Effects of Thermal Radiation on a 3D Sisko Fluid over a Porous Medium Using Cattaneo-Christov Heat Flux Model

Transcription

1 Commun. Theor. Phys Vol. 70 No. 2 August Effects of Thermal Radiation on a 3D Sisko Fluid over a Porous Medium Using Cattaneo-Christov Heat Flux Model Deog-Hee Doh 2 M. Muthtamilselvan 1 E. Ramya 1 and P. Revathi 1 1 Department of Mathematics Bharathiar University Coimatore Tamilnadu India 2 Division of Mechanical Engineering College of Engineering Korea Maritime and Ocean University Busan South Korea Received Decemer ; revised manuscript received January Astract This paper investigates the three-dimensional flow of a Sisko fluid over a idirectional stretching sheet in a porous medium. By using the effect of Cattaneo-Christov heat flux model heat transfer analysis is illustrated. Using similarity transformation the governing partial differential equations are transferred into a system of ordinary differential equations that are solved numerically y applying Nachtsheim-Swigert shooting iteration technique along with the 6-th order Runge-Kutta integration scheme. The effect of various physical parameters such as Sisko fluid ratio parameter thermal conductivity porous medium radiation parameter Brownian motion thermophoresis Prandtl numer and Lewis numer are graphically represented. DOI: / /70/2/230 Key words: Sisko fluid Cattaneo-Christov heat flux model thermal radiation porous medium Nomenclature u v w Velocity components of the fluid in x y and z directions x y z Cartesian coordinates a n Material constant of a Sisko fluid c d Stretching rate A Sisko fluid k Permeaility of the porous medium K Porous medium µ f Dynamic viscosity of the fluid ρ Density of the fluid T Temperature T w Temperature of the fluid near the plate T Amient temperature of the fluid kt Thermal conductivity k Thermal conductivity of the fluid far away from the sheet surface q r Radiative heat flux C p Specific heat of the fluid at constant pressure ϵ Small parameter known as thermal conductivity R Radiation parameter σ Stefan-Boltzman constant k Mean asorption coefficient C Concentration C Amient concentration of the fluid D T Thermophoretic diffusion coefficient Nt Thermophoresis parameter N Brownian motion ρ f Density of the ase fluid Re a Re Reynolds numers P r Prandtl numer Le Lewis numer α Stretching ratio parameter α m Thermal diffusivity of the fluid C fx Skin friction coefficient for x direction C fy Skin friction coefficient for y direction Nu x Local nusselt numer Supported y Basic Science Research Program through the National Research Foundation of Korea NRF Funded y the Korea Government under Grant Nos. 2015H1 and C1A the MSIP No. 2015R1A2A2A No. 2017R1A2B the Program of Small and Medium Business y SMBA WC300 R and D S and Department of Science and Technology India through INSPIRE Junior Research Fellowship under Grant No. IF Corresponding author muthtamill@yahoo.co.in c 2018 Chinese Physical Society and IOP Pulishing Ltd

2 No. 2 Communications in Theoretical Physics Introduction Most of the studies descried the flow of viscous fluid y using classical Newtonian model. The greater part of the fluids in industry does not hold frequently estalished supposition of a linear relationship amongst stress and rate of strain and consequently descried as non- Newtonian fluids. Biological fluids Polymeric liquids luricating oils liquid crystals drilling mud and paints are the rheological complex fluids which has vescoelastic manner and cannot e represented just as Newtonian fluids. The non-newtonian fluid flow is usually more complex and particularly non-linear this may get more troules utilizing numerical methods to concentrate on such flows are investigated y Neofytou. [1] Nadeem et al. [2] studied the cause of nanoparticles for Jeffrey fluid flow over a stretching sheet. Sajid and Hayat [3] exhiited an examination to research the wire covering investigation of Sisko fluid withdrawal from a ath. Wang et al. [4] discussed for the MHD peristaltic flow qualities of a Sisko fluid in a symmetric or asymmetric channel. Khan et al. [5] investigated the 3-D Sisko fluid flow over a idirectional stretching surface with Cattaneo-Christov heat flux model. They concluded that the concentration dissemination was reduced with the improvement of the heterogeneous and homogeneous reactions in oth case of shear thickening and shear thinning fluids. Molati et al. [6] discussed on the unidirectional incompressile MHD flow of a Sisko fluid. Akyildiz et al. [7] focussed the thin film flow of a Sisko fluid on a moving elt. The outcomes otained reveals many interesting conducts that warrant investigation of the conditions identified with non-newtonian fluid phenomena particularly the shear-thinning phenomena. Shear-thinning lessens the wall shear stress. Hayat et al. [8] investigated the effect of nanoparticles and magnetic field in the 3D flow of Sisko fluid. The flow is caused y a idirectional stretching sheet. Valipoura et al. [9] examined the Buongiorno Model is applied for melting heat transfer effect on nanofluid heat transfer intensification etween two horizontal parallel plates in a rotating system. Many applications in engineering disciplines involve high permeaility porous media. In such situation Darcy equation fails to give satisfactory results. Therefore use of non-darcy models which takes care of oundary and inertia effects is of fundamental and practical interest to otain accurate results for high permeaility porous media. Khan et al. [10] presented the numerical solutions for the flow of an MHD Sisko fluid past a porous medium. A hypothetical non-linear model for the move through a porous medium has een produced y utilizing modified Darcy s law. Sheikholeslami and Ganji [11] exhiited twodimensional laminar constrained convection nanofluid flow through a stretching porous surface. Doh et al. [12] studied the transient heat and mass transfer flow of a micropolar fluid etween porous vertical channel with oundary conditions of third kind. Recently various researcher pulished papers aout the nanofluid flow through a porous mediam. [13 20] In matter thermal motion of charge particles generate an electromagnetic radiation is denoted as thermal radiation. Some fields of applications are Medicine X- ray radiography food safety and smoke detector. Hayat et al. [21] examined the effect of warm radiation on peristaltic transport of nanofluid in a channel with convective limit conditions. Zeeshan et al. [22] investigated the effects of heat transfer and thermal radiation on the flow of ferro-magnetic fluid on a stretching surface. Bhatti and Rashidi [23] acquired the impact of thermo-diffusion and thermal radiation on Williamson nanofluid past a permeale stretching/shrinking surface. Majeed et al. [24] considered two-dimensional heat transfer of unsteady ferromagnetic liquid and oundary layer flow of magnetic dipole with given heat flux. Akar et al. [25] analyzed the peristaltic consistent of three diverse nanoparticles with water as ase liquid affected y slip oundary conditions through a vertical asymmetric porous channel within the presence of MHD. Waqas et al. [26] studied the characteristics of generalized Burgers fluid over a stretched surface y using Cattaneo-Christov heat flux model. Malik et al. [27] investigated the convective flow of Sisko fluid with the thought of Cattaneo-Christov heat flux model and thermal relaxation time. Liu et al. [28] explored the time and space partial Cattaneo-Christov constitutive model to portray heat conduction. Hayat et al. [29] addressed aout the stagnation point stream of Jeffrey liquid towards a stretching cylinder. In the current study the flow of three-dimensional Sisko fluid past in a porous medium is considered. Here the Brownian motion and thermophoresis are taken into account. By using the similarity transformation the partial differential equations are reduced to ordinary differential equations are solved numerically y applying Nachsheim-swigert shooting iteration technique along with the 6-th order Runge-Kutta integration scheme. The effects of various physical parameters on the velocity temperature and concentration are graphically represented. 2 Mathematical Formulation We consider the steady three-dimentional layer flow of an incompressile Sisko fluid past a porous medium which is outlined in Fig. 1. The sheet is thought to e stretched along x and y-directions with linear velocities u = cx and v = dy respectively where c d > 0 are the stretching rates and flow happens in the space z > 0. Radiation Brownian motion and thermophoresis effects are also there. The oundary layer equations governing the

3 232 Communications in Theoretical Physics Vol. 70 three dimensional Sisko fluid with heat and mass transfer are expressed elow [5] u x + v y + w z = 0 1 u u x + v u y + w u z = a 2 u ρ z 2 u n µ f ρ z z k u 2 u v v v +v +w x y z = a 2 v ρ z 2 + u n 1 v ρ z z z µ f k v 3 u T x + v T y + w T [ z δ E u 2 2 T x 2 + v2 2 T y 2 + w2 2 T z 2 +2uv 2 T x y + 2vw 2 T x z + u u x + v u y + w u T z x + u v v v T +v +w x y z y + u w w w T ] +v +w x y z z = KT 2 T ρc p z 2 1 q r ρc p u C x + v C y + w C z = D B z 4 2 C z 2 + D T 2 T T z 2 5 where u = U w x v = V w y = dy w = 0 T = T w then C D B z + D T T T z = 0 at z = 0 6 u 0 v 0 T T C C as z 7 where the velocity components are u v w in the x y and z directions respectively the material constant of the Sisko fluid is a n 0 represents the shear rate viscosity consistency index and power-law index respectively. Involvement of power index n provides an edge to Sisko fluid as shear thinning n < 1 shear thickening n > 1 the temperature T the ratio diffusion coefficient δ E the density of the ase fluid ρ f with the specific heat of fluid at constant temperature c f. Fig. 1 Schematic diagram of the prolem. The thermal conductivity of the fluid is assumed to vary linearly with temperature as [ kt = k 1 + ε T T ] 8 T w T where k signifies the thermal conductivity of the fluid far from the sheet surface and ε is a small parameter known as the thermal conductivity parameter. We now use the following dimensionless variales u = cxf η v = cyg η c n 2 1/n+1 w = c ρ f / [ 2n f + 1 n ] 1 + n ηf + g x n 1/n+1 θη = T T ϕη = C C T w T C c 2 n 1/n+1 η = z [x 1 n/1+n ]. 9 /ρ f Making use of the transformations Eq. 9Eq. 1 is identically satisfied and Eqs. 2 7 having in mind Eq. 8 leads to the following forms 2n Af + n f n 1 f Kf f 2 + ff + f g = 0 10 Ag + f n 1 g n 1g f f n 2 Kg + 2n 1 + εθθ + Rθ + P r ϕ + Nt θ + P rleϕ 2n N fθ + P rgθ P rλ E [ 2n 2n fg g 2 + gg = n f + g f + g 2n 2θ ] θ + f + g = 0 12 f + P rlegϕ = 0 13 f0 = 0 g0 = 0 f 0 = 1 g 0 = α θ0 = 1 Nϕ 0 + Ntθ 0 = 0 14 f 0 g 0 θ 0 ϕ 0 as η. 15 From the aove equations prime denotes the differentiation with respect to η the material parameter of Sisko fluid is A the porous medium is K the local Reynolds numers are denoted as Re a and Re the radiation parameter is denoted as R the generalized Prandtl numer is P r the stretching ratio parameter is α the relaxation time of the heat flux is denoted as λ E the thermophorosis parameter is denoted as Nt the Brownian motion is de-

4 No. 2 Communications in Theoretical Physics 233 noted as N the Lewis numer is denoted as Le. These parameters are stated as follows A = Re2/n+1 Re a = Re U wxρ a a Re = U w 2 n x n ρ N = ρc pd B c ρ f ρc f a R = 16σT 3 3k k P r = ρc pxu w Re 2/n+1 k K = µ f x ku w Le = α m D B α m = k ρc p α = d c Nt = ρc pd T T w T ρ f. 16 ρc f T a Local Nusselt numer and skin friction coefficients are given y Re 1/n+1 C fx = Af 0 f 0 n 17 Re 1/n+1 C fy = V w Ag 0 + f 0 n 1 g 0 18 U w Re 1/n+1 Nu x = θ It is noted that the dimensionless mass flux represented y the Sherwood numer Sh x is identically zero. 3 Numerical Solution The nonlinear ordinary differential Eqs with the oundary conditions in Eqs are solved numerically y using Runge-kutta method along with Nachtheim-Swigert shooting iteration technique. According to the major requirements of this numerical method the main steps of the technique are given as follows. [5] Let f = y 1 f = y 1 = y 2 f = y 2 = y 3 f = y 3 g = y 4 g = y 4 = y 5 g = y 5 = y 6 g = y 6 θ = y 7 θ = y 7 = y 8 θ = y 8 ϕ = y 9 ϕ = y 9 = y 10 ϕ = y 10 y 3 = y 2 2 2n/y 1 y 3 y 3 y 4 + Ky 2 A + n y 3 n 1 20 y 6 = y n 1y 6 y 3 y 3 n 2 + Ky 5 [2n/]y 1 y 6 y 4 y 6 A + y 3 n 1 21 y 8 = P r[2n/]y 1y 8 P ry 8 y 4 + P rλ E [2n/y 1 + y 4 ][2n/y 2 + y 5 ]y εy 7 + R P rλ E [2n/y 1 + y 4 ] 2 22 y 10 = Nt 2n N y 8 P rle y 1 y 10 P rley 4 y and the oundary conditions ecome y 1 0 = 0 y 2 0 = 1 y 2 = 0 24 y 4 0 = 0 y 5 0 = α y 5 = 0 25 y 7 0 = 1 y 7 = 0 26 N y Nty 7 0 = 0 y 9 = Numerical Results and Discussion The predominant focus of this article is to analyze the characteristics of Cattaneo-Christov heat flux model for the Sisko fluid flow past a porous medium. The effect of various physical parameters like stretching ratio parameter α = 0.0 to 1.5 taken as α = 0 and α = 1 that represent the unidirectional and axisymmetric stretching Sisko fluid A = 0.0 to 1.5 thermal conductivity parameter ϵ = 0.0 to 1.5 porous medium K = 0.0 to 1.5 radiation parameter R = 0.0 to 1.5 Brownian motion N = 0.1 to 1.5 thermophoresis parameter Nt = 0.1 to 1.5 chosen as Nt < 0 and Nt > 0 that physically represent the heated and cold surface respectively Prandtl numer P r = 0.1 to 1.7 taken as P r > 0 that represent the oils relaxation time of the heat transfer λ E = 0.0 to 0.3 and Lewis numer Le = 0.1 to 1.5 on a dimensionless velocities f η g η temperature distriution θη and concentration distriution ϕη are studied numerically. Fig. 2 Comparative study of velocity profiles for different values of Sisko fluid A. The effect of different values of Sisko fluid A is compared with the present study and the previous study of Khan et al. [5] and etter agreement is found. This is shown in Fig. 2. Effect of Sisko fluid parameter A on the velocity profiles f η g η temperature θη and concentration ϕη distriutions are plotted in Figs. 3 6

5 234 Communications in Theoretical Physics Vol. 70 respectively. Here the velocity profiles f η g η are increased at the same time the temperature and concentration profiles are decreased when the Sisko fluid parameter A is increased. Here increment in Sisko fluid parameter towards a low viscosity at high shear rate leads to a decline in oth temperature and concentration profiles and the related oundary layer thickness. the ratio parameter α is increased the velocity profiles increases and temperature and concentration profiles decreases. Fig. 6 Concentration profile ϕη for different values of Sisko fluid A. Fig. 3 Velocity profile f η for different value of Sisko fluid A. Fig. 7 Velocity profile f η for different values of the stretching ratio parameter α. Fig. 4 Velocity profile g η for different value of Sisko fluid A. Fig. 8 Velocity profile g η for different values of the stretching ratio parameter α. Fig. 5 Temperature profile θη for different values of Sisko fluid A. Figures 7 10 present the variations in the velocities f η g η temperature θη and concentration ϕη distriutions for different values of ratio parameter α. When Through Figs the enhancement of temperature θη and concentration ϕη profiles are shown for the higher values of thermal conductivity parameter ϵ. This increase is a direct result of thermal conductivity of the fluids for higher values of the small scalar parameter ϵ

6 No. 2 Communications in Theoretical Physics 235 arisen in the variale thermal conductivity. In addition more heat is transferred from sheet to the liquid and eventually the temperature dispersion is expanded. concentration and oundary layer thickness ut the velocity profiles f η and g η are reduced. Figures depict an enhancement ehaviour of the temperature and concentration profiles and their oundary layer for larger values of the radiation parameter R. For large values of radiation parameter generates a significant amount of heating to the Sisko fluid which enhances the Sisko fluid temperature and concentration oundary layer thickness. Figures display that the temperature θη concentration ϕη and oundary layer thickness drops when the Prandtl numer is increased. Fig. 9 Temperature profile θη for different values of stretching ratio parameter α. Fig. 12 Concentration profile ϕη for the different values of thermal condutivity parameter ϵ. Fig. 10 Concentration profile ϕη for the different values of stretching ratio parameter α. Fig. 13 Velocity profile f η for different values of the porous medium K. Fig. 11 Temperature profile θη for different values of thermal conductivity parameter ϵ. The porous medium K on the velocity temperature θη and concentration ϕη distriutions are plotted in Figs respectively. The increment of the porous medium give rise to the increment in the temperature Fig. 14 Velocity profile g η for different values of the porous medium K.

7 236 Communications in Theoretical Physics Vol. 70 The temperature and concentration profiles of the Sisko fluid flow are seen to decrease with the addition of thermal relaxation parameter λ E as depicted in Figs In a physical sense additional time is important for the heat transfer for molecule to molecule of the liquid. In the result the temperature and concentration oundary layer are diminished in the Sisko fluids. Moreover for λ E = 0 that is for traditional Fourier s law where the temperature is higher when contrasted with the Cattaneo- Christov model. This is ecause of heat transfer through material right away. Figure 23 shows the effect of the Brownian motion parameter N on the concentration ϕη profile. Here increase in the Brownian motion parameter leads to the decrease in the concentration profile. Generally for higher values of Brownian motion have a tendency to heat the fluid in the oundary layer due to this seen that declines in the concentration profile. Fig. 18 Concentration profile ϕη for different values of radiation parameter R. Fig. 15 Temperature profile θη for different values of porous medium K. Fig. 19 Temperture profile θη for different values of Prandtl numer P r Fig. 16 Concentration profile ϕη for different values of porous medium K. Fig. 20 Concentration profile ϕη for different values of Prandtl numer P r. Fig. 17 Temperature profile θη for different values of radiation parameter R. Figure 24 descries the influence of thermophoresis parameter N t on the concentration profile ϕη. When

8 No. 2 Communications in Theoretical Physics 237 the thermophoresis parameter increases the concentration profile is also increases. Generally improving the values of thermophoresis parameter generates a force leads to move the particles from the hotter region to the colder regions for which there is a gain in the heat transfer rates. values of Lewis numer Le causes a decrease in the concentration distriution ϕη. Fig. 24 Concentration profile ϕη for different values of thermophoresis parameter Nt. Fig. 21 Temperture profile θη for different values of the relaxation time parameter λ E. Fig. 25 Concentration profile ϕη for different values of Lewis numer Le. Fig. 22 Concentration profile ϕη for different values of the relaxation time parameter λ E. Fig. 23 Concentration profile ϕη for different values of Brownian motion N. Figure 25 descries the impact of Lewis numer Le on the concentration profile ϕη. It shows that the higher 5 Conclusion The steady three-dimensional flow of Sisko fluid past a porous medium is investigated numerically. The main results of the current work are listed elow i Increase in the Sisko fluid A tends to the increment in the velocity profiles f η and g η and reduction in the temperature and concentration distriutions. ii Higher values of the ratio parameter α increases the magnitude of the velocity f η g η and decreases the temperature and concentration profiles. iii The temperature and concentration distriutions are increased for the higher thermal conductivity ϵ radiation R and it is decreased for the larger values of Prandtl numer P r. vi Increment in the porous medium K creates the reduction in the velocity and rise in the temperature and concentration. v Concentration profile is reduced for the increasing values of the Brownian motion N and Lewis numer Le respectively. vi Large values of the thermophoresis parameter N t tends to the enhancement in concentration field.

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