Open access peer-reviewed chapter

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

By Taza Gul and Shakeela Afridi

Submitted: February 28th 2018Reviewed: June 26th 2018Published: November 5th 2018

DOI: 10.5772/intechopen.79772

Downloaded: 168

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.

Keywords

  • thin film spray
  • Casson nanofluid
  • stretched cylinder
  • heat and mass transfer
  • thermophoresis
  • HAM

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 Uw, and temperature at the surface of cylinder is taken Tw. The uniform ambient temperature is considered Tbsuch that TwTb>0for assisting flow and TwTb<0for opposing flow.

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

ur+ur+wz=0,E1
uwr+wwz=υ1+1β12wr2+1rwr+gβTTb1Cb+1ρρρCCb,E2
uur+wuz=1ρPr+υ1+1β12ur2+1rurur2,E3
uTr+wTz=α2Tr2+1rTr+ρcpρcpDBTrϕr+DTTbTr2,E4
uCr+wCz=DB2Cr2+1rCr+DTTb2Tr2+1rTr,E5

where urzand wrzare velocity components; ρis density; υis kinematic viscosity; β1is the constant characteristic to Casson fluid; βis the coefficient of thermal expansion; gis the gravitational acceleration along z-axis; T,Tb,Cand Cbdetermine the temperature, ambient temperature, concentration, and ambient concentration, respectively; and α,DT,DBstands for the thermal diffusivity, thermophoresis diffusion coefficient, and Brownian diffusion coefficient.

The suitable boundary conditions are

u=Uw,w=Ww,T=Tw,C=Cwatr=a,E6
μwr=Tr=Cr=0,u=wdzatr=b.E7

Here Uw=carepresents the suction and injection velocity, and Ww=2czis the stretching velocity such that crepresents the stretching parameter and δis the thickness of fluid film.

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

u=ca fηη,w=2cz df,Tz=TbTrefcz2υnfθη,Cz=CbCrefcz2υnfϕη,E8
whereη=ra2.

In the case of the outer radius bof the flow, η=ba2.

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

1+1β1η3fη3+2fη2+Ref2fη2fη2+λθ+Nrϕ=0,E9
η2θη2+θη+Pr.Refθη2fηθ+ηθηNtθη+Nbϕη=0,E10
η2ϕη2+ϕη+Le.Refϕη2fηϕ+NtNbη2θη2+θη=0,E11

where

Re=ca22υnf,λ=gβaTwT1CWw2,Nr=ρρCwCρβTwT1C,Pr=μcpk,Nt=ρcpDTΔTρcpαTb,Le=υDB,Nb=ρcpDBΔCρcpα.E12

In Eq. (12) Restands for the Reynolds number, λis the buoyancy parameter or in other word it is the natural convection parameter, Nrstands for the buoyancy ratio, Prrepresents the Prandtl number, Ntis used to represent thermophoresis parameter, Leis Lewis number, and Nbis Brownian motion parameter.

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

f1=f1η=θ1=ϕ1=1,E13
2fβη2=fβ=θβη=ϕβη=0,E14

where βis the thickness of liquid film sprayed on the outer surface of the cylinder.

Integrating Eq. (3) for pressure term

ppbμc=Reηf221+1β1fη.E15

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

2fβη2=0.E16

The shear stress on the cylinder is

τ=ρυ4czaf1=4cμzaf1E17

The deposition velocity Vis written as

V=cafββ.E18

Mass flux m1is in association with the deposition per axial length which is

m1=V2πbE19

The normalized mass flux m2is

m2=m12πa2c=m14πυnfRe=fβE20

The flow, temperature, and concentration rates are

Sf=2υnfWwwrr=a,Nu=akTwTbTrr=a,Sh=a2CwCbCrr=a.E21

The nondimensional forms for the abovementioned physical properties are

zReaSf=2f1η2,Nu=2kθ1η,Sh=ϕ1η.E22

3. Solution by homotopy analysis method

Initially guessed values for f,θ, and ϕat η=1are

f0η=β2β13η33βη236βη+23β+η,θ0η=η22+βη1+32,ϕ0η=η22+βη1+32.E23

The linear operators for the given functions Lf,Lθand Lϕare selected as

Lf=4fη4,Lθ=2θη2andLϕ=2ϕη2,E24
which satisfies the following general solution:
LfA1+A2η+A3η2+A4η3=0,LθA5+A6η=0andLϕA7+A8η=0,E25

where Aii=18are constants of general solution.

The corresponding nonlinear operators Nf, Nθ, and Nϕare defined as

Nffξpθξp=1+1β1η3fηpη3+2fηpη2+Refηp2fηpη2fηpη2+λθηp+Nrϕηp=0,E26
Nθfξpθξp=η2θηpη2+θηpη+Pr.Refηpθηpη2fηpηθηp+ηθηpηNtθηpη+Nbϕηpη=0,E27
Nϕfηpϕηp=η2ϕηpη2+ϕηpη+Le.Refηpϕηpη2fηpηϕηp+NtNbη2θηpη2+θηpη=0,E28

where p01is embedded parameter.

4. Zeroth-order deformation problem

The equations of zeroth-order deformation problem are obtained as

1pLffηpf0η=phfNffηp,E29
1pLθθηpθ0η=phθNθfηpθηp,E30
1pLϕϕηpϕ0η=phϕNϕfηpϕηp.E31

Here hf,hθand hϕare auxiliary nonzero parameters. The corresponding boundary conditions are written as

fηpη=1=1fηpηη=1=1,θηpη=1=1ϕηpη=1=1,E32
2fηpηη=β=0,θηpηη=β=0,ϕηpηη=β=0.E33

Since

p=0fη0=f0η=η,θη0=θ0η=1,ϕη0=ϕ0η=1,E34

p=1fη1=fη,θη1=θη,ϕη1=ϕη.E35

Using the Taylor’s expansions of fηp,θηpand ϕηpabout p=0in Eqs. (28)(31), we obtained

fηp=f0η+w=1fwηpw,E36
θηp=θ0η+w=1θwηpw,E37
ϕηp=ϕ0η+w=1ϕwηpw,E38

where

fwη=1w!wfηpηwp=0,θwη=1w!wθηpηwp=0,ϕwη=1w!wϕηpηwp=0.E39

The convergence of series depends on hf,hθ, and hφ. So let us suppose that series converges at p=1for some values of hf,hθ, and hφ, then Eqs. (35)(37) become

fη=f0η+w=1fwη,E40
θη=θ0η+w=1θwη,E41
ϕη=ϕ0η+w=1ϕwη.E42

5. wthorder deformation problem

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

Lffwηχwfw1η=ћfwfη,E43
Lθθwηχwθw1η=ћθwfη,E44
Lϕϕwηχwϕw1η=ћϕwfη,E45

where

χw={1,ifp>10,ifp1.
wfη=1+1β1ηfw1+fw1+Rej=0w1fw1jfjfw1jfj+Grtθw1+Grcϕw1,E46
wθη=ηθw1+θw1+Pr.Rej=0w1fw1jθj2fw1jθj+ηθw1j=0w1Ntθw1+Nbϕw1,E47
wϕη=ηϕw1+ϕw1+Le.Rej=0w1fw1jϕj2fw1jϕj+NtNbηθw1+θw1.E48

The related boundary conditions are

fw1=fw1=θw1=ϕw1=1,fwβ=θwβ=ϕwβ=0.E49

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

fwη=e1+e2η+e3η2+fwη,θwη=e4+e5η+θwη,ϕwη=e6+e7η+ϕwη.E50

Here fwξ,gwξand θwξrepresent the particular solutions, and the constant Aii=18are 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 λand buoyancy ratio Nron velocity field is prescribed in Figure 4. Velocity grows with the rising values of λbecause the natural convection parameter λand momentum boundary layer are in direct relation. The similar effect for the rising values of Nrcan be seen in Figure 4. The effect of thickness parameter βand Casson fluid parameter β1versus velocity field is shown in Figure 5. Increasing values of βgenerate 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 βdeclines the flow motion. The similar effect of the larger amount of the Casson fluid parameter β1is shown in Figure 5. The rising values of the parameter β1imply a decline in the yield stress of the Casson fluid. In Figure 6, the behavior of the thermophoretic parameter Ntand Reynolds number Reis observed over the field of temperature. The larger amount of thermophoresis parameter Ntdepreciates temperature profile because rising values of Ntenhance 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 Rereduces temperature field. Rising values of Reynolds number Reenhance 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 Ntand Brownian motion parameter Nbin concentration field. The larger amount of Nbdisplays a falling performance against concentration field. The parameter Nbis 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 Ntenhance the concentration field. The reason behind this is that Ntis in direct relation with concentration pitch, while the Nbis 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 Nr and λ .

Figure 5.

Variation of velocity with β and β 1 .

Figure 6.

Variation of Nt and Re .

Figure 7.

Variation of Nt and Nb .

Figure 8 represents the behavior of the concentration field with respect to Reynolds number Reand Lewis number Le. The larger amount of Reimproves the concentration field. The reason is that larger values of Regenerate the enhancement in the inertial forces to rise concentration field. The concentration boundary layer is falling with the rising value of Lewis number Le.

Figure 8.

Variation of Re and Le .

Figure 9 shows the relationship between pressure distribution over the stretching surface versus Reynolds number Reand Casson fluid parameter β1. The larger amount of β1increases the viscous forces, and more pressure are required at the surface. Thus the larger amount of β1decreases the pressure distribution. The larger amount of the Reynolds number Redecreases 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 β. The thick boundary layer increases friction force and improves the cooling effect. Therefore, the Nussselt and Sherwood numbers are increased. The Reynolds number Redecreases the fluid flow due to inertial forces. Due to this reason, the larger quantity of Reenhances the f1,Θ1and ϕ1. Similar effect for rising values of the Casson parameter β1has been shown in Table 1. The reason is that the viscous forces become dominant with the larger amount of β1to enhance the f1,Θ1and ϕ1. The comparison of present work and published work has been shown in Tables 2 and 3, and closed agreement for f1,Θ1and ϕ1has been achieved.

βReβ1f1θ1ϕ1
1.50.81.20.01493420.7856460.212503
1.60.01509640.9166340.260143
1.70.016739591.033380.309284
0.80.01493420.7856460.212503
0.90.01514220.8815670.240775
1.00.01496530.9769280.269410
1.20.01493420.7856460.212503
1.30.01549060.7856910.212518
1.40.01600150.7857330.212531

Table 1.

Numerical values for skin friction coefficient, local Nusselt number, and Sherwood number for various physical parameters when h=0.5,Pr=0.5,β1=1.2,β=1.5,Nt=0.5,Nb=1,Nr=0.6,Re=0.8,Le=0.5.

Re[30][31][4]Present
0.50.882200.88270.886910.886942
1.01.177761.17811.179531.72926
2.01.593901.59411.594343.26714
5.02.417452.41752.41756.75053
10.03.344453.34453.3444710.1078

Table 2.

Values of f1for various Reynolds numbers when h=0.5,Pr=0.5,β1=1.2,β=1.1,Nt=0.5,Nb=1,Nr=0.6,Le=0.5.

Pr[30][31][4]Present
0.71.5681.56831.568781.56846
2.03.0353.03603.035963.68121
7.06.1606.15926.158137.24452
10.010.777.46687.4647710.2634

Table 3.

Values of θ1for various Prandtl numbers when h=0.5,β1=1.2,β=1.1,Nt=0.5,Nb=1,Nr=0.6,Re=0.8,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 Re, and as a result the velocity of the fluid flow reduces, while the upsurge values of Reenhance the f1. Similarly, the Nusselt number and the Sherwood number are also increasing.

  • The larger amount of the thickness parameter βof 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 β. In fact, the cooling effect increases with the rising values of βto enhance the friction force.

  • The greater amount of the Brownian motion parameter Nbdeclines the concentration field. The reason is that the rising values of Nbimprove 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 Nt, while the concentration field falls to reduce with the larger amount of Nt, 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.

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Taza Gul and Shakeela Afridi (November 5th 2018). The Heat and Mass Transfer Analysis During Bunch Coating of a Stretching Cylinder by Casson Fluid, Fluid Flow Problems, Farhad Ali, IntechOpen, DOI: 10.5772/intechopen.79772. Available from:

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