The Heat and Mass Transfer Analysis During Bunch Coating of a Stretching Cylinder by Casson Fluid

Abstract

The aim of this study is to coat a stretching cylinder with the help of a liquid film spray. The Casson fluid has been chosen for the coating phenomena. The thickness of the liquid film has been used as variable, and the influence of heat and mass transmission under the impact of thermophoresis has been encountered in the flow field. The required pressure term for the spray pattern during variable thickness has mainly been focused. Using the suitable similarity transformations, the basic flow equations for the fluid motion have been converted into high-order nonlinear coupled differential equations. Series solutions of subsequent problem have been obtained using controlling procedure optimal approach. Important physical constraints of skin friction, Nusselt number, and Sherwood number have been calculated numerically and discussed. Other physical parameters involved in the problem, i.e., Reynolds number Re , Casson fluid parameter β 1 , Prandtl number Pr , Lewis number Le , Brownian motion parameter N b , and thermophoresis parameter N t have been illustrated. The skin friction effect and its physical appearance are also included in this work. The convergence is checked by plotting h-curves. The emerging parameters are discussed by plotting graphs. The recent work is also compared with the published work.

1. Introduction

The relation among pressure and flow is an important phenomena which plays a vital role to understand the circulation of the blood in the human body and its sustainability. The approach of pressure [1] can bring a partial barrier in some areas of the smaller vessels due to the thrilling change of the yield stress. The blood is equally a mixture of two fluids Casson and the other one is Newtonian fluid and studied by Srivastava and Saxena [2]. They focused on the effect of resistance created by the viscosity term and wall shear stresses. Later on, this fluid is studied by many researchers on the stretching surfaces for other industrial and engineering usages [3]. Mahdy [4] have examined the fluid motion over an extending cylinder considering Casson fluid. Hayat et al. [5] have examined the third-order fluid motion over an extending tube within the effect of MHD. Qasim et al. [6] have considered the slip flow of sighted ferrofluid over an extending cylinder. Sheikholeslami [7] has studied the suction idea considering nanofluid over an extended cylinder. Manjunatha et al. [8] have examined the radiation effect in a porous space using dusty fluid and stretching cylinder. Abdulhameed et al. [9] have examined the oscillatory flow phenomena using circular cylinder. Hakeem et al. [10] have studied the flow of Walter’s B fluid over an extending sheet. Pandey et al. [11] have studied the Walter’s B viscoelastic nanofluid film energetic from below. The interesting and fruitful applications of thin film are the wire and fiber coating, processing of food stuff, extrusion of polymer and metal, drawing of plastic sheets, continuous casting, fluidization of reactor, and chemical processing equipment. On the basis of these applications, researchers did a lot of work on it. Wang [12] was the main researcher who investigated liquid film on an unstable extending surface. Recently Tawade et al. [13] have investigated liquid film flow over an unstable extending surface with thermal radiation, in the existence of continuous magnetic field using numerical method. The liquid film flow considering non-Newtonian fluids proliferates in many life geographies which is used mostly in cylindrical shapes. Several researchers [14, 15, 16, 17] investigated power-law fluid with unsteady extending surface using different cases. Megahe [18] and Abolbashari et al. [19] have scrutinized thin film flow of Casson fluid using slip boundary conditions. Recently Qasim et al. [20] have examined the liquid film flow of nanofluid considering Buongiorno’s model.

The liquid film spray on a stretching sheet has also an important phenomena to coat the metals and increase their life. The idea of spray on the stretching surface is the study of Wang [21]. Recently Noor et al. [22] considered the thin film spray of nanofluid on a stretching cylinder. They compared their results with the experimental data and found the impact of the physical parameters during flow phenomena. They also discussed the application of their work in detail. Most of the mathematical problems in the field of engineering are composite in their nature, and the exact resolution is very tough or even not conceivable. The solution of these problems is tackled through numerical and analytical methods. Homotopy analysis method is one of the popular techniques for the solution of such complex problems. Liao [22, 23, 24, 25, 26] investigates this series solution technique for the solution of nonlinear problems. The other important feature of this method is that its solution contains all the embedded parameters involved in the problem and also the range of the embedded parameters. The high nonlinear problems have been solved by Abbasbandy [27] due to the fast convergence of this method. Alshomrani and Gul [28], Gul [29] have studied the solution of nonlinear differential equations through HAM arises in the field of engineering and industry.

2. Formulation

Consider the thin film flow of Casson nanofluid elegantly through a circular cylinder of radius “a.” The cylinder is supposed to be stretched along with radial direction with velocity $U_w$ , and temperature at the surface of cylinder is taken $T_w$ . The uniform ambient temperature is considered $T_b$ such that $T_w-T_b>0$ for assisting flow and $T_w-T_b<0$ for opposing flow.

The governing equations of continuity, heat transfer, and mass transfer are

$$\frac{\partial u}{\partial r}+\frac{u}{r}+\frac{\partial w}{\partial z}=0,$$ E1

$$u\frac{\partial w}{\partial r}+w\frac{\partial w}{\partial z}=\upsilon \left(1+\frac{1}{{\beta }_1}\right)\left(\frac{{\partial }^{2}w}{\partial r^2}+\frac{1}{r}\frac{\partial w}{\partial r}\right)+g{\beta }^{\ast }\left(T-T_b\right)\left(1-C_b\right)+\frac{1}{\rho }\left({\rho }^{\ast }-\rho \right)\left(C-C_b\right),$$ E2

$$u\frac{\partial u}{\partial r}+w\frac{\partial u}{\partial z}=-\frac{1}{\rho }\frac{\partial P}{\partial r}+\upsilon \left(1+\frac{1}{{\beta }_1}\right)\left(\frac{{\partial }^{2}u}{\partial r^2}+\frac{1}{r}\frac{\partial u}{\partial r}-\frac{u}{r^2}\right),$$ E3

$$u\frac{\partial T}{\partial r}+w\frac{\partial T}{\partial z}=\alpha \left(\frac{{\partial }^{2}T}{\partial r^2}+\frac{1}{r}\frac{\partial T}{\partial r}\right)+\frac{{\rho }^{\ast }{c}_p^{\ast }}{\rho c_p}\left(D_B\frac{\partial T}{\partial r}\frac{\partial \varphi }{\partial r}+\frac{D_T}{T_b}{\left(\frac{\partial T}{\partial r}\right)}^2\right),$$ E4

$$u\frac{\partial C}{\partial r}+w\frac{\partial C}{\partial z}=D_B\left(\frac{{\partial }^{2}C}{\partial r^2}+\frac{1}{r}\frac{\partial C}{\partial r}\right)+\frac{D_T}{T_b}\left(\frac{{\partial }^{2}T}{\partial r^2}+\frac{1}{r}\frac{\partial T}{\partial r}\right),$$ E5

where $u\left(r,z\right)$ and $w\left(r,z\right)$ are velocity components; $\rho$ is density; $\upsilon$ is kinematic viscosity; ${\beta }_1$ is the constant characteristic to Casson fluid; ${\beta }^{\ast }$ is the coefficient of thermal expansion; $g$ is the gravitational acceleration along z-axis; $T,T_b,C$ and $C_b$ determine the temperature, ambient temperature, concentration, and ambient concentration, respectively; and $\alpha ,D_T,D_B$ stands for the thermal diffusivity, thermophoresis diffusion coefficient, and Brownian diffusion coefficient.

The suitable boundary conditions are

$$u=U_w,w=W_w,T=T_w,C=C_w\mathit{at}r=a,$$ E6

$$\mu \frac{\partial w}{\partial r}=\frac{\partial T}{\partial r}=\frac{\partial C}{\partial r}=0,u=w\frac{\mathit{d\delta }}{\mathit{dz}}\mathit{at}r=b.$$ E7

Here $U_w=-\mathit{ca}$ represents the suction and injection velocity, and $W_w=2\mathit{cz}$ is the stretching velocity such that $c$ represents the stretching parameter and $\delta$ is the thickness of fluid film.

The similarity transformations are used to alter the basic Eqs. (1)–(7) used in [22] as

$$u=-\mathit{ca} \frac{f\left(\eta \right)}{\sqrt{\eta }},w=2\mathit{cz} \frac{\mathit{df}}{\mathit{d\eta }},T\left(z\right)=T_b-T_{\mathit{ref}}\left(\frac{{\mathit{cz}}^2}{{\upsilon }_{\mathit{nf}}}\right)\theta \left(\eta \right),C\left(z\right)=C_b-C_{\mathit{ref}}\left(\frac{{\mathit{cz}}^2}{{\upsilon }_{\mathit{nf}}}\right)\varphi \left(\eta \right),$$ E8

where$$\eta ={\left(\frac{r}{a}\right)}^2.$$

In the case of the outer radius $b$ of the flow, $\eta ={\left(\frac{b}{a}\right)}^2.$

Using these transformations in Eqs. (1), (2), (4)–(7), we obtained a set of dimensionless equations which is

$$\left(1+\frac{1}{{\beta }_1}\right)\left(\eta \frac{{\partial }^{3}f}{\partial {\eta }^3}+\frac{{\partial }^{2}f}{\partial {\eta }^2}\right)+\text{Re}\left(f\frac{{\partial }^{2}f}{\partial {\eta }^2}-{\left(\frac{\partial f}{\partial \eta }\right)}^2+\lambda \left(\theta +\mathit{Nr\varphi }\right)\right)=0,$$ E9

$$\eta \frac{{\partial }^2\theta }{\partial {\eta }^2}+\frac{\partial \theta }{\partial \eta }+\text{Pr}.\text{Re}\left(f\frac{\partial \theta }{\partial \eta }-2\frac{\partial f}{\partial \eta }\theta \right)+\eta \frac{\partial \theta }{\partial \eta }\left(N_t\frac{\partial \theta }{\partial \eta }+N_b\frac{\partial \varphi }{\partial \eta }\right)=0,$$ E10

$$\eta \frac{{\partial }^2\varphi }{\partial {\eta }^2}+\frac{\partial \varphi }{\partial \eta }+\mathit{Le}.\text{Re}\left(f\frac{\partial \varphi }{\partial \eta }-2\frac{\partial f}{\partial \eta }\varphi \right)+\frac{N_t}{N_b}\left(\eta \frac{{\partial }^2\theta }{\partial {\eta }^2}+\frac{\partial \theta }{\partial \eta }\right)=0,$$ E11

where

$$\begin{array}{l}\text{Re}=\frac{{\mathit{ca}}^2}{2{\upsilon }_{\mathit{nf}}},\lambda =\frac{g{\beta }^{\ast }a\left(T_w-T_{\infty }\right)\left(1-C_{\infty }\right)}{W_w^2},\mathit{Nr}=\frac{\left(\rho -{\rho }^{\ast }\right)\left(C_w-C_{\infty }\right)}{{\mathit{\rho \beta }}^{\ast }\left(T_w-T_{\infty }\right)\left(1-C_{\infty }\right)},\\ \text{Pr}=\frac{\mu c_p}{k},N_t=\frac{{\rho }^{\ast }{c}_p^{\ast }{D}_T\mathrm{\Delta }T}{\rho c_p\alpha T_b},\mathit{Le}=\frac{\upsilon }{D_B},N_b=\frac{{\rho }^{\ast }{c}_p^{\ast }{D}_B\mathrm{\Delta }C}{\rho c_p\alpha }.\end{array}$$ E12

In Eq. (12) $\text{Re}$ stands for the Reynolds number, $\lambda$ is the buoyancy parameter or in other word it is the natural convection parameter, $\mathit{Nr}$ stands for the buoyancy ratio, $\text{Pr}$ represents the Prandtl number, $N_t$ is used to represent thermophoresis parameter, $\mathit{Le}$ is Lewis number, and $N_b$ is Brownian motion parameter.

Physical conditions for momentum, thermal, and concentration fields are transformed as

$$f\left(1\right)=\frac{\partial f\left(1\right)}{\partial \eta }=\theta \left(1\right)=\varphi \left(1\right)=1,$$ E13

$$\frac{{\partial }^{2}f\left(\beta \right)}{\partial {\eta }^2}=f\left(\beta \right)=\frac{\partial \theta \left(\beta \right)}{\partial \eta }=\frac{\partial \varphi \left(\beta \right)}{\partial \eta }=0,$$ E14

where $\beta$ is the thickness of liquid film sprayed on the outer surface of the cylinder.

Integrating Eq. (3) for pressure term

$$\frac{p-p_b}{\mathit{\mu c}}=\frac{\text{Re}}{\eta }{f}^2-2\left(1+\frac{1}{{\beta }_1}\right)\frac{\partial f}{\partial \eta }.$$ E15

At the outer surface, the shear stress of the liquid film is zero, i.e.,

$$\frac{{\partial }^{2}f\left(\beta \right)}{\partial {\eta }^2}=0.$$ E16

The shear stress on the cylinder is

$$\tau =\frac{\mathit{\rho \upsilon }4\mathit{cz}}{a}{f}^{″}\left(1\right)=\frac{4\mathit{c\mu z}}{a}{f}^{″}\left(1\right)$$ E17

The deposition velocity $V$ is written as

$$-V=-\mathit{ca}\frac{f^{\hspace{0.17em}}\left(\beta \right)}{\sqrt{\beta }}.$$ E18

Mass flux $m_1$ is in association with the deposition per axial length which is

$$m_1=V2\mathit{\pi b}$$ E19

The normalized mass flux $m_2$ is

$$m_2=\frac{m_1}{2\pi a^{2}c}=\frac{m_1}{4{\mathit{\pi \upsilon }}_{\mathit{nf}}\text{Re}}=f\left(\beta \right)$$ E20

The flow, temperature, and concentration rates are

$$S_f=\frac{2{\upsilon }_{\mathit{nf}}}{W_w}{\left(\frac{\partial w}{\partial r}\right)}_{r=a},\mathit{Nu}=-\frac{\mathit{ak}}{\left(T_w-T_b\right)}{\left(\frac{\partial T}{\partial r}\right)}_{r=a},\mathit{Sh}=-\frac{a}{2\left(C_w-C_b\right)}{\left(\frac{\partial C}{\partial r}\right)}_{r=a}.$$ E21

The nondimensional forms for the abovementioned physical properties are

$$\frac{z\text{Re}}{a}{S}_f=\frac{{\partial }^{2}f\left(1\right)}{\partial {\eta }^2},\mathit{Nu}=-2k\frac{\partial \theta \left(1\right)}{\partial \eta },\mathit{Sh}=-\frac{\partial \varphi \left(1\right)}{\partial \eta }.$$ E22

3. Solution by homotopy analysis method

Initially guessed values for $f,$ $\theta$ , and $\varphi$ at $\eta =1$ are

$$\begin{array}{l}{f}_0\left(\eta \right)=\frac{\beta }{2{\left(\beta -1\right)}^3}\left[{\eta }^3-3{\mathit{\beta \eta }}^2-\left(3-6\beta \right)\eta +\left(2-3\beta \right)\right]+\eta ,\\ {\theta }_0\left(\eta \right)=\frac{-{\eta }^2}{2}+\beta \left(\eta -1\right)+\frac{3}{2},{\varphi }_0\left(\eta \right)=\frac{-{\eta }^2}{2}+\beta \left(\eta -1\right)+\frac{3}{2}.\end{array}$$ E23

The linear operators for the given functions $L_f,L_{\theta }$ and $L_{\varphi }$ are selected as

$$L_f=\frac{{\partial }^{4}f}{\partial {\eta }^4},L_{\theta }=\frac{{\partial }^2\theta }{\partial {\eta }^2}\text{and}{L}_{\varphi }=\frac{{\partial }^2\varphi }{\partial {\eta }^2},$$ E24

which satisfies the following general solution:

$$L_f\left(A_1+A_2\eta +A_3{\eta }^2+A_4{\eta }^3\right)=0,L_{\theta }\left(A_5+A_6\eta \right)=0\text{and}{L}_{\varphi }\left(A_7+A_8\eta \right)=0,$$ E25

where $A_i\left(i=1-8\right)$ are constants of general solution.

The corresponding nonlinear operators $N_f$ , $N_{\theta }$ , and $N_{\varphi }$ are defined as

$$N_f\left[f\left(\xi ;p\right),\theta \left(\xi ;p\right)\right]=\left(1+\frac{1}{{\beta }_1}\right)\left(\eta \frac{{\partial }^{3}f\left(\eta ;p\right)}{\partial {\eta }^3}+\frac{{\partial }^{2}f\left(\eta ;p\right)}{\partial {\eta }^2}\right)+\text{Re}\left(f\left(\eta ;p\right)\frac{{\partial }^{2}f\left(\eta ;p\right)}{\partial {\eta }^2}-{\left(\frac{\partial f\left(\eta ;p\right)}{\partial \eta }\right)}^2\right)+\left(\mathit{\lambda \theta }\left(\eta ;p\right)+\mathit{Nr\varphi }\left(\eta ;p\right)\right)=0,$$ E26

$$N_{\theta }\left[f\left(\xi ;p\right),\theta \left(\xi ;p\right)\right]=\eta \frac{{\partial }^2\theta \left(\eta ;p\right)}{\partial {\eta }^2}+\frac{\partial \theta \left(\eta ;p\right)}{\partial \eta }+\text{Pr}.\text{Re}\left(f\left(\eta ;p\right)\frac{\partial \theta \left(\eta ;p\right)}{\partial \eta }-2\frac{\partial f\left(\eta ;p\right)}{\partial \eta }\theta \left(\eta ;p\right)\right)+\eta \frac{\partial \theta \left(\eta ;p\right)}{\partial \eta }\left(N_t\frac{\partial \theta \left(\eta ;p\right)}{\partial \eta }+N_b\frac{\partial \varphi \left(\eta ;p\right)}{\partial \eta }\right)=0,$$ E27

$$N_{\varphi }\left[f\left(\eta ;p\right),\varphi \left(\eta ;p\right)\right]=\eta \frac{{\partial }^2\varphi \left(\eta ;p\right)}{\partial {\eta }^2}+\frac{\partial \varphi \left(\eta ;p\right)}{\partial \eta }+\mathit{Le}.\text{Re}\left(f\left(\eta ;p\right)\frac{\partial \varphi \left(\eta ;p\right)}{\partial \eta }-2\frac{\partial f\left(\eta ;p\right)}{\partial \eta }\varphi \left(\eta ;p\right)\right)+\frac{N_t}{N_b}\left(\eta \frac{{\partial }^2\theta \left(\eta ;p\right)}{\partial {\eta }^2}+\frac{\partial \theta \left(\eta ;p\right)}{\partial \eta }\right)=0,$$ E28

where $p\in \left[0,1\right]$ is embedded parameter.

4. Zeroth-order deformation problem

The equations of zeroth-order deformation problem are obtained as

$$\left(1-p\right)L_f\left[f\left(\eta ;p\right)-f_0\left(\eta \right)\right]={\mathit{ph}}_f{N}_f\left[f\left(\eta ;p\right)\right],$$ E29

$$\left(1-p\right)L_{\theta }\left[\theta \left(\eta ;p\right)-{\theta }_0\left(\eta \right)\right]={\mathit{ph}}_{\theta }{N}_{\theta }\left[f\left(\eta ;p\right),\theta \left(\eta ;p\right)\right],$$ E30

$$\left(1-p\right)L_{\varphi }\left[\varphi \left(\eta ;p\right)-{\varphi }_0\left(\eta \right)\right]={\mathit{ph}}_{\varphi }{N}_{\varphi }\left[f\left(\eta ;p\right),\varphi \left(\eta ;p\right)\right].$$ E31

Here $h_f,h_{\theta }$ and $h_{\varphi }$ are auxiliary nonzero parameters. The corresponding boundary conditions are written as

$${\left.f{\left(\eta ;p\right)}_{\eta =1}=1,\frac{\partial f\left(\eta ;p\right)}{\partial \eta }\right|}_{\eta =1}=1,\theta \left(\eta ;p\right){\left|{}_{\eta =1}=1,\varphi \left(\eta ;p\right)\right|}_{\eta =1}=1,$$ E32

$${\left.\frac{{\partial }^{2}f\left(\eta ;p\right)}{\partial \eta }\right|}_{\eta =\beta }=0,{\left.\frac{\partial \theta \left(\eta ;p\right)}{\partial \eta }\right|}_{\eta =\beta }=0,{\left.\frac{\partial \varphi \left(\eta ;p\right)}{\partial \eta }\right|}_{\eta =\beta }=0.$$ E33

Since

$$p=0\Rightarrow f\left(\eta ;0\right)=f_0\left(\eta \right)=\eta ,\theta \left(\eta ;0\right)={\theta }_0\left(\eta \right)=1,\varphi \left(\eta ;0\right)={\varphi }_0\left(\eta \right)=1,$$ E34

$$p=1\Rightarrow f\left(\eta ;1\right)=f\left(\eta \right),\theta \left(\eta ;1\right)=\theta \left(\eta \right),\varphi \left(\eta ;1\right)=\varphi \left(\eta \right).$$ E35

Using the Taylor’s expansions of $f\left(\eta ;p\right),\theta \left(\eta ;p\right)$ and $\varphi \left(\eta ;p\right)$ about $p=0$ in Eqs. (28)–(31), we obtained

$$f\left(\eta ;p\right)=f_0\left(\eta \right)+\sum _{w=1}^{\infty }{f}_w\left(\eta \right)p^w,$$ E36

$$\theta \left(\eta ;p\right)={\theta }_0\left(\eta \right)+\sum _{w=1}^{\infty }{\theta }_w\left(\eta \right)p^w,$$ E37

$$\varphi \left(\eta ;p\right)={\varphi }_0\left(\eta \right)+\sum _{w=1}^{\infty }{\varphi }_w\left(\eta \right)p^w,$$ E38

where

$${\left.f_w\left(\eta \right)=\frac{1}{w!}\frac{{\partial }^{w}f\left(\eta ;p\right)}{\partial {\eta }^w}\right|}_{p=0},{\left.{\theta }_w\left(\eta \right)=\frac{1}{w!}\frac{{\partial }^w\theta \left(\eta ;p\right)}{\partial {\eta }^w}\right|}_{p=0},{\left.{\varphi }_w\left(\eta \right)=\frac{1}{w!}\frac{{\partial }^w\varphi \left(\eta ;p\right)}{\partial {\eta }^w}\right|}_{p=0}.$$ E39

The convergence of series depends on $h_f,h_{\theta }$ , and $h_{\phi }$ . So let us suppose that series converges at $p=1$ for some values of $h_f,h_{\theta }$ , and $h_{\phi }$ , then Eqs. (35)–(37) become

$$f\left(\eta \right)=f_0\left(\eta \right)+\sum _{w=1}^{\infty }{f}_w\left(\eta \right),$$ E40

$$\theta \left(\eta \right)={\theta }_0\left(\eta \right)+\sum _{w=1}^{\infty }{\theta }_w\left(\eta \right),$$ E41

$$\varphi \left(\eta \right)={\varphi }_0\left(\eta \right)+\sum _{w=1}^{\infty }{\varphi }_w\left(\eta \right).$$ E42

5. ${\mathit{w}}^{\mathit{th}}$ order deformation problem

By taking $w$ times derivatives of Eqs. (28)–(32) and then dividing by $w!$ as well as substituting $p=0$ , we obtained the following equations:

$$L_f\left[f_w\left(\eta \right)-{\chi }_w{f}_{w-1}\left(\eta \right)\right]={\mathit{ћ}}_f{\mathrm{\Re }}_w^f\left(\eta \right),$$ E43

$$L_{\theta }\left[{\theta }_w\left(\eta \right)-{\chi }_w{\theta }_{w-1}\left(\eta \right)\right]={\mathit{ћ}}_{\theta }{\mathrm{\Re }}_w^f\left(\eta \right),$$ E44

$$L_{\varphi }\left[{\varphi }_w\left(\eta \right)-{\chi }_w{\varphi }_{w-1}\left(\eta \right)\right]={\mathit{ћ}}_{\varphi }{\mathrm{\Re }}_w^f\left(\eta \right),$$ E45

where

$${\chi }_w=\{{}_{1,\text{if}p>1}^{0,\text{if}p\le 1}.$$

$${\mathrm{\Re }}_w^f\left(\eta \right)=\left(1+\frac{1}{{\beta }_1}\right)\left(\eta f_{w-1}^{‴}+f_{w-1}^{″}\right)+\text{Re}\sum _{j=0}^{w-1}\left(f_{w-1-j}{f}_j^{″}-f_{w-1-j}^{\prime }{f}_j^{\prime }\right)+\left({\mathit{Gr}}_t{\theta }_{w-1}+{\mathit{Gr}}_c{\varphi }_{w-1}\right),$$ E46

$${\mathrm{\Re }}_w^{\theta }\left(\eta \right)=\eta {\theta }_{w-1}^{″}+{\theta }_{w-1}^{\prime }+\text{Pr}.\text{Re}\sum _{j=0}^{w-1}\left(f_{w-1-j}{\theta }_j^{\prime }-2f_{w-1-j}^{\prime }{\theta }_j\right)+\eta {\theta }_{w-1}^{\prime }\sum _{j=0}^{w-1}\left(N_t{\theta }_{w-1}^{\prime }+N_b{\varphi }_{w-1}^{\prime }\right),$$ E47

$${\mathrm{\Re }}_w^{\varphi }\left(\eta \right)=\eta {\varphi }_{w-1}^{″}+{\varphi }_{w-1}^{\prime }+\mathit{Le}.\text{Re}\sum _{j=0}^{w-1}\left(f_{w-1-j}{\varphi }_j^{\prime }-2f_{w-1-j}^{\prime }{\varphi }_j\right)+\frac{N_t}{N_b}\left(\eta {\theta }_{w-1}^{″}+{\theta }_{w-1}^{\prime }\right).$$ E48

The related boundary conditions are

$$\begin{array}{c}{f}_w\left(1\right)=f_w^{\prime }\left(1\right)={\theta }_w\left(1\right)={\varphi }_w\left(1\right)=1,\\ f_w^{\prime \prime }\left(\beta \right)={\theta }_w^{\prime }\left(\beta \right)={\varphi }_w^{\prime }\left(\beta \right)=0.\end{array}$$ E49

The general solution of Eqs. (42)–(44) is given by

$$\begin{array}{cc}{f}_w\left(\eta \right)& =e_1+e_2\eta +e_3{\eta }^2+f_w^{\ast }\left(\eta \right),\\ {\theta }_w\left(\eta \right)& =e_4+e_5\eta +{\theta }_w^{\ast }\left(\eta \right),\\ {\varphi }_w\left(\eta \right)& =e_6+e_7\eta +{\varphi }_w^{\ast }\left(\eta \right).\end{array}$$ E50

Here $f_w^{\ast }\left(\xi \right),g_w^{\ast }\left(\xi \right)$ and ${\theta }_w^{\ast }\left(\xi \right)$ represent the particular solutions, and the constant $A_i\left(i=1-8\right)$ are determined from boundary conditions (49).

6. Discussion about graphical results

The purpose of this study is to enhance the heat and mass diffusion by choosing a thin-layer spray of the Casson nanofluid over a stretching cylinder. The physical configuration of the problem is shown in Figure 1. The solution of the problem has been obtained using the homotopy approach, and the main features for the convergence (h-curves) of homotopy analysis method (HAM) have been shown in Figures 2 and 3. These figures demonstrate the h-curves for velocity, temperature, and concentration fields, respectively. The impact of buoyancy parameter $\lambda$ and buoyancy ratio $\mathit{Nr}$ on velocity field is prescribed in Figure 4. Velocity grows with the rising values of $\lambda$ because the natural convection parameter $\lambda$ and momentum boundary layer are in direct relation. The similar effect for the rising values of $N_r$ can be seen in Figure 4. The effect of thickness parameter $\beta$ and Casson fluid parameter ${\beta }_1$ versus velocity field is shown in Figure 5. Increasing values of $\beta$ generate friction force and decline the velocity field because the thicker flow creates hurdles in fluid motion, while the thin layer is comparatively fast flowing. Therefore, larger amount of $\beta$ declines the flow motion. The similar effect of the larger amount of the Casson fluid parameter ${\beta }_1$ is shown in Figure 5. The rising values of the parameter ${\beta }_1$ imply a decline in the yield stress of the Casson fluid. In Figure 6, the behavior of the thermophoretic parameter $N_t$ and Reynolds number $\text{Re}$ is observed over the field of temperature. The larger amount of thermophoresis parameter $N_t$ depreciates temperature profile because rising values of $N_t$ enhance the concentration profile due to its direct relation and its product in the model equation increases the cooling effect to reduce the temperature field. The larger quantity of $\text{Re}$ reduces temperature field. Rising values of Reynolds number $\text{Re}$ enhance the inertial forces. The powerful inertial forces kept the fluid particles tightly closed, and more heat energy is required to break down the bonds among these atoms. In other words the inertial forces raise the boiling point of the fluid, and more heat energy is required to enhance the temperature. Figure 7 shows the influences of thermophoretic parameter $N_t$ and Brownian motion parameter $N_b$ in concentration field. The larger amount of $N_b$ displays a falling performance against concentration field. The parameter $N_b$ is owing to the thinning of boundary layer because the random flow of liquid particles makes the decline in the concentration. The rising values of thermophoresis parameter $N_t$ enhance the concentration field. The reason behind this is that $N_t$ is in direct relation with concentration pitch, while the $N_b$ is in inverse relation to the concentration field.

Figure 1.

Physical geometry of the problem.

Figure 2.

h-curve for velocity profile.

Figure 3.

Combined h-curve for temperature and concentration fields.

Figure 4.

Variation of velocity with $\mathit{Nr}$ and $\lambda$ .

Figure 5.

Variation of velocity with $\beta$ and ${\beta }_1$ .

Figure 6.

Variation of $\mathit{Nt}$ and $\text{Re}$ .

Figure 7.

Variation of $\mathit{Nt}$ and $\mathit{Nb}$ .

Figure 8 represents the behavior of the concentration field with respect to Reynolds number $\text{Re}$ and Lewis number $\mathit{Le}$ . The larger amount of $\text{Re}$ improves the concentration field. The reason is that larger values of $\text{Re}$ generate the enhancement in the inertial forces to rise concentration field. The concentration boundary layer is falling with the rising value of Lewis number $\mathit{Le}$ .

Figure 8.

Variation of $\text{Re}$ and $\mathit{Le}$ .

Figure 9 shows the relationship between pressure distribution over the stretching surface versus Reynolds number $\text{Re}$ and Casson fluid parameter ${\beta }_1$ . The larger amount of ${\beta }_1$ increases the viscous forces, and more pressure are required at the surface. Thus the larger amount of ${\beta }_1$ decreases the pressure distribution. The larger amount of the Reynolds number $\text{Re}$ decreases the pressure distribution. The strong inertial effects packed the fluid particle tightly, and as a result the pressure distribution decreases.

Figure 9.

Variation of pressure term.

Table 1 shows the numerical values of the skin friction coefficient, local Nusselt number, and Sherwood number of various physical parameters. The skin friction coefficient rises with the growth of thickness parameter $\beta$ . The thick boundary layer increases friction force and improves the cooling effect. Therefore, the Nussselt and Sherwood numbers are increased. The Reynolds number $\text{Re}$ decreases the fluid flow due to inertial forces. Due to this reason, the larger quantity of $\text{Re}$ enhances the $f^{″}\left(1\right),{\mathrm{\Theta }}^{\prime }\left(1\right)$ and ${\varphi }^{\prime }\left(1\right)$ . Similar effect for rising values of the Casson parameter ${\beta }_1$ has been shown in Table 1. The reason is that the viscous forces become dominant with the larger amount of ${\beta }_1$ to enhance the $f^{″}\left(1\right),{\mathrm{\Theta }}^{\prime }\left(1\right)$ and ${\varphi }^{\prime }\left(1\right)$ . The comparison of present work and published work has been shown in Tables 2 and 3, and closed agreement for $f^{″}\left(1\right),{\mathrm{\Theta }}^{\prime }\left(1\right)$ and ${\varphi }^{\prime }\left(1\right)$ has been achieved.

$\beta$	$\mathbf{Re}$	${\beta }_1$	$f^{″}\left(1\right)$	${\theta }^{\prime }\left(1\right)$	${\varphi }^{\prime }\left(1\right)$
1.5	0.8	1.2	0.0149342	0.785646	0.212503
1.6			0.0150964	0.916634	0.260143
1.7			0.01673959	1.03338	0.309284
	0.8		0.0149342	0.785646	0.212503
	0.9		0.0151422	0.881567	0.240775
	1.0		0.0149653	0.976928	0.269410
		1.2	0.0149342	0.785646	0.212503
		1.3	0.0154906	0.785691	0.212518
		1.4	0.0160015	0.785733	0.212531

Table 1.

Numerical values for skin friction coefficient, local Nusselt number, and Sherwood number for various physical parameters when $h=-0.5,\mathit{Pr}=0.5,{\beta }_1=1.2,\beta =1.5,\mathit{Nt}=0.5,\mathit{Nb}=1,\mathit{Nr}=0.6,\text{Re}=0.8,\mathit{Le}=0.5$ .

$\mathbf{Re}$	[30]	[31]	[4]	Present
0.5	0.88220	0.8827	0.88691	0.886942
1.0	1.17776	1.1781	1.17953	1.72926
2.0	1.59390	1.5941	1.59434	3.26714
5.0	2.41745	2.4175	2.4175	6.75053
10.0	3.34445	3.3445	3.34447	10.1078

Table 2.

Values of $f^{″}\left(1\right)$ for various Reynolds numbers when $h=-0.5,\mathit{Pr}=0.5,{\beta }_1=1.2,\beta =1.1,\mathit{Nt}=0.5,\mathit{Nb}=1,\mathit{Nr}=0.6,\mathit{Le}=0.5$ .

$\mathbf{Pr}$	[30]	[31]	[4]	Present
0.7	1.568	1.5683	1.56878	1.56846
2.0	3.035	3.0360	3.03596	3.68121
7.0	6.160	6.1592	6.15813	7.24452
10.0	10.77	7.4668	7.46477	10.2634

Table 3.

Values of $-{\theta }^{\prime }\left(1\right)$ for various Prandtl numbers when $h=-0.5,{\beta }_1=1.2,\beta =1.1,\mathit{Nt}=0.5,\mathit{Nb}=1,\mathit{Nr}=0.6,\text{Re}=0.8,\mathit{Le}=0.5$ .

7. Conclusion

The heat and mass transfer effect of a thin film over the extended surface of a cylinder has been explored in the recent research. The spray phenomenon has been studied in the form of velocity, temperature, concentration, and pressure distribution profiles, respectively. The similarity transformation has been used to alter the governing equations into the set of nonlinear differential equations. The solution of the problems has been obtained through the homotopy analysis method (HAM). The impact of the embedded parameters has been examined and discussed. The outcomes of the recent study have been pointed out as:

• The inertial forces become stronger with the larger amount of Reynolds number $\text{Re}$ , and as a result the velocity of the fluid flow reduces, while the upsurge values of $\text{Re}$ enhance the $f^{″}\left(1\right)$ . Similarly, the Nusselt number and the Sherwood number are also increasing.

• The larger amount of the thickness parameter $\beta$ of the thin film produces hurdles in the spray phenomenon, and as a result the velocity of the fluid decreases. On the other hand, the skin friction, Nusselt number, and the Sherwood number grow with the larger values of $\beta$ . In fact, the cooling effect increases with the rising values of $\beta$ to enhance the friction force.

• The greater amount of the Brownian motion parameter $N_b$ declines the concentration field. The reason is that the rising values of $N_b$ improve the thinning of the fluid layer and as a result the concentration profile reduces.

• The temperature field increases with the rising value of the thermophoresis parameter $N_t$ , while the concentration field falls to reduce with the larger amount of $N_t$ , because the thermophoresis parameter is in inverse relation to the concentration profile.

• The comparison of the present study with the published work authenticates the obtained result.