Analytical Study of Radiative CassonNanoliquid Flow with Heat Absorption

K. Loganathan1(B), K. Tamilvanan2, Amelec Viloria3,4, Noel Varela4,and Omar Bonerge Pineda Lezama5

1 Department Mathematics, Faculty of Engineering,Karpagam Academy of Higher Education, Coimbatore 641021, Tamilnadu, India

loganathankaruppusamy304@gmail.com2 Department of Mathematics, Government Arts College for Men,

Krishnagiri 635001, Tamilnadu, India3 Universidad de la Costa, Barranquilla, Colombia

4 Universidad Peruana de Ciencias Aplicadas, Lima, Peru5 Universidad Tecnolgica Centroamericana (UNITEC), San Pedro Sula, Honduras

Abstract. The divergence of thermally radiative MHD flow of a Cas-son nanofluid over a stretching paper alongside heat absorption. Thegoverning non linear equations are remodeled into a nonlinear ODE’s.The HAM is adopted to find the series solution. The changes of pertinentparameters are analyzed with diagrams and tables. The fluid velocity iscontrolled by suction and it develops with injection. The local Nusseltnumber rapidly suppresses with increasing the magnetic field parameterin heat generation case.

Keywords: Casson nanoliquid · Heat absorption · Magnetic field ·Thermal radiation

1 Introduction

Most of the engineering and industrial processes depend on heat transfer mecha-nism, because they have cooling and heating processes. In general, the ordinaryfluids are transfer less amount heat because they owing poor thermal conduc-tivity. Various researchers are tried to increase the fluid thermal conductivity indifferent ways. One of the simplest method is nanosized particles are suspendedinto an ordinary fluids to raise the fluid thermal conductivity. Applications ofnanofluids are investigated by many authors for both the nanofluids with New-tonian or non-Newtonian base with different geometrical shapes. One of thebase fluid is Casson fluid and which posses yield stress. After applying the shearstress, Casson fluid performs as a solid when low shear stress and it moves whenhigher shear stress compared to the yield stress. Example of these fluids aresoup, blood, jelly, tomato sauce, etc. Some important studies in this directionsare [1–10].

c© Springer Nature Switzerland AG 2020Y. Tan et al. (Eds.): ICSI 2020, LNCS 12145, pp. 678–685, 2020.https://doi.org/10.1007/978-3-030-53956-6_63

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Analytical Study of Radiative Casson Nanoliquid Flow 679

2 Governing Equations

The flow system is modeled with

1. Incompressible flow2. Casson nanoliquid3. thermal radiation4. Magnetic field5. Buongiorno nanofluid model6. Stretching sheet with linear velocity.7. heat absorption

∂u

∂x+

∂v

∂y= 0 (1)

u∂u

∂x+ v

∂u

∂y= v

(1 +

1β

)∂2u

∂y2−

(σB2n

ρ

)(2)

u∂T

∂x+ v

∂T

∂y= αT

∂2T

∂y2− 16σsT

2∞

3keρCp+

ρ∗C∗pρCp

[DB

∂C

∂y

∂T

∂y+

DTT∞

∂T

∂y

2]

+Q

ρCp(T − T∞) (3)

u∂C

∂x+ v

∂C

∂y= DB

∂2C

∂y2+

DTT∞

∂2T

∂y2(4)

The boundary points of the above system are:

u = uw(x) = ax, v = vw, at y = 0 (5)u → 0, v → 0, T → T∞, C → C∞, at y → ∞ (6)

where β (=Casson fluid parameter), σ (=electrical conductivity), ρ (=densityof fluid), αT (=thermal diffusivity), Cp (=specific heat), DB(= Browniandiffusion), DT (+ thermophoretic diffusion coefficient), Q (=heat absorp-tion/generation coefficient).

Consider the transformations:

η =√

a

νy, v = −√aνf(η), u = axf ′(η), θ = T − T∞

Tw − T∞ ; θ =C − C∞Cw − C∞ (7)

The following ODEs are retreived from the governing system using above trans-formations,(

1 +1β

)f

′′′(η) + f

′′(η)f(η) − f ′2(η) − Mf ′(η) = 0 (8)

(1 +

43Rd

)θ′′(η) + Prθ

′(η)f(η) + PrNbθ′(η)φ′(η) + PrNtθ

′2(η)

+PrHgθ = 0 (9)

φ′′(η) + Sc(fφ′) +Nt

Nbθ′′(η) = 0 (10)

680 K. Loganathan et al.

The boundary points of f, θ, φ becomes

f(0) = fw, f′(0) = 1, f

′(∞) = 0, θ(∞) = 0, φ(∞) = 0 (11)

where M = σB2n

ρ [=magnetic field parameter], Pr = ν/αT [=Prandtl num-

ber], Nb = (ρ∗c∗pρcp

DB(CW − C∞))/ν [=Brownian motion] parameter, Nt =(ρ

∗c∗pρcp

DT (TW −T∞))/(νT∞) [=thermophoresis parameter], Hg = Q/ρcpa [=heatgeneration/absorption parameter], Sc = νDB [= Schmidt number], Rd =43

σsT3∞

keρcpαT[=radiation parameter].

The surface drag force and heat transfer rate can be defined as:

12Cf

√Re = f

′′(0)

(1 +

1β

)and

Nu√Re

= −θ′(0)(

1 +43Rd

)

Table 1. Order of approximations

Order −Fηη(0) −θη(0) −φη(0)1 −1.875 −0.229472 −0.8256415 −2.07069 −0.235657 −0.631038

10 −2.07751 −0.234869 −0.6157515 −2.07737 −0.235076 −0.62118820 −2.07737 −0.235057 −0.62060725 −2.07737 −0.235014 −0.62010430 −2.07737 −0.235064 −0.62043735 −2.07737 −0.235032 −0.62034840 −2.07737 −0.235041 −0.620326

3 Results and Discussion

The present nonlinear system was solve through HAM technique. The HAM wascomputed via MATHEMATICA software. Figure 1 sketched for the convergentsolutions of the current study. Appoximation orders of HAM is shown in Table 1.The examinations are complete for various range of the relevant parametersintricate in this study. It is clear from Fig. 2 the velocity profile enhances formagnetic parameter (M) whereas it reduces for suction/ injection parameter

Analytical Study of Radiative Casson Nanoliquid Flow 681

Table 2. The surface drag force and heat transfer ratevalues of β, Fw, M, Nb, Nt, Rd,Hg

β Fw M Nb Nt Rd Hg12Cf

√Re Nu√

Re

0.3 0.2 0.3 −0.2 0.1 0.2 0.2 −2.23314 −0.3296930.1 −1.96458 −o.3295521.5 −1.81318 −0.3283950.1 −1.75965 −0.3278740.5 0.2 0.3 0.1 0.1 0.3 −0.2 −1.74059 −0.275054

−0.5 −1.83052 −0.3128420.5 −1.92547 −0.3548110.1 −1.97484 −0.400887

0.5 0.2 0.3 0.1 0.1 0.3 −0.2 −1.83493 −0.6404820 −1.91934 −0.6337210.5 −2.15183 −0.6337210.1 −2.22367 −0.615345

0 −1.87965 −0.6211882 −1.81318 −0.6108980.1 −1.75965 −0.602256

0.5 0.2 0.3 0.1 0.1 0.3 −0.2 −0.329107 −0.7917910 −0.328067 −0.6211882 −0.327535 −0.4516930.1 −0.326996 −0.28316

0.5 0.2 0.3 0.1 0.1 0.3 −0.2 −0.434391 −0.6075220 −0.45963 −0.6369942 −0.64889 −0.6395230.1 −0.67117 −0.64451

0.5 0.2 0.3 0.1 0.1 0.3 −0.2 −0.340084 −0.616351−0.5 −0.333235 −0.6270370.5 −0.329107 −0.6340880.1 −0.271261 −0.65124

(fw). In Fig. 3 temperature (θ(η)) increases for the radiation (Rd), Brownianconstant (Nb), heat absorption constant (Hg), thermophoresis constant (Nt)and Casson parameter (β) and it decays for suction/injection parameter (fw)and Prandtl number (Pr). The concentration(φ(η)) rises with higher (β) andNt but it diminishes with upsurge in (fw) and Sc (see Fig. 4) (Table 2).

682 K. Loganathan et al.

(a) (b)

(c)

Fig. 1. h-curves for hf,θ,φ

(a) (b)

Fig. 2. Influence of fw and M on velocity

Analytical Study of Radiative Casson Nanoliquid Flow 683

(a) (b)

(c) (d)

(e) (f)

(g)

Fig. 3. Influence of Rd, Nb, Nt, β, Hg, Pr and fw on temperature

684 K. Loganathan et al.

(a) (b)

(c) (d)

Fig. 4. Influence of β, Nb, fw and Sc on concentration

4 Conclusion

The key features of the present study is given below:

– Temperature profile enhances while increasing Rd, Nb, Nt, and Hg.– Casson parameter (β) enhances the concentration and temperature profiles.– Higher range of Prandtl number (Pr) deduce the thermal boundary.

References

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Analytical Study of Radiative Casson Nanoliquid Flow 685

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7. Loganathan, K., Sivasankaran, S., Bhuvaneswari, M., Rajan S.: Dufour and Soreteffects on MHD convection of Oldroyd-B liquid over stretching surface with chem-ical reaction and radiation using Cattaneo-Christov heat flux. IOP: Mater. Sci.Eng. 390, 012077 (2018)

8. Bhuvaneswari, M., Eswaramoorthi, S., Sivasankaran, S., Rajan, S., Saleh Alshom-rani, A.: Effects of viscous dissipation and convective heating on convection flow ofa second-grade liquid over a stretching surface: an analytical and numerical study.Scientia Iranica B 26(3), 1350–1357 (2019)

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https://doi.org/10.1007/s10973-020-09847-whttps://doi.org/10.1007/s10973-020-09847-whttps://doi.org/10.1007/s10973-020-09414-3https://doi.org/10.1007/s10973-020-09751-3

Analytical Study of Radiative Casson Nanoliquid Flow with Heat Absorption1 Introduction2 Governing Equations3 Results and Discussion4 ConclusionReferences