Three-Dimensional Unsteady Mixed Convection Flow of Non-Newtonian Nanofluid with Consideration of Retardation Time Effects

Abstract

The advances in nanotechnology led to the development of new kinds of engineered fluids called nanofluids. Nanofluids have several industrial and engineering applications, such as solar energy systems, heat conduction processes, nuclear systems, chemical processes, etc. The motivation of the present work is to analyze and explore the thermal and dynamic behaviors of a non-Newtonian fluid flow under time retardation effects. The flow is unsteady and caused by a bidirectional, periodically moving surface. In addition to the convective heat transfer and fluid flow, the radiation and chemical reactions have also been considered. The governing equations are established based on the modified Cattaneo–Christov heat flux formulation. It was found that the bidirectional velocities oscillate periodically, and that the magnitude of the oscillation increases with the retardation time. Higher temperatures occur when the porosity parameter is increased, and lower concentrations are encountered for higher values of the concentration relaxation parameter. The current results can be applied in thermal systems, heat transfer enhancement, chemical synthesis, solar systems, power generation, medical applications, the automotive industry, process industries, refrigeration, etc.

1. Introduction

Several studies on nanofluid applications have recently been presented. Leading to more compact systems and higher performances, nanofluids have become widely used in heat transfer systems, the production and enhancement of energy resources, the extrusion process, the nuclear industry, and many other applications. When nanoparticles are suspended in a base fluid, the suspension becomes characterized by enhanced thermophysical properties. Nanoparticles are particles with dimensions in the nanometer range, typically between 1 and 100 nanometers in size. The size of nanoparticles can vary depending on their composition, shape, and method of synthesis. In general, nanoparticles smaller than 10 nanometers tend to have unique physical and chemical properties due to their high surface-area-to-volume ratio, while those larger than 100 nanometers start to exhibit bulk-like behavior. These nanoparticles are uniformly distributed in the base liquid to enhance the thermal impact of the base materials. Such predictions were first experimentally confirmed by Choi [1]. Buongiorno [2] presented a detailed description of nanofluid convective heat transfer with a focus on the thermophoretic and Brownian motion effects. Hayat et al. [3] investigated the hydrothermal behavior of a nanofluid flow under the effect of an external magnetic field. Sui et al. [4] studied the Cattaneo–Christov double diffusive convection of a Maxwell nanofluid past a stretching sheet. Hsiao [5] presented a numerical study on the micropolar nanofluid flow by considering the MHD and viscous dissipation effects. Turkyilmazoglu [6] considered free and circular jet cooling using nanofluids. Ahmed et al. [7] presented a study on the stagnation point of nanofluid flow past a rotating disk under the effect of a heat source. Turkyilmazoglu [8] used the Buongiorno model to study the nanofluid flow in an asymmetric channel. Tlili et al. [9] studied the bioconvective non-Newtonian nanofluid flow past a stretching cylinder by considering the effect of the activation energy. Abbasi et al. [10] studied the effect of using a hybrid nanofluid on the flow over a curved channel. Kiranakumar et al. [11] presented a comprehensive review of the electrical properties of graphene oxide nanoparticles. Waqas et al. [12] studied the effect of applying an exponential heat flux on the bioconvection of a non-Newtonian nanofluid past a moving surface. Chu et al. [13] used the Keller box method to study the radiative heat transfer of various kinds of hybrid nanofluids. Habib et al. [14] investigated the combined EHD, MHD, and activation energy effects on the bioconvective, time-dependent nanofluid flow caused by an extending sheet. Xia et al. [15] studied the entropy generation caused by the natural bioconversion of Eyring–Powell nanofluids. Waqas et al. [16] studied the bioconvective micropolar nanofluid flow under the impacts of a magnetic field, radiation, and the Joule effect. Liu et al. [17] performed a molecular dynamics study on the effect of using CuO nanoparticles on the phase-change process of a PCM. Mekheimer et al. [18] studied numerically the nanoparticle drug injection in blood to detect diseased organs.

The characteristics of non-Newtonian materials are very important in industrial frameworks and engineering processes. For example, non-Newtonian polymers have several applications in manufacturing processes and chemical industries. Based on their fluid behavior, non-Newtonian liquids are classified into diverse categories. The Maxwell model describes non-Newtonian viscous flow on a long timescale. However, retardation time features are novel rheological consequences that are observed in some non-Newtonian liquids. Such characteristics are identified with the help of the Oldroyd-B fluid model. Kumar et al. [19] presented a study on the rheological behavior of Oldroyd-B fluid with consideration of viscous dissipation and the Joule and radiation effects. Sajid et al. [20] numerically investigated the mixed convection of Oldroyd-B liquid under the effects of viscous dissipation and an external magnetic field. Irfan et al. [21] tested the effect of considering a variable thermal conductivity on the double diffusive convection of Oldroyd-B nanofluid. Bai et al. [22] presented a numerical study on the transient Oldroyd-B double diffusive flow. Roy and Pop [23] conducted an investigation on the mixed convection of Oldroyd-B nanofluid past a shrinking surface. Mabood et al. [24] predicted the effect of radiation and chemical reactions on the rheological behavior of Oldroyd-B nanofluid.

The main objective of the current study is to observe the thermal and hydrodynamic behaviors of Oldroyd-B nanofluid flow caused by a bidirectionally oscillating porous surface. To perform this study, the unsteady mathematical model governing the coupled heat and mass transfer phenomena is developed using the Cattaneo–Christov thermal flux model, and HAM is used to find the semi-analytical solution. The effects of chemical reactions and radiation on the flow and heat and mass transfer are presented in terms of their velocity, temperature, and concentration profiles.

2. Problem Formulation

The Oldroyd-B nanofluid flow over a bidirectional, periodically oscillating porous surface is considered in the current study. The three-dimensional flow is caused by a moving surface with time-dependent velocity. The bidirectional surface is maintained at horizontal and vertical velocities expressed as $u=u_w=ax\sin \omega t$ and $v=v_w=by\sin \omega t$, respectively. Here, $\omega$ and $b$ are the angular frequencies and stretching rate, respectively. Following the cartesian coordinates, $x$ and $y$ are being continued in the surface direction while the $z$ axis is toward the normal direction. For oscillatory phenomena, it has been assumed that the magnitude of oscillation is small in such a manner that the flow regime is kept laminar [25,26,27]. The heat and concentration equations are established via the modified heat and mass flux theories. In addition, the chemical reaction effect is considered in the concentration equation. Based on the above-mentioned assumptions, the governing equations are expressed as follows [25,26,27]:

$$\frac{\partial v}{\partial y}+\frac{\partial u}{\partial x}+\frac{\partial w}{\partial z}=0,$$

(1)

$$\begin{array}{l}\left(\frac{\partial u}{\partial y}\right)v+\left(\frac{\partial u}{\partial x}\right)u+\frac{\partial u}{\partial t}+\left(\frac{\partial u}{\partial z}\right)w+{\lambda }_1\left(\begin{array}{l}2\left(w\frac{{\partial }^{2}u}{\partial z\partial t}+vw\frac{{\partial }^{2}u}{\partial y\partial z}\right)+2\left(uv\frac{{\partial }^{2}u}{\partial x\partial y}\right)\\ +\frac{{\partial }^{2}u}{\partial t^2}+v^2\frac{{\partial }^{2}u}{\partial y^2}+u^2\frac{{\partial }^{2}u}{\partial x^2}+w^2\frac{{\partial }^{2}u}{\partial z^2}\\ +2\left(v\frac{{\partial }^{2}u}{\partial y\partial t}\right)+2\left(uw\frac{{\partial }^{2}u}{\partial x\partial z}+u\frac{{\partial }^{2}u}{\partial x\partial t}\right)\end{array}\right)-\nu \frac{{\partial }^{2}u}{\partial {\tilde{z}}^2}\\ +\nu {\lambda }_2\left(\frac{{\partial }^{2}v}{\partial z^2}\frac{\partial u}{\partial y}+\frac{{\partial }^{2}w}{\partial z^2}\frac{\partial u}{\partial z}+\frac{{\partial }^{2}u}{\partial z^2}\frac{\partial u}{\partial x}\right)\\ =\nu {\lambda }_2\left(\frac{{\partial }^{3}u}{\partial z^2\partial t}+v\frac{{\partial }^{3}u}{\partial y\partial z^2}+u\frac{{\partial }^{3}u}{\partial x\partial z^2}+w\frac{{\partial }^{3}u}{\partial z^3}\right)-\frac{\nu u}{k_p},\end{array}$$

(2)

$$\begin{array}{l}u\left(\frac{\partial v}{\partial x}\right)+v\left(\frac{\partial v}{\partial y}\right)+w\left(\frac{\partial v}{\partial z}\right)+\frac{\partial v}{\partial t}+{\lambda }_1\left(\begin{array}{l}2\left(w\frac{{\partial }^{2}v}{\partial z\partial t}\right)+2\left(uv\frac{{\partial }^{2}v}{\partial x\partial y}+vw\frac{{\partial }^{2}v}{\partial y\partial z}\right)\\ +\frac{{\partial }^{2}v}{\partial t^2}+v^2\frac{{\partial }^{2}v}{\partial y^2}+w^2\frac{{\partial }^{2}v}{\partial z^2}+u^2\frac{{\partial }^{2}v}{\partial x^2}\\ +2\left(v\frac{{\partial }^{2}v}{\partial y\partial t}\right)+2\left(uw\frac{{\partial }^{2}v}{\partial x\partial z}+u\frac{{\partial }^{2}v}{\partial x\partial t}\right)\end{array}\right)-\nu \frac{{\partial }^{2}v}{\partial z^2}\\ +\nu {\lambda }_2\left(\frac{\partial v}{\partial y}\frac{{\partial }^{2}v}{\partial z^2}+\frac{{\partial }^{2}w}{\partial z^2}\frac{\partial v}{\partial \tilde{z}}+\frac{\partial v}{\partial \tilde{x}}\frac{{\partial }^{2}u}{\partial {\tilde{z}}^2}\right)=\nu {\lambda }_2\left(\frac{{\partial }^{3}v}{\partial z^2\partial t}+w\frac{{\partial }^{3}v}{\partial z^3}+v\frac{{\partial }^{3}v}{\partial y\partial z^2}+u\frac{{\partial }^{3}v}{\partial x\partial z^2}\right)-\frac{\nu u}{k_p},\end{array}$$

(3)

$$\begin{array}{l}u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}+w\frac{\partial T}{\partial z}+\frac{\partial T}{\partial t}+{\delta }_2\left[\begin{array}{l}2u\frac{{\partial }^{2}T}{\partial x\partial t}+w^2\frac{{\partial }^{2}T}{\partial z^2}+v^2\frac{{\partial }^{2}T}{\partial y^2}+u^2\frac{{\partial }^{2}T}{\partial x^2}+2w\frac{{\partial }^{2}T}{\partial z\partial t}+w\frac{\partial u}{\partial z}\frac{\partial T}{\partial x}\\ +v\left(\frac{\partial T}{\partial z}\frac{\partial w}{\partial y}+\frac{\partial T}{\partial y}\frac{\partial v}{\partial y}\right)+\frac{\partial w}{\partial t}\frac{\partial T}{\partial z}+u\left(\frac{\partial T}{\partial y}\frac{\partial v}{\partial x}+\frac{\partial T}{\partial x}\frac{\partial u}{\partial x}+\frac{\partial T}{\partial z}\frac{\partial w}{\partial x}\right)\\ +\frac{\partial u}{\partial t}\frac{\partial T}{\partial x}+\frac{\partial v}{\partial t}\frac{\partial T}{\partial y}+\frac{{\partial }^{2}T}{\partial t^2}+v\frac{\partial u}{\partial y}\frac{\partial T}{\partial x}+w\frac{\partial v}{\partial z}\frac{\partial T}{\partial y}\\ +2v\left(u\frac{{\partial }^{2}T}{\partial x\partial y}+w\frac{{\partial }^{2}T}{\partial y\partial z}+\frac{{\partial }^{2}T}{\partial y\partial t}\right)+w\left(2u\frac{{\partial }^{2}T}{\partial x\partial z}+\frac{\partial w}{\partial z}\frac{\partial T}{\partial z}\right)\end{array}\right]\\ =\left({\alpha }_m+\frac{16{\sigma }_s{T}_{\infty }^3}{3k^{\ast }{\left(\rho c\right)}_f}\right)\frac{{\partial }^{2}T}{\partial z^2}+{\sigma }_{\ast }\left[\frac{D_T}{T_{\infty }}{\left(\frac{\partial T}{\partial z}\right)}^2+D_B\frac{\partial C}{\partial z}\frac{\partial T}{\partial z}\right],\end{array}$$

(4)

$$\begin{array}{l}\left(\frac{\partial C}{\partial y}\right)v+\left(\frac{\partial C}{\partial x}\right)u+\frac{\partial C}{\partial t}+\left(\frac{\partial C}{\partial z}\right)w+{\delta }_2\left[\begin{array}{l}2u\frac{{\partial }^{2}C}{\partial x\partial t}+w^2\frac{{\partial }^{2}C}{\partial z^2}+v^2\frac{{\partial }^{2}C}{\partial y^2}+u^2\frac{{\partial }^{2}C}{\partial x^2}\\ +2w\frac{{\partial }^{2}C}{\partial z\partial t}+w\frac{\partial u}{\partial z}\frac{\partial C}{\partial x}+v\frac{\partial v}{\partial y}\frac{\partial C}{\partial y}\\ +v\frac{\partial w}{\partial y}\frac{\partial C}{\partial z}+\frac{\partial v}{\partial t}\frac{\partial C}{\partial y}\\ +u\left(\frac{\partial C}{\partial y}\frac{\partial v}{\partial x}+\frac{\partial u}{\partial x}\frac{\partial C}{\partial x}+\frac{\partial C}{\partial z}\frac{\partial w}{\partial x}\right)\\ +\frac{\partial u}{\partial t}\frac{\partial C}{\partial x}+\frac{{\partial }^{2}C}{\partial t^2}+v\frac{\partial u}{\partial y}\frac{\partial C}{\partial x}\\ +\frac{\partial w}{\partial t}\frac{\partial C}{\partial \tilde{z}}+w\frac{\partial v}{\partial z}\frac{\partial C}{\partial y}\\ +2v\left(u\frac{{\partial }^{2}C}{\partial x\partial y}+w\frac{{\partial }^{2}C}{\partial y\partial z}+\frac{{\partial }^{2}C}{\partial y\partial t}\right)\\ +w\left(2u\frac{{\partial }^{2}C}{\partial x\partial z}+\frac{\partial w}{\partial z}\frac{\partial C}{\partial z}\right)\end{array}\right]\\ =\left(\frac{D_T}{T_{\infty }}\right)\frac{{\partial }^{2}T}{\partial z^2}+D_B\frac{{\partial }^{2}C}{\partial z^2}-k_{\ast }\left(C-C_{\infty }\right).\end{array}$$

(5)

The boundary conditions are [25,26,27]:

$$u=u_w=ax\sin \omega t, w=0, C=C_w, v=v_w=by \sin \omega t, T=T_w, z=0,$$

(6)

$$u\to 0, C\to C_{\infty }, T\to T_{\infty }, v\to 0, as z\to \infty .$$

(7)

The used dimensionless variables are [25,27]:

$$\left.\begin{array}{l}v=ya{g}_{\xi }(\xi ,\tau ), \mathrm{w}=-\sqrt{va}(+g(\xi ,\tau )+f(\xi ,\tau )),u=xa{f}_{\xi }(\xi ,\tau )\\ \xi =\sqrt{\frac{a}{v}z}, \tau =t\omega , \theta (\xi ,\tau )=\frac{T-T_{\infty }}{T_w-T_{\infty }}, \varphi (\xi ,\tau )=\frac{C-C_{\infty }}{C_w-C_{\infty }}.\end{array}\right\}$$

(8)

Using these new variables, the governing equations become:

$$f_{\xi \xi \xi }+f{f}_{\xi \xi }+g{f}_{\xi \xi }-\left(f_{\xi }{}^2+S{f}_{\xi \tau }\right)-Ha{f}_{\xi }-{\beta }_1\left[\begin{array}{l}{S}^2{f}_{\xi \tau \tau }+2S{f}_{\xi }{f}_{\xi \tau }+{\left(f+g\right)}^2{f}_{\xi \xi \xi }\\ -2\left(f+g\right)\left(S{f}_{\xi \xi \tau }+f_{\xi }{f}_{\xi \xi }\right)\end{array}\right]+{\beta }_2\left[\begin{array}{l}S{f}_{\xi \xi \xi \tau }+f_{\xi \xi }\left(g_{\xi \xi }+f_{\xi \xi }\right)\\ -\left(f+g\right)f_{\xi \xi \xi \xi }\end{array}\right],$$

(9)

$$\begin{array}{l}{g}_{\xi \xi \xi }+\left(g+f\right)g_{\xi \xi }-Ha{f}_{\xi }-\left(g_{\xi }{}^2+S{g}_{\xi \tau }\right)+{\beta }_2\left[\begin{array}{l}{g}_{\xi \xi }\left(f_{\xi \xi }+g_{\xi \xi }\right)\\ +S{g}_{\xi \xi \xi \tau }-\left(g+f\right)g_{\xi \xi \xi \xi }\end{array}\right]\\ -{\beta }_1\left[\begin{array}{l}{S}^2{g}_{\xi \tau \tau }+g_{\xi \xi \xi }{\left(f+g\right)}^2+2S{g}_{\xi }{g}_{\xi \tau }\\ -2\left(S{g}_{\xi \xi \tau }+g_{\xi }{g}_{\xi \xi }\right)\left(f+g\right)\end{array}\right]=0,\end{array}$$

(10)

$$\frac{\left(1+Rd\right)}{\mathrm{Pr}}{\theta }_{\xi \xi }+\left(Nb{\theta }_{\xi }{\varphi }_{\xi }-S{\theta }_{\tau }+Nt{\theta }_{\xi }{}^2+\left(g+f\right){\theta }_{\xi }\right)-{\delta }_T\left(\begin{array}{l}{S}^2{\theta }_{\tau \tau }-S\left(f_{\tau }{\theta }_{\xi }+g_{\tau }{\theta }_{\xi }\right)\\ -2S\left(g+f\right){\theta }_{\tau \xi }+{\left(g+f\right)}^2{\theta }_{\xi \xi }\\ +\left(f_{\xi }+g_{\xi }\right)\left(f+g\right){\theta }_{\xi }\end{array}\right)=0,$$

(11)

$${\varphi }_{\xi \xi }+Le\mathrm{Pr}\left[(g+f){\varphi }_{\xi }-S{\varphi }_{\tau }-Kr\varphi \right]+\frac{Nt}{Nb}{\theta }_{\xi \xi }-Sc{\delta }_c\left(\begin{array}{l}{S}^2{\varphi }_{\tau \tau }-S\left(f_{\tau }{\varphi }_{\xi }+g_{\tau }{\varphi }_{\xi }\right)\\ -2S\left(g+f\right){\varphi }_{\tau \xi }+{\left(g+f\right)}^2{\varphi }_{\xi \xi }\\ +\left(f_{\xi }+g_{\xi }\right)\left(f+g\right){\varphi }_{\xi }\end{array}\right)=0.$$

(12)

The dimensionless boundary conditions are as follows:

$$\left.\begin{array}{c}g(0,\tau )=0, f(0,\tau )=0, f_{\xi }(0,\tau )=\sin \tau , g_{\xi }(0,\tau )=\gamma \sin \tau , \theta (0,\tau )=1\\ \varphi (0,\tau )=1, g_{\xi }(\infty ,\tau )=0, f_{\xi }(\infty ,\tau )=0, \theta (\infty ,\tau )=0, \varphi (\infty ,\tau )=0.\end{array}\right\}$$

(13)

The dimensionless parameters ${\beta }_1,$${\beta }_2$ (Deborah numbers), $Nb$ (Brownian motion), $Le$ (Lewis number), $Kr$ (chemical reaction), $\gamma$ (stretching ratio constant), $Kr$ (reaction constant), ${\delta }_T$ (thermal relaxation constant), ${\delta }_c$ (concentration relaxation constant), S (ratio of frequency to stretching rate), Pr (Prandtl number), and $Nt$ (thermophoresis) are defined as:

$$\left.\begin{array}{c}Le=\frac{{\alpha }_m}{D_B}, Kr=\frac{k_{\ast }}{a}, {\beta }_1={\lambda }_{1}a, {\beta }_2={\lambda }_{2}a, \mathrm{Pr}=\frac{\nu }{{\alpha }_m}, S=\frac{\omega }{a}, Nb=\frac{{\sigma }_{\ast }{D}_B(C_w-C_{\infty })}{\nu },\\ R=\frac{16{\sigma }_s{T}_{\infty }^3}{3k{k}^{\ast }},{\delta }_c={\delta }_{2}a, \gamma =\frac{b}{a} Nt=\frac{{\sigma }_{\ast }{D}_T(T_w-T_{\infty })}{\nu T_{\infty }}, Ha=\frac{\nu }{a{k}_p}, {\delta }_T={\delta }_{1}a.\end{array}\right\}$$

(14)

The local Sherwood and Nusselt numbers are expressed by the following relations [27]:

$${({\mathrm{Re}}_x)}^{-1/2}N{u}_x=-\left(1+R\right){\theta }_{\xi }(0,\tau ), S{h}_x{({\mathrm{Re}}_x)}^{-1/2}=-{\varphi }_{\xi }(0,\tau ).$$

(15)

where ${\mathrm{Re}}_x$ is the local Reynolds number.

3. Homotopy Analytical Method

The resolutions of the formulated model are performed analytically with the implementation of the homotopy analysis scheme. The motivations for using the homotopy analysis method are justified, as this scheme provides high accuracy and little error. It should also be mentioned that the HAM method does not have any limitations related to the fixation of any small or large parameters [28,29,30]. The initial guesses for performing simulations are:

$$\left.\begin{array}{l}{f}_0\left(\xi ,\tau \right)=\left(1-e^{-\xi }\right)\sin \tau , g_0\left(\xi ,\tau \right)=\gamma \left(1-e^{-\xi }\right)\sin \tau ,\\ {\theta }_0\left(\xi ,\tau \right)=e^{-\xi }, {\varphi }_0\left(\xi ,\tau \right)=e^{-\xi }.\end{array}\right\}$$

(16)

The use of the linear operators $\left(\Upsilon \right)$:

$$\begin{array}{cccc}{\Upsilon }_f=\frac{{\partial }^3}{\partial {\xi }^3}-\frac{\partial }{\partial \xi },& {\Upsilon }_g=\frac{{\partial }^3}{\partial {\xi }^3}-\frac{\partial }{\partial \xi },& {\Upsilon }_{\theta }=\frac{{\partial }^2}{\partial {\xi }^2}-1,& {\Upsilon }_{\varphi }=\frac{{\partial }^2}{\partial {\xi }^2}-1.\end{array}$$

(17)

gives the following equations:

$$\left.\begin{array}{c}\begin{array}{cc}{l}_f\left[{\kappa }_1+{\kappa }_2{e}^{\xi }+{\kappa }_3{e}^{-\xi }\right]=0,& l_g\left[{\kappa }_4+{\kappa }_5{e}^{\xi }+{\kappa }_6{e}^{-\xi }\right]=0,\end{array}\\ \begin{array}{cc}{l}_{\theta }\left[{\kappa }_7{e}^{\xi }+{\kappa }_8{e}^{-\xi }\right]=0,& l_{\varphi }\left[{\kappa }_9{e}^{\xi }+{\kappa }_{10}{e}^{-\xi }\right]=0.\end{array}\end{array}\right\}$$

(18)

4. Convergence of HAM

The convergence regime is estimated with the proper values of auxiliary factors $h_f,h_{\theta },h_g$ and $h_{\varphi }$. The feasible region for ensuring the solution validity is highlighted with plotted h-curves. Figure 1 is presented to predict this convergence region. The convergence regime is defined by $-1.4\le h_f\le 0.2,$ $-1.4\le h_{\theta }\le -0.2,$ $-1.3\le h_g\le -0.1$ and $-1.3\le h_{\varphi }\le -0.35.$

This section is dedicated to exploring the effects of the governing parameters on the flow, temperature, and concentration fields. The formulated mathematical problem is associated with the theoretical flow assumptions instead of any experimental data. Therefore, physical analysis is performed for ranges of governing parameters as follows: $0\le Ha\le 0.3,$ $0\le {\beta }_2\le 1.4,$ $0.1\le {\delta }_T\le 1.2,$ $0.1\le \mathrm{Pr}\le 1.2,$ $0.1\le Nb\le 1.2,$ $0.1\le Rd\le 1.2,$ and $0.2\le {\delta }_C\le 0.8$, $0.0\le Kr\le 0.5.$ Figure 2a,b illustrate the velocity component profiles $f_{\xi }$ and $g_{\xi }$ for various values of the porosity parameter $Ha.$ A declining behavior in both velocity components is observed when $Ha$ is increased. Physically, this reduction is due to the reduced permeability of the porous media. Figure 3a,b present the effects of the retardation time factor ${\beta }_2$ on $f_{\xi }$ and $g_{\xi }$. Both velocity components have larger values for higher values of ${\beta }_2$. This increase is due to the intensification of the flow caused by the retardation time. In fact, the retardation time is associated with the rest position attained by fluid particles. Figure 4a,b show the effect of the retardation parameter ${\beta }_2$ on $f_{\xi }$ and $g_{\xi }$. The increase in ${\beta }_2,$ leads to an increase in velocity. In both graphs, a phase shift of smaller magnitude is noticed.

The results illustrated in Figure 5a report the effects of ${\delta }_T$ on the behavior of the temperature profile $\theta$. Lower temperatures are encountered when the thermal relaxation factor ${\delta }_T$ is increased. This decrease leads to a lower heat transfer rate. Figure 5b illustrates the temperature profile $\theta$ for various Prandtl number $\mathrm{Pr}$ values. The Prandtl number is inversely proportional to the thermal diffusivity. Thus, the increase in $\mathrm{Pr}$ values, leads to a reduction in temperature. In Figure 5c, the effect of increasing the Brownian motion constant $N_b$ on the temperature field is presented. Higher temperatures occur for larger values of $N_b$. Physically, Brownian motion is based on the random movement of heated particles, some of which collide with each other. This collision enhances thermal transport. The effect of the radiation parameter $Rd$ on the temperature $\theta$ profile is exposed in Figure 5d. The increase in $Rd$ leads to higher temperature values. This is due to the enhancement of heat transfer by the combined effects of convective and radiative heat transfers. Figure 6a–e illustrate the changes in concentration profile $\varphi$ caused by the variation in the porosity parameter Ha, the Lewis number $Le,$ the concentration relaxation parameter ${\delta }_C$, the reaction parameter $Kr$, and the Brownian parameter $Nb.$ As presented in Figure 6a, due to the change in the porosity parameter, an increase in the concentration is exhibited. Physically, such observations are due to the permeability of the porous space. From Figure 6b, it is concluded that the increase in $Le$ reduces the concentration value $\varphi .$ The results presented in Figure 6c–e demonstrate that the increases in ${\delta }_C$, $Kr$, and $Nb$ lead to lower values of concentration.

Table 1. Evaluation of Nusselt number at various governing parameters.

$\mathbf{Pr}$	$\mathbf{Le}$	${\mathbf{N}}_{\mathit{b}}$	${\mathit{N}}_{\mathit{t}}$	$\frac{\mathit{N}{\mathit{u}}_{\mathit{x}}}{{({\mathbf{Re}}_{\tilde{\mathit{x}}})}^{1/2}}$
0.2	0.1	0.3	0.3	0.51425
0.4				0.56789
0.8				0.60670
0.3	0.4			0.62317
	0.6			0.64209
	0.8			0.66324
		0.2		0.48359
		0.4		0.4556
		0.6		0.427857
			0.2	0.48053
			0.6	0.46475
			0.8	0.428857

presents the effect of the governing parameters on the Nusselt number. It is noticed that the increase in $\mathrm{Pr}$ leads to an enhancement of the heat transfer rate, while the opposite occurs for $Nb$ and $Nt$. Similarly,

Table 2. Evaluation of Sherwood number at various governing parameters.

$\mathit{L}\mathit{e}$	${\mathbf{N}}_{\mathit{b}}$	${\mathit{N}}_{\mathit{t}}$	$\frac{\mathit{S}{\mathit{h}}_{\tilde{\mathit{x}}}}{{({\mathbf{Re}}_{\tilde{\mathit{x}}})}^{1/2}}$
0.2	0.3		0.754231
0.6			0.83544
0.8			0.93567
0.3	0.2		0.64567
	0.6		0.72675
	0.8		0.74534
		0.2	0.63556
		0.4	0.58324
		0.6	0.53677

shows the effects of the governing parameters on the Sherwood number. An important enhancement of the mass transfer is noticed for higher values of $Le$ and $N_b$.

5. Conclusions

The heat and mass transfer characteristics of Oldroyd-B nanofluid are analyzed in consideration of chemical reactions and thermal radiation effects. The flow is generated by the bidirectional periodic oscillating surface. The HAM computations are performed for solution assessment. The main findings can be summarized as follows:

❖

A reduction in the flow intensity due to the increase in the porosity parameter is noticed.

❖

No phase shift of the oscillating velocity is encountered for all the considered parameters.

❖

An augmentation of the velocity magnitude occurs when the oscillation frequency ratio is increased.

❖

The retardation parameter causes an increase in velocity.

❖

The increase in the thermal relaxation factor causes a reduction in temperature.

❖

The porosity parameters enhance the thermal and concentration profiles.

❖

An augmentation of the heat transfer occurs when the radiative parameter is increasing.

❖

Lower concentrations are encountered for higher values of the Prandtl and Lewis numbers.

❖

The local Nusselt number increases with the Lewis and Prandtl numbers.

❖

The current analysis can be extended by modifying the model, evaluating the entropy generation, performing a sensitivity analysis, considering hybrid nanofluids, using an artificial neural network, etc.