Introduction

The dispersion of nanoparticles in the basefluid is modern way to increase its heat transportation performance. Basefluids include organic liquid, oils, polymer solution, water, biofluids, etc., while nanoparticles are made from metallic oxides (Al2O3, CuO, TiO2, ZnO), metals (Ag, Au, Cu), nitrate ceramics (AIN, TiC, SIC), carbides and CNTs (carbon nanotubes). Nanoparticles comprise size from 1 to 100 nm. The mixture or combination of basefluid and nanoparticles is referred as nanofluid. Thermal features of basefluid can be highly affected by submersion of nanoparticles. Initial step in this domain was taken by Choi [1]. Nanofluids have vital applications in solar cells, drug delivery systems, computer devices, refrigerants, solar collectors, solar thermoelectric devises, cooling and heating of modern systems, etc. Radiation impact in melting flow of CNTs with chemical reactions is explored by Hayat et al. [2]. Hosseini et al. [3] studied heat source, magnetic effect and entropy production in flow of nanofluid. Melting effect in flow of CNTs by numerical approach is presented by Hayat et al. [4]. Entropy production in flow of non-Newtonian fluid is elaborated by Khan et al. [5]. Nanofluid (Cu + water) flow by a down-point rotating cone is presented by Dinarvand and Pop [6]. Hayat et al. [7] examined nanofluid during peristaltic flow with temperature-dependent viscosity. Convective flow of Jeffrey nanofluid between two infinite parallel plates is elaborated by Hayat et al. [8]. Some relevant analysis in this domain can be seen in Refs. [9,10,11,12,13,14,15,16,17,18,19].

Recently, scientists and engineers have performed various experiments on submersion of two or more nanosized particles in same basefluid. Such mixture or combination of nanoparticles and baseliquid is known by hybrid nanofluid. Hybrid nanofluid has exceptional characteristics as compared to nanofluid. A brief study on hybrid nanomaterial is presented by Sarkar et al. [20]. Sajid and Ali [21] analyzed thermal conductance of hybrid nanomaterial. In order to study features of elliptical tube via hybrid nanomaterial, a numerical investigation is given by Huminic and Huminic [22]. Hayat and Nadeem [23] studied heat transport feature via hybrid nanofluid. An experimental analysis on hybrid nanomaterial is performed by Sun et al. [24]. Muhammad et al. [25] performed a comparative analysis of hybrid nanofluid, basefluid and nanofluid in the presence of stagnation point.

Nowadays squeezed flow comprising between two parallel plates is an area of great attention for the scientists and engineers. Squeezed flow is generated due to the motion of the plates toward each other. Applications of squeezed flow in industrial as well as engineering fields include lubrications, metal molding, polymer processing, injection modeling, compression, food processing, etc. Initial analysis in this direction is made by Stefan [26]. Melting impact in rotatory squeezing flow of CNTs is expressed by Hayat et al. [27]. Singeetham and Puttanna [28] elaborated squeezed flow of a non-Newtonian fluid. Slip condition in squeezed flow with double stratification is analyzed by Ahmed et al. [29]. Few recent articles on the topic can be studied in Refs. [30,31,32,33].

Existing information on the topic witnessed that very little analysis is yet made about flow of hybrid nanofluid between two parallel plates. Motivation behind this work is to elaborate entropy production in squeezed flow of hybrid nanomaterial. CNTs (SWCNTs, MWCNTs) and CuO are utilized as nanoparticles in water-basefluid. Heat transportation features are explored via viscous dissipation and melting effect. Shooting method (bvp4c) is implemented for solution development. Comparative study for basefluid (water), nanofluid (CNTs (SWCNTs, MWCNTs) + water) and hybrid nanofluid (CNTs (SWCNTs, MWCNTs) + CuO + water) is performed by graphical method.

Formulations

Assume unsteady squeezed flow of hybrid nanofluid bounded between two parallel plates such that the upper plate moves toward the lower fixed plate. The upper plate lies at \(y = h(t) = \sqrt {\tfrac{{\upsilon_{\text{f}} (1 - bt)}}{a}}\) while the lower plate at \(y = 0.\) Both plates are separated by a distance \(h(t) = \sqrt {\tfrac{{\upsilon_{\text{f}} (1 - bt)}}{a}} .\) CNTs and CuO are treated as first and second nanoparticles, respectively, while water is taken as basefluid. In Cartesian coordinates, flow is along x-axis. Here y-axis is perpendicular to the x-axis (see Fig. 1). Flow field expressions under mentioned assumptions are:

Fig. 1
figure 1

Schematic diagram for the squeezed flow

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$$\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0,$$
(1)
$$\frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y} = - \frac{1}{{\rho_{\text{hnf}} }}\frac{\partial p}{\partial x} + \upsilon_{\text{hnf}} \left( {\frac{{\partial^{2} u}}{{\partial x^{2} }} + \frac{{\partial^{2} u}}{{\partial y^{2} }}} \right),$$
(2)
$$\frac{\partial v}{\partial t} + u\frac{\partial v}{\partial x} + v\frac{\partial v}{\partial y} = - \frac{1}{{\rho_{\text{hnf}} }}\frac{\partial p}{\partial y} + \upsilon_{\text{hnf}} \left( {\frac{{\partial^{2} v}}{{\partial x^{2} }} + \frac{{\partial^{2} v}}{{\partial y^{2} }}} \right),$$
(3)
$$\frac{\partial T}{\partial t} + u\frac{\partial T}{\partial x} + v\frac{\partial T}{\partial y} = \alpha_{\text{hnf}} \left( {\frac{{\partial^{2} T}}{{\partial x^{2} }} + \frac{{\partial^{2} T}}{{\partial y^{2} }}} \right) + \frac{{\mu_{\text{hnf}} }}{{(\rho c_{\text{p}} )_{\text{hnf}} }}\left( {\frac{\partial u}{\partial y} + \frac{\partial v}{\partial x}} \right)^{2} + \frac{{\mu_{\text{hnf}} }}{{(\rho c_{\text{p}} )_{\text{hnf}} }}4\left( {\frac{\partial u}{\partial x}} \right)^{2} .$$
(4)
$$\begin{aligned}& u\left( {x,y,t} \right) = U_{\text{w}} \left( {x,t} \right) = 0,T = T_{\text{m}} {\text{ at }}y = 0 \hfill \\& u\left( {x,y,t} \right) = 0,v\left( {x,y,t} \right) = \frac{{{\text{d}}h(t)}}{{{\text{d}}t}},T = T_{\text{h}} {\text{ at }}y = h(t). \hfill \\ \end{aligned}$$
(5)

Melting condition is [34]:

$$k_{\text{hnf}} \left( {\frac{\partial T}{\partial y}} \right) = \rho_{\text{hnf}} (\lambda_{1} + C_{{s}} (T_{\text{m}} - T_{0} ))v\;{\text{at }}y = 0.$$
(6)

Pressure gradient is eliminated from Eqs. (1) and (2) by differentiating Eq. (1) w.r.t y and Eq. (2) w.r.t x.

We consider the transformations for converting the above expressions (PDEs) into ODEs [34]:

$$\eta = \frac{y}{h(t)},h(t) = \sqrt {\frac{{\upsilon_{\text{f}} (1 - bt)}}{a}} ,u = \frac{{axf^{\prime } (\eta )}}{1 - bt},v = - \sqrt {\frac{{\upsilon_{\text{f}} a}}{1 - bt}} f(\eta ),\theta = \frac{{T - T_{\text{m}} }}{{T_{\text{f}} - T_{\text{m}} }}.$$
(7)

Condition for incompressibility is verified while other expressions become

$$\frac{{A_{11} }}{{(1 - \phi_{1} )^{2.5} (1 - \phi_{2} )^{2.5} }}f^{(\text{iv})} + ff^{{{\prime \prime \prime }}} - f^{{\prime }} f^{{\prime \prime }} - \frac{3}{2}{\text{Sq}}f^{{\prime \prime }} - \frac{\text{Sq}}{2}\eta f^{{{\prime \prime \prime }}} = 0,$$
(8)
$$\frac{{\kappa_{\text{hnf}} }}{{\kappa_{\text{f}} }}\theta^{{\prime \prime }} + B_{11} (\mathop {\Pr }\limits {\text{Sq}}(f\theta^{{\prime }} - \frac{\text{Sq}}{2}\eta \theta^{{\prime }} ) + \frac{\Pr }{{(1 - \phi_{1} )^{2.5} (1 - \phi_{2} )^{2.5} }}({\text{Ec}}_{{x}} (f^{{\prime \prime }} )^{2} + 4{\text{Ec}}(f^{{\prime }} )^{2} )) = 0,$$
(9)
$$f^{{\prime }} (0) = 0,\theta (0) = 0,f(1) = \frac{\text{Sq}}{2},f^{{\prime }} (1) = 0,\theta (1) = 1,\frac{{\kappa_{\text{hnf}} }}{{\kappa_{\text{f}} }}M\theta^{{\prime }} \left( 0 \right) + \frac{\Pr }{{A_{11} }}f(0) = 0.$$
(10)

Here

$$A_{11} = \frac{1}{{\left( {1 - \phi_{2} } \right)\,\left( {\left( {1 - \phi_{1} } \right) + \phi_{1} \tfrac{{\rho_{\text{CNT}} }}{{\rho_{\text{f}} }}} \right) + \phi_{2} \tfrac{{\rho_{\text{CuO}} }}{{\rho_{\text{f}} }}}},$$
(11)
$$B_{11} = \left( {1 - \phi_{2} } \right)\left( {\left( {1 - \phi_{1} } \right) + \phi_{1} \frac{{(\rho c_{\text{p}} )_{\text{CNT}} }}{{(\rho c_{\text{p}} )_{\text{f}} }}} \right) + \phi_{2} \frac{{(\rho c_{\text{p}} )_{\text{CuO}} }}{{(\rho c_{\text{p}} )_{\text{f}} }}.$$
(12)

Associated physical parameters are defined by

$$\begin{aligned} {\text{Sq}} = & \frac{b}{a},M = \frac{{C_{\text{pf}} \left( {T_{\infty } - T_{\text{m}} } \right)}}{{\lambda_{1} + C_{\text{s}} \left( {T_{\text{m}} - T_{0} } \right)}},{\text{Ec}}_{\text{x}} = \frac{{a^{2} x^{2} }}{{c_{\text{pf}} \left( {T_{\text{f}} - T_{\text{m}} } \right)(1 - bt)^{2} }},\mathop {\Pr }\limits = \frac{{\nu_{\text{f}} }}{{\alpha_{\text{f}} }}, \\ {\text{Ec}} = & \frac{{a\upsilon_{\text{f}} }}{{c_{\text{pf}} \left( {T_{\text{f}} - T_{\text{m}} } \right)(1 - bt)}},\varOmega = \frac{{T_{\text{f}} - T_{\text{m}} }}{{T_{\text{h}} }}. \\ \end{aligned}$$
(13)

Nusselt number (\({\text{Nu}}_{\text{x}} (Re_{\text{x}} )^{{ - \tfrac{1}{2}}}\)) expression

In dimensional and dimensionless form of (\({\text{Nu}}_{\text{x}} (Re_{\text{x}} )^{{ - \tfrac{1}{2}}}\)) is

$${\text{Nu}}_{\text{x}} = \frac{{xq_{\text{w}} }}{{k(T_{\text{f}} - T_{\text{h}} )}},\;{\text{with}}\;q_{\text{w}} = - k_{\text{hnf}} \left( {\frac{\partial T}{\partial y}} \right)_{{{{y}} = 0,}}$$
(14)

and

$${\text{Nu}}_{\text{x}} ({Re}_{\text{x}} )^{{ - \tfrac{1}{2}}} = - \frac{{\kappa_{\text{hnf}} }}{{\kappa_{\text{f}} }}\theta^{{\prime }} (0),$$
(15)

where \(Re_{\text{x}} = \sqrt {\tfrac{{\left( {1 - {\text{ct}}} \right)\,\upsilon_{\text{f}} }}{a}}\) is local Reynolds number.

Entropy production rate (Ns) and Bejan number (Be) expressions

Total entropy production rate (SGT) is

$${\text{SG}}_{\text{T}} = {\text{SG}}_{\text{H}} ({\text{Entropy production through heat transfer}}) + {\text{SG}}_{\text{F}} ({\text{Entropy production through fluid friction}}).$$

Thus, we have

$${\text{SG}}_{\text{T}} = \frac{{\kappa_{\text{hnf}} }}{{T_{\text{h}}^{2} }}\left( {\left( {\frac{\partial T}{\partial x}} \right)^{2} + \left( {\frac{\partial T}{\partial y}} \right)^{2} } \right) + \frac{{\mu_{\text{nf}} }}{{T_{\text{h}} }}\left( {4\left( {\frac{\partial u}{\partial x}} \right)^{2} + \left( {\frac{\partial v}{\partial x} + \frac{\partial u}{\partial y}} \right)^{2} } \right).$$
(16)

In non-dimensional form, entropy production rate (Ns) is

$${\text{Ns}} = \frac{{{\text{SG}}_{\text{T}} }}{{{\text{SG}}_{0} }} = \theta^{{{\prime }2}} + \frac{\Pr }{{\varOmega (\tfrac{{\kappa_{\text{hnf}} }}{{\kappa_{\text{f}} }})\left( {1 - \phi_{1} } \right)^{2.5} \left( {1 - \phi_{2} } \right)^{2.5} }}(4{\text{Ec}}f^{{{\prime }2}} + {\text{Ec}}_{1} f^{{{\prime \prime }2}} ),$$
(17)

where \({\text{SG}}_{{_{0} }}\) represents rate of characteristic entropy production and defined by

$$S_{{{\text{G}}_{0} }} = \frac{{\kappa_{\text{hnf}} (T_{\text{m}} - T_{\text{h}} )}}{{T_{\text{h}}^{ 2} {\text{h}}^{2} }}.$$
(18)

Bejan number (Be) is

$${\text{Be}} = \frac{{{\text{SG}}_{\text{H}} }}{{{\text{SG}}_{\text{T}} }}$$
(19)

while Be in dimensionless form is

$${\text{Be}} = \frac{{\theta^{\prime 2} }}{\text{Ns}}.$$
(20)

Model for nanofluid

Expressions for hybrid nanofluid using Hamilton–Crosser model are [12]:

$$ \begin{aligned} &\rho_{\text{hnf}} = \left( {1 - \phi_{2} } \right)\left( {\left( {1 - \phi_{1} } \right)\rho_{\text{f}} + \phi_{1} \rho_{\text{CNT}} } \right) + \phi_{2} \rho_{\text{CuO}} , \\ &(\rho c_{\text{p}} )_{\text{hnf}} = \left( {1 - \phi_{2} } \right)\left( {\left( {1 - \phi_{1} } \right)(\rho c_{\text{p}} )_{\text{f}} + \phi_{1} (\rho c_{\text{p}} )_{\text{CNT}} } \right) + \phi_{{2(\rho {\text{c}}_{\text{p}} ){\text{CuO}}}} , \\ &\frac{{\kappa_{\text{hnf}} }}{{\kappa_{\text{nf}} }} = \frac{{\kappa_{\text{CuO}} + (n - 1)\kappa_{\text{nf}} - (n - 1)\phi_{2} (\kappa_{\text{nf}} - \kappa_{\text{CuO}} )}}{{\kappa_{\text{CuO}} + (n - 1)\kappa_{\text{nf}} + \phi_{2} (\kappa_{\text{nf}} - \kappa_{\text{CuO}} )}}, \\ &\frac{{\kappa_{\text{nf}} }}{{\kappa_{\text{f}} }} = \frac{{\kappa_{\text{CNT}} + (n - 1)\kappa_{\text{f}} - (n - 1)\phi_{1} (\kappa_{\text{f}} - \kappa_{\text{CNT}} )}}{{\kappa_{\text{CNT}} + (n - 1)\kappa_{\text{f}} + \phi_{1} (\kappa_{\text{f}} - \kappa_{\text{CNT}} )}}, \\ &\mu_{\text{hnf}} = \frac{{\mu_{\text{f}} }}{{\left( {1 - \phi_{1} } \right)^{2.5} \left( {1 - \phi_{2} } \right)^{2.5} }},\upsilon_{\text{hnf}} = \frac{{\mu_{\text{hnf}} }}{{\rho_{\text{hnf}} }}. \\ \end{aligned} $$

For nanofluid, the Hamilton–Crosser expressions are

$$\begin{aligned}& \rho_{\text{nf}} = \left( {1 - \phi_{1} } \right)\rho_{\text{f}} + \phi_{1} \rho_{\text{CNT}} , \\ &(\rho c_{\text{p}} )_{\text{nf}} = \left( {1 - \phi_{1} } \right)(\rho c_{\text{p}} )_{\text{f}} + \phi_{1} (\rho c_{{p}} )_{\text{CNT}} , \\ &\frac{{\kappa_{\text{nf}} }}{{\kappa_{\text{f}} }} = \frac{{\kappa_{\text{CNT}} + (n - 1)\kappa_{\text{f}} - (n - 1)\phi_{1} (\kappa_{\text{f}} - \kappa_{\text{CNT}} )}}{{\kappa_{\text{CNT}} + (n - 1)\kappa_{\text{f}} + \phi_{1} (\kappa_{\text{f}} - \kappa_{\text{CNT}} )}}, \\ &\mu_{\text{nf}} = \frac{{\mu_{\text{f}} }}{{\left( {1 - \phi_{1} } \right)^{2.5} }},\upsilon_{\text{nf}} = \frac{{\mu_{\text{nf}} }}{{\rho_{\text{nf}} }}. \\ \end{aligned}$$

Here n is shape parameter, i.e., n = 6 represents that nanoparticles are of tube like or cylindrical shape (Table 1).

Table 1 Thermal characteristics of CNTs, CuO and water [28]
Full size table

Numerical solution

After implementing transformations given in Eq. (7), the transformed flow field expressions are then solved by shooting method (a numerical technique with RK-4 algorithm). Shooting technique with RK-4 is applied only for IBVPs of first order, thus reducing the flow expression into first order as [14]:

$$\begin{aligned} &f_{11} = f,f_{12} = f_{11}^{{\prime }} = f^{{\prime }} ,f_{13} = f_{12}^{{\prime }} = f^{{\prime \prime }} ,f_{14} = f_{13}^{{\prime }} = f^{{{\prime \prime \prime }}} ,\\&f_{21} = \theta ,f_{22} = f_{21}^{{\prime }} = \theta^{{\prime }} , f_{1} = f_{14}^{{\prime }} = f^{(iv)} = - A_{11} \left( {1 - \phi_{1} } \right)^{2.5}\\&\quad\!\qquad\times\, \left( {1 - \phi_{2} } \right)^{2.5} (f_{11} f_{14} - f_{12} f_{13} - \frac{3}{2}{\text{Sq}}f_{13} - \frac{\text{Sq}}{2}\eta f_{14} ), \\ &f_{2} = f_{22}^{{\prime \prime }} = \theta^{{\prime \prime }} = - \frac{{B_{11} }}{{\tfrac{{k_{\text{hnf}} }}{{k_{\text{f}} }}}}\left(\mathop {\Pr }\limits Sq(f_{11} f_{22} - \frac{Sq}{2}\eta f_{22} \right)\\&\qquad\; + \frac{\Pr }{{\left( {1 - \phi_{1} } \right)^{2.5} \left( {1 - \phi_{2} } \right)^{2.5} }}({\text{Ec}}_{\text{x}}\,f_{13}^{2} + 4{\text{Ec}}f_{12}^{2} )), \\ &f_{12} (0) = 0,f_{21} (0) = 0,\frac{{k_{\text{hnf}} }}{{k_{\text{f}} }}f_{22} (0) + \frac{\Pr }{{{\text{MA}}_{11} }}f_{11} (0) = 0,\\&f_{13} (0) = 0,f_{2} (0) = 1,f_{14} (0) = 0. \\ \end{aligned}$$

Analysis

The vital theme behind this section is to visualize comparative study among basefluid (water), nanofluid (CNTs (water)) and hybrid nanofluid (CNTs + CuO (water)). Velocity of fluid (\(f^{{\prime }} (\eta )\)), rate of entropy production (Ns), temperature of fluid (\(\theta (\eta )\)) and Bejan number (Be) are studied against higher estimations of \({\text{Sq}},\phi_{1} ,\phi_{2} ,M\) and Ec in Figs. 2a–4r). The graphical visualization is performed as follows:

Fig. 2
figure 2figure 2

a \(f^{{\prime }} (\eta )\) vs. Sq (SWCNTs + CuO + water), b \(f^{{\prime }} (\eta )\) vs. Sq (MWCNTs + CuO + water), c \(f^{\prime}(\eta )\) vs. Sq (comparison), d \(f^{{\prime }} (\eta )\) vs. \(\phi_{1}\) (SWCNTs + CuO + water), e \(f^{{\prime }} (\eta )\) vs. \(\phi_{1}\) (MWCNTs + CuO + water), f \(f^{{\prime }} (\eta )\) vs. \(\phi_{1}\) (comparison), g \(f^{{\prime }} (\eta )\) vs. M (SWCNTs + CuO + water), h \(f^{{\prime }} (\eta ) f^{\prime}(\eta )\) vs. M (MWCNTs + CuO + water). i \(f^{{\prime }} (\eta )\) vs. M (comparison), j \(f^{{\prime }} (\eta )\) vs. \(\phi_{2}\) (SWCNTs + CuO + water), k \(f^{{\prime }} (\eta )\) vs. \(\phi_{2}\) (MWCNTs + CuO + water), l \(f^{{\prime }} (\eta )\) vs. \(\phi_{2}\) (comparison)

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  1. 1.

    First graph is plotted for comparative analysis of nanofluid (using SWCNTs + water) and hybrid nanofluid (using SWCNTs + CuO  + water) against each physical parameter.

  2. 2.

    Second graph is plotted for comparative study of nanofluid and hybrid nanofluid by replacing SWCNTs by MWCNTs.

  3. 3.

    In third graph, the comparative study among nanofluid (using CNTs (SWCNTs, MWCNTs) + water), hybrid nanofluid (using CNTs (SWCNTs, MWCNTs) + CuO + water) and basefluid (using water) is visualized against each pertinent parameter. Here we take \({\text{Sq}} = M = 0.1,\phi_{1} = \phi_{2} = 0.5,{\text{Ec}} = {\text{Ec}}_{\text{x}} = 0.5.\)

Analysis of velocity (\(f^{{\prime }} (\eta )\))

In Fig. 2a and b, the variations in \(f^{{\prime }} (\eta )\) during comparative study of nanofluid (SWCNTs  + water, MWCNTs + water) and hybrid nanofluid (SWCNTs + CuO + water, using MWCNTs + CuO + water) are sketched against higher estimations of Sq. Clearly, it is detected that \(f^{{\prime }} (\eta )\) is an increasing function of Sq and impact of hybrid nanofluid is more than nanofluid. Physically, increment in Sq leads to larger squeezing force experienced by fluid particles. Thus, \(f^{\prime}(\eta )\) intensifies. Velocity of fluid (\(f^{{\prime }} (\eta )\)) via \(\phi_{1}\) in comparative study of nanofluid (SWCNTs + water, MWCNTs + water) and hybrid nanofluid (SWCNTs + CuO + water, MWCNTs + CuO + water) is labeled in Fig. 2d and e. Increment in \(f^{{\prime }} (\eta )\) is noticed via higher \(\phi_{1}\), and conspicuous impact is detected for hybrid nanofluid. Impact of M on \(f^{{\prime }} (\eta )\) during flow of nanofluid (using SWCNTs + water, using MWCNTs + water) and hybrid nanofluid (SWCNTs + CuO + water, MWCNTs + CuO + water) is portrayed in Fig. 2g and h. Direct variations in \(f^{{\prime }} (\eta )\) are seen against higher M, and dominant trend is noticed for hybrid nanofluid. Indeed higher M leads to more convective flow from melting surface toward hot fluid. Hence, \(f^{{\prime }} (\eta )\) increases. Figure 2j and k shows the impact of \(\phi_{2}\) on \(f^{{\prime }} (\eta )\) during flow of nanofluid (SWCNTs + water, MWCNTs + water) and hybrid nanofluid (SWCNTs + CuO + water, MWCNTs + CuO + water). As expected, no impact on \(f^{{\prime }} (\eta )\) is seen for higher \(\phi_{2}\) during flow of nanofluid while \(f^{{\prime }} (\eta )\) intensifies during flow of hybrid nanofluid. Impact of hybrid nanofluid is also dominant. Figure 2c, f, i and l is plotted for impact of basefluid (water), nanofluid (CNTs (SWCNTs + water, MWCNTs + water)) and hybrid nanofluid (CNTs (SWCNTs, MWCNTs) + CuO + water) on \(f^{{\prime }} (\eta )\) when \({\text{Sq}} = \phi_{1} = M = \phi_{2} = 0.1\), respectively. It can be seen clearly that hybrid nanofluid shows effective behavior than that of nanofluid as well as basefluid.

Analysis of temperature (\(\theta (\eta )\))

Temperature (\(\theta (\eta )\)) of fluid against Sq during flow of nanofluid (SWCNTs + water, MWCNTs + water) and hybrid nanofluid (SWCNTs + CuO + water, MWCNTs + CuO + water) is presented in Fig. 3a and b. Temperature (\(\theta (\eta )\)) decay with higher estimations of Sq and behavior of hybrid nanofluid is prominent. Physically, higher Sq leads to stronger squeezing force which results in closeness of both plates. Thus, decay in kinematic velocity leads to reduction in \(\theta (\eta )\). Figure 3d and e depicts the impact of \(\phi_{1}\) on \(\theta (\eta )\) during comparative study of nanofluid (using SWCNTs + water, using MWCNTs + water) and hybrid nanofluid (using SWCNTs + CuO + water, using MWCNTs + CuO + water). \(\theta (\eta )\) decays with higher \(\phi_{1}\), and hybrid nanofluid shows effective behavior. \(\theta (\eta )\) against M is sketched in Fig. 3g and h during comparative study of nanofluid (SWCNTs + water, MWCNTs + water) and hybrid nanofluid (SWCNTs + CuO + water, MWCNTs + CuO + water). \(\theta (\eta )\) reduces with higher M, and behavior of hybrid nanofluid is prominent when compared with nanofluid. Indeed higher M leads to addition of cold fluid particles from melting surface to heated fluid. Thus, \(\theta (\eta )\) decays. Figure 3j and k presents impact of \(\phi_{2}\) on \(\theta (\eta )\) during flow of nanofluid (SWCNTs + water, MWCNTs + water) and hybrid nanofluid (SWCNTs + CuO + water, MWCNTs + CuO + water). Decrease in \(\theta (\eta )\) is observed for higher \(\phi_{2}\), and clearly, hybrid nanofluid shows effective trend. \(\theta (\eta )\) against Ec during flow of nanofluid (SWCNTs + water, MWCNTs + water) and hybrid nanofluid (SWCNTs + CuO + water, MWCNTs + CuO + water) is plotted in Fig. 3m and n. Here \(\theta (\eta )\) directly varies with Ec, and hybrid nanofluid shows overriding trend. Physically, higher Ec leads to production of more drag force between fluid particles and so \(\theta (\eta )\) intensifies. Figure 3c, f, i, l, o is labeled to examine \(\theta (\eta )\) during flow of basefluid (water), nanofluid (using CNTs (SWCNTs, MWCNTs) + water) and hybrid nanofluid (using CNTs (SWCNTs, MWCNTs) + CuO + water) when \({\text{Sq}} = \phi_{1} = M = \phi_{2} = {\text{Ec}} = 0.1.\) As expected, better performance is shown by hybrid nanofluid than nanofluid and basefluid, respectively.

Fig. 3
figure 3figure 3figure 3

a \(\theta (\eta )\) vs. \(Sq\) (SWCNTs + CuO + water), b \(\theta (\eta )\) vs. \(Sq\) (MWCNTs + CuO + water), c \(\theta (\eta )\) vs. Sq (comparison), d \(\theta (\eta )\) vs. \(\phi_{1}\) (SWCNTs + CuO + water), e \(\theta (\eta )\) vs. \(\phi_{1}\) (MWCNTs + CuO + water), f \(\theta (\eta )\) vs. \(\phi_{1}\) (comparison), g \(\theta (\eta )\) vs. M (SWCNTs + CuO + water), h \(\theta (\eta )\) vs. M (MWCNTs + CuO + water), i \(\theta (\eta )\) vs. M (comparison), j \(\theta (\eta )\) vs. \(\phi_{2}\) (SWCNTs + CuO + water), k \(\theta (\eta )\) vs. \(\phi_{2}\) (MWCNTs + CuO + water), l \(\theta (\eta )\) vs. \(\phi_{2}\) (comparison), m \(\theta (\eta )\) vs. Ec (SWCNTs + CuO + water), n \(\theta (\eta )\) vs. Ec (MWCNTs + CuO + water), o \(\theta (\eta )\) vs. Ec (comparison)

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Analysis for rate of entropy production (Ns) and Bejan number (Be)

Entropy production (Ns) against Sq during comparative study of nanofluid (using SWCNTs + water, using MWCNTs + water) and hybrid nanofluid (SWCNTs + CuO + water, MWCNTs + CuO + water) is depicted in Fig. 4a and b. Ns intensifies with higher Sq, and prominent impact is observed for nanofluid. Further entropy production (Ns) is higher at the walls. Ns via higher estimations of M in comparative analysis of nanofluid (using SWCNTs + water, using MWCNTs + water) and hybrid nanofluid (SWCNTs + CuO + water, MWCNTs + CuO + water) is labeled in Fig. 4d and e. Intensification in Ns is observed, and nanofluid shows prominent behavior. Also entropy production (Ns) is larger near the walls. Figure 4g and h is portrayed for Ns against Ec during flow of nanofluid (using SWCNTs + water, using MWCNTs + water) and hybrid nanofluid (using SWCNTs + CuO + water, using MWCNTs + CuO + water). Clearly, Ns intensifies with larger Ec and impact of nanofluid is dominant. Entropy production Ns during comparative study of basefluid (water), nanofluid (using CNTs (SWCNTs, MWCNTs) + water) and hybrid nanofluid (using CNTs (SWCNTs, MWCNTs) + CuO + water) when \({\text{Sq}} = M = {\text{Ec}} = 0.1\) is studied in Fig. 4c, f, i, respectively. Impact of nanofluid is dominant which is followed by hybrid nanofluid and basefluid. In order to study comparison between entropy production through heat transfer (\(N_{\text{H}}\)) and through fluid friction (\(N_{\text{F}}\)), Bejan number (Be) against \(\eta\) is plotted. Be lies in between 0 and 1. \(N_{\text{H}}\) dominates over \(N_{\text{F}}\) when \({\text{Be}} \in (0.5,\,1]\) while \(N_{\text{F}}\) dominates over \(N_{\text{H}}\) when \({\text{Be}} \in [0.0.5).\) Be versus higher estimations of Sq during flow of nanofluid (SWCNTs + water, MWCNTs + water) and hybrid nanofluid (SWCNTs + CuO + water, MWCNTs + CuO + water) is presented in Fig. 4j and k. Be intensifies with increment in Sq, and impact of hybrid nanofluid is more than nanofluid. Further, it is clear that \(N_{\text{F}}\) is dominant over \(N_{\text{H}} .\) Figure 4m and n is labeled for Be against M during comparative study between nanofluid (SWCNTs + water, MWCNTs + water) and hybrid nanofluid (SWCNTs + CuO + water, MWCNTs + CuO + water). Decay in Be is noticed, and nanofluid shows effective trend when compared with hybrid nanofluid. \(N_{\text{F}}\) also dominates over \(N_{\text{H}} .\) Be via higher estimations of Ec during flow of nanofluid (SWCNTs (water), MWCNTs (water)) and hybrid nanofluid (SWCNTs + CuO + water, MWCNTs + CuO + water) is portrayed in Fig. 4p and q. Increment in Be is detected while hybrid nanofluid shows overriding trend. \(N_{\text{F}}\) dominates over \(N_{\text{H}} .\) Be during comparative analysis of basefluid (water), nanofluid nanofluid (using CNTs (SWCNTs, MWCNTs) + water) and hybrid nanofluid (using CNTs (SWCNTs, MWCNTs) + CuO + water) when \({\text{Sq}} = M = {\text{Ec}} = 0.1\) is visualized in Fig. 4c, f, i, l, o, r. As expected, better trend is seen for hybrid nanofluid followed by nanofluid and basefluid, respectively.

Fig. 4
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a Ns vs. Sq (SWCNTs + CuO + water, b Ns vs. Sq (MWCNTs + CuO + water), c Ns vs. Sq (comparison), d Ns vs. Ns (SWCNTs + CuO + water), e Ns vs. M (MWCNTs + CuO + water), f Ns vs. M (comparison), g Ns vs. Ec (SWCNTs + CuO + water), h Ns vs. Ec (MWCNTs + CuO + water), i Ns vs. Ec (comparison), j Be vs. Sq (SWCNTs + CuO + water), k Be vs. Sq (MWCNTs + CuO + water), l Be vs. Sq (comparison), m Be vs. M (SWCNTs + CuO + water), n Be vs. M (MWCNTs + CuO + water), o Be vs. M (comparison), p Be vs. Ec (SWCNTs + CuO + water), q Be vs. Ec (MWCNTs + CuO + water), r Be vs. Ec (comparison)

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Analysis of Nusselt number (\({\text{Nu}}_{\text{x}} (Re_{\text{x}} )^{{ - \tfrac{1}{2}}}\))

Nusselt number (\({\text{Nu}}_{\text{x}} (Re_{\text{x}} )^{{ - \tfrac{1}{2}}}\)) against higher estimations of \(\phi_{1} ,\phi_{2} ,\) M and Sq during flow basefluid (water), nanofluid (using CNTs (SWCNTs, MWCNTs) + water) and hybrid nanofluid (using CNTs (SWCNTs, MWCNTs) + CuO + water) is evaluated in Table 2. It is founded that \({\text{Nu}}_{\text{x}} (Re_{\text{x}} )^{{ - \tfrac{1}{2}}}\) intensify with higher estimations of mentioned physical parameters. It is analyzed that effect of hybrid nanofluid is efficient which is followed by nanofluid and basefluid, respectively.

Table 2 Evaluation of Nusselt number (\(\text{Nu}_{\text{x}} (Re_{\text{x}} )^{{ - \tfrac{1}{2}}}\)) for various estimations of \(\phi_{1} ,\phi_{2} ,M\) and Sq when \({\text{Ec}} = {\text{Ec}}_{\text{x}} = 0.1\)
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Comparison of current analysis with previously published work on nanofluid using Buongiorno model

In this section, we have compared our theoretical analysis on hybrid nanofluid with previously published work on nanfluid by Farooq et al. [35]. Excellent agreements are founded for the covering parameters. Same impacts of Sq and Mon \(f^{{\prime }} (\eta )\) as well as \(\theta (\eta )\) are founded. Similarly, impacts of Sq and Mon Nusselt number in both published and current analysis have a great agreement (Fig. 5).

Fig. 5
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a Comparison of current analysis and published work [35] during impact of Sq on \(f^{{\prime }} (\eta )\). b Comparison of current analysis and published work [35] during impact of M on \(f^{{\prime }} (\eta )\). c Comparison of current analysis and published work [35] during impact of Sq on \(\theta (\eta )\). d Comparison of current analysis and published work [35] during impact of M on \(\theta (\eta )\)

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Conclusions

Velocity of fluid \((f^{{\prime }} (\eta ))\) directly varies with higher estimations of Sq \(\phi_{1} ,\) M and \(\phi_{2} .\) As anticipated, better performance is detected for hybrid nanofluid (CNTs (SWCNTs, MWCNTs) + CuO + water) followed by nanofluid (CNTs (SWCNTs, MWCNTs  + water) and basefluid (water), respectively. An increase in temperature \((\theta (\eta ))\) is seen for larger Ec while it reduces with higher estimations of Sq \(\phi_{1} ,\) M and \(\phi_{2} .\) Increment in Sq, \(\phi_{1} ,\) M and \(\phi_{2}\) leads to rise in Nusselt number (\({\text{Nu}}_{\text{x}} (Re_{\text{x}} )^{{ - \tfrac{1}{2}}}\)). Entropy production rate (Ns) intensifies with larger of Sq M and Ec. An increment in Bejan number (Be) is detected with rise of Sq and Ec while it reduces through higher M. Impacts of hybrid nanofluid (CNTs (SWCNTs, MWCNTs  + CuO + water) are more effective when compared with nanofluid (CNTs (SWCNTs, MWCNTs  + water) and basefluid (water).