Newtonian Heating Effects of Oldroyd-B Liquid Flow with Cross-Diffusion and Second Order Slip

Abstract

The current study highlights the Newtonian heating and second-order slip velocity with cross-diffusion effects on Oldroyd-B liquid flow. The modified Fourier heat flux is included in the energy equation system. The present problem is modeled with the physical governing system. The complexity of the governing system was reduced to a nonlinear ordinary system with the help of suitable transformations. A homotopy algorithm was used to validate the nonlinear system. This algorithm was solved via MATHEMATICA software. Their substantial aspects are further studied and reported in detail. We noticed that the influence of slip velocity order two is lower than the slip velocity order one.

1 Introduction

Heat transport through non-Newtonian fluids is the significant study in recent times because of its industrial and engineering applications. Oldroyd-B fluid is one of the types of non-Newtonian fluids. This fluid contains viscoelastic behaviour. Loganathan et al. [1] exposed the 2nd-order slip phenomena of Oldroyd-B fluid flow with cross diffusion impacts. Hayat et al. [2] performed the modified heat flux impacts with multiple chemical reactions on Oldroyd-B liquid flow. Eswaramoorthi et al. [3] studied the influence of cross-diffusion on viscoelastic liquid induced by an unsteady stretchy sheet. Elanchezhian et al. [4] examined the important facts of swimming motile microorganisms with stratification effects on Oldroyd-B fluid flow. Loganathan and Rajan [5] explored the entropy effects of Williamson nanoliquid caused by a stretchy plate with partial slip and convective surface conditions. The innovative research articles on non-Newtonian fluid flow with different geometry’s and situations are studied in ref’s [6,7,8,9,10].

As far as our survey report the Newtonian heating effects along with slip order two on Oldroyd-B liquid flow is not examined yet. The present study incorporates the cross diffusion and modified Fourier heat flux into the problem. The eminent homotopy technique [11,12,13] is employed for computing the ODE system and the results are reported via graphs.

2 Modeling

We have constructed the Oldroyd-B liquid flow subjected to below stated aspects:

• 1. Incompressible flow

• 2. Second-order velocity slip

• 3. Magnetic field

• 4. Binary chemical reaction

• 5. Stretching plate with linear velocity.

• 6. Cross-diffusion effects

• 7. Modified Fourier heat flux

Figure 1 represents graphical illustration of physical problem. The governing equations are stated below:

Fig. 1.

Schematic diagram

$$\begin{aligned}&\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0, \end{aligned}$$

(1)

$$\begin{aligned}&\frac{\partial u}{\partial x} u+\frac{\partial u}{\partial y} v+A_1 \left( u^2\frac{\partial ^2 u}{\partial x^2}+v^2 \frac{\partial ^2 u}{\partial y^2}+2uv \frac{\partial ^2 u}{\partial x \partial y} \right) = \mu \frac{\partial ^2 u}{\partial y^2} \nonumber \\&\quad \quad \quad -\mu A_2 v \frac{\partial ^3 u}{\partial y^3}-\frac{\partial u}{\partial x}\frac{\partial ^2 u}{\partial y^2}-\frac{\partial u}{\partial y}\frac{\partial ^2 v}{\partial y^2}+\bigg (u \frac{\partial ^3 u}{\partial x \partial y^2} \bigg )-\frac{\sigma B^2_{0}}{\rho } \bigg (u+A_1 v \frac{\partial u}{\partial y} \bigg ), \nonumber \\ \end{aligned}$$

(2)

$$\begin{aligned}&\frac{\partial T}{\partial x} u +\frac{\partial T}{\partial y} v =\frac{k}{\rho c_p}\frac{\partial ^2 T}{\partial y^2}-\frac{1}{\rho c_p}\frac{\partial q_{r}}{\partial y}+\frac{D_m k_T}{c_s c_p}\frac{\partial ^2 C}{\partial y^2} \end{aligned}$$

(3)

$$\begin{aligned}&\frac{\partial C}{\partial x} u +\frac{\partial C}{\partial y} v =D_m \frac{\partial ^2 C}{\partial y^2}+\frac{D_m k_T}{T_m}\frac{\partial ^2 T}{\partial y^2}-k_m(C-C_\infty ) \ \end{aligned}$$

(4)

The boundary points are

$$\begin{aligned}&u=u_{w}+u_{slip}=ax+\lambda _1 \frac{\partial u}{\partial y}+\lambda _2 \frac{\partial ^2 u}{\partial y^2}, \ \ \ v=0, \nonumber \\&-k\frac{\partial T}{\partial y}=h_{f} T, \ \ \ C=C_{w} \ \ \ {\hbox {at}} \ {y=0}, \end{aligned}$$

(5)

$$\begin{aligned}&u (\rightarrow 0), \ \ v (\rightarrow 0), \ \ \ T (\rightarrow T_{\infty }),\ \ \ C (\rightarrow C_{\infty }) \ \ \text {as} \ {y(\rightarrow \infty }), \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \end{aligned}$$

(6)

where $A_1$ (= relaxation time), $A_2$ (= retardation time), $B_0$ (= constant magnetic field), a (= stretching rate), $c_p$ (= specific heat), $c_\infty$ (= ambient concentration), $c_w$ (= fluid wall concentration), $D_m$ (= diffusion coefficient), k (= thermal conductivity), $T_\infty$ (= ambient temperature), $T_w$ (= convective surface temperature), u, v (= Velocity components), $u_w$ (= velocity of the sheet), $\lambda _1$ (= first order slip velocity factor), $\lambda _2$ (= second order slip velocity factor), $\mu$ (= kinematic viscosity), $\rho$ (= density), $\sigma$ (= electrical conductivity), $\gamma$ (= dimensionless thermal relaxation time). The energy equation updated with Cattaneo-Christov heat flux is defined as:

$$\begin{aligned}&u \frac{\partial T}{\partial x}+v \frac{\partial T}{\partial y} + 2uv \frac{\partial T^2}{\partial x \partial y}+ \lambda \Bigg ( u^2 \frac{\partial ^2 T}{\partial x^2}+ v^2 \frac{\partial ^2 T}{\partial y^2} +\left( u \frac{\partial u}{\partial x}+v \frac{\partial u}{\partial y} \right) \frac{\partial T}{\partial x} \nonumber \\&\quad \quad \quad \quad \quad +\left( u \frac{\partial v}{\partial x}+v \frac{\partial v}{\partial y} \right) \frac{\partial T}{\partial y}\Bigg )=\frac{k}{\rho c_p}\frac{\partial ^2 T}{\partial y^2}-\frac{1}{\rho c_p}\frac{\partial q_{r}}{\partial y}+\frac{D_m k_T}{c_s c_p}\frac{\partial ^2 C}{\partial y^2} \end{aligned}$$

(7)

The transformations are

$$\begin{aligned}&\psi =\sqrt{a\mu }xf(\eta ), u=\frac{\partial \psi }{\partial y}, v=-\frac{\partial \psi }{\partial x}, \eta =\sqrt{\frac{a}{\mu }}y \nonumber \\&v=-\sqrt{a\mu }f(\eta ), \ \ u= axf'(\eta ), \ \ \phi (\eta )=\frac{C-C_{\infty }}{C_{w}-C_{\infty }}, \theta (\eta )=\frac{T-T_{\infty }}{T_{w}-T_{\infty }}, \ \ \end{aligned}$$

(8)

From the above transformations we derive the ODE system as follows,

$$\begin{aligned}&f'''+\beta \left( f''^{2}-ff^{iv}\right) +ff''-f'^2+\alpha \left( 2ff'f''-f^2f''' \right) -M\left( f'-\alpha ff'' \right) =0 \ \ \ \ \ \end{aligned}$$

(9)

$$\begin{aligned}&f \theta ' -\gamma \left( f^2\theta ''+ff'\theta ' \right) +\frac{1}{Pr}(1+\frac{4}{3}Rd)\theta ''+D_f\phi ''=0 \end{aligned}$$

(10)

$$\begin{aligned}&\frac{1}{Sc}\phi ''+f \phi '-Cr\phi +Sr\theta ''=0 \end{aligned}$$

(11)

with boundary points

$$\begin{aligned}&f(0)=0, \ f'(0)=1+\epsilon _1f''(0)+\epsilon _2f'''(0), \ \ \theta '(0)=-Nw(1+\theta (0)),\ \ \phi (0)=1 \nonumber \\&f'(\infty )=0, \ \ \theta (\infty )=0, \ \ \phi (\infty )=0, \end{aligned}$$

(12)

The variables are defined as:

$\epsilon _1$ = (first order velocity constant) = $\lambda _1 \sqrt{a/\mu }$; $\epsilon _2$ = (second order velocity constant)$\,=\,\lambda _2 \frac{a}{\mu }\frac{h_f}{k} \sqrt{\mu /a}$; $\alpha$ = (relaxation time constant) = $A_1 a; \quad \beta$ = (retardation time constant) = $A_2 a$; M = (magnetic field constant) = $\frac{\sigma B_{0}^2}{\rho a}$; Pr = (Prandtl number) = $\frac{\rho C_p}{k}$; Rd = (radiation constant) = $\frac{4 \sigma ^* T_{\infty }^3}{kk^*}$; $\gamma \,=\,\lambda a$; $D_f$ = (Dufour number) = $\frac{D_m k_T}{\mu c_s c_p}\frac{c_w-c_\infty }{T_w-T_\infty }$; Cr = (chemical reaction constant) = $\frac{k_m}{ a}$; Sc = (Schmidt number) = $\frac{\mu }{D_m}$; Sr = (Soret number) = $\frac{D_m k_T}{\mu T_m} \frac{T_w-T_\infty }{c_w-c_\infty }$.

3 Solution Methodology

We using the homotopy technique for validate the convergence of the nonlinear systems. The basic guesses and linear operators are defined as:

$$\begin{aligned}&f_{0}=\eta {e^{-\eta }}+\frac{3\epsilon _2-2\epsilon _1}{\epsilon _2-1-\epsilon _1}*e^{-\eta }-\frac{3\epsilon _2-2\epsilon _1}{\epsilon _2-1-\epsilon _1}, \ \ \ \ \phi _{0}=e^{-\eta }, \ \ \ \ \theta _{0}=\frac{Nw*e^{-\eta }}{1-Nw} \\&L_{f}=f'(f''-1), \ \ \ \ \ \ \ \ L_{\phi }=(\phi '')-(\phi ), L_{\theta }=(\theta '')-(\theta ). \end{aligned}$$

which satisfies the property

$$\begin{aligned} L_{f}\left[ D_1+D_2 e^{\eta }+D_3 e^{-\eta }\right] =0, \ \ \ \ L_{\phi }\left[ D_6e^{\eta }+D_7 e^{-\eta }\right] =0, \ \ \ \ L_{\theta }\left[ D_4e^{\eta }+D_5 e^{-\eta }\right] =0, \ \ \ \ \end{aligned}$$

where $D_k (k=1--7)$ are constants. The special solutions are

$$\begin{aligned}&f_{m}(\eta )=f^{*}_{m}(\eta )+D_{1}+D_{2}e^\eta +D_{3}e^{-\eta } \\&\phi _{m}(\eta )=\phi ^{*}_{m}(\eta )+D_{6}e^{\eta }+D_{7}e^{-\eta }\\&\theta _{m}(\eta )=\theta ^{*}_{m}(\eta )+D_{4}e^{\eta }+D_{5}e^{-\eta }. \end{aligned}$$

In Fig. 2 the straight lines are named as h-curves. The permissible range of $h_f$, $h_\theta$ & $h_\phi$ are $-1.7 \le h_f \le -0.6, -1.2 \le h_\theta \le -0.2, -1.2 \le h_\phi \le -0.2$, respectively. Order of convergent series is depicted in Table 1. Table 2 depicts $f''(0)$ in the special case $M=\beta =0$. It is noted that the $f''(0)$ values are well matched with the previous reports [14,15,16].

Fig. 2.

h-curves for $h_f, h_\theta , h_\phi$

Fig. 3.

$f^{'}(\eta )$ for various range of parameters ($\alpha$, $\beta$, $\epsilon _1$, $\epsilon _2$).

Fig. 4.

$\theta (\eta )$ for various range of parameters ($\gamma$, $R_d$, $\epsilon _1$, $\epsilon _2$, Nw and $D_{F}$).

Fig. 5.

$\phi (\eta )$ for various range of parameters (Cr and Sr).

Table 1. Approximations for convergence

4 Results and Discussion

Physical Characteristics of rising parameters versus, Concentration $\phi (\eta )$, velocity $f(\eta )$ and temperature $\theta (\eta )$ are investigated in Figs. 3, 4 and 5. Figure 3 depicted the velocity distribution $f(\eta )$ for different range of $\alpha$, $\beta$, $\epsilon _1$, $\epsilon _2$. It is noted that the velocity reduces for $\beta$ and $\epsilon _1$, while it increases for $\alpha$ and $\epsilon _2$. The temperature distribution $\theta (\eta )$ for different range of $\gamma$, $R_d$, $\epsilon _1$, $\epsilon _2$, Nw and $D_{F}$ are sketched in Fig. 4. Thermal boundary layer decays with increasing the $\gamma$ and $\epsilon _2$ values. Larger values of $R_d$, $\epsilon _1$ and $D_{F}$ boosts the temperature distribution $\theta (\eta )$. Figure 5 shows the influence on $\phi (\eta )$ for various values of Cr and Sr. These parameters shows the opposite effect in $\phi (\eta )$.

Table 2. Validation of $f''(0)$ in the specific case for various $\alpha$ when $\beta =M=0$

5 Conclusion

The salient outcomes the flow problem is given below:

• 1. Retardation time parameter $\left( \beta \right)$ is inversely proportional to the relaxation time parameter $\left( \alpha \right)$ is in velocity profile.

• 2. Thermal boundary layer enhances due to increasing the $R_{d}$,Nw, $D_{F}$ whereas it decays for higher $\epsilon _{1}$ and $\gamma$.

• 3. Higher Soret number values enhance the solutal boundary thickness.