0.8801	0.7632	0.4831
$n$
0.0	0.3800	0.7722	0.9562	0.8622	1.8771
0.5	0.3650	0.7522	0.9888	0.8338	1.8046
1.0	0.3576	0.7426	1.0054	0.8199	1.7694

Figure 2.

Radial component of velocity for fixed values of: $A=0.09$ , $\mathrm{Re}=0.9$ , $\mathrm{Pr}=6.4$ , $R=0.5$ , $\zeta =0.5$ , $\xi =0.1$ , $\alpha =1.2$ , $\epsilon =0.3$ , $R_3=0.7$ , $r=0.2$ , $\mathrm{Rem}=1.5$ , and $n=1.5$ .

Figure 3.

Radial component of velocity for fixed value of: $A=0.09$ , $\mathrm{Re}=0.9$ , $\mathrm{Pr}=6.4$ , $R=0.5$ , $\zeta =0.5$ , $\xi =0.1$ , $\alpha =1.2$ , $\epsilon =0.3$ , $R_3=0.7$ , $r=0.2$ , $\mathrm{Rem}=1.5$ , and $n=1.5$ .

Figure 4.

Azimuthal component of velocity for fixed values of: $A=0.09$ , $\mathrm{Re}=0.9$ , $\mathrm{Pr}=6.4$ , $R=0.5$ , $\zeta =0.5$ , $\xi =0.1$ , $\alpha =1.2$ , $\epsilon =0.3$ , $R_3=0.7$ , $r=0.2$ , $\mathrm{Rem}=1.5$ , and $n=1.5$ .

Figure 5.

Azimuthal component of velocity for fixed values of: $A=0.09$ , $\mathrm{Re}=0.9$ , $\mathrm{Pr}=6.4$ , $R=0.5$ , $\zeta =0.5$ , $\xi =0.1$ , $\alpha =1.2$ , $\epsilon =0.3$ , $R_3=0.7$ , $r=0.2$ , $\mathrm{Rem}=1.5$ , and $n=1.5$ .

Radial velocity profile

Figure 2(a)–(d) shows the behavior of radial velocity component for different controlling physical parameters. The radial velocity component is significantly increasing near the disk surface, and boundary layer becomes thinner for increasing values of parameters $\zeta$ , $\epsilon$ , and $\alpha$ ; this effect can be seen in Figure 2(a)–(c). A large value of the disk thickness parameter $\zeta$ as shown in Figure 2(a) leads to a large mass transfer effect, which also results in reduction of surface sheer stress as shown in Figure 2(b). Due to the structure of constant number $\epsilon$ it is obvious that for small radius $R_0$ , the value of $\epsilon$ increases which consequently increases the radial velocity component as shown in Figure 2(c). This is due the fact that a very small contact between the disk surface and the fluid particles is taken place and less opposite force is produced which lead to an increase in radial velocity. Similarly, the larger value of $\alpha$ enhances the thickness coefficient of the disk so that the velocity increases. The magnetic force is used to control the turbulence nature of the fluid and makes it as laminar. The effect of magnetic force strength $R_3$ is depicted in Figure 2(d). As expected, for increasing value of $R_3$ , the radial velocity component gets decelerated and shows no significant change in boundary layer. The Lorentz force is produced due to the infliction of magnetic field (vertical) on conducting fluid, which have greater tendency to reduce flow velocity around the disk. As a result, the axial, azimuthal and as well as the radial velocity of fluid decrease. It is worth to be noted that the magnetic effects produce resistance on fluid particles, which generate some amount of heat; as a result, fluid temperature also improved. Thus, we can conclude that radial velocity experiences maximum effect of $R_3$ near the disk surface, while it has very less effect on temperature of the fluid. The thickness of the boundary is consequentially decreases with increasing value of power law exponent $n$ as shown in Figure 3(a). This result also verifies Andersson’s result. The same behavior of the radial velocity component is depicted in Figure 3(b).

Tangential velocity profile

The effects of different physical parameters are next exhibited on the tangential velocity component in Figure 4(a)–(d). The tangential component of velocity shows the same behavior as radial component with the changing values of the disk thickness index $\zeta$ . Figure 4(a) shows that the tangential component increases with increasing value of $\zeta$ ; as a result, the boundary layer gets thicker. Similar behavior can be seen for tangential component as the constant number $\epsilon$ increases. Tangential velocity is a function of rotational speed $\mathrm{\Omega }$ . So as expected, the stretching parameter $A$ increases as the tangential velocity is retarded as evident from Figure 4(c). Because when the disk stretches the fluid particle above the disk surface, it gets some space and becomes relaxed for a while; as a result, their tangential velocity and temperature reduce, which causes to decrease the fluid velocity and the average temperature of fluid. The thickness of the boundary layer is reduced with increasing value of power index $n$ ; as a result, the tangential velocity decreases which affects the axial velocity as well. These observations can be seen in Figure 4(d) and Figure 6(d), respectively. The parameter $R_3$ is the axial component of applied magnetic force. Figure 5(a) shows that as $R_3$ increases, the azimuthal velocity decreases and as a result reduction of boundary layer thickness. A slightly same result occurs under the influence of rotational Reynolds number $\mathrm{Re}$ as shown in Figure 5(b).

Figure 6.

Axial component of velocity for fixed values of: $A=0.09$ , $\mathrm{Re}=0.9$ , $\mathrm{Pr}=6.4$ , $R=0.5$ , $\zeta =0.5$ , $\xi =0.1$ , $\alpha =1.2$ , $\epsilon =0.3$ , $R_3=0.7$ , $r=0.2$ , $\mathrm{Rem}=1.5$ , and $n=1.5$ .

Axial velocity profile

To discuss the effect of disk thickness index $\zeta$ , magnetic base strength $R_3$ , constant parameter $\epsilon$ , and power law index $n$ on axial velocity component as shown in Figure 6(a)–(d) are displayed. The effect of disk thickness $\zeta$ is illustrated in Figure 6(a). Here, an increase in $\zeta$ results in the axial component ( $-h(\xi )$ ) of velocity. This is due to the fact that keeping $r$ as constant the denominator becomes dominant by increasing $\zeta$ which is responsible for the increase in $-h(\xi )$ . As usually, the force based on magnetic strength $R_3$ plays its role which reduced the axial component of velocity. This affect can be seen in Figure 6(b). Figure 6(c) illustrates an interesting result of axial velocity with respect to the constant parameter $\epsilon$ . It is observed that with the increasing values of $\epsilon$ the axial velocity near the disk surface increases while it decreases when the fluid approaches to ambient velocity. The impact of power law exponent $n$ on axial velocity is illustrated in Figure 6(d). It is observed that the magnitude of the axial velocity increases for larger values of $n$ . The reason is that the thickness of disk is reduced by taking a large value of $n$ , so that the fluid velocity increases.

Axial magnetic velocity profile

The disk thickness index $\zeta$ has marked the effect on the axial magnetic velocity $m(\xi )$ . As disk thickness increases, the axial magnetic velocity increases, as shown in Figure 7(a). Hence, the disk thickness index can be used to increase the axial magnetic velocity. Magnetic Reynolds number is the ratio of induction to diffusion. Increasing of magnetic Reynolds number $R{e}_m$ means to decrease the diffusion rate, which results in the reduction of magnetic velocity. This fact can be observed in Figure 7(b).

Figure 7.

Axial velocity component of magnetic field for fixed values of: $A=0.09$ , $\mathrm{Re}=0.9$ , $\mathrm{Pr}=6.4$ , $R=0.5$ , $\varphi =0.08$ , $\zeta =0.5$ , $\xi =0.1$ , $\alpha =1.2$ , $\epsilon =0.3$ , $M=0.1$ , $r=0.2$ , and $n=1.5$ .

Temperature profile

The variation of temperature profile is shown in Figure 8(a)–(d). Figure 8(a) depicts that the thickness of thermal boundary layer enlarges with the increasing value of disk thickness index $\zeta$ . Its means that heat transfer process slows down for larger values of $\zeta$ . Prandtl number is the ratio of momentum diffusivity to thermal diffusivity. Thus, Prandtl number is increasing with reduction of thermal diffusivity. That is why the temperature distribution as well as the disk thickness of boundary layer decrease as Prandtl number increases. This fact can be observed in Figure 8(b). This is due to the fact that higher values of Prandtl number result in reduction of thermal conductivity which consequently increase heat transfer at the surface of the disk that is the reason which decreases the thickness of thermal boundary layer. Figure 8(c) illustrates the effect of stretching parameter $A$ on temperature profile. The temperature decreases with increasing values of $A$ . This is due to the fact that a very small rotational velocity is produced when $A$ increases. Thus, small resistance is created between fluid particles disk surface and consequently temperature is reduced. Figure 8(d) depicts that when radiation parameter $R$ increases the temperature distribution is enhanced. As the increase in radiation is responsible to the reduction of mean absorption coefficient, it results in the increase in the rate of radiation heat transfer to the fluid. The parameter’s ranges are selected to achieve acceptable error by the PCM method. Their upper and lower bounds are chosen in a such a way that the minimum error is reached. For this purpose error versus various parameters are plotted in Figures 9 and 10. The domain in each sub-figure is taken from the previous data used throughout this analysis. If the present finding are used for engineering purposes, the correct physical parameters up to the desire accuracy must be chosen from these ranges.

Figure 8.

Temperature distribution $\theta$ for fixed values of: $A=0.09$ , $\mathrm{Re}=0.9$ , $\mathrm{Pr}=6.4$ , $R=0.5$ , $\varphi =0.08$ , $\zeta =0.5$ , $\xi =0.1$ , $\alpha =1.2$ , $\epsilon =0.3$ , $M=0.1$ , $r=0.2$ , and $n=1.5$ .

Figure 9.

Parameters range for the physical paremeters $\xi$ , $\epsilon$ , $\alpha$ , and $R_3$ .

Figure 10.

Parameters range for the physical paremeters $\mathrm{Re}$ , $R_{\mathrm{em}}$ , $\mathrm{Pr}$ , and $n$ .

Conclusion

A variable thickness flow over a stretchable rotating disk is investigated in this article. Navier–Stokes equations together with Maxwell and heat transfer equations are transformed to a system of nonlinear ODEs. The arrangement of ODEs is further processed through a numerical scheme of PCM. The present investigation enables us to conclude that non-flatness of the stretching disk has a significant effect on the development of boundary layer. Moreover, the following conclusions can be drawn from the present work:

• Magnetic effects produce resistance on fluid particles, which generate some amount of heat, as a result fluid temperature also improved. While radial velocity also experience maximum effect of $R_3$ near the disk surface for rising credit of magnetic field strength.

• Magnetic field can be used to protect the rotatory body from excessive heat.

• A growing value of disk thickness parameter leads to an excessive mass transfer.

• The increasing values of constant parameter $\epsilon$ positively affect the axial velocity near the surface, while it decreases when the fluid approaches to ambient velocity.

• The heat transfer process slows down for large value of no-flatness parameter $\zeta$ .

• Prandtl number increases heat transfer rate at the surface of disk.

• Magnetic force can be used to control the turbulence nature of the fluid.