Researchers are using different types of nanomaterials for the enhancement of the thermal performance of regular fluids such as water, kerosene oil, etc. However, these days, the researchers are more interested in hybrid nanomaterials. The purpose of this communication is to examine the stability analysis of Cu–Al2O3/water hybrid nanofluid over a non-linear shrinking sheet. The hybrid nanomaterials are composed of Cu and Al2O3. These hybridized nanomaterials are then dissolved in water taken as base fluid to form Cu–Al2O3//water hybrid nanofluid. Mathematical analysis and modeling have been attended in the presence of viscous dissipation and suction/injection effects. The governing equations of mathematical models are transformed into self-similar solutions in the form of ODEs by using similarity transformation. Solutions of the non-linear ODEs are created by employing of three-stage Lobatto IIIa formula which is built-in BVP4C function in the MATLAB software. A comparison of the current study has been done with the preceding published literature. The distributions of velocity, temperature profiles, coefficient of skin friction and heat transfer rate are presented graphically and conferred for numerous significant parameters entering into the problem. Results revealed the existence of dual solutions for a certain range of the suction/blowing parameter. Stability analysis is also done in order to obtain dual solutions stability. The smallest eigenvalues suggest that the first solution is stable from the second solution. Hybrid nanomaterials have a high scope toward nurturing our day-to-day life.

velocity components

skin friction coefficient

copper

alumina

local Nusselt number

thermal conductivity of the nanofluid

temperature

a constant

variable temperature at the sheet

ambient temperature

positive constant

suction and blowing parameter

local Nusselt number

Eckert number

smallest eigenvalue

Stability transformed variable

thermal conductivity of the hybrid nanofluid

thermal diffusivity

Prandtl number

suction/injection velocity

velocity of shrinking surface

heat capacitance of the hybrid nanofluid

effective density of the hybrid nanofluid

effective dynamic viscosity of the hybrid nanofluid

transformed variable

nanoparticle volume fraction

In the present century, enhancement of heat transfer is a very interesting subject to discuss due to its various applications in engineering and industries. Many applications of industries are required high thermal conductivity but there is no such regular fluid. Fluids like oil, ethylene glycol and water possess low thermal conductivity and cannot be used in many applications. Due to this drawback, a new kind of fluid has been introduced by Choi and Eastman [1] in 1995 which is nanofluid to improve thermal efficiency. It is composed of the dispersed nanometer-sized particles and base liquid. The examination of nanofluids is useful in heat exchangers, thermal engineering, biomedicine, cancer therapy, electronic chemical process, etc. Nanofluids have good heat transfer characteristics as compared to ordinary regular fluids and possess the new kind of production coolants. It is proved now that the thermal conductivity of nanofluid is significantly high and therefore important applications to microscale cooling are noticeable. Especially the nanofluids model of single-phase flow is significant in lubricants, coolants, and also in daily life applications such as mobile computer processors, air-conditioning, refrigeration, microelectronics, etc. In industrial science, heating development can be easily gotten from solar thermal energy which is friendly and convenient. A process that changes the radiations of solar into heat energy is known as a solar collector. For boosting up the effectiveness of these collectors, oil, ethylene glycol and water are substituted in nanofluids. In the last two decades are witnessed the great developments in nanofluids, many researchers studied nanofluid numerically and experimentally. Gherasim et al. [2] examined the nanofluid experimentally by considering the water as a base fluid and particles of alumina as nanoparticles. They found that heat transfer is increased in radial flow cooling systems. Chandrasekar et al. [3] investigated the thermal conductivity of alumina water-based nanofluid by using of Maxwell and Einstein models. They kept the diameters of nanoparticles 43nm and found that viscosity and thermal conductivity of nanofluid are directly proportional to each other. Nanofluids containing graphene oxide nanosheets were studied by Yu et al. [4] in which it was seen that thermal conductivity increased in the range 5.0–30.2vol% when base fluid was distilled water. Dalkılıç et al. [5] carried out the study on water-based hybrid nanofluid in which they considered graphite and alumina as the solid particles. TiO2–MWCNTs–water hybrid nanofluid was experimentally examined by Akhgar and Toghraie [6]. They used two models and found that thermal conductivity is maximized in 38.7%. Moreover, Esfahani and Toghraie [7] investigated silica/water–ethylene glycol and found that maximum thermal conductivity is produced when volume fraction is 5%. Some more related experimental and numerical works on nanofluids and hybrid nanofluids and their applications can be seen in these articles [8–19].

As it is mentioned that many researchers considered the numerical approach to understanding the behavior of nanofluids. There are two mathematical models in which the numerical approach can be used easily, the first one is Tiwari and Das's model [20] and the second one is Buongiorno's model [21]. Hayat et al. [22] used Tiwari and Das's model to investigate the stagnation point flow of nanofluid in presence of thermal radiation effect. They considered water as a base fluid and carbon nanotubes as a solid material and noticed that temperature and melting parameter are inversely proportional to each other. Alrashed et al. [23] also considered same model and examined nanofluid by numerical approach. Further, temperature of surface reduced for higher values of Reynolds number. Other related articles of Tiwari and Das's model [20] have been used by many researchers in their studies (refer [24–35]). Farooq et al. [36] considered Buongiorno's model in their studies for non-Newtonian nanofluid with effect of thermal radiation and magnetic field. It is stated that thickness of concentration profiles decreased as Brownian motion of materials are enhanced. The same model is considered for cross nanofluid by Khan et al. [37] in which they also investigated the effect of activation energy on the various profiles. Further, Hayat et al. [38–44], Khan et al. [45–47], Rasool and Zhang [48], Khan et al. [49], Lund et al. [50], Ullah et al. [51], Rasool et al. [52], Lund et al. [53], Sheremet et al. [54] used Buongiorno's model in their studies and investigated the effect of Brownian motion and thermophoresis effect on the temperature of various nanofluids. A few more interesting works on Buongiorno's model can be seen in these articles [55–57]. It is worth to highlight that most of the above studies have been done for single solutions so far. The present study aims to determine the multiple solutions by considering Tiwari and Das's model. In recent years, it can be seen in the published literature of heat transfer enhancement, researchers are interested to find the better kind of fluids instead of single-phase and two-phase nanofluids. Regarding this, in order to get high thermal conductivity a new kind of nanofluid has been introduced namely ‘hybrid nanofluid’. Hybrid nanofluid can be defined as it is the kind of nanofluid in which two different kinds of nanoparticles dissolve in the single base fluid. It has been observed that it produces excellent thermal conductivity as compared to base fluid and nanofluid containing single nanoparticles. These kinds of fluids can be used in many fields of heat transfer such as generator cooling, electronic cooling, nuclear system cooling, the coolant in machining, drug reduction, etc. Due to the interesting behavior of hybrid nanofluids in rising characteristics of heat conductivity attracts the many researchers to consider hybrid nanofluids in their studies to deal with real problems of the word. Sheikholeslami et al. [58] considered the hybrid nanofluid in the circular cavity. Arasteh et al. [59] investigated the hybrid nanofluid and found that increments in metal foams rise the temperature gradient. Devi and Devi [60] considered the hybrid nanofluid over the stretching surface in the presence of the magnetic effect. After one year, Devi and Devi [61] examined another hybrid nanofluid by a mathematical approach. They compared their results with experimental results and found in the excellent agreement. In their study, water is base fluid and alumina and copper are solid nanoparticles and they found that heat transfer of hybrid nanofluid is higher than the nanofluid containing single nanoparticles. Hybrid nanofluid for multiple solutions was studied by Waini et al. [62] in which they successfully found the dual solutions. Further, they also performed stability analysis and determined that only the first solution is the stable one. In the same year, Waini et al. [63] also considered another kind of hybrid nanofluid over the vertical thin needle and obtained the dual solutions.

The motivation behind this investigation is the above studies in which numerous researchers considered different base fluid along with different nanoparticles and found interesting behavior of heat characteristics. Very few researchers attempted to find the multiple solutions of hybrid nanofluids. In this study, we examine the problem of steady flow and heat transfer of hybrid nanofluid over the non-linear shrinking surface with the effect of viscous dissipation and wall mass suction. For that purpose, Tiwari and Das [20] have been employed in the equations of nanofluid by considering water as a base fluid and two different nanoparticles namely alumina and copper. The main aims of this paper are to advance the steady flow of hybrid nanofluid toward the non-linear shrinking surface into five directions. Firstly, to obtain multiple solutions by employing of numerical method in MATLAB software. Secondly, to predict the effect of hybrid nanofluid in both solutions. Thirdly, to examine the effect of viscous dissipations. Fourth, to analyze the nanoparticles (alumina and copper) effects for pure water base fluid. Fifth, to perform stability analysis for indicating the stable solution.

2Problem formulationLet us consider the steady incompressible hybrid nanofluid flow on a shrinking surface with the effect of viscous dissipation. Water is considered as a base fluid copper and alumina nanoparticles. Fig. 1 shows the physical model of the problem. The velocity of shrinking surface is uw(x)=−cxm where c is positive constant. On the other hand, mass flux velocity is also considered as vw(x)=−bcϑx(m−1)/2. Temperature inside the boundary layer is Tw=T∞+T0x2m and outside the boundary layer is T∞. Hybrid nanofluid is considered in which the size of nanoparticles is uniform. By considering the above assumptions, boundary layer equations can be written in the model of Tiwari and Das [20] as follows

The subjected boundary conditions are

where respective velocities of x-axis and y-axis are u and v and T stands for temperature. Further, khnf, μhnf, ρhnf and (ρcp)hnf are the thermal conductivity, the dynamic viscosity, the density, and heat capacity of the hybrid nanofluid respectively. Further, subscript hnf indicates the hybrid nanofluid thermophilic properties. In this study, we follow the special form of thermophysical properties of Waini et al. [63] in order to examine the flow model. Table 1 is given for the thermophysical properties of hybrid nanofluid which will be used in this study. Moreover, Table 2 is constructed for the thermophysical properties of nanoparticles and base fluid.Thermophysical properties of hybrid nanofluid [63,77].

Properties | Hybrid nanofluid |
---|---|

Dynamic viscosity | μhnf=μf(1−ϕCu)2.5(1−ϕAl2O3)2.5 |

Density | ρhnf=(1+ϕAl2O3)[(1−ϕCu)ρf+ϕCuρCu]+ϕAl2O3ρAl2O3 |

Thermal conductivity | khnf=kAl2O3+2knf−2ϕAl2O3(knf−kAl2O3)kAl2O3+2knf+ϕAl2O3(knf−kAl2O3)×(knf)where knf=kCu+2kf−2ϕCu(kf−kCu)kCu+2kf+ϕCu(kf−kCu)×(kf) |

Heat capacity | (ρcp)hnf=(1+ϕAl2O3)[(1−ϕCu)(ρcp)f+ϕCu(ρcp)Cu]+ϕAl2O3(ρcp)Al2O3 |

The thermo physical properties of the base fluid (water) and the nanoparticles [63,78].

Fluids | ρ (kg/m3) | cp (J/kgK) | k (W/mK) |
---|---|---|---|

Alumina (Al2O3) | 3970 | 765 | 40 |

Copper (Cu) | 8933 | 385 | 400 |

Water (H2O) | 997.1 | 4179 | 0.613 |

Now, following similarity transformations are used to convert PDEs into ODEs.

Substituting Eq. (5) in Eqs. (1)–(3), Eq. (1) is automatically satisfied and Eqs. (2) and (3) take the following dimensionless form

Subject to boundary conditions

Here, prime denotes differentiation due to η; the Prandtl number Pr=ϑf/αf and Eckert number Ec=c2ρf/T0(ρcp)f, b is the suction parameter (b<0) and injection parameter (b>0).

The physical quantities of interest are the coefficient of skin friction Cf and Nusselt number Nux which can be defined as

By applying Eq. (9) in Eq. (10), we have

where Re is local Reynold number.3Stability analysisThe study of stability analysis was carried out by Merkin [64] and kept continue by many other researchers in their studies [65–69]. According to them, the ordinary (similarity) equations possess multiple solutions where first (second) or upper (lower) solutions are always stable (unstable). In order to test these features, the governing Eqs. (2) and (3) are changed to unsteady form by introducing new time-dependent similarity variable τ=cxm−1t, we get

The new similarity variables are now

By substituting Eq. (14) into Eqs. (12) and (13), we get

Subject to boundary conditions

To obtain the steady flow solution's stability f(η)=f0(η), and θ(η)=θ0(η) satisfying Eqs. (6) and (7), we have these functions (refer [70–74])

where F(η, τ) and G(η, τ) are the small relative with f0(η), θ0(η) and γ is an unknown eigenvalue parameter. It should be noted that there are an infinite set of eigenvalues γ1<γ2<γ3… By putting Eq. (18) into Eqs. (15) and (16) subject to the boundary conditions (17), we get the following linear eigenvalue problemWe follow the Weidman et al. [75] and keep τ=0, and F=F0 and G=G0 in Eqs. (19) and (20) in order to determine the initial growth or decay of solutions. Henceforth, we get following linearized eigenvalues problems

Subject to corresponding boundary conditions

In order to get values of the smallest eigenvalue, we solved Eqs. (21) and (22) subject to boundary conditions (23). The signs of the smallest eigenvalue predict that either flow is stable or not. If the sign of the smallest eigenvalue is positive (negative) then flow is said to be stable (unstable).

4Results and discussionThe system of transformed ODEs (6) and (7) along with corresponding boundary conditions (8) has been solved numerically by employing a three-stage Lobatto IIIa formula with the help of the MATLAB software for various values of the nanoparticle volume fractions ϕAl2O3, ϕCu, suction/injection parameter S and Eckert number Ec. It is worth to highlight that when ϕAl2O3=ϕCu=0 then our Eq. (6) is reduced to a simple viscous fluid case. The exact solutions of the reduced equation are already given in the paper of Yasin et al. [76] (see Eq. (22)). Further, the results of the present study are compared to the results of Yasin et al. [76] in Table 3 and found in excellent agreement. Table 4 is also constructed in order to validate the results of the coefficient of skin friction for Al2O3−Cu/H2O hybrid nanofluid for different values of ϕCu when ϕAl2O3=0.1, S=0, Pr=6.135, for stretching surface (f′(0)=1) with Waini et al. [62] and Devi and Devi [61]. For other properties of the fluid and solid, we follow the Devi and Devi [61] in which ρf=997, cpf=4180, kf=0.607. The present results show a favorable agreement with those gotten by Waini et al. [62] and Devi and Devi [61]. The results of the present study for the coefficient of skin friction and heat transfer rate are given in Table 5.

The compression of results (f″(0)) of present numerical method with Yasin et al. [76].

m | b | Exact solutions | Present results | ||
---|---|---|---|---|---|

1st solution | 2nd solution | 1st solution | 2nd solution | ||

1 | −2 | 1 | – | 1.0001 | – |

−3 | 2.61803 | 0.38197 | 2.61803 | 0.38195 | |

−4 | 3.37205 | 0.26795 | 3.73205 | 0.26795 | |

1.5 | −3 | – | – | 2.18308 | 0.11939 |

2.5 | −3 | – | – | 1.47109 | 0.49794 |

The results of f″(0) and −θ′(0) over the shrinking surface where Pr=6.2 and b=−3.

m | Ec | ϕCu | ϕAl2O3 | f″(0) | −θ′(0) |
---|---|---|---|---|---|

1 | 0.1 | 0 | 0.05 | 2.65107872 | 14.1234276 |

1 | 0.1 | 0.01 | 0.05 | 2.77281027 | 13.6122442 |

1 | 0.1 | 0.05 | 0.05 | 3.17252159 | 11.7696479 |

1 | 0.1 | 0.1 | 0.05 | 3.49494958 | 9.84447959 |

1 | 0.1 | 0.1 | 0 | 3.77446217 | 11.6647986 |

1 | 0.1 | 0.1 | 0.1 | 3.19641917 | 8.33053511 |

1 | 0 | 0.1 | 0.1 | 3.19641917 | 9.37865329 |

1.5 | 0 | 0.1 | 0.1 | 2.71203378 | 8.03940995 |

1.5 | 0.1 | 0.1 | 0.1 | 2.71203378 | 7.09939876 |

2.5 | 0.1 | 0.05 | 0.1 | 1.80512347 | 6.44830162 |

2.5 | 0.1 | 0.05 | 0.05 | 1.96672019 | 7.85309519 |

Variation of f″(0) with suction/injection parameter b and volume fraction parameter ϕAl2O3 for base fluid water was drawn in Fig. 2. The skin friction coefficient advances for small values of ϕAl2O3 and increasing behavior is noticed for lower values of suction parameter b in the first solution. Physically, suction produces more resistance in the fluid flow and therefore the coefficient of skin friction decreases in the stable solution. On the other hand, the opposite behavior is noticed for the second solution. It is worth noticing that there is a range of dual solutions and no solution which depends upon the critical values where dual solutions exist. When ϕAl2O3=0, 0.01, and 0.1, the respective dual solutions ranges are b≤−1.9561, −1.9686, and −2.1066. Fig. 3 demonstrates the variation of f″(0) with b for various values of ϕCu. It can be noticed that the critical values b (b≤−2.2995) for the Al2O3 water-based nanofluid is a bit lesser than the critical values b (b≤−2.267 for ϕCu=0.01; b≤−2.1713 for ϕCu=0.05) for the hybrid nanofluid. On the other hand, the f″(0) increases for higher values of ϕCu and the decreasing trend can be seen for lower values of the suction parameter b in the first solution. On the other hand, the opposite behavior is noticed for the second solutions.

Variations of −θ′(0) with b for increasing values of ϕAl2O3 and ϕCu are plotted in Figs. 4 and 5 respectively. From these plots, it can be observed that the rate of heat transfer reduces in both solutions. This reduction helps the fluid to transfer coolant easily inside the boundary layer and therefore, the temperature of fluid decreases. The profile of velocity f′(η) for several values of m is revealed in Fig. 6. This figure displays that there are two distinct profiles of the velocity of fluid for the various values of non-negative constant. It is observed that when m=0, there exists only a single solution. It is noticed that the enhancing of m leads to the rise of the velocity of hybrid nanofluid in the first solution, while opposite trend is noticed in the second solution. On the other hand, velocity distributions of ϕAl2O3=ϕCu=0.01 are higher than the ϕAl2O3=ϕCu=0.1. The profile of velocity for Cu–Al2O3/water hybrid nanofluid is sketched in Fig. 7. It is noticed that the fluid velocity boundary layer thickness is higher for the non-linear surface in a stable solution as expected. The velocity profile demonstrates a declining nature for higher values of ϕCu in the first solution. However, the increasing and decreasing behavior of the velocity profile is found in the second solution, the same behavior of velocity profile is examined in Fig. 8 when we checked the effect of ϕAl2O3. For the first solution, the thickness of the velocity boundary layer improves for the influence of nanoparticle volume fraction ϕAl2O3. Practically, this is because a higher nanoparticle volume fraction ϕAl2O3 parameter produces less skin friction and as a resulting thickness of the velocity boundary layer and velocity of hybrid nanofluid flow increase.

Fig. 9 exhibits the variation of nanoparticle volume fraction ϕCu on the distribution of temperature for linear and non-linear cases. The higher value of ϕCu reduces the convective heat transfer and as a result, heat is transferred from heated fluid to hot surface due to which temperature increases in both solutions. The same behavior is noticed in both solutions when the effect of nanoparticle volume fraction ϕAl2O3 is checked in temperature profile in Fig. 10. Fig. 11 is sketched for temperature distribution for the influence of the power index m. The thickness of the thermal boundary layer enhances for a stable solution when m is increased, while the dual behavior of temperature is found in the second solution. It is worth mentioning that thickness of thermal boundary layer is lesser when ϕAl2O3=ϕCu=0.01 which means that a large amount of heat transfer is possible as compared to ϕAl2O3=ϕCu=0.1. The influence of a Eckert number Ec on temperature profile θ(η) is revealed in Fig. 12. The thermal boundary layer thickness and temperature rise in both solutions. Practically, this rise of temperature because of the lower effect of enthalpy difference because Eckert number is the ratio of the kinetic energy flow and an enthalpy difference of boundary layer. It advances the converting of kinetic energy into internal energy by differing fluid stresses. It is worth to mention that cooling of the surface or in other words loss of temperature from the surface to the fluid flow is possible for 0<Ec≪1.

For several values of b, the values of the smallest eigenvalue γ have been taken and demonstrated graphically in Fig. 13. This figure shows that positive values of γ denote to a disturbance initial decay and the flow becomes stable. On the other hand, the negative value of γ represents a disturbance initial growth and the flow is known as in a stable flow. It can be noticed that the values of the smallest eigenvalue γ approach to zero from both solutions as the values of b are tending to bc.

5ConclusionThe steady flow of Cu–Al2O3/water hybrid nanofluid on a non-linear shrinking sheet of variable thickness is examined with the effect of viscous dissipation. With the help of Tiwari and Das's model, the system of governing equations of hybrid nanofluid is taken. The system of PDEs is changed to a system of non-linear ODEs by employing similarity transformations. The resultant boundary value problems of ODEs have been solved by using BVP4C in MATLAB. Derivations of equations of stability analysis also have been done. The key points of this study can be presented as follows.

- 1.
The flow in the first solution accelerates for increasing values of volume fraction parameter ϕAl2O3. Further, velocity of fluid increases more when m>1 as compared to m≤1. Velocity of fluid decelerates for the stable solution for higher values of volume fraction parameter ϕCu.

- 2.
Increasing volume fraction (ϕCu and ϕAl2O3) decreases the heat transfer rate in both solutions.

- 3.
At high Prandtl number, the thickness of thermal boundary layer decreases in both solutions of hybrid nanofluid as expected.

- 4.
Profile of temperature profile θ(η) increases in both solutions for advance values volume fraction parameters ϕCu, ϕAl2O3 and Eckert number Ec.

- 5.
Dual solutions exist in the ranges of suction/injection parameter.

- 6.
Only first solution can be visualized and is said be stable solution.

The authors declare no conflicts of interest.

The authors would like to extend their sincere appreciation to the Deanship of Scientific Research at King Saud University for its funding of this research through Research Group Project No. RG-1439-029.

*et al*.

*et al*.