1. Introduction

The flows over a moving frame have captured the attention due to the various ramifications in multiple processes of engineering.. Having such important implications in view, Bachok et al. [¹,²] discussed the nanofluids flows in the presence of a moving surface. Turkyilmazoglu [³] executed a time-dependent convection analysis of the flow induced by a vertical surface under heat transportation. Thumma et al. [⁴] studied the natural convective flow of MHD nanofluids over a plate by considering the temperature gradients with suction. The topic of boundary-driven flows on moving surfaces has been elaborated by many; see, for example, [⁵-⁹].

At present, nanomaterials have been subject of increasing interest. The concept of hybrid nanofluid is the latest advancement in the theory of nanofluids to improve the rate of heat transportation in engineering, industrial, and technological processes. These hybrid nanofluids have multiple appliances across fields such as acoustics, medicine, transportation, hybrid-powered procedures, micro-fluidics, naval structures, solar accumulators, energy and engine cells, etc. In accordance with such powerful implications, Jana et al. [¹⁰] investigated the heat transport in hybrid nano-fluidics. Suresh et al. [¹¹] analyzed the Cu-Al₂O₃-water hybrid nanofluid under the transportation of heat. Vafaei et al. [¹²] investigated the evolution of thermal conductivity in hybrid nanomaterials. Later on, distinct experimental and theoretical models have been developed to examine the multidimensional behavior of hybrid nanofluids [¹³-¹⁷].

Modern efforts have focused on introducing and developing more efficient and affordable technologies for energy storage. For example, nanomaterials are regarded as the best coolants in, among others, aerospace, optics, transport, nuclear, and aircraft industries. In this direction, Rashidi et al. [¹⁸] reported the magnetic nanomaterials flow by considering radiation and the presence of the buoyancy effect. Makinde [¹⁹] addressed the convected flow of nanomaterial induced by a porous plate by accounting the thermal radiation and mass transportation. Hayat et al. [²⁰] executed a study of 3D non-Newtonian nanomaterial flow under radiative phenomena. Some featured works in revolution of nanofluid theory can be consulted through [²¹-³⁰].

Here, our primary motive is to evaluate the aspects of heat transport and flow of a hybrid nanofluid under moving frame by considering spherically-shaped particles. The nonlinear phenomenon of thermal radiation is also taken into account; thus, a more realistic setting is proposed in comparison to the linearized thermal radiation case. According to literature review, no investigation with such physical flow features is reported yet. Elucidations are made with the help of RungeKutta-Fehlberg-45 method.

2. Mathematical modeling

Consider a steady-state flow and heat transportation of a hybrid nanofluid in a moving frame. The plate is stretched along the opposite or same direction having free-stream velocity U _∞ and constant velocity U _w . The nonlinear thermal radiation aspects are also taken into account. The flow is confined to y > 0 and the sheet coincides with the plane y = 0. The ambient fluid temperature is represented by T _∞.

Governing equations of hybrid nanofluid are defined as:

where u and v denote the velocity components of the nanofluid along the x and y directions, respectively, σ ^∗ the Stefan-Boltzmann constant, k ^∗ the mean absorption coefficient and q _r the radiated surface heat flux.

The expressions for viscosity, thermal conductivity, specific heat and density of the hybrid nanofluid are as follows [16]:

where n = 3 represents the case for spherical nanoparticles. The basic thermo-physical features of nanofluid at 25◦C are given in Table I.

Interrelated boundary conditions are as follows:

Now, we introduced the similarity transformation as

where U= 𝑈 𝑤 + 𝑈 ∞ , θw=(T∞/Tw),θw>1.

Making use of Eq. (8), Eq. (1) is identically satisfied and Eqs. (2) and (3) take the form:

Note that the Eqs. (9) and (10) are transformed form of Eqs. (2) and (3). These expressions are obtained by putting the values of similarity variables given in Eq. (8).

The boundary conditions are converted into the form:

where A=Uw/U∞ the parameter of velocity ratio, Pr=(μCp)f/kf the Prandtl number, Ec=Uw2/(Tw-T∞)Cpf) the Eckert number and R=(16σ*T∞3)/(3khnfk*) is the radiation parameter. The friction factor (C _fx ) and local Nusselt number (N _ux ) are prescribed as:

The friction factor (C _fx ) and local Nusselt number (Nu _x ) are prescribed as:

where τ _w the shear stress and q _w the radiative heat flux and can be expressed as:

After Utilization of Eq. (8), C _fx and Nu _x can be transformed as

where Re = U _x /v _f is local Reynolds number.

3. Least square technique

This section elucidates the procedure of least square method. Least square scheme is applied to found the numerical iterations of the problem. This technique is very efficient and straight forward and has the following steps [²⁶]:

Step 1: Equations (9) and (10) can be rewritten as

Step 2: To interpret the solutions of the Eqs. (12) and (13), the following trial-based solutions are suggested:

In the above expressions, N indicates the number of approximations. N is large for more accurate solutions. The ansatz of Eqs. (14) and (15) must satisfy the boundary conditions associated with Eqs. (9) and (10). The solutions can be written as:

Step 3: The residual vectors for each expression in Eqs. (12) and (13) are further developed, by using the reduced trial solution, to obtain

Step 4: The reduced form of the trial solutions is the exact solutions of Eqs. (14)-(17) by vanishing the residual vectors. The construction of subsequent relations is made fout of the residuals

Here the problem domain is represented by Γ. A minimum of Eq. (18) can be achieved by the derivatives of Eq. (18) with respect to the unknowns A _i , which must vanish, so to have

where the factor 2 can be ignored.

Step 5: To obtain the values of all A^’ _i s, we solved the system of algebraic expressions obtained from Eqs. (19) and (20). The substitution of the values of the A^’ _i s into the reduced trial solution yields the desired solutions of Eqs. (11)-(14).

4. Results and discussion

The objective of the present study is to extend the work of nanofluids to hybrid nanofluids in the presence of a moving plate by considering nonlinear radiative heat transfer and viscous dissipation. The model equations are transformed with the help of similarity transformations. Further, the obtained similarity equations have been solved numerically. The variations of embedded parameters on friction factor and heat transfer characteristics are shown numerically and graphically.

Fluctuation in skin friction coefficient is presented in Fig. 1 for distinct values of Pr and A for both nanofluid and hybrid nanofluid. Here one can see that an increment in Pr and A leads to an enhancement of the friction factor for both the nanofluid and the hybrid nanofluid. Further, the nanofluid shows overriding performance compared to the hybrid nanofluid The variations in Nu _x through Pr and Ec for both nano and hybrid nanofluids are described in Fig. 2. The higher Pr scale back the Nu _x but in the other hand, Nu _x increases for ascending values of Ec. A comparative analysis shows that the reduction in Nu _x is higher for the hybrid nanofluid case than the nanofluid. The influence of R and Pr on Nu _x is portrayed in Fig. 3. This plot shows that the heat transport rate reduces for higher values of Pr, while a contradiction is recognized for increasing value of R.

FIGURE 1 Variations of A and Pr on C _fx . 

FIGURE 2 Variations of Ec and Pr on Nu _x . 

FIGURE 3 Variations of R and Pr on Nu _x . 

The profile of θ(η) versus η is observed in Fig. 4. This figure illustrates that θ(η) is improved rapidly as R is incremented. Physically, a higher value of R corresponds to a higher thickness of the thermal layer. The larger R leads to a decrement in the coefficient of mean absorption.

FIGURE 4 Variations of R on θ(η) 

Hence, the heat amount transfer to liquid is stirring. Fluctuations in θ(η) via θ _w are depicted in Fig. 5. An increase in θ _w corresponds to a higher temperature and thicker thermal layer. The larger θ _w provides more energy to working liquid due to which presence of θ _w give peak to temperature profile. Figure 6 exhibits the role of Pr on the profile of θ(η). A raise in values of Pr diminish the temperature.

FIGURE 5 Variations of θw on θ(η). 

FIGURE 6 Variations of Pr on θ(η). 

Figure 7 presents the influence of Ec on the θ(η) profile. Here, it can be noticed that the temperature and its interrelated thickness of layer is improved for increasing Ec. Basically, the existence of viscous heat in heat expressions acts as an internal heat agent. Hence, the thermal energy booming in the fluid. The effect of (₂ on f(η) and θ(η) are demonstrated through Figs. 8 and 9. Figure 8 represents that for larger (₂, the flow is faster for both the nanomaterial and the mixture of nanomaterials. The temperature θ(η) is found to increase for higher (₂ (see Fig. 9). The nanoparticles generate warmness in liquid due to presence of photo-catalytic aspect. Hence, the addition of more nanoparticles leads to a temperature enhancement.

FIGURE 7 Variations of Ec on θ(η). 

FIGURE 8 Variations of ϕ₂ on f’(η). 

FIGURE 9 Variations of ϕ₂ on θ(η). 

The effect of A on θ(η) is portrayed in Figs. 10 and 11. From Fig. 10, it is noticeable that the liquid is close to the surface of the solid on which the liquid flows due to a higher absolute value of A(< 0). Physically, the convection current is improved by an increase in λ. From Fig. 11, one can observe that thetemperature of the fluid scales back for larger values of A(> 0). The behavior of Φ₁ on both f’(η) and θ(η) are plotted in Figs. 12 and 13. From Fig. 12, it is observed that f’(η) and its interrelated concentration layer become larger as a function of Φ₁. Figure 13 indicates that the increasing behavior of θ(η) and its interrelated concentration layer is reduced due to the parameter Φ₁. Figures 14 and 15 show the stream line graphs for different parameters.

FIGURE 10 Variations of A on θ(η). 

FIGURE 11 Variations of A on θ(η). 

FIGURE 12 Variations of ϕ₁ on f’(η). 

FIGURE 13 Variations of ϕ₁ on θ(η). 

FIGURE 14 a) Stream lines for nanofluid, b) Stream lines for hybrid nanofluid. 

FIGURE 15 a) Stream lines of nanofluid for A = 1, b) Stream lines of hybrid nanofluid for A = 1. 

Numeric values of C _fx and Nu _x for distinct values of influential constraints are shown in Table II. It was observed that both C _fx and Nu _x increase for greater values of A (see Table II).

Moreover Nu _x rises with R and θ _w . Furthermore, from Table II one can see that Nu _x decreases with Ec. It can be noted from Table II that the thermal conductivity of hybrid nanomaterials high in comparison to a regular nanomaterial.

5. Conclusions

The impact of hybrid nanoparticles on fluid flow on a moving plate in the presence of thermal radiation is evaluated. The nonlinear thermal radiation is considered in the energy expression. Further, we visualized that the admixture of distinct nanomaterials corresponds to a high heat capacity in comparison to a regular nanofluid. The most notable outcomes of this study are: