Energy transport features of Oldroyd-B nanofluid flow over bidirectional stretching surface subject to Cattaneo–Christov heat and mass fluxes

By considering the Cattaneo–Christov approach and Buongiorno’s model, the thermal transport model is formulated for the flow of Oldroyd-B nanofluid over a bidirectional stretching surface. The flow profile of Oldroyd-B nanofluid is examined for various physical parameters, and the effects of heat source/sink are also utilized to explore the thermal transport properties subject to thermal relaxation time. Governing mathematical models are developed on the basis of basic laws and presented in the form of Partial differential equations (PDEs). The governing partial differential equations are transformed into ordinary differential equations considering suitable dimensionless transformations. The homotopic method is applied to study the feature of heat and velocity components in fluid flow. The influence of each physical parameter over the thermal and concentration profile is displayed graphemically. It is noticed that thermal transport is decreasing with increment in thermal relaxation time. The mass transfer becomes weak with magnifying values of the stretching strength parameter. Moreover, the larger thermophoretic parameter regulates the heat transfer during fluid flow.

1 Introduction

The heating and cooling fluids play a significant role in the advancement and development of engineering and industries. In general, usual fluids such as water,glycol, and ethylene have the incompetent ability of thermal conduction. However, metals possess a three-time greater ability of thermal transport compared to usual fluids. Therefore, it was required to make the mixture of a usual fluid and metal particle to enhance the thermal conductivity of the fluid. This mixture is called nanofluid. This concept of nanoparticle addition in a based fluid was introduced by Choi (1995). Choi et al. (2001) experimentally proved that the thermal transport of nanofluid is high, followed by a base fluid. Due to their high thermal conductivity, nanofluids have numerous applications in, for example, the advancement of thermal machines and the microelectromechanical system. The influence of the Brownian motion parameter on the heat transfer of nanofluid is studied by Evans et al. (2006). Sheikholeslami and Ganji (2013) investigated the flow of Cu-H₂O nanofluid between flat surfaces and concluded that the concentration of nanoparticles was enhanced by magnifying the Nusselt number. Sheikholeslami et al. (2014) examined the effect of magnetic force over the convective energy transport during nanofluid flow and noticed that, by increasing the Hartmann number, the heat diffusion in the system rises. The flow of Burgers nanofluid under the influence of stretching surface is examined by Khan et al. (2016a). For a greater radiation parameter, the heat profile of the Burgers nanofluid is ascending. Earlier research (Hayat et al., 2016; Hayat et al., 2017a; Hayat et al., 2018; Hayat et al., 2019) studied the convective transport of nanofluid by considering slip mechanism over a surface. Moreover, it analyzed the ferromagnetic and cross nanofluid flows considering various physical influences. The thermal transport rate magnifies for a greater ratio of velocity. Hayat et al. (2017b) discussed the two-dimensional flow of the Burgers nanofluid with a modified heat and mass flux approach. They concluded that an increment in thermal relaxation time sufficiently decreases the thermal transport. Some studies (Khan et al., 2019; Khan et al., 2020a; Khan and Alzahrani, 2020) analyzed the transport of nanofluid considering porous media; the magnifying influence of the Eckert number gives strength to the heat conduction of the nanofluid. Bhattacharyya et al. (2019) considered the flow of nanofluid over the rotating and stretchable surface. Based on this study, the higher Reynolds number minimizes the magnitude of radial velocity. Sher et al. (2019) explored the features of the Burgers nanofluid model formulation for nanofluid with a stretching surface with an improved thermal flux theory, which was investigated by Ahmad et al. (2019). Khan et al. (2020b) proposed a new flow model for the Burgers nanofluid and investigated the thermal transport. They summarized that, for increasing magnitudes of stagnation, the parameter flow velocity of nanofluid is improved. Amjid et al. (2020) approximated the transport of Casson nanofluid over a curve surface. They examined that flow is decreasing because of magnifying the Casson constant. Muhammad et al. (2020) computed the bidirectional flow of nanofluid applying thermal radiation and concluded that the increment in thermophoresis force increases the heat conduction in the nanofluid flow. Khan and Alzahrani (2021), Khan et al. (2021), and Khan and Puneeth (2021) presented some current significant studies about nanofluids.

Thermal transport is a significant natural phenomenon with multiple applications in industries. The theory of heat transport was firstly introduced by Fourier (Grattan-Guinness, 2005) and introduced Fourier’s law of thermal transport. Laterally, based on Fourier’s law, numerous researchers studied the features of thermal transformation in different fluid flow phenomena (Rashidi et al., 2014; Ahmed et al., 2019; Nadeem and Khan, 2019; Khan et al., 2020; Khan et al., 2022; Raza et al., 2022). Cattaneo (1948) made amendments to Fourier’s law and stated that thermal transport depends on the time called the thermal relaxation time. Christov (2009) introduced further modifications to the Cattaneo model. The Cattaneo–Christov approach is proposed to investigate heat transformation using familiarizing thermal relaxation time. After that, researchers applied this Cattaneo–Christov approach to study the thermal transformation phenomena in different kinds of fluid flows. Straughan (2010) presented a study based on the Cattaneo–Christov theory. Haddad (2014) applied the Cattaneo–Christov thermal transport theory to examine the instability. Shehzad et al. (2016) applied Cattaneo–Christov to study the Darcy flow of viscoelastic fluid and concluded that the flow profile of such fluid changes by varying the values of the Forchheimer parameter. Saleem et al. (2017) employed Cattaneo–Christov approach to investigate the Maxwell fluid flow and found that thermal conduction decreased with ascending thermal relaxation time. Waqas et al. (2018) applied the Cattaneo–Christov approach to inquiry the thermal conduction in Burgers fluid and reported that a greater thermal relaxation time constant reduced the thermal flow of the fluid. Rasool and Zhang (2019) examined the Darcy flow of nanofluid over the surface by applying the Cattaneo–Christov approach. Shehzad et al. (2019) studied the fluid flow by applying the Cattaneo–Christov approach of thermal and mass transfer and investigated that the ascending values of the Prandtl parameter enhanced the thermal curve. Hafeez et al. (2020) investigated chemical reactions, applied the Cattaneo–Christov approach to study the fluid flow using rotating surfaces, and used the Midrich technique to examine the influence of different parameters. Ahmed et al. (2020) applied the Cattaneo–Christov approach to analyze fluid transport over a stretching sheet. Recently, Iqbal et al. (2020) analyzed the heat transfer while the Burgers nanofluid flow based on the Cattaneo–Christov approach. They found that the magnifying values of the thermal relaxation constant results minimized the thermal flow rate. Rehman et al. (2016) constructed the solution of the set of equations to study the heat transfer phenomena. Currently, Ali et al. (2021) applied a finite differencing approach to analyze the heat transfer. Some other studies are carried out to solve such problems for examining thermal transport (Kamran et al., 2015; Kamran et al., 2016a; Kamran et al., 2016b; Rehman and Kamran, 2019).

The heat transport during the nanofluid flow using stretched surfaces has been examined by numerous researchers (Khan et al., 2016a; Hayat et al., 2019; Khan et al., 2020b). The Cattaneo–Christov approach has been applied by researchers to study heat transfer in the Burgers fluid (Khan et al., 2016b; Khan and Khan, 2016). Still, no mathematical formulation has been developed to analyze the thermal transport by applying the Cattaneo–Christov theory for the flow of the Oldroyd-B nanofluid and Buongiorno’s model. In this article, we proposed a new model of heat transport applying the Cattaneo–Christov approach and Buongiorno’s model to investigate the thermal transport in the Burgers nanofluid under the influence of a stretching sheet. The homotopic algorithm is applied to solve governing problems.

2 Mathematical formulation

This section presents a mathematical formulation for three-dimensional convective transport of Oldroyd-B nanofluid employed by a bidirectional stretching sheet. In-compressible and steady flow phenomena are studied in this article. A model combining the Cattaneo–Christov approach with Buongiorno’s model is employed to present a mathematical formulation for solutal and thermal conduction in the flow of the Oldroyd-B nanofluid. A cartesian coordinate system was applied to formulate the proposed model. Axial velocities in the x and y direction are u = ax and v = by, where a and b denote constants. The flow region is taken as z > 0. The velocity vector is V = [u, v, w], T_w denotes temperature, and C_w represents concentration, respectively. The flow pattern is presented in Figure 1.

FIGURE 1

FIGURE 1. The flow configuration.

The rheology of Oldroyd-B fluid is represented by the following equation:

$\left(1+{\lambda }_1\frac{D}{Dt}\right)\mathbf{S}=\mu \left(1+{\lambda }_2\frac{D}{Dt}\right){\mathbf{A}}_1,(1)$

where S is tensor representing the extra stress, μ is dynamic viscosity,${\mathbf{A}}_1=\left(\nabla \mathbf{V}\right)+{\left(\nabla \mathbf{V}\right)}^T$ denotes the Rivlin–Ericksen tensor, and $\frac{D}{Dt}$ is upper convective derivative.

The continuity and momentum governing model by neglecting pressure are given as follows (khan et al., 2020; Iqbal et al., 2020):

$\nabla \cdot \mathbf{V}=0,(2)$

${\rho }_f\frac{d\mathbf{V}}{dt}=div\mathbf{S}.(3)$

For the current problem, the law of energy conservation is (Iqbal et al., 2020)

$\frac{dT}{dt}-\left[D_B\nabla C\cdot \nabla T+\frac{D_T}{D_{\infty }}{\left(\nabla T\right)}^2\right]=-\frac{1}{\left(\rho c_p\right)}\left(\nabla \cdot \mathbf{q}\right),(4)$

where q denotes the thermal flux and satisfies the equation

$\mathbf{q}+\left[\frac{\partial \mathbf{q}}{\partial t}+(\nabla \cdot \mathbf{V})\mathbf{q}+\mathbf{V}\cdot \nabla \mathbf{q}-\mathbf{q}\cdot \nabla \mathbf{V}\right]{\lambda }_t=-k\nabla T.(5)$

The mass transfer model is as follows (Iqbal et al., 2020; Iqbal et al., 2021)

$\frac{dC}{dt}-\frac{D_T}{D_{\infty }}{\nabla }^{2}T=-\nabla \cdot \mathbf{J},(6)$

where J indicates mass flux and satisfies the following relation:

$\mathbf{J}+{\lambda }_c\left[\frac{\partial \mathbf{J}}{\partial t}+\mathbf{V}\cdot \nabla \mathbf{J}-\mathbf{J}\cdot \nabla \mathbf{V}+(\nabla \cdot \mathbf{V})\mathbf{J}\right]=-D_B\nabla C.(7)$

Here, λ_t, λ_c, and λ₁ denote the thermal and the mass and fluid relaxation time, respectively. λ₂ is the Oldroyd-B fluid parameter, ν denotes kinematics viscosity, T denotes temperature, and C) represents the concentration, respectively. D_B denotes diffusion coefficient, whereas q is thermal flux and J represents mass flux, respectively.

Eliminating S from $\left(1\right)$ and $\left(3\right)$, q from Eqs. $\left(4\right)$ and $\left(5\right)$m and J between Eqs. $\left(6\right)$ and $\left(7\right)$, we arrive at following governing PDEs:

$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{\partial z}=0,(8)$

$\begin{array}{lll}\hfill & u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}+w\frac{\partial u}{\partial z}+{\lambda }_1\left[2uv\frac{{\partial }^{2}u}{\partial x\partial y}+2wv\frac{{\partial }^{2}u}{\partial y\partial z}+2wu\frac{{\partial }^{2}u}{\partial x\partial z}\right.\hfill & \hfill \\ \hfill & \left.+,u^2\frac{{\partial }^{2}u}{\partial x^2}+v^2\frac{{\partial }^{2}u}{\partial y^2}+w^2\frac{{\partial }^{2}u}{\partial z^2}\right]=\nu \frac{{\partial }^{2}u}{\partial z^2}-\frac{\sigma B_0^2}{\rho }\left(u+{\lambda }_{1}w\frac{\partial u}{\partial z}\right)\hfill & \hfill \\ \hfill & +\nu {\lambda }_2\left[u\frac{{\partial }^{3}u}{\partial x\partial z^2}+v\frac{{\partial }^{3}u}{\partial y\partial z^2}+w\frac{{\partial }^{3}u}{\partial z^3},-,\frac{\partial u}{\partial x},\frac{{\partial }^{2}u}{\partial z^2}\right.\hfill & \hfill \\ \hfill & \left.-\frac{\partial u}{\partial z}\frac{{\partial }^{2}w}{\partial z^2}-\frac{\partial u}{\partial y}\frac{{\partial }^{2}v}{\partial z^2}\right],\hfill & \hfill \end{array}(9)$

$\begin{array}{lll}\hfill & u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}+w\frac{\partial v}{\partial z}+{\lambda }_1\left[2uv\frac{{\partial }^{2}v}{\partial x\partial y}+2wu\frac{{\partial }^{2}v}{\partial x\partial z}+2wv\frac{{\partial }^{2}v}{\partial y\partial z}\right.\hfill & \hfill \\ \hfill & \left.+,w^2\frac{{\partial }^{2}v}{\partial z^2}+u^2\frac{{\partial }^{2}v}{\partial x^2}+v^2\frac{{\partial }^{2}v}{\partial y^2}\right]=\nu \frac{{\partial }^{2}v}{\partial z^2}-\frac{\sigma B_0^2}{\rho }\left(v+{\lambda }_{1}w\frac{\partial v}{\partial z}\right)\hfill & \hfill \\ \hfill & +\nu {\lambda }_2\left[u\frac{{\partial }^{3}v}{\partial x\partial z^2}+v\frac{{\partial }^{3}v}{\partial y\partial z^2}+w\frac{{\partial }^{3}v}{\partial z^3}-\frac{\partial v}{\partial z}\frac{{\partial }^{2}w}{\partial z^2}-\frac{\partial v}{\partial x}\frac{{\partial }^{2}u}{\partial z^2}-\frac{\partial v}{\partial y}\frac{{\partial }^{2}v}{\partial z^2}\right.,\hfill & \hfill \end{array}(10)$

$\begin{array}{ccc}\hfill & u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}+w\frac{\partial T}{\partial z}-\tau \left[D_B\left(\frac{\partial C}{\partial z}\frac{\partial T}{\partial z}\right)+\frac{D_T}{T_{\infty }}{\left(\frac{\partial T}{\partial z}\right)}^2\right]\hfill & \hfill \\ \hfill & +{\lambda }_t\left[\begin{array}{c}\hfill u^2\frac{{\partial }^{2}T}{\partial x^2}+v^2\frac{{\partial }^{2}T}{\partial y^2}+w^2\frac{{\partial }^{2}T}{\partial z^2}+2uv\frac{{\partial }^{2}T}{\partial x\partial y}+2vw\frac{{\partial }^{2}T}{\partial y\partial z}+2uw\frac{{\partial }^{2}T}{\partial x\partial z}+u\frac{\partial u}{\partial x}\frac{\partial T}{\partial x}+u\frac{\partial v}{\partial x}\frac{\partial T}{\partial y}\hfill \\ \hfill +u\frac{\partial w}{\partial x}\frac{\partial T}{\partial z}+v\frac{\partial u}{\partial y}\frac{\partial T}{\partial x}+v\frac{\partial v}{\partial y}\frac{\partial T}{\partial y}+u\frac{\partial w}{\partial x}\frac{\partial T}{\partial z}+w\frac{\partial u}{\partial z}\frac{\partial T}{\partial x}+w\frac{\partial v}{\partial z}\frac{\partial T}{\partial y}+w\frac{\partial w}{\partial z}\frac{\partial T}{\partial z}\hfill \end{array}\right]\hfill & \hfill \\ \hfill & -{\lambda }_t\tau D_B\left[u\frac{{\partial }^{2}C}{\partial x\partial z}\frac{\partial T}{\partial z}+u\frac{\partial C}{\partial z}\frac{{\partial }^{2}T}{\partial x\partial z}+v\frac{{\partial }^{2}C}{\partial y\partial z}\frac{\partial T}{\partial z}+v\frac{\partial C}{\partial z}\frac{{\partial }^{2}T}{\partial y\partial z}+w\frac{{\partial }^{2}C}{\partial z^2}\frac{\partial T}{\partial z}+w\frac{\partial C}{\partial z}\frac{{\partial }^{2}T}{\partial z^2}\right]\hfill & \hfill \\ \hfill & -2{\lambda }_t\tau \frac{D_T}{T_{\infty }}\left[u\frac{\partial T}{\partial z}\frac{{\partial }^{2}T}{\partial x\partial z}+v\frac{\partial T}{\partial z}\frac{{\partial }^{2}T}{\partial y\partial z}+w\frac{\partial T}{\partial z}\frac{{\partial }^{2}T}{\partial z^2}\right]={\alpha }_1\left[\frac{{\partial }^{2}T}{\partial z^2}\right]\hfill & \hfill \\ \hfill & +{\lambda }_t\frac{Q_0}{\rho c_p}\left(u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}+w\frac{\partial T}{\partial z}\right)+Q_0\left(T-T_w\right)\hfill & \hfill \end{array}(11)$

$\begin{array}{lll}\hfill & u\frac{\partial C}{\partial x}+v\frac{\partial C}{\partial y}+w\frac{\partial C}{\partial z}+{\lambda }_c\hfill & \hfill \\ \hfill & \left[\begin{array}{c}\hfill w^2\frac{{\partial }^{2}C}{\partial z^2}+u^2\frac{{\partial }^{2}C}{\partial x^2}+v^2\frac{{\partial }^{2}C}{\partial y^2}+2uv\frac{{\partial }^{2}C}{\partial x\partial y}+2uw\frac{{\partial }^{2}C}{\partial x\partial z}+2vw\frac{{\partial }^{2}C}{\partial y\partial z}\hfill \\ \hfill +u\frac{\partial u}{\partial x}\frac{\partial C}{\partial x}+u\frac{\partial w}{\partial x}\frac{\partial C}{\partial z}+v\frac{\partial u}{\partial y}\frac{\partial C}{\partial x}+u\frac{\partial v}{\partial x}\frac{\partial C}{\partial y}+v\frac{\partial v}{\partial y}\frac{\partial C}{\partial y}\hfill \\ \hfill +v\frac{\partial w}{\partial y}\frac{\partial C}{\partial z}+w\frac{\partial v}{\partial z}\frac{\partial C}{\partial y}+w\frac{\partial u}{\partial z}\frac{\partial C}{\partial x}+w\frac{\partial w}{\partial z}\frac{\partial C}{\partial z}\hfill \end{array}\right]\hfill & \hfill \\ \hfill & -\frac{D_T}{T_{\infty }}{\lambda }_c\left[w\frac{{\partial }^{3}T}{\partial z^3}+v\frac{{\partial }^{3}T}{\partial y\partial z^2}+u\frac{{\partial }^{3}T}{\partial x\partial z^2}\right]=\frac{D_T}{T_{\infty }}\left(\frac{{\partial }^{2}T}{\partial z^2}\right)+D_B\left[\frac{{\partial }^{2}C}{\partial z^2}\right],\hfill & \hfill \end{array}(12)$

with the boundary conditions

$u=u_w=ax,v=v_w=by,w=0,T=T_w,C=C_w\text{\hspace{0.17em}at\hspace{0.17em}}z=0,(13)$

$u\to 0,v\to 0,T\to T_{\infty },C\to C_{\infty }\text{\hspace{0.17em}as\hspace{0.17em}}z\to \infty \text{.}(14)$

Here, ${\alpha }_1=\left(\frac{k}{\rho c_p}\right)$ indicates thermal diffusivity, p and ρ denote pressure and density, respectively, c_p is the specific heat capacity, k is thermal conductivity, amd C_∞ and T_∞ denote the ambient concentration and temperature, respectively.

See the following transformations (Ahmad et al., 2019; Iqbal et al., 2021):

$\begin{array}{lll}\hfill & u=ax{f}^{\prime }\left(\eta \right),v=ay{f}^{\prime }\left(\eta \right),w=-\sqrt{a\nu }\left(f+g\right),\eta =z\sqrt{\frac{a}{\nu }}\hfill & \hfill \\ \hfill & \varphi \left(\eta \right)=\frac{C-C_{\infty }}{C_w-C_{\infty }},\theta \left(\eta \right)=\frac{T-T_{\infty }}{T_w-T_{\infty }},\hfill & \hfill \end{array}(15)$

By invoking these transformations, Eq. 8 is satisfied and Eqs. (9) − (12) take the following forms:

$\begin{array}{ll}\hfill & f^{\prime \prime \prime }+\left(1+M^2{\beta }_1\right)\left(f+g\right)f^{\prime \prime }-{\left(f^{\prime }\right)}^2+{\beta }_1\left[2\left(f+g\right)f^{\prime }{f}^{\prime \prime }\right.\hfill \\ \hfill & \left.{-\left(f+g\right)}^2{f}^{\prime \prime \prime }\right]+{\beta }_2\left[\left(f^{\prime \prime }+g^{\prime \prime }\right)f^{\prime \prime }-\left(f+g\right)f^{iv}\right]-M^2{f}^{\prime }=0,\hfill \end{array}(16)$

$\begin{array}{ll}\hfill & g^{\prime \prime \prime }+\left(1+M^2{\beta }_1\right)\left(f+g\right)g^{\prime \prime }-{\left(g^{\prime }\right)}^2+{\beta }_1\left[2\left(f+g\right)g^{\prime }{g}^{\prime \prime }\right.\hfill \\ \hfill & \left.{-\left(f+g\right)}^2{g}^{\prime \prime \prime }\right]+{\beta }_2\left[\left(f^{\prime \prime }+g^{\prime \prime }\right)g^{\prime \prime }-\left(f+g\right)g^{iv}\right]=0,\hfill \end{array}(17)$

$\begin{array}{lll}\hfill & {\theta }^{\prime \prime }+\mathrm{Pr}\left(f+g\right){\theta }^{\prime }+\mathrm{Pr}\left(N_b{\varphi }^{\prime }{\theta }^{\prime }+N_t{\theta }^{\prime 2}\right)-\mathrm{Pr}{\beta }_t\left[\left(f^{\prime }+g^{\prime }\right)\right.\hfill & \hfill \\ \hfill & \left.\left.\left(f+g\right){\theta }^{\prime }+{\left(f+g\right)}^2{\theta }^{\prime \prime }\right)\right]-{\beta }_t{N}_b\mathrm{Pr}\left[\left(f+g\right){\theta }^{\prime \prime }{\varphi }^{\prime }+\left(f+g\right){\varphi }^{\prime \prime }{\theta }^{\prime }\right]\hfill & \hfill \\ \hfill & -2N_t{\beta }_t\mathrm{Pr}\left(f+g\right){\theta }^{\prime \prime }{\theta }^{\prime }+\mathrm{Pr}\delta \left[\theta +{\beta }_t\left(f+g\right){\theta }^{\prime }\right]=0,\hfill & \hfill \end{array}(18)$

$\begin{array}{lll}\hfill & {\varphi }^{\prime \prime }+Le\mathrm{Pr}\left(f+g\right){\varphi }^{\prime }+\frac{N_t}{N_b}Le\mathrm{Pr}{\theta }^{\prime \prime }-Le\mathrm{Pr}{\beta }_c\left[\left(f+g\right)\left(f^{\prime }+g^{\prime }\right)\right.\hfill & \hfill \\ \hfill & \left.{\varphi }^{\prime }+{\left(f+g\right)}^2{\varphi }^{\prime \prime }\right]-Le\mathrm{Pr}\frac{N_t}{N_b}\left(f+g\right){\theta }^{\prime \prime \prime }=0,\hfill & \hfill \end{array}(19)$

with transformed boundary conditions are follows:

$\begin{array}{ll}\hfill & \hfill f\left(0\right)=0,g\left(0\right)=0,f^{\prime }\left(0\right)=1,g^{\prime }\left(0\right)=R,\theta \left(0\right)=1,\\ \hfill & \hfill \varphi \left(0\right)=1,\end{array}(20)$

$f^{\prime }\left(\infty \right)=0,g^{\prime }\left(\infty \right)=0,\theta \left(\infty \right)=0,\varphi \left(\infty \right)=0.(21)$

Here, β₁ is the relaxation time, β₂ the Oldroyd-B fluid material parameter, Pr the Prandtl number, Le the Lewis number, β_t the thermal relaxation time constraint, and β_c the mass relaxation time. Thermophoresis, Brownian motion, and ratio of stretching rates are (N_t), $\left(N_b\right)$, and $\left(R\right)$, respectively, are define as follows:

$\begin{array}{c}\hfill {\beta }_1={\lambda }_1\frac{U_0}{l},{\beta }_2={\lambda }_2\frac{U_0}{l},{\beta }_t={\lambda }_t\frac{U_0}{l},{\beta }_c={\lambda }_c\frac{U_0}{l},R=\frac{b}{a}\hfill \\ \hfill Le=\frac{{\alpha }_1}{D_B},\mathrm{Pr}=\frac{\nu }{{\alpha }_1},N_t=\frac{\tau D_T\left(T_w-T_{\infty }\right)}{\nu T_{\infty }},N_b=\frac{\tau D_B\left(C_w-C_{\infty }\right)}{\nu },\hfill \end{array}(22)$

3 Homotopic solution approach

In this manuscript,the homotopic analysis approach (Hayat et al., 2017b; Khan et al., 2020b; Khan et al., 2020) is applied to solve the Eqs 16–19 with boundary conditions defined in Eqs 20, 21. Initial guess and linear operators are needed to proceed with this approach. Hence, for analytic solutions’ linear operators $\left({\mathit{\pounds }}_f,{\mathit{\pounds }}_{\theta },{\mathit{\pounds }}_{\varphi }\right)$ and the initial guesses (f₀, θ₀, ϕ₀) used, the models of energy and the concentration diffusion are

$f_0\left(\eta \right)=1-e^{-\eta },g_0\left(\eta \right)=\alpha \left(1-e^{-\eta }\right),{\theta }_0\left(\eta \right)=e^{-\eta },{\varphi }_0\left(\eta \right)=e^{-\eta },(23)$

${\mathit{\pounds }}_f\left[f\left(\eta \right)\right]=f^{\prime \prime \prime }-f^{\prime },{\mathit{\pounds }}_{\theta }\left[\theta \left(\eta \right)\right]={\theta }^{\prime \prime }-\theta ,{\mathit{\pounds }}_{\varphi }\left[\varphi \left(\eta \right)\right]={\varphi }^{\prime \prime }-\varphi .(24)$

4 Physical analysis of results

The three-dimensional steady-state thermal transfer phenomena of nanofluid by a bidirectional stretching surface is analyzed. For the Oldroyd-B nanofluid flow features, the Cattaneo–Christov approach and Buongiorno’s model are employed together. The homotopic approach is used to analyze the impact of various physical parameters involved in Eqs. (16 – (19) with boundary conditions (20) and (21).

The axial velocity components $f^{\prime }\left(\eta \right)$, $g^{\prime }\left(\eta \right)$, thermal distribution $\theta \left(\eta \right)$, and solutal distribution $\varphi \left(\eta \right)$ are examined under the impact of physical parameters such as relaxation timer (β₁), Oldroyd B fluid $\left({\beta }_2\right)$, thermophoresis $\left(N_t\right)$, thermal relaxation $\left({\beta }_t\right)$, and Brownian motion (N_b), mass relaxation $\left({\beta }_c\right)$ and explained through Figs. 2–8. Moreover, for leading constraints, magnitudes are fixed as R = 0.2, β₁ = 0.5, β₂ = 0.15, Pr = 5.0, M = 1.5, β_t = 0.3, β_c = 0.5, Le = 4.5, N_b = 0.5, and N_t = 0.7. The influence of physical parameters is graphemically presented and also theoretically discussed. $f^{\prime }\left(\eta \right)$ and $g^{\prime }\left(\eta \right)$ represent the x-axis and y-axis, respectively. $\theta \left(\eta \right)$ is the thermal profile and $\varphi \left(\eta \right)$ denotes solutal curve of nanofluid.

Figures 2A–D represent the effect of stretching strength (R) over $f^{\prime }\left(\eta \right)$, $g^{\prime }\left(\eta \right)$, thermal profile $\theta \left(\eta \right)$, and solutal profile $\varphi \left(\eta \right)$ of the Oldroyd-B nanofluid. It is clear that the $f^{\prime }\left(\eta \right)$ is falling for magnifying magnitudes of R but the magnitude of the normal velocity component $g^{\prime }\left(\eta \right)$ is increasing for larger R. As R denotes stretching ratio defined as $R=\frac{b}{a}$, where a and b represent stretching rate along x-direction and y-direction, respectively. This implies that, for higher R, the stretching strength along the y-axis is greater than stretching in the x-direction. Therefore, the flow velocity $f^{\prime }\left(\eta \right)$ is decreasing, whereas the curves for $g^{\prime }\left(\eta \right)$ are escalating for greater values of R. The impact of the Deborah parameter β₁ (relaxation time) over the axial-velocity components $f^{\prime }\left(\eta \right)$, $g^{\prime }\left(\eta \right)$, thermal profile $\theta \left(\eta \right)$, and solutal profile $\varphi \left(\eta \right)$ of the Oldroyd-B nanofluid is represented in Figures 3A–D. The graphs in Figures 3A–D indicate that the flow profiles of the Oldroyd-B nanofluid are showing a falling behavior for increasing values of β₁, whereas the heat and solutal energy profile during flow are growing for magnifying values of β₁. The increment in the relaxation time constraint physically increases resistance to flow, so the velocity profiles decrease. However, because the resistance collision of particles increases, it directly increases the thermal energy of the system. Hence, resistance results in additional heat generation, which increases the thermal transport and solutal profile. The influence of thermal source/sink constraint (δ) over the thermal and solutal profiles is presented in Figures 3A,B. The increment in the source parameter (δ > 0) increases the thermal profile, but it falls for heat sink constraint (δ < 0). The magnification of the source parameter (δ > 0) generates extra heat, which magnifies the heat transport, whereas the increment in sink parameter (δ < 0) absorbs heat which obviously refers to minimizing the heat transfer. Figures 4A–D shows the influence of the Oldroyd-B fluid parameter (β₂) over velocity components $f^{\prime }\left(\eta \right)$, $g^{\prime }\left(\eta \right)$, thermal transport $\theta \left(\eta \right)$, and solutal transport $\varphi \left(\eta \right)$. These graphs indicate that $f^{\prime }\left(\eta \right)$ and $g^{\prime }\left(\eta \right)$ fall for greater values of β₂, whereas the thermal and solutal profiles of nanofluid are growing. The impact of the thermophoretic force parameter (N_t) heat and concentration profile of the nanofluid is presented in Figures 5A,B. Graphical results indicate that heat and concentration profiles of the nanofluid are showing a growth trend for magnifying values of N_t and the thermal thickness of the boundary also magnifies for greater N_t. In the thermophoresis process, fluid particles move from a warm to a cool region. This particle motion produces extra heat, due to which the solutal profile of the Oldroyd-B nanofluid escalates. Figures 6A,B represents the effect of (N_b) over the thermal and solutal distributions of the Oldroyd-B nanofluid. These graphs show that the thermal transport magnifies, whereas the solutal transport de-escalates for magnifying values of N_b. Basically, the Brownian motion parameter controls the motion of fluid particles. For high values of parameter N_b, the random motion of fluid particles is high and increases the kinetic energy, and particles collide with each other, due to which the thermal energy also increases. Due to this collision, fluid particles’ mass transport profile decays. The influence of parameters (β_t) and (β_c) over the heat and mass transport profile is presented in Figures 7A,B. The thermal and concentration distributions represent the decaying trend for magnifying values of β_t and β_c, respectively. The thermal relaxation time parameter represents the time duration required by a particle to transfer heat from one part to another. As known physically, with the passage of time, thermal transport becomes slow due to the decrease in the thermal gradient. Therefore, magnifying thermal relaxation time implies that the thermal transport slows down, and the heat transport profile is falling. Similarly, the mass profile also decreases for ascending solutal relaxation time parameter.

FIGURE 2

FIGURE 2. (A–D) Impact of R on f′, g′, θ, and ϕ.

FIGURE 3

FIGURE 3. (A–D) Impact of β₁ on f′, g′, θ, and ϕ.

FIGURE 4

FIGURE 4. (A–D) Impact of β₂ on f′, g′, θ, and ϕ.

FIGURE 5

FIGURE 5. (A,B) Effect of N_t on θ and ϕ.

FIGURE 6

FIGURE 6. (A,B) Effect of N_b on θ and ϕ.

FIGURE 7

FIGURE 7. (A,B) Effect of β_t and β_c on θ and ϕ.

5 Validation of homotopic outcomes

In this section, approximated – f′′(0), − g′′(0), and − θ′(0) for different α are displayed in Tables 1, 2, and the obtained values are compared with other studies. This comparison indicates that our approximations are accurate and the adopted homotopic technique is a valid approach.

TABLE 1

TABLE 1. An evaluation table for − f′′(0) and − g′′(0) for several magnitudes of R in reduced case when β₁ = β₂ = M = 0.

TABLE 2

TABLE 2. An assessment table for − θ′(0) for pertinent ranges of R in reducing case when β₁ = β₂ = δ = β_t = N_t = N_b = 0 and Pr = 1.

Concluding remarks

The Cattaneo–Christov approach and Buongiorno’s model were employed together to investigate the momentum and energy transport features of the three-dimensional flow of the Oldroyd-B nanofluid over a bidirectional stretched sheet. After utilizing the homotopy analysis method on the governing equation to solve them, we found some important results, which are noted here.

We found that, for magnifying stretching strength R, the vertical velocity in the y-direction magnifies, but it decreases in the x-direction. The flow curves of the Oldroyd-B nanofluid for the relaxation time parameter show an opposite behavior compared to the Oldroyd-B material parameter.

For varying values of the Brownian motion parameter, the solutal transport decays, whereas the thermal conduction shows enhancing trend for it. Moreover, the thermal and concentration transport = escalates for larger magnitudes of the thermophoretic force parameter. In addition, a larger thermal relaxation time parameter deteriorates the thermal transport. The mass transfer profile decreases for a greater solutal relaxation time parameter. The larger stretching strength decreases the thermal energy transport of the nanoliquid. Figures 8A,B clearly elucidate that the thermal profile is augmented for larger heat source parameter while it declines for heat sink parameter.

FIGURE 8

FIGURE 8. (A,B) Impact of δ on θ.

Data availability statement

The original contributions presented in the study are included in the article/supplementary materials. Further inquiries can be directed to the corresponding authors.

Author contributions

All authors listed have made a substantial, direct, and intellectual contribution to the work and approved it for publication.

Funding

This work is supported by National Natural Science Foundation of China (12071408), Natural Science foundation of Jiangsu (BK20201149) and Jiangsu Funding Program for excellent Postdoctoral Talent, Grant No: 2022ZB675. The authors would like to thank the Deanship of Scientific Research at Umm Al-Qura University for supporting this work by Grant Code:22UQU4331317DSR67.

Acknowledgments

The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through Large Groups (RGP.2/94/43). The authors would like to thank the Deanship of Scientific Research at Umm Al-Qura University for supporting this work by Grant Code: 22UQU4331317DSR67.

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s note

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Nomenclature

u, v, w Velocity components (ms⁻¹)

μ Dynamic viscosity (kg m⁻¹ s⁻¹)

x, y, z Cartesian coordinates (ms⁻¹)

N_b Brownian motion parameter

α₁ Thermal diffusivity (m² s⁻¹)

β₁ Fluid relaxation time parameter

k Thermal conductivity $\left(\frac{W}{MK}\right)$

C_w Concentration at surface

l Specific length (m)

D_B The mass diffusion coefficient (m² s ⁻¹)

q Heat flux

σ Fluid electric conductivity (Sm⁻¹)

C_∞ Ambient concentration

c_p Specific heat capacity (JK⁻¹ kg⁻¹)

C Concentration

S Extra stress tensor

ν Kinematic viscosity (m² s⁻¹)

U₀ Reference velocity (ms⁻¹)

Le Lewis number

λ₁ Fluid relaxation time (s)

λ₂ The material parameter of Oldroyd-B fluid (s²)

ρ Fluid density (kg m⁻³)

θ Dimensionless temperature

J Mass flux

T_w Surface temperature (K)

T_∞ Ambient temperature (K)

D_T Thermophoresis coefficient (m² s⁻¹)

β_t Thermal relaxation time parameter (s)

h_f Heat convection coefficient $\left(\frac{W}{K{M}^2}\right)$

η Dimensionless similarity variable

ax, by linear stretching velocities (ms⁻¹)

T Fluid temperature (K)

A₁ First Rivlin–Ericksen tensor

f′ Dimensionless fluid velocity component

Pr Prandtl number

β_c Solutal relaxation time parameter (s)

ϕ Dimensionless concentration