1 Introduction

Energy has special importance in building and development of human society. The global energy is shortened in the past few decades due to rapid growth in the development of human society. The scientists and engineers across the globe are trying to find the new energy sources and new energy technologies just to meet the desired energy consumption of human beings. It is the most important source of energy which is available freely through sun. The renewable sources are geothermal energy, biomass from plants, hydropower from water and solar energy. There are serious issues in the usage of fossil fuel energy and other ordinary sources of energy to obtain the global energy requirement for the present and future. Solar energy directly comes from sun, and it can be converted into electricity and heat. Solar energy is the best candidate for renewable energy because the energy produced by it is enough for billions of years. Further, the solar energy is 2000 times higher than the consumption of human society. Hence, the implementation of solar energy has gained the special attention recently. Nanomaterial is introduced as a new energy material due to size of its particle that is smaller or equal than the de Broglie or coherent waves. Nanoparticle has an ability to absorb the incident radiation directly from sun. Such radiative characteristics of nanofluids enhance its use in thermal power systems. The implementation of nanofluids in solar system highly depends on the radiative motion of nanoparticles. There are many advantages and applications of solar energy. The renewable source of energy that never ends is solar power. Solar power has special importance in our daily life because it is a natural process of gaining water, heat and electricity with help of nature. The best source of renewable energy with minimal environmental impact is solar energy. The nanoparticles enhance the radiative properties of liquids that lead to an increase in the efficiency of direct absorption of solar collectors. Investigations through solar radiation have gained the special focus of recent researchers (Turkyilmazoglu and Pop 2013; Zheng et al. 2013; Lin et al. 2014; Shehzad et al. 2014; Mushtaq et al. 2014; Kandasamy et al. 2014; Hussain et al. 2015; Sheikholeslami et al. 2015; Hayat et al. 2015a, b).

A new class of energy transport fluids is developed by the mixture of base liquids and nanoparticles. This class of liquids is called nanofluids and has popularity due to their higher thermal performance. The thermal conductivity of ordinary base fluid may be enhanced by adding the solid nanosized particles. The recent researchers are not only engaged to explore thermal properties of nanoliquids but also to develop mechanisms behind the increasing thermal efficiency that may be helpful in next-generation coolants for computers and for nuclear reactors. The mixture of nanoliquid with biotechnological components has tremendous applications in pharmaceuticals, agriculture, biological sensors, etc. The biotechnological processes involve different types of nanomaterials including nanofibers, nanowires, nanostructures, nanomachines and many others. The magnetonanoliquids are the fluids that have both magnetic and liquid characteristics. These materials have key role in optical gratings, modulators, optical switches and optical fiber filters. The magnetic nanoparticles are quite prominent in medicine, cancer therapy, tumor analysis, sink float separation, etc. The recent works on magneto nanofluids are presented in the Refs. Sheikholeslami et al. (2014a, b) Shehzad et al. (2014), Khan and Makinde (2014), Rashidi et al. (2014), Lin et al. (2015), Zhang et al. (2015), Abbasi et al. (2015), Hayat et al. (2015), Gireesha et al. (2016).

To explore the role of stratification in heat and mass transfer is current topic among the recent investigators. Stratification of fluid arises due to change in temperature and variation in concentration. It also occurs due to the presence of different types of fluids. Investigation on thermal and solutal stratification of oxygen and hydrogen in rivers is very important because these may be directly affected by the growth rate of all cultured species. The analysis of thermal stratification in solar engineering is very essential because better stratification gives high energy efficiency. The performance of system may be enhanced by the implementation of thermal stratification in energy devices. Stratified fluids are quite obvious in nature, and these occur in any heterogenous fluid body. Few examples of stratification include stratification in oceans and reservoirs, rivers, heterogenous mixture in industrial process, etc. Ibrahim and Makinde (2013) reported the impact of thermal and solutal stratification in natural convection flow of nanofluid over a flat plate. Doubly stratified flow of viscous liquid in a porous space with Soret and Dufour effects has been examined by Srinivasacharya and Surender (2014). Rashad et al. (2014) provided an analysis to explore the characteristics of thermally and solutally stratified flow of micropolar liquid in the presence of chemical reaction. Thermally radiative flow of doubly stratified Jeffrey nanofluid over a moving sheet has been addressed by Hayat et al. (2014).

The aim of this attempt is to develop a mathematical model for the renewable energy. For this purpose, we consider the solar radiation in two-dimensional flow of an Oldroyd-B fluid with nanoparticles over a stretched sheet. Solar energy at present is an important source of clean and renewable energy. The nanofluid is colloidal suspension and is uniformly dispersed in ordinary base liquid. We also incorporate the effects of double stratification. The governing nonlinear mathematical model is solved by development of homotopic algorithm (Liao 2009; Turkyilmazoglu 2010; Zheng et al. 2013; Abbasbandy et al. 2014; Abbasi et al. 2015; Hayat et al. 2015d). The results are plotted and discussed in detail for multiple values of arising parameters. The proper values of these parameters are quite essential in the industrial processes and manufacturing. The values of Deborah numbers De 1 and De 2 lie within the range of 0–1 to explore the characteristics of liquid. On the other hand side, the zero value of thermal and solutal stratification parameter corresponds to the unstratified medium. In addition, the temperature and concentration at the boundary are zero when we fix S T  = 0 = S C . The values of local Nusselt and Sherwood numbers are computed numerically for multiple values of physical parameters.

2 Governing problems

An incompressible flow of an Oldroyd-B fluid over a thermally and solutally stratified stretching sheet is assumed. The considered flow is electrically conductive under the action of applied magnetic field B 0. The role of thermal radiation is encountered in the energy equations. The effects of nanoparticles are taken into account in the presence of mixed convection and heat source. The governed mathematical equations under boundary layer approach can be expressed as

$$ \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0, $$
(1)
$$ \begin{aligned} u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y} + \lambda_{1} \left( {u^{2} \frac{{\partial^{2} u}}{{\partial x^{2} }} + v^{2} \frac{{\partial^{2} u}}{{\partial y^{2} }} + 2uv\frac{{\partial^{2} u}}{\partial x\partial y}} \right) = & \nu \left( {\frac{{\partial^{2} u}}{{\partial y^{2} }} + \lambda_{2} \left( {\begin{array}{*{20}l} {u\tfrac{{\partial^{3} u}}{{\partial x\partial y^{2} }} + v\tfrac{{\partial^{3} u}}{{\partial y^{3} }}} \\ { - \tfrac{\partial u}{\partial x}\tfrac{{\partial^{2} u}}{{\partial y^{2} }} - \tfrac{\partial u}{\partial y}\tfrac{{\partial^{2} u}}{{\partial y^{2} }}} \\ \end{array} } \right)} \right)\, \\ & - \frac{{\sigma B_{0}^{2} }}{\rho }u + g\beta_{T} ((T - T_{\infty } ) + \beta_{C} (C - C_{\infty } )), \\ \end{aligned} $$
(2)
$$ u\frac{\partial T}{\partial x} + v\frac{\partial T}{\partial y} = \alpha \frac{{\partial^{2} T}}{{\partial y^{2} }} + \tau \left( {D_{B} \frac{\partial C}{\partial y}\frac{\partial T}{\partial y} + \frac{{D_{T} }}{{T_{\infty } }}\left( {\frac{\partial T}{\partial y}} \right)^{2} } \right) - \frac{1}{{\left( {\rho c} \right)_{f} }}\frac{{\partial q_{r} }}{\partial y} + \frac{Q}{{\left( {\rho c} \right)_{f} }}(T - T_{\infty } ), $$
(3)
$$ u\frac{\partial C}{\partial x} + v\frac{\partial C}{\partial y} = D_{B} \frac{{\partial^{2} C}}{{\partial y^{2} }}\,+\, \frac{{D_{T} }}{{T_{\infty } }}\frac{{\partial^{2} T}}{{\partial y^{2} }}, $$
(4)

where u and v denote the velocity components in the x- and y-directions, ρ the fluid density, λ 1 the relaxation time, λ 2 the retardation time, g the gravitational acceleration, β T the thermal expansion coefficient, β C the concentration expansion coefficient, σ the electrical conductivity, T the fluid temperature, α the thermal diffusivity, D B the Brownian diffusion coefficient, D T the thermophoretic coefficient, q r the radiative heat flux, Q the heat source/sink parameter and C the fluid concentration.

The subjected boundary conditions are

$$ u = u_{w} (x) = cx,\, \, v = 0,\, \, T = T_{w} = T_{0} + bx,\,\;C = C_{w} = C_{0} + {\text{d}}x\;{\text{at }}y = 0, $$
(5)
$$ u \to 0,\, \, T \to T_{\infty } = T_{0} + ax,\, \, C \to C_{\infty } = C_{0} + {\text{ex as }}y \to \infty . $$
(6)

In Eqs. (5) and (6), c denotes the stretching rate, a, b, d, e the dimensional constants and T 0 and C 0 are the reference temperature and concentration, respectively.

The radiative flux via Rosseland assumption is

$$ q_{r} = - \frac{{4\sigma^{ * } }}{{3k^{ * } }}\frac{{\partial T^{4} }}{\partial y}, $$
(7)

where \( \sigma^{*} \) is the Stefan–Boltzmann constant and \( k^{*} \) the mean absorption coefficient. The differences in temperature within the flow are considered to be small such that T 4 can be described as a linear function of temperature. Expanding T 4 about T ∞ through Taylor’s theorem and neglecting the higher order terms, one has

$$ T^{4} \cong T_{\infty }^{4} + (T - T_{\infty } )4T_{\infty }^{3} = 4T_{\infty }^{3} T - 3T_{\infty }^{4} . $$
(8)

Now Eq. (4) has the following form

$$ u\frac{\partial T}{\partial x} + v\frac{\partial T}{\partial y} = \alpha \frac{{\partial^{2} T}}{{\partial y^{2} }} + \tau \left( {D_{B} \frac{\partial C}{\partial y}\frac{\partial T}{\partial y} + \frac{{D_{T} }}{{T_{\infty } }}\left( {\frac{\partial T}{\partial y}} \right)^{2} } \right) + \frac{1}{{(\rho c)^{f} }}\frac{{16\sigma^{*} T_{\infty }^{3} }}{{3k^{*} }}\frac{{\partial^{2} T}}{{\partial y^{2} }} + \frac{Q}{{(\rho c)_{f} }}(T - T_{\infty } ). $$
(9)

Setting

$$ u = cxf^{{\prime }} (\eta ),\quad v = - \sqrt {c\nu } f(\eta ),\quad \eta = y\sqrt {\frac{c}{\nu }},\quad \theta (\eta ) = \frac{{T - T_{\infty } }}{{T_{w} - T_{0} }},\quad \phi(\eta) = \frac{{C - C_{\infty}}}{C_{w} - C_{0}}.$$
(10)

Equation (1) is satisfied identically, and other Eqs. become

$$ f^{\prime\prime\prime\prime} + De_{1}\left( {2ff^{\prime} f^{\prime\prime} - f^{2} f^{\prime\prime\prime} } \right) + ff^{\prime\prime} - f^{\prime2} + Ha^{2} \beta_{1} ff^{\prime\prime} + De_{2} \left( {f^{\prime\prime2} - f^{2}f^{iv} } \right) - Ha^{2} f^{\prime} + \lambda (\theta + N\phi ) = 0, $$
(11)
$$ \left( {1 + \frac{4}{3}T_{R} } \right)\theta^{\prime\prime} + \Pr f\theta^{\prime} + \Pr N_{b} \theta^{\prime} \phi^{\prime} + \Pr N_{T} \theta^{\prime2} - \Pr f^{\prime}\theta - \Pr S_{T} f^{\prime} = 0, $$
(12)
$$ \phi^{\prime\prime} + \mathop {\Pr }\limits Lef\phi^{\prime} + \left( {N_{T} /N_{B} } \right)\,\theta^{\prime\prime} - \mathop {\Pr }\limits Lef^{\prime} \phi - \mathop {\Pr }\limits LeS_{C} f^{\prime} = 0, $$
(13)
$$ f = 0, \quad f^{\prime} = 1,\; \theta = 1 - S_{T} ,\; \phi = 1 - S_{C} {\text{ at }} \; \eta = 0, $$
(14)
$$ f^{\prime} \to 0,\, \, \theta \to 0,\, \, \phi \to 0{\text{ as }} \; \eta \to \infty , $$
(15)

where De 1 = λ 1 c is the Deborah number with respect to relaxation time, De 2 = λ 2 c is the Deborah number with respect to retardation time, Ha 2 = σB 20 /ρc the Hartman number, \( \lambda = Gr_{x} /\text{Re}_{x}^{2} \) the thermal buoyancy parameter with Gr x  = gβ T (T − T 0)x 3/ν 2 the local Grashof number and \( Re_{x} = u_{w} (x)x/\nu \) the local Reynolds number, Pr = ν/α the Prandtl number, \( Rd = \tfrac{{4\sigma^{ * } T_{\infty }^{3} }}{{kk^{ * } }} \) the thermal radiation parameter, S T  = a/b the thermal stratification parameter, S C  = e/d the solutal stratification parameter, Le = ν/D B the Schmidt number and f, θ and ϕ the dimensionless velocity, temperature and nanoparticle concentration, respectively.

The dimensionless forms of local Nusselt and local Sherwood numbers can be expressed as follows:

$$ Nu_{x} /Re_{x}^{1/2} = - \left( {1 + \frac{4}{3}T_{R} } \right)\,\theta^{\prime} (0),\quad Sh_{x} /Re_{x}^{1/2} = - \phi^{\prime} (0). $$
(16)

3 Computations

To develop the homotopic procedure (Liao 2009; Turkyilmazoglu 2010; Zheng et al. 2013; Abbasbandy et al. 2014; Abbasi et al. 2015; Hayat et al. 2015d), we choose the initial guesses and operators in the forms given below:

$$ f_{0} (\eta ) = (1 - \exp ( - \eta )),\, \, \theta_{0} (\eta ) = (1 - S_{T} )\exp ( - \eta ),\, \, \phi_{0} (\eta ) = (1 - S_{C} )\exp ( - \eta ), $$
(17)
$$ L(f) = f^{\prime\prime\prime\prime} - f^{\prime} ,\quad L(\theta ) = \theta^{\prime\prime} - \theta ,\quad L(\phi ) = \phi^{\prime\prime} - \phi , $$
(18)

with

$$ \begin{aligned} L(f)(B_{1} + B_{2} e^{\eta } + B_{3} e^{ - \eta } ) = & 0,\, \, L(\theta )(B_{4} e^{\eta } + B_{5} e^{ - \eta } ) = 0,\, \\ L(\phi )(B_{5} e^{\eta } + B_{7} e^{ - \eta } ) = & 0, \\ \end{aligned} $$
(19)

where B i (i = 1–7) are the arbitrary constants.

4 Analysis and discussion

Homotopy analysis method has been implemented to find the solution expressions of coupled nonlinear ordinary differential systems. It is well known that the convergence of obtained solutions is highly reliable on the proper selection of auxiliary parameters ℏ f , ℏ θ and ℏ ϕ . To select the proper region of convergence, we plotted the ℏ-curves of functions f(η), θ(η) and ϕ(η) at 21st order of approximations. The selected region of convergence lies within the ranges −1.45 ≤ ℏ f  ≤ −0.05, −1.40 ≤ ℏ θ  ≤ −0.30 and − 1.25 ≤ ℏ ϕ  ≤ −0.30 (Fig. 1). Table 1 also shows that the computed homotopic solutions are convergent in the whole region of η when ℏ f  = −0.75 = ℏ θ  = ℏ ϕ .

Table 1 Convergence of homotopy solution for different order of approximations when De 1 = 0.3, De 2 = 0.2, Ha = 0.6, λ = 0.3 = N = T R , S = 0.2, Pr = 1.0 = Le, T S  = 0.4 = C S and ℏ f  = −0.75 = ℏ θ  = ℏ ϕ
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Fig. 1
figure 1

Plot of ℏ-curves for f(η), θ(η) and ϕ(η) at 22th-order of deformations when De 1 = 0.3, De 2 = 0.2, Ha = 0.6, λ = 0.3 = N = T R , S = 0.2, Pr = 1.0 = Le, T S  = 0.4 = C S and N B  = 0.2 = N T

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The main theme of Figs. 2, 3, 4, 5, 6, 7, 8, 9 and 10 is to visualize the role of Deborah numbers De 1 and De 2, Hartman number Ha, mixed convection parameter λ, buoyancy ratio parameter N, heat generation parameter S, radiation parameter T R , thermal stratification parameter T S and Brownian motion parameter N B on the temperature distribution θ(η). Figure 2 shows the characteristics of Deborah number De 1 on θ(η). From this Fig., we evaluated that the temperature θ(η) and thermal boundary layer are enhanced corresponding to the increasing values of De 1. Physically, Deborah number De 1 appeared due to relaxation time, and this relaxation time is increased when De 1 enhances. Such larger relaxation time gives rise to temperature and thermal boundary layer thickness. On the other hand side, the temperature profile decays for higher values of Deborah number De 2 (see Fig. 3). This situation is arising due to appearance of retardation time in Deborah number De 2. The larger values of De 2 imply to higher retardation time that shows a reduction in temperature and thermal boundary layer thickness. The results of Maxwell fluid flow case can be retrieved by setting De 2 = 0. Further, the present analysis can be reduced to viscous fluid flow problem by setting De 1 = 0 = De 2.

Fig. 2
figure 2

Impact of De 1 on θ(η) when De 2 = 0.2, Ha = 0.6, λ = 0.3 = N = T R , S = 0.2, Pr = 1.0 = Le, T S  = 0.4 = C S and N B  = 0.2 = N T

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Fig. 3
figure 3

Impact of De 2 on θ(η) when De 1 = 0.3, Ha = 0.6, λ = 0.3 = N = T R , S = 0.2, Pr = 1.0 = Le, T S  = 0.4 = C S and N B  = 0.2 = N T

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Fig. 4
figure 4

Impact of Ha on θ(η) when De 1 = 0.3, De 2 = 0.2, λ = 0.3 = N = T R , S = 0.2, Pr = 1.0 = Le, T S  = 0.4 = C S and N B  = 0.2 = N T

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Fig. 5
figure 5

Impact of λ on θ(η) when De 1 = 0.3, De 2 = 0.2, Ha = 0.6, N = 0.3 = T R , S = 0.2, Pr = 1.0 = Le, T S  = 0.4 = C S and N B  = 0.2 = N T

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Fig. 6
figure 6

Impact of N on θ(η) when De 1 = 0.3, De 2 = 0.2, Ha = 0.6, λ = 0.3 = T R , S = 0.2, Pr = 1.0 = Le, T S  = 0.4 = C S and N B  = 0.2 = N T

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Fig. 7
figure 7

Impact of S on θ(η) when De 1 = 0.3, De 2 = 0.2, Ha = 0.6, λ = 0.3 = N = T R , Pr = 1.0 = Le, T S  = 0.4 = C S and N B  = 0.2 = N T

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Fig. 8
figure 8

Impact of T R on θ(η) when De 1 = 0.3, De 2 = 0.2, Ha = 0.6, λ = 0.3 = N, S = 0.2, Pr = 1.0 = Le, T S  = 0.4 = C S and N B  = 0.2 = N T

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Fig. 9
figure 9

Impact of T S on θ(η) when De 1 = 0.3, De 2 = 0.2, Ha = 0.6, λ = 0.3 = N = T R , S = 0.2, Pr = 1.0 = Le, C S  = 0.4 and N B  = 0.2 = N T

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Fig. 10
figure 10

Impact of N B on θ(η) when De 1 = 0.3, De 2 = 0.2, Ha = 0.6, λ = 0.3 = N = T R , S = 0.2, Pr = 1.0 = Le, T S  = 0.4 = C S and N T  = 0.2

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The variations in temperature θ(η) for multiple values of Ha are shown in Fig. 4. Here, we investigated that the presence of Ha enhances the temperature θ(η) and its associated boundary layer thickness. The temperature profile boosts for larger Ha. This occurs due to Lorentz force which is involved in Hartman number Ha. Due to stronger Lorentz force, the temperature θ(η) is enhanced. The hydrodynamic flow case can be obtained in the absence of Ha. Figures 5 and 6 are drawn to examine the impact of mixed convection parameter λ and buoyancy ratio parameter N on the temperature θ(η). These Figs. clearly interpret that the temperature and its associated boundary layer thickness are smaller for larger values of λ and N. The buoyance force occurs in the definition of mixed convection and buoyancy ratio parameters. This buoyancy force is stronger corresponding to the higher mixed convection and buoyancy ratio parameters that leads to lower temperature and thinner thermal boundary layer thickness. From Fig. 7, it is revealed that an increase in heat generation parameter S enhances the temperature field significantly. More heat is produced in fluid due to an enhancement in S which gives rise to temperature and thermal boundary layer. The temperature is lower in the absence of thermal radiation (see Fig. 8). The presence of thermal radiation parameter corresponds to the thicker boundary layer thickness due to more heat absorbed by the fluid. Impact of thermal stratification S T parameter on temperature field is shown in Fig. 9. The temperature is decreased rapidly via an increase in S T . Higher temperature is achieved at S T  = 0, i.e., unstratified situation. The presence of thermal stratification reduces the effective temperature difference between the sheet and ambient fluid that shows a reduction in temperature. Figure 10 shows that the larger values of Brownian motion parameter correspond to higher temperature profile. The Brownian motion parameter appeared due to involvement of nanoparticles. Insertion of nanoparticles enhances the thermal properties of fluid due to which temperature is an increasing function of Brownian motion parameter.

The influences of Deborah numbers De 1 and De 2, Hartman number Ha, Lewis number Le, solutal stratification parameter C S , Brownian motion parameter N B and thermophoresis parameter N T on nanoparticle concentration ϕ(η) are shown in Figs. 11, 12, 13, 14, 15, 16 and 17. The Deborah number De 1 enhances the nanoparticle concentration, while Deborah number De 2 reduces it (see Figs. 11, 12). It is due to reason that Deborah number De 1 is dependent on relaxation time, while Deborah number De 2 is related to retardation time. The presence of relaxation and retardation times in Deborah numbers corresponds to opposite behavior of De 1 and De 2 on nanoparticle concentration. Figure 13 shows the variations in nanoparticle concentration for multiple values of Hartman number. The nanoparticle concentration is weaker for hydrodynamic case in comparison with magnetohydrodynamic situation. The larger values of Lewis number lead to smaller nanoparticle concentration field and weaker boundary layer thickness. Physically the mass transfer at the sheet increases as we enhance the Lewis number due to which lower nanoparticle concentration profile is noticed in Fig. 14. The increase in solutal stratification parameter corresponds to a reduction in the nanoparticle concentration. Figure 15 also shows that the curve of nanoparticle concentration is higher when solutal stratification parameter is zero. From Fig. 16, we examined that the impact of Brownian motion parameter on nanoparticle concentration is highly significant for smaller values of N B . Such variation is very weak for higher values of Brownian motion parameter. In Fig. 16, we incorporate the influence of thermophoresis parameter on nanoparticle concentration. Here, we examined that the nanoparticle concentration is enhanced gradually when we give rise to the values of thermophoresis parameter.

Fig. 11
figure 11

Impact of De 1 on ϕ(η) when De 2 = 0.2, Ha = 0.6, λ = 0.3 = N = T R , S = 0.2, Pr = 1.0 = Le, T S  = 0.4 = C S and N B  = 0.2 = N T

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Fig. 12
figure 12

Impact of De 2 on ϕ(η) when De 1 = 0.3, Ha = 0.6, λ = 0.3 = N = T R , S = 0.2, Pr = 1.0 = Le, T S  = 0.4 = C S and N B  = 0.2 = N T

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Fig. 13
figure 13

Impact of Ha on ϕ(η) when De 1 = 0.3,  De 2 = 0.2, λ = 0.3 = N = T R , S = 0.2, Pr = 1.0 = Le, T S  = 0.4 = C S and N B  = 0.2 = N T

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Fig. 14
figure 14

Impact of Le on ϕ(η) when De 1 = 0.3,  De 2 = 0.2,  Ha = 0.6, λ = 0.3 = N = T R ,  S = 0.2,  Pr = 1.0,  T S  = 0.4 = C S and N B  = 0.2 = N T

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Fig. 15
figure 15

Impact of C S on ϕ(η) when De 1 = 0.3,  De 2 = 0.2,  Ha = 0.6, λ = 0.3 = N = T R ,  S = 0.2,  Pr = 1.0 = Le,  T S  = 0.4 and N B  = 0.2 = N T

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Fig. 16
figure 16

Impact of N B on ϕ(η) when De 1 = 0.3, De 2 = 0.2, Ha = 0.6, λ = 0.3 = N = T R , S = 0.2, Pr = 1.0 = Le, T S  = 0.4 = C S and N T  = 0.2

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Fig. 17
figure 17

Impact of N T on ϕ(η) when De 1 = 0.3, De 2 = 0.2, Ha = 0.6, λ = 0.3 = N = T R , S = 0.2, Pr = 1.0 = Le, T S  = 0.4 = C S and N B  = 0.2

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Table 1 shows the convergent values of \( f^{\prime \prime } \left( 0 \right) \), \( \theta^{\prime } \left( 0 \right) \) and \( \phi^{\prime } \left( 0 \right) \) when De 1 = 0.3, De 2 = 0.2 = S, Ha = 0.6, λ = 0.3 = N = T R , Pr = 1.0 = Le, T S  = 0.4 = C S , N T  = 0.2 = N B and ℏ f  = −0.75 = ℏ θ  = ℏ ϕ . This Table clearly indicates that the 21st order of HAM deformations leads to the convergent solutions. In Table 2, we computed the numerical values of local Nusselt number \( \left( {1 + \tfrac{4}{3}T_{R} } \right)\,\theta^{\prime}(0) \) and local Sherwood number \( \phi^{\prime } \left( 0 \right) \) for multiple values of De 1, De 2, Ha, λ, N and S when S C  = 0.4 = S T , N B  = 0.2 = N T , Le = 1.0 = Pr and T R  = 0.3. From this Table, we investigated that the values of local Nusselt and Sherwood numbers are smaller for larger values of Deborah number De 1, Hartman number Ha and heat source parameter S. These values are enhanced when we use the larger values of Deborah number De 2, mixed convection parameter λ and buoyancy ratio parameter N. Table 3 shows the values of \( \left( {1 + \tfrac{4}{3}T_{R} } \right)\,\theta^{\prime}(0) \) and \( \phi^{\prime } \left( 0 \right) \) for various values of S C , S T , N B , N T , Le, Pr and T R when De 1 = 0.3, De 2 = 0.2 = S, Ha = 0.6 and λ = 0.3 = N. Here, we examined that the values of \( \left( {1 + \tfrac{4}{3}T_{R} } \right)\,\theta^{\prime}(0) \) and \( \phi^{\prime } \left( 0 \right) \) are decreased for larger values of thermal stratification parameter S T and solutal stratification parameter S C . However, reverse behavior of Nusselt number is observed for the increasing values of thermal radiation parameter T R .

Table 2 Numerical computations of \( \left( {1 + \tfrac{4}{3}T_{R} } \right)\,\theta^{\prime}(0) \) and \( \phi^{\prime } \left( 0 \right) \) for multiple values of De 1, De 2, Ha, λ, N and S when S C  = 0.4 = S T , N B  = 0.2 = N T , Le = 1.0 = Pr and T R  = 0.3
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Table 3 Numerical computations of \( \left( {1 + \tfrac{4}{3}T_{R} } \right)\,\theta^{\prime}(0) \) and \( \phi^{\prime } \left( 0 \right) \) for multiple values of S C , S T , N B , N T , Le, Pr and T R when De 1 = 0.3, De 2 = 0.2 = S, Ha = 0.6 and λ = 0.3 = N
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5 Conclusions

We explored the role of solar radiation in magnetohydrodynamic flow of an Oldroyd-B nanofluid in this work. The flow analysis is performed over thermally and solutally stratified moving surface. The heat source/sink effect is also analyzed. The governing equations of momentum, energy and mass species are coupled due to the presence of mixed convection. We found that the temperature profile is reverse for increasing values Deborah numbers De 1 and De 2. This reverse situation appeared due to the dependence of De 1 and De 2 on the relaxation and retardation times. The presence of Lorentz force in Hartman number leads to the higher temperature and nanoparticle concentration profiles. It is revealed that the effects of mixed convection parameter λ on temperature are more dominant in comparison with buoyancy ratio parameter N. The temperature and thermal boundary layer thickness are enhanced rapidly when we increase the values of thermal radiation parameter T R . We also investigated that an increase in thermal and solutal stratification parameter leads to lower temperature and nanoparticle concentration profiles. Hence, the temperature and concentration are higher in case of unstratified surface. It is examined that the presence of nanoparticles enhances the thermal conductivity of fluid that corresponds to higher temperature.