IMPACT OF DOUBLE-DIFFUSION AND SLIP OF ORDER 2 ONCONVECTION OF CHEMICALLY REACTING OLDROYD-B LIQUID

WITH CATTANEO-CHRISTOV DUAL FLUXby

Fouad Othman M. MALLAWI1, Sheniyappan ESWARAMOORTHI 2,Marimuthu BHUVANESWARI 3 and Sivanandam SIVASANKARAN1,∗

1Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia2Department of Mathematics, Dr. N.G.P. Arts and Science College, Coimbatore - 641048, Tamilnadu, India

3Department of Mathematics, Kongunadu Polytechnic College, D.Gudalur, Dindigul, Tamilnadu, India∗Email: sd.siva@yahoo.com (corresponding author)

AbstractThis article express the outcomes of mixed convective flow of a chemically reacting Oldroyd-B liquid(OBL) with Cattaneo-Christov double flux (CCDF) under the consequence of second order slip (SS), heatabsorption (HA)/heat generation (HG) and Newtonian cooling (NC)/Newtonian heating (NH). The gov-erning PDEs are converted into ODEs using suitable variables. The homotopy analysis method (HAM)is employed to solve these resultant equations. The outcomes of diverse physical parameters, like, re-laxation time, retardation time, Richardson number, buoyancy ratio, Prandtl number, radiation, heatabsorption/generation, Schmidt number, chemical reaction, suction/injection, slip and Newtonian heat-ing are discussed.Keywords: Oldroyd-B liquid; Cattaneo-Christov double flux; Newtonian heating; Homotopy analysismethod; heat generation; Second order slip.

1 Introduction

The non-Newtonian liquids play a vital role in industry, engineering, pharmaceuticals etc. Example ofsuch liquids are shampoos, sugar solutions, polymeric liquids, blood, inks and it cannot illustrated as a linearconstitutive model. Many liquid models were developed to exhibit the features of non-Newtonian liquids.Usually non-Newtonian liquids can be segregated as liquids of rate, differential and integral types. Amongthese classification, rate type liquids were considered for memory and elastic effects. One of the simplest ratetype of liquid is OBL and this liquid predicts the retardation and relaxation time characteristics. This liquidwas initiated by Oldroyd [1] in 1950. It is useful in chemical and process industry when they encounter boththe elastic and memory effects exhibited by most biological and polymers liquids. Rajagopal and Bhatnagar[2] derived the exact solution of simple OBL. Analytical solution of 3D OBL with Soret and Dufour effectswere derived by Farooq et al. [3]. Several studies about OBL flow are found in under different conditionsare Fetecau et al. [4], Liu et al. [5], Jamil et al. [6] and Motsa and Ansari [7].

Heat transfer mechanism is a natural phenomenon and it occurs due to variations of temperature withinthe same object or between bodies and this is very useful in many industrial processes, like, cooling ofnuclear reactor, power generation, electronic devices cooling and magnetic drugs targeting. Fourier [8]initiated “Fouriers law of heat conduction" and there is no material satisfy this law. Then, Cattaneo [9] madesome modification by including a relaxation time parameter for heat flux in order to avoid the paradox ofheat conduction. After that, Christov [10] improved the Cattaneo model by introducing the thermal relaxation

1

time with Oldroyds upper-convected derivatives to reach the material invariant formulation. The outcome ofupper-convected Maxwell fluid with CC model with heat generation was investigated by Asmadi et al. [11].They noticed that the lower amount of heat transfer gradient gets in the presence of thermal relaxation timeparameter. The CC heat flux model was successfully used in OBL by Mustafa [12], in Maxwell nanofluid bySui et al. [13], in nanofluid by Dogonchi and Ganji [14], in second grade fluid by Mallawi et al. [15] and inMaxwell fluid by Khan et al. [16].

The transfer of energy from the surface is proportional to the temperature local surface is treated asNewtonian heating. The NH has significant applications in engine cooling, thermal energy storage, nuclearturbines, heat exchanger, petroleum industry, solar radiation etc. Newtonian heating of a MHD OBL flow wasinvestigated by Mehmood et al. [17]. They found that the energy enhances with enhancing the Newtonianheating parameter. Few significant important studies in this area are found in Refs. ([18] -[20]). No-slipboundary conditions are not sufficient in micro-electro mechanical systems. Because there is no relativemotion between the surface and liquid immediately in contact with the surface. But slip flow model exhibitsthe relation between the velocity of the surface and velocity gradient normal to the surface. The flow ofchemically reactive nanofluid with radiation and second-order slip was portrayed by Zhu et al. [21]. Theyfound the fluid temperature rises with the presence of second-order velocity slip and the reverse trend wasobtained with first-order velocity slip case. Velocity slip of a Oldroyd-B nanoliquid flow over a permeablestretching sheet with heat generation/absorption was examined by Das et al. [22]. They showed that the heattransfer gradient declines when raising the velocity slip parameter. Slip flow of a MHD Jeffrey nanofluidflow with radiation and triple stratification was pedantically solved by Jagan et al. [23]. They proved that thefluid velocity reduces when rising the velocity slip parameter.

From the above literature studies indicate that, chemically reacting fluid flow of an OBL with CCDFflux and second order velocity slip has not studied yet. This analysis carried out the impact of second ordervelocity slip and Newtonian heating on OBL over a stretchy plate with heat generation/absorption, chemicalreaction and Cattaneo-Christov double flux.

2 Mathematical Formulation

We consider the 2D flow of an OBL towards a stretchy plate. Let Uw = Ax be the plate velocity, A > 0is the constant. The surface of the plate is adopted with partial slip condition. The plate is kept at unvaryingtemperature Tw and concentration Cw which is higher than the free stream temperature T∞ and concen-tration C∞. The heat transfer phenomenon is calculated in the presence of Newtonian cooling/heading. Itis assumed that the liquid is a gray, absorbing and emitting. The Rosseland diffusion model is employedfor thermal radiation. Thermal and solutal buoyancy effects are taken into account. Time independent de-structive and constructive chemical reactions are considered here, see ([24] - [27]). The CCDF model wasincorporated for analyzing the variations of mass and hear transfer. In addition, the energy equation is mod-eled with heat absorption/generation. The governing boundary layer equations under CCDF are given by,

∂u1∂x

+∂u2∂y

= 0 (1)

u1∂u1∂x

+ u2∂u1∂y

+ λ1

(u21

∂2u1

∂x2+ u22

∂2u1

∂y2+ 2u1u2

∂2u1∂x∂y

)= g

(βT

[T − T∞

]+ β

C

[C − C∞

])+ν

(∂2u1

∂y2+ λ2

[u1

∂3u1

∂y2∂x+ u2

∂3u1

∂y3− ∂2u1

∂y2

∂u1∂x

− ∂2u2

∂y2

∂u1∂y

])(2)

u1∂T

∂x+ u2

∂T

∂y+ λ3ΩT

= α∂2T

∂y2+ α

16σ∗T 3∞

3kk∗∂2T

∂y2+

Q

ρCp

(T − T∞

)(3)

2

u1∂C

∂x+ u2

∂C

∂y+ λ4ΩC

= DB∂2C

∂y2− k1

[C − C∞

](4)

where (u1, u2)- velocity components in (x, y) coordinates, ν - kinematic viscosity, (λ1, λ2) - relaxation andretardation time parameters, g - acceleration due to gravity, (β

T, β

C- coefficients of thermal and concen-

tration expansions, (λ3, λ4) - relaxation time of heat and mass fluxes, α - thermal diffusivity, ρ - density,Cp- specific heat capacity, σ∗ - Stefan Boltzmann constant, k - thermal conductivity, k∗ - mean absorptioncoefficient, Q- heat absorption/generation coefficient, DB - mass diffusivity and k1 - chemical reaction rate.

The relaxation time of heat and mass fluxes are

ΩT

= u1∂u1∂x

∂T

∂x+ u2

∂u2∂y

∂T

∂y+ u21

∂2T

∂x2+ u22

∂2T

∂y2+ 2u1u2

∂2T

∂x∂y+ u1

∂u2∂x

∂T

∂y+ u2

∂u1∂y

∂T

∂x

ΩC

= u1∂u1∂x

∂C

∂x+ u2

∂u2∂y

∂C

∂y+ u21

∂2C

∂x2+ u22

∂2C

∂y2+ 2u1u2

∂2C

∂x∂y+ u1

∂u2∂x

∂C

∂y+ u2

∂u1∂y

∂C

∂x

with the boundary conditions

u1 = Ax+ L1∂u1∂y

+ L2∂2u1

∂y2, u2 = Vw(x),

∂T

∂y= −hcT , C = Cw at y = 0,

u1 → 0,∂u1∂y

→ 0, T → T∞, C → C∞ at y → ∞ (5)

Now, the following dimensionless variables are introduced:

η = y

√A

ν, u1 = xAF ′, u2 = −

√vAF, Θ =

T − T∞

T∞, Φ =

C − C∞

Cw − C∞, (6)

F′′′ − F

′2+ F

′′F + α1

[2F

′′F

′F − F

′′′F 2]+ α2

[F

′′2 − F ivF]+Ri [Θ +NΦ] = 0 (7)

1

Pr

[4

3R+ 1

]Θ

′′+Θ

′F − ΓT

[Θ

′F

′F +Θ

′′F 2]+HgΘ = 0 (8)

1

ScΦ

′′+Φ

′F − ΓC

[Φ

′F

′F +Φ

′′F 2]− CrΦ = 0 (9)

with the conditions

F (0) = Fw,F′(0) = 1 +∆1F

′′(0) + ∆2F

′′′(0),

Θ′(0) = −NT [1 + Θ(0)] ,Φ(0) = 1, F

′(∞) = 0,Θ(∞) = 0,Φ(∞) = 0. (10)

where α1 = λ1A is the relaxation time constant, α2 = λ2A is the retardation time constant, Ri = GrxRe2x

is the

Richardson number, Grx =gβ

T (Tw−T∞)x3

ν2is the thermal buoyancy parameter, Rex = Ax2

ν is the Reynolds

number, N = Gr∗xGrx

=gβ

C(Cw−C∞)

gβT(Tw−T∞)

is the ratio of concentration to thermal buoyancy, Gr∗x =gβ

C(Cw−C∞)x3

ν2

is the concentration buoyancy parameter, Pr = να is the Prandtl number, R = 4σ∗T 3

∞kk∗ is the thermal radiation

parameter, ΓT

= λ3A is the non-dimensional thermal relaxation time parameter, Hg = QρCpA

is the heat

generation/absorption parameter, Sc = ν

DBis the Schmidt number, Γ

C= λ4A is the non-dimensional solutal

relaxation time parameter, Cr = k1A is the chemical reaction parameter, ∆1 = L1

√Aν is the first order slip

parameter, ∆2 = L2Aν is the second order slip parameter and NT = hc

√νA is the conjugate parameter.

3

The non-dimensional form of the skin friction coefficient, local Nusselt and Sherwood numbers are

1

2Cf√Rex =

1 + α1

1 + α2F ′′(0);

Nu√Rex

= BT

(1 +

4

3R

)(1 +

1

Θ′(0)

)and

Sh√Rex

= −Φ′(0).

-2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

hF,hQ & hF

F''H0

L,Q

'H0L&

F'H0

LF'H0LQ'H0LF''H0L

Figure 1 – h curves of F ′′(0), Θ′(0) and Φ′(0) with α1 = 0.1, α2 = 0.1, Ri = 0.5, N = 0.2, Pr = 1.5, R = 0.3,ΓT = 0.1, Hg = −0.3, Sc = 1.0, ΓC = 0.2, Cr = 1.0, Fw = 0.2, β1 = 0.1, β2 = 0.2 and NT = 0.1.

Table 1 – Comparison of −f ′′(0) for distinct values of α1 with α2 = 0.

α1 Abel et al.[29] Wagas et al.[30] Present Study0.0 0.99998 1.00000 1.00000

0.4 1.10185 1.10190 1.10190

0.8 1.19669 1.19671 1.19671

1.2 1.28525 1.28536 1.28536

1.6 1.36864 1.36876 1.36873

3. HAM Solutions

The obtained equations (7)-(9) with boundary conditions (10) are solved using HAM. We assume theinitial approximations as follows,F0(η) = Fw + ηExp(−η) + 3∆2−2∆1

∆2−1−∆1Exp(−η)− 3∆2−2∆1

∆2−1−∆1,

Θ0(η) =NT

1−NTExp(−η) and Φ0(η) = Exp(−η).

The linear operators areLF = d3F

dη3− dF

dη ; LΘ = d2Θdη2

−Θ and LΦ = d2Φdη2

− Φ with the conditionsLF [E1 + E2Exp(η) + E3Exp(−η)] = 0, LΘ [E4Exp(η) + E5Exp(−η)] = 0,LΦ [E6Exp(η) + E7Exp(−η)] = 0,where Ek(k = 1 − 7) are arbitrary constants. After substituting the Mth order HAM equations, we get,FM (η) = F o

M (η) + C1 + C2Exp(η) + C3Exp(−η),ΘM (η) = Θo

M (η) + C4Exp(η) + C5Exp(−η), ΦM (η) = ΦoM (η) + C6Exp(η) + C7Exp(−η),

where F oM (η),Θo

M (η) and ΦoM (η) are the particular solutions. In general, the HAM solutions have auxiliary

parameter, namely hF , hΘ and hΦ which control the solution convergence, see Bhuvaneswari et al.[28].Figure 1 shows the range values of hF , hΘ and hΦ and the ranges are −1.6 ≤ hF ≤ −0.3, −2 ≤ hΘ ≤ 0.0

and −1.7 ≤ hΦ ≤ −0.2. We choose hF = hΘ = hΦ = −1 for the present calculation with 15th orderof approximations. Table 1 shows the numerical value of −f ′′(0) for distinct values of α1 with α2 = 0compared with Abel et al. [29] and Wagas et al. [30] and we proved that our results are in excellentagreement.

4

3 Results and Discussion

The outcomes of physical parameters, like, α1, α2, R,ΓT ,Hg,ΓC , Cr, Fw, ∆1, ∆2 and NT , for fixedvalues of Richardson number (Ri = 0.5), buoyancy ratio parameter (N = 0.2), Prandtl number (Pr = 1.5)and Schmidt number (Sc = 1.0) are described in this section. Figures 2(a-b) give the impact of α1 andα2 for velocity profile. It is found that the stream speed becomes smaller when increasing the values ofα1. Physically, larger values of α1 reveals to higher relaxation time component which against the liquidspeed. This causes to suppress the liquid speed and thinner the associated boundary layer thickness. On thecontrary, the higher values of α2 lead to enhance the liquid speed. The variations of R on heat absorbing andgenerating liquid with Newtonian heating and cooling cases are illustrated in Figures 3(a-b). It is seen thatthe temperature grows for Newtonian heating case and it diminishes for Newtonian cooling case. Physically,bigger thermal conjugate parameter is strengthening the heat transfer coefficient, which transfer more amountof energy from hotter place to cooler place. This means that increase the liquid temperature and thickenthermal boundary layer thickness. On the contrary, negative values of NT reduce the heat transfer coefficientand there by decreasing the liquid temperature. In addition, the thermal boundary layer thickness is higherin heat generating liquid compared to the heat absorbing liquid. Because, the positive values of Hg (heatgeneration) is to increase the liquid thermal state and transfer more heat between liquid particles. Thisimplies to enrich the thickness of thermal boundary layer. On the other hand, the negative values of Hg (heatabsorption) is to create the opposite behavior and causes to reduction of thermal boundary layer thickness.

Α1 = 0.0, 0.3, 0.5

Second Order Slip

First Order Slip

0 1 2 3 40.0

0.2

0.4

0.6

0.8

1.0

1.2

Η

F'HΗ

L

Α2 = 0.0, 0.1, 0.3

Second Order Slip

First Order Slip

0 1 2 3 40.0

0.2

0.4

0.6

0.8

1.0

1.2

Η

F'HΗ

L

(a) (b)Figure 2 – Impact of α1(a) and α2 (b) on velocity profile.

R = 0.0, 0.5, 1.0, 1.5, 2.0

R = 0.0, 0.5, 1.0, 1.5, 2.0

Newtonian HeatingNewtonian Cooling

0 1 2 3 4

-0.2

-0.1

0.0

0.1

0.2

0.3

Η

QHΗ

L

R = 0.0, 0.5, 1.0, 1.5, 2.0

R = 0.0, 0.5, 1.0, 1.5, 2.0

Newtonian Heating

Newtonian Cooling

0 2 4 6 8 10

-1.0

-0.5

0.0

0.5

1.0

1.5

Η

QHΗ

L

(a) (b)Figure 3 – Impact of R on temperature profile for Hg = −0.3 (a) and Hg = 0.3 (b).

Figures 4(a-b) portray the impact of ΓT on temperature profile with and without thermal radiation forNewtonian heating and cooling cases. It is observed that the liquid temperature and its corresponding bound-

5

ary layer thickness are suppressed by enhancing the thermal relaxation time parameter in Newtonian heatingcase. The quite opposite behavior is obtained in Newtonian cooling case. In addition, the thermal boundarylayer thickness is high in the presence of radiation compared to the absence of radiation. Physically, higheramount of thermal radiation leads to increase the energy transport between particles and thereby increasingthe thermal boundary layer thickness. The effects of Hg on temperature profile for Newtonian heating andcooling cases in the presence/absence of radiation are illustrated in Figures 5(a-b). It is found that the liquidtemperature is an enhancing function of heat generation/absorption parameter in Newtonian heating case andthe opposite trend is obtained in Newtonian cooling case. Also, the thicker boundary layer gets at R = 0.3.

Newtonian HeatingNewtonian Cooling

GT = 0.0, 0.1, 0.2, 0.3, 0.5

GT = 0.0, 0.1, 0.2, 0.3, 0.5

0.0 0.5 1.0 1.5 2.0 2.5

-0.05

0.00

0.05

Η

QHΗ

L

GT = 0.0, 0.1, 0.2, 0.3, 0.5

GT = 0.0, 0.1, 0.2, 0.3, 0.5

Newtonian Heating

Newtonian Cooling

0.0 0.5 1.0 1.5 2.0 2.5 3.0

-0.05

0.00

0.05

0.10

ΗQ

HΗL

(a) (b)Figure 4 – Impact of ΓT on temperature profile for R = 0 (a) and R = 0.3 (b).

Hg = -1.0, - 0.5, 0.0, 0.1, 0.2

Hg = -1.0, - 0.5, 0.0, 0.1, 0.2

Newtonion HeatingNewtonion Cooling

0.0 0.5 1.0 1.5 2.0 2.5

-0.10

-0.05

0.00

0.05

0.10

Η

QHΗ

L

Hg = -1.0, - 0.5, 0.0, 0.1, 0.2

Hg = -1.0, - 0.5, 0.0, 0.1, 0.2

Newtonion HeatingNewtonion Cooling

0.0 0.5 1.0 1.5 2.0 2.5 3.0

-0.10

-0.05

0.00

0.05

0.10

0.15

Η

QHΗ

L

(a) (b)Figure 5 – Impact of Hg on temperature profile for R = 0 (a) and R = 0.3 (b).

Figures 6(a-b) explain the variations of ΓC in concentration profile for different types of chemical re-actions and boundary conditions. It is seen that the liquid concentration and its associated boundary layerthickness are suppressed when raising the concentration relaxation time for both boundary conditions. Ingenerative chemical reaction, the liquid concentration overshoots near the surface and it starts to decay andfinally tends to minimum (zero) value at η → ∞. The effects of Cr on concentration profile for differentboundary conditions with Fourier and non-Fourier mass fluxes are presented in Figures 7(a-b). It is seenthat the solutal boundary layer thickness gets decay for enhancing the chemical reaction parameter. Also,we observed that boundary layer gets thicken for non-Fourier case than the Fourier case. In addition, thedecreasing rate is high in generative chemical reaction case compared to the destructive chemical reactioncase.

Figures 8(a-b) exhibit the comparative view of local Nusselt number for various values of ΓT and Hgwith Newtonian heating, cooling, second order slip and first order slip parameters. It is found that the heattransfer gradient rises with raising the values of ΓT and opposite behavior can be found for large valuesof Hg. The local Nusselt number is an enhancing function of thermal radiation parameter and the reverse

6

trend is witnessed for higher values of Hg, see Figures 9(a-b). In addition, the higher depression rate of heattransfer gradient is found when Hg ∈ (0.6, 0.8). The larger heat transfer gradient is obtained in Newtonianheating case compared to the cooling case. The similar trend is obtained for both second order slip and firstorder slip. The impact of ΓC and Cr with heat generating, absorbing, second order slip and first order slipparameter is shown in Figures 10(a-b). We proved from this figures that, the mass transfer gradient enhancesfor larger values of ΓC and Cr for all cases. The increasing rate of mass transfer is high in Fourier casecompared to the non-Fourier case. It also observed that the mass transfer rate gets maximum in generativechemical reaction case.

GC = 0.0, 0.2, 0.4, 0.6

Second Order Slip

First Order Slip

0.0 0.5 1.0 1.5 2.0 2.50.0

0.2

0.4

0.6

0.8

1.0

Η

FHΗ

L GC = 0.0, 0.2, 0.4, 0.6

Second Order Slip

First Order Slip

0 5 10 150

20

40

60

80

100

ΗF

HΗL

(a) (b)Figure 6 – Impact of ΓC on concentration profile for Cr = 1 (a) and Cr = −1 (b).

Cr = - 0.4, - 0.2, 0.0, 0.2, 0.4

Second Order Slip

First Order Slip

0 2 4 6 80.0

0.2

0.4

0.6

0.8

1.0

Η

FHΗ

L Cr = - 0.4, - 0.2, 0.0, 0.2, 0.4

Second Order SlipFirst Order Slip

0 1 2 3 4 50.0

0.2

0.4

0.6

0.8

1.0

Η

FHΗ

L

(a) (b)Figure 7 – Impact of Cr on concentration profile for ΓC = 0.0 (a) and ΓC = 0.2 (b).

The increment/decrement of skin friction coefficient for distinct values of Hg, ∆1, ∆2 and NT is pro-vided in Figures 11 (a-b). It is noted that the rate of skin friction coefficient decreases with respect to Hg, ∆1

and ∆2 for Newtonian heating case, see Figure 11(a). Higher amount of decreasing rate (189%) is observedfor heat generating liquid with first order slip condition and Newtonian heating case when Fw = −0.2.In Newtonian cooling case, we gained opposite trend, i.e., higher amount of increment (2%) is attained atFw = −0.2. Skin friction coefficient vanishes at Fw = 0.0 for heat generating liquid with first order slipand Newtonian cooling. Figures 12(a-d) demonstrate the consequence of local Nusselt number with respectto ΓT and R with SSHG, FSHG, SSHA and FSHA for Newtonian heating and cooling cases. It is noticedthat rate of change of local Nusselt number enhances when the values of ΓT is rising, see Figure 12(a-b).Also found that the higher amount of increment (17.4%) is obtained in Newtonian cooling case with SSHGat ΓT = 0.4. From Figure 12(c-d), the local Nusselt number gets enhancement for heat absorption case andit decays in the heat generating case. In addition, the decreasing percentage is close to zero when R → 1.At R = 0.3, we obtained lager decrement (21.7%) in Newtonian cooling. The variation of local Sherwood

7

number for different values of ΓC and Cr with SSHG, FSHG, SSHA and FSHA for Newtonian heatingand cooling cases is provided in Figures 13(a-d). It is seen that the mass transfer gradient is an escalatingfunction of ΓC and Cr for all cases. From these figures, higher amount of mass transfer rate is observed inthe presence of destructive chemical reaction. It is also observed that mass transfer rate declines when in-creasing the chemical reaction parameter for all the cases considered. The second-order slip provides highermass transfer rate for all vales of ΓC and both cases of Newtonian heating and Newtonian cooling.

(a) (b)Figure 8 – Variations of Nu√

Refor different values of Hg and ΓT for second order slip (a) and first order slip (b)

with Newtonian heating (upper profile) and Newtonian cooling(lower profile).

4 Conclusions

The purpose of this research paper is to investigate the outcomes of combined convective flow of achemically reacting OBL with CCDF in the presence of second-order slip, heat absorption/generation andNewtonian cooling/heating. After our investigations, we found that he stream speed reduces with enhancingthe velocity relaxation time parameter and it increases when raising the velocity retrardation time parameter.The thicker thermal boundary layer is obtained in the presence of Newtonian heating parameter and quiteopposite trend is obtained in Newtonian cooling parameter. The decreasing rate of solutal boundary layerthickness is high in generative chemical reaction case compared to the destructive chemical reaction case.The larger amount of heat transfer gradient is obtained in Newtonian heating case compared to the coolingcase. The increasing rate of mass transfer is high in Fourier case compared to the non-Fourier case.

(a) (b)Figure 9 – Variations of Nu√

Refor different values of Hg and R for second order slip (a) and first order slip (b)

with Newtonian heating (upper profile) and Newtonian cooling(lower profile).

8

(a) (b)Figure 10 – Variations of Sh√

Refor different values of Cr and ΓC for heat generating (a) and heat absorption (b)

with second order slip (lower profile) and first order slip(upper profile).

(a) (b)Figure 11 – Variation percentage of skin friction coefficient (SFC) for various values of Fw with heat absorp-tion/generation for Newtonian heating (a) and Newtonian cooling (b).

(a) (b)

(c) (d)Figure 12 – Variation percentage of local Nusselt number (NN) for various values of ΓT (a-b) and R (c-d) withheat absorption/generation for Newtonian heating (a,c) and Newtonian cooling (b,d).9

Nomenclature

A positive constant C fluid concentration(kgm−3)Cf skin friction coefficient Cp specific heat(Jkg−1K−1)

Cw wall concentration DB mass diffusivityF ′ non-dimensional velocity g acceleration due to gravityFw suction/injection parameter Gr∗x concentration buoyancy numberGrx thermal buoyancy number Hg heat generation/absorption parameterk thermal conductivity(Wm−1K−1) k∗ mean absorption coefficientk1 chemical reaction rate NT Newtonian heating parameterPr Prandtl number Q heat generation/absorption coefficientR radiation parameter Rex Reynolds numberRi Richardson number Sc Schmidt numberT fluid temperature(K) Tw wall temperature(K)u&v velocity components(ms−1) x&y direction coordinates(m)

Greek symbolsα thermal diffusivity(m2s−1) α1 relaxation time constantα2 retardation time constant σ∗ Stefan Boltzmann constant(Wm−2K−4)

βC coefficient of concentration expansion(kg−1m3) βT coefficient of thermal expansion(K−1)

C∞ free stream concentration η dinensional variableΓC mass relaxation parameter ΓT thermal relaxation parameterλ1 relaxation time of heat flux λ2 relaxation time of mass fluxν kinematic viscosity(m2s−1) Φ non-dimensional concentrationρ fluid density(kgm−3) Θ non-dimensional temperature

AbbreviationsCCDF Cattaneo-Christov double flux HA heat absorptionHAM homotopy analysis method HG heat generationNC Newtonian cooling NH Newtonian heatingOBL Oldroyd B liquid ODE ordinary differential equationSS second order slip PDE partial differential equation

(a) (b)

(c) (d)Figure 13 – Variation percentage of local Sherwood number (SN) for various values of ΓC (a-b) and Cr (c-d)with heat absorption/generation for Newtonian heating (a,c) and Newtonian cooling (b,d).

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Acknowledgments

This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University,Jeddah, Saudi Arabia, under Grant No. (D-257-130-1440) The authors, therefore, acknowledge with thanksDSR technical and financial support.

References

[1] Oldroyd, J.G. The formulation of rheological equations of state, Proceedings of the Royal Society ofLondon Series, 200 (1950), pp. 523-541.

[2] Rajagopal, K.R., Bhatnagar, P.K., Exact solutions for some simple flows of an Oldroyd-B fluid, ActaMechanica, 113 (1995), pp. 233-239.

[3] Farooq, A., et al., Soret and Dufour effects on three dimensional Oldroyd-B fluid, Physica A: Statisti-cal Mechanics and its Applications, 503 (2018), pp. 345-354.

[4] Fetecau, C., et al., On the energetic balance for the flow of an Oldroyd-B fluid due to a flat platesubject to a time-dependent shear stress, Computers & Mathematics with Applications, 60 (2010), pp.74-82.

[5] Liu, Y., et al., Unsteady helical flow of a generalized Oldroyd-B fluid with fractional derivative, Inter-national Journal of Mathematics Trends and Technology, 5 (2014), pp. 66-76.

[6] Jamil M., et al., Unsteady helical flows of Oldroyd-B fluids, Communications in Nonlinear Scienceand Numerical Simulation, 16 (2011), pp. 1378-1386.

[7] Motsa, S.S., Ansari, M.S., Unsteady boundary layer flow and heat transfer of Oldroyd-B nanofluidtowards a stretching sheet with variable thermal conductivity, Thermal Science, 19 (2015), pp. S239-S248.

[8] Fourier, J.B.J., Theorie Analytique De La Chaleur, Chez Firmin Didot. Paris. 1822.

[9] Cattaneo, C., Sulla conduzione del calore. Atti. Semin. Mat. Fis Univ. Modena Reggio Emilia, 3 (1948),pp. 83-101.

[10] Christov, C.I., On frame indifferent formulation of the Maxwell-Cattaneo model of finite speed heatconduction, Mechanics Research Communications, 36 (2009), pp. 481-486.

[11] Asmadi, M.S., et al., Finite difference solution of convective flow of upper-convected Maxwell fluidover a horizontal wedge with suction and heat generation using Cattaneo-Christov heat flux model,Thermal Science, 1 (2019), pp 1-13.

[12] Mustafa, M., An analytical treatment for MHD mixed convection boundary layer flow of Oldroyd-Bfluid utilizing non-Fourier heat flux model, International Journal of Heat and Mass Transfer, 113(2013), pp. 1012-1020.

[13] Sui, J., et al., Boundary layer heat and mass transfer with CattaneoeChristov double-diffusion inupper-convected Maxwell nanofluid past a stretching sheet with slip velocity, International Journal ofThermal Sciences, 104 (2016), pp. 461-468.

[14] Dogonchi, A.S., Ganji, D.D., Impact of Cattaneo-Christov heat flux on MHD nanofluid flow and heattransfer between parallel plates considering thermal radiation effect, Journal of the Taiwan Instituteof Chemical Engineers, 80 (2017), pp. 52-63.

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[15] Mallawi, F.M., et al., Impact of double-stratification on convective flow of a non-Newtonian liquid in aRiga plate with Cattaneo-Christov double-flux and thermal radiation, Ain Shams Engineering Journal,In Press.

[16] Khan, S.U., et al., Heat transfer analysis based on Cattaneo-Christov heat flux model and convectiveboundary conditions for flow over an oscillatory stretching surface, Thermal Science, 23 (2019), pp.443-455.

[17] Mehmood, T., et al., Transverse thermopherotic MHD Oldroyd-B fluid with Newtonian heating, Re-sults in Physics, 8 (2018), pp. 686-693.

[18] Ramzan, M., et al., Three dimensional flow of an Oldroyd-B fluid with Newtonian heating, Interna-tional Journal of Numerical Methods for Heat & Fluid Flow, 25 (2015), pp. 68-85.

[19] Patil, P.M., et al., Unsteady thermal radiation mixed convection flow from a moving vertical plate in aparallel free stream: Effect of Newtonian heating, International Journal of Heat and Mass Transfer,62 (2013), pp. 534-540

[20] Lesnic, D., et al., Free convection boundary-layer flow along a vertical surface in a porous mediumwith Newtonian heating, International Journal of Heat and Mass Transfer, 42 (1999), 4, pp. 2621-2627.

[21] Zhu, T., et al., Second-order slip MHD flow and heat transfer of nanofluids with thermal radiation andchemical reaction, Applied Mathematics and Mechanics (English Edition), 36(9) (2015), pp. 1131-1146.

[22] Das, K., et al., Effect of magnetic field on Oldroyd-B type nanofluid flow over a permeable stretchingsurface, Propulsion and Power Research , 7 (2018), pp. 238-246.

[23] Jagan, K., et al., Soret and Dufour effect on MHD Jeffrey nanofluid flow towards a stretching cylinderwith triple stratification, radiation and slip, Defect and Diffusion Forum, 387 (2018), pp. 523-533.

[24] Kasmani, R.M., et al., Effect of chemical reaction on convective heat transfer of boundary layer flowin nanofluid over a wedge with heat generation/absorption and suction", Journal of Applied FluidMechanics, 9 (2016), pp. 379-388.

[25] Niranjan, H., et al., Chemical reaction, Soret and Dufour effects on MHD mixed convection stagnationpoint flow with radiation and slip condition, Scientia Iranica. Transaction B, Mechanical Engineering,24(2) (2017), 698-706.

[26] Eswaramoorthi, S., et al., Soret and Dufour effects on viscoelastic boundary layer flow, heat andmass transfer in a stretching surface with convective boundary condition in the presence of radiationand chemical reaction, Scientia Iranica. Transaction B, Mechanical Engineering, 23(6) (2016), pp.2575-2586.

[27] Sivasankaran, S., et al., Chemical reaction, radiation and slip effects on MHD mixed convectionstagnation-point flow in a porous medium with convective boundary condition, International Jour-nal of Numerical Methods for Heat & Fluid Flow, 27(2) (2017), pp. 454-470

[28] Bhuvaneswari, M., et al., Effect of viscous dissipation and convective heating on convection flowof a second grade fluid over a stretching surface: Analytical and numerical study, Scientia Iranica.Transaction B, Mechanical Engineering, 26(3) (2019), pp. 1350-1357.

[29] Abel, M.S., et al., MHD flow and heat transfer for the upper-convected Maxwell fluid over a stretchingsheet, Meccanica, 47 (2012), pp. 385-393.

[30] Waqas, M., et al., Stratified flow of an Oldroyd-B nanoliquid with heat generation, Results in Physics,7 (2017), pp. 2489-2496.

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