Stability analysis of C u - C 6 H 9 N a O 7 and Ag - C 6 H 9 Na O 7 nanofluids with effect of viscous dissipation over stretching and shrinking surfaces using a single phase model.

A mathematical analysis is performed to study the flow and heat transfer phenomena of Casson based nanofluid with effects of the porosity parameter and viscous dissipation over the exponentially permeable stretching and shrinking surface. The considered nanofluid comprises Casson as a base fluid that contains silver ( A g ) and copper ( C u ) solid nanoparticles. The system of the nonlinear governing partial differential equations (PDEs) are converted into ordinary differential equations (ODEs) by applying similarity transformation. The obtained ODEs are solved by using shooting technique in Maple software. Numerically obtained results reveal dual solutions for various values of pertinent parameters. Due to occurrence of dual solutions, the stability analysis is done in order to find stable solution. Positive signs of smallest eigenvalues point out that the first solution is stable and second unstable. The variation of the velocity and the temperature profiles with coefficient of the skin friction and the Nusselt number are shown graphically. Both temperature profiles and its boundary layer thicknesses increase as volume fraction of nanoparticles of A g and C u are increased in the Casson fluid. Velocity profiles and corresponding boundary layer thicknesses decrease by suspension of nanoparticles of silver and copper, whereas the silver A g nanoparticles show the greater rate of heat transfer enhancement as compared to copper C u nanoparticles when suspended in Casson fluid.

Introduction

It is worth to note that we are living in the period of the science and technology. While at present, the most modern rapid growing technology is a nanotechnology that possess huge applications in different fields such as in transportation, atomic reactors, electronics and biomedicine. After discovery of carbon nanotubes by Morinobu Endo [1] and the potential of their importance in the nano-science discussed by Iijima [2], the nano-technology have achieved a great importance to the various fields of research. The nanofluid is one of the interesting topics of nano-science. It may be defined as, the suspension of the nano-meter size particles with 1nm–100nm in the common fluid is named as nanofluid [3]. The nanoparticles that are mostly used in preparation of the nanofluids are ceramics (, ), pure metals (, , , etc), nitrides (, ), metal carbides (), nano-tubes of carbon, non-metals of Graphite, coated of metals (, ) and some functionalized nanoparticles Masuda, Ebata & Teramae [4]. Whereas the base fluids includes water, oils, bio-fluids, lubricants, ethylene, acetone, decent, polymer solutions and other coolants.

Choi and Eastman [5] were the first to introduce such a new kind of the modern fluid. It is observed the material that is mostly used to prepare the nanofluids are ceramic particles. Because of these particles are more stable in solution as compare to others [6]. The ceramics can be divided into three different classes, such as oxides (i.e. zirconia and alumina), non-oxides (i.e, silicide's, nitrides and carbides) and mixtures of non-oxides and oxides. Each one of them possess unique material property. The nanofluids prepared by the combination of pure metal and alloy with common base fluid possess high thermal conductivity as compare to prepared by oxides, but these fluids are seen in less quantity in literature. The alloy nanofluids are prepared by the inert gas condensation process or by the mechanical alloying. The carbon based nanofluids have large intrinsic thermal conductivity. The examples of nanofluids prepared from carbon are carbon nanotubes that are single wall nanotubes, multi wall nanotubes, and ultra-dispersed diamond in the different fluids. Thermal conductivity performance in nanofluids have been reported up to 150% by Choi et al. [7] and critical heat flux enhancement up to 200% by You et al. [8] when compared to the corresponding properties and critical heat flux values of common base fluids.

The study about the flow over stretching and shrinking surfaces have taken much importance due to their huge applications in engineering and industries like capillary effect in the small pores, hydraulic properties of the agricultural clay soil, rising of the shrinking balloons, the shrinking-swell behavior and shrinking films. Miklavčič and Wang [9] studied different types of the flows on shrinking surfaces first time. Flow is more resisted in shrinking surfaces as compare to the stretching surfaces. Since the flow is probably not going to occur in the shrinking surface easily, it presents sufficient suction on the boundary to conflict vorticity in the boundary layer. Recently, many researchers considered the stretching/shrinking surfaces such as Lund et al. [10], Salleh et al. [11], Ahmad et al. [12], Dero et al. [13], Ismail et al. [14], Selimefendigil et al. [15]. Hayat et al. [16] investigated the Tiwari and Das's model on the exponentially stretching surfaces and most of them found the single solutions. There are many types of the non-Newtonian fluid, the Casson fluid is one of important type of non-Newtonian fluid that acts like solid if shear stress is lesser than yield stress. It shows deformation when yield stress becomes lesser than shear stress. Recent study about Casson fluid flow can be found in [17, 18]. The well-known examples of the Casson fluids are blood and chocolate.

In this study, there have been extended the work of Hatami et al. [19] with multiple solutions for and nanofluids over a stretching/shrinking surfaces. To best of author's knowledge, the stability analysis in occurrence of the multiple solutions in and nanofluids flow on the exponential permeable stretching/shrinking surface is not examined by other researchers in past years. The aim of present research is to find multiple solutions along stability analysis and to find the effect of silver and copper nanoparticles in Casson fluid flow. The fluid is vital for enhancement of the heat transfer and numerous techniques are introduced to improve it. Rohni et al [20] claimed that execution of conventional fluid in the upgrade of heat transfer is extremely poor. Considering the conclusion of all previous studies, this study considered non-Newtonian model. The motivation behind this work is that the Tiwari and Das's model has many applications for Casson based nanofluids and to consider all multiple solutions. There is hoped that this study will provide a fruitful help to the researchers which are interested to work in field of the nanotechnology theoretically and practically.

Problem formulation

Consider steady two-dimensional incompressible flow and heat transfer of Casson base nanofluid over the exponentially stretching and shrinking surface with viscous dissipations in the porous medium as shown in Figure 1. It is supposed the rheological equation of the state to the isotropic and the Casson fluid flow is written as by Nakamura and Sawada [21]:where plastic dynamical viscosity relative to non-Newtonian fluid is indicated by , denotes fluid yield stress, denotes product of the rate of the deformation of components with itself, where π = is ()-th component of the rate of deformation and the critical value of is denoted by . The surface is exponentially stretched in axis and the vertical to the y axis. Further, the system of the governing equations in the present analysis is written ashere, velocity components are denoted by u and v along the x and y axes respectively, is the effective density, is effective dynamic viscosity, is heat capacitance, is thermal diffusivity and is thermal conductivity of nanofluid. Furthermore, is porous medium where is constant of porous medium. Here, nf indicates the nanofluid thermophilic properties, subscript f indicates the base fluid whereas s nanosized solid particles. The subjected boundary conditions are

Figure 1

Physical model of flow.

The following similarity transformations are used to obtain similarity solutions.The velocity components and are defined as

By applying above transformations, Eqs. (3) and (4) along the boundary conditions (5) take the following forms.subject to boundary condition

Moreover, is the porosity parameter, is Prandtl number, is Eckert number and is suction/injection parameter.

The skin-friction coefficient and the local Nusselt number is defined asUsing Eq. (6) in above relations, we getwhere is a local Reynolds number.

Stability analysis

According to requirement of stability analysis, there is need to convert Eqs. (3) and (4) in unsteady case in order to perform stability analysis of solutions. Thus, we havewhere t is the time. Now, a new dimensionless variable is introduced, then equation [6] can be written as followsBy applying Eq. (13) in Eqs. (11) and (12), we getThe boundary conditions become

To check the stability of solutions of and , that satisfy boundary value problems (7)(8) and (9) we haveWhere both the and are the small relative with the respectively, is the unknown eigenvalue that provides an infinite set of the eigenvalues The stability of solution depends upon the sign of the smallest eigenvalue . If the value of is positive that means that flow is the stable and there takes place the initial decay, conversely, the negative value of shows the unstable flow which indicates initial growth of disturbance. By substituting Eq. (17) in Eqs. (14), (15), and (16), we getsubject to boundary conditions

To achieve stability analysis of the solution, Eqs. (18) and (19) of linear eigenvalue problem along the boundary conditions (20) have been solve by applying BVP4c solver function in the MATLAB. Further, we kept the values of following parameters as and vary the values of and .

Result and discussion

In present paper, two dimensional Casson base nanofluids prepared by two different kinds of the solid nanoparticles of Silver and Copper are studied. The effects of the solid volume fractions are examined in a range of with value of the . The thermo-physical properties of the base fluid and nanoparticles are presented in Table 1.

Table 1

Thermo-physical properties of base fluid (Casson) and nanoparticles.

Physical properties	ρ(kgm^−3)	C_p(kg^−1k^−1)	k(Wm^−1k^−1)
C_6H_9NaO_7	989	4175	0.6376
Cu	8933	385	401
Ag	10500	235	429

The numerical results of Eqs. (7) and (8) subjecting to the boundary conditions (9) are obtained by applying the shooting technique with Maple software. Dual solutions are found for the skin friction coefficient and local Nusselt number for the different applied parameters at the different initial guesses. In order to validate the obtained numerical results, the present results are compared to the results obtained by Ghosh & Mukhopadhyay [22] that are shown in Table 2. The comparison shows very good agreement in both solutions. Furthermore, to clarify the silent features concerned to the flow and the heat transfer phenomena, the obtained numerical results are shown in Figures 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, and 21, graphically. Another technique which is the three stage Lobatto IIIa formula has been developed in the bvp4c with help of the finite difference code. After that, stability analysis has been done by applying bvp4c solver function. As indicated by Rehman et al. [23] “this collocation formula and the collocation polynomial provides a continuous solution that is fourth order accurate uniformly in . Mesh selection and error control are based on the residual of the continuous solution”. The smallest eigenvalues shown in Table 3 express the first (second) solution is the stable (unstable) and the physically feasible (unrealizable) because smallest eigenvalue is positive (negative).

Table 2

Comparison of results for and for,, and

Sudipta Ghosh1 and Swati Mukhopadhyay [22]				Present study
First solution___		Second solution__		First solution_____		Second solution___
f''(0)	−θ'(0)	f''(0)	−θ'(0)	f''(0)	−θ'(0)	f''(0)	−θ'(0)
2.39082	1.77124	-0.97223	0.84832	2.39095	1.77271	-0.97362	0.85083

Figure 2

Graph of with different values of and for working fluid.

Figure 3

Graph of with different values for and for working fluid.

Figure 4

Graph of with different values of and for working fluid.

Figure 5

Graph of with different values of and for working fluid.

Figure 6

Graph of with different values of suction for different values of working fluid.

Figure 7

Graph of with different values of suction for different values of in working fluid.

Figure 8

Graph of rate of heat transfer with different values of suction for different values of in working fluid.

Figure 9

Graph of rate of heat transfer with different values of suction for different values of in working fluid.

Figure 10

Graph of heat transfer rate with the different values of and working fluid.

Figure 11

Graph of Skin frictions with the different values of in and working fluid.

Figure 12

Effect of on the velocity distributions for working fluid.

Figure 13

Effect of on the velocity distributions for working fluid.

Figure 14

The combined effect of on the velocity distribution for and working fluid.

Figure 15

Effect of on the temperature distribution for working fluid.

Figure 16

Effect of on the temperature distribution for working fluid.

Figure 17

The combined effect of on the temperature distribution for and working fluid.

Figure 18

Effect of on the temperature distribution for working fluid.

Figure 19

Effect of on the temperature distribution for working fluid.

Figure 20

Effect of on the temperature distribution for working fluid.

Figure 21

Effect of on the temperature distribution for working fluid.

Table 3

Smallest eigenvalues at different values of and for and nanoparticles when ,, and .

			Cu−(ε)		Ag−(ε)
f_w	β	γ	1st solution	2nd solution	1st solution	2nd solution
2.8	2	0.5	2.4763	-2.8293	2.8181	-1.8969
3	2.5	0.5	2.7237	-2.2835	2.9201	-1.8971
3	2	0.2	2.6011	-1.9091	2.9207	-1.8906
3	2	0.5	2.5941	-1.8975	2.8881	-1.8969

Figures 2, 3, 4, and 5 demonstrate the change in skin friction coefficient and Nusselt number for solid nanoparticles and Casson base nanofluids with variation of stretching/shrinking parameter at various values of the suction parameter keeping values of the other parameters , and fixed. Furthermore, these Figs also show the stretching and shrinking parameter is one of the main reason behind the existence of the multiple (dual) solutions in this problem. It is observed that there are two solutions for case of the , unique solution for whereas in case of the no solution exits. It should be noted that the indicates critical point of the parameter where solution exists. The critical values of the are for -Casson base fluid and -1.65721 for -Casson base fluid for respectively that are demonstrated in Figures 2, 3, 4, and 5 of the paper. Figures 2 and 3 show the increasing value of suction parameter along the variation of stretching/shrinking parameter decreases rate of skin friction coefficient for the stretching case and opposite result is examined for the shrinking case when in first solution for both and Casson base nanofluids. Whereas in the second solution, increasing value of suction parameter decreases the value of skin friction coefficient throughout the flow over both stretching and shrinking surfaces. The effect of the suction parameter along variation of at the Nusselt number or rate of heat transfer for and Casson base nanofluids is depicted in Figures 4 and 5 respectively. From these Figures it is clearly examined that the increment in the value of suction parameter increase the heat transfer rate in first solutions and reverse to it, in the second solutions heat transfer rate decrease. Same type of findings about skin friction coefficient and heat transfer rate along the suction parameter for the different values of nanoparticles volume fraction for the and Casson base nanofluid are shown in Figures 6, 7, 8, and 9. For particular parameters, these Figs represents the regions of no solutions for , and and, dual solutions for , and . Figures 6 and 7 show the increasing the quantity of the solid volumetric fraction of the silver and copper nanoparticles in the based fluid with respect to suction parameter , the skin friction rate increases in the first solution whereas in the second solution it decreases. This is because of the increasing rate of the solid nanoparticles volume fraction increases the thermal conductivity of the Casson fluid that leads to further thinning of the thickness of the velocity boundary layer. Figures 8 and 9 illustrate the rate of the heat transfer at various values of solid volume fraction of nanoparticles for and Casson base nanofluid respectively, along the variation of suction parameter . The results of the both Figs clarify that the increasing rate of volume fraction of both types of nanoparticles in Casson fluid as the rate of heat transfer decreases. Figure 10 indicates the differentiation in heat transfer rate with suspension of the and nanoparticles in Casson fluid due to variation of . From the Figure it is observed that nanoparticles show more rate of heat enhancement compared to copper nanoparticles in Casson fluid. Whereas in Figure 11 same type of the findings are observed when suction parameter is replaced by the volumetric fraction parameter. Figures 12 and 13 illustrate the effect of Casson parameter at velocity profiles of and Casson base nanofluids for two values of porosity parameter ( ) respectively. From Figure 12, it can be observed the velocity and boundary layer thickness decrease in first solution that indicates the magnitude of velocity is greater in the Casson based fluid as compare to viscous fluids. But in second solution, at start, it is increasing but after a certain point it is decreasing as in the first solution where the value of is increased for both values of parameter In case of silver nanoparticles suspended in Casson fluid, Figure 13 also shows the trend of velocity and boundary layer thickness of the -Casson base nanofluid remained same as it is already observed in -Casson nanofluid. Similar observation mentioned by Gangadhar et al. [24]. The effect of Casson parameter for two different values of the porosity parameter ( ) on the velocity profile of the and Casson base nanofluid are shown in Figure 14. From this Fig, it is observed that the flow of the -Casson nanofluid is comparatively faster than the flow of -Casson base nanofluid in first solution but in second solution at start, the flow of Casson base nanofluid is slightly faster as compared to Casson base nanofluid but after a point result show same trend as it is already observed in the first solution by decreasing the numerical value of . Figure 15 indicates the temperature profile of -Casson base nanofluid, where the increasing value of Casson parameter, the temperature of the flowing nanofluid decreases in the first solution clearly for both values of porosity parameter . In the second solution it is reverse for but for the temperature increases at start while after a point reverse result is observed. Furthermore, in Figure 16 the temperature profile of Casson base nanofluid with increasing value of Casson parameter also show the same trend of the result as it is shown in temperature profile of the suspended nanofluid. Fig. 17 indicates the combined result of the temperature profiles for and Casson base nanofluids. This show a slight difference in thermal enhancement in variation of the nanoparticles in base fluid, the -Casson nanofluid show nearly more heat transfer enhancement as compare to the base nanofluid in first solution. Whereas, in second solution at start, the heat transfer enhancement of -Casson base nanofluid is noticed more than -Casson base nanofluid but after a point result remained same as it is observed in first solution. Figures 18 and 19 show that in both solutions, the temperature and thermal boundary layer thickness increases as Eckert number increases in and Casson based nanofluids respectively, whether the porosity exists or not The main reason behind rising of thermal boundary layer thickness is rising of the viscous resistance at increasing value of the . As a results, more heat energy is accumulated within the fluid particles near to the boundary. It can be clearly examined from Figures 20 and 21 that the greater value of the Prandtl number decrease the temperature of and Casson based nanofluids in both solutions respectively, for both values of the porosity parameter . Generally, the greater Prandtl number possesses lower thermal diffusivity. Therefore, an increase in Prandtl number will decrease the thermal boundary layer thickness and the temperature of the nanofluid.

Conclusion

In present paper, the boundary layer flow and heat transfer of the Casson based nanofluid with two metallic nanoparticles copper and silver over the exponential permeable stretching/shrinking surface is examined. A single phase fluid model proposed by the Tiwari and Das [20] is used. The dual solutions are found for particular values of pertinent parameters. Stability analysis is performed to find the stable solution. The main findings are given below:

The dual solutions are found for both stretching and shrinking surface.

The increasing rate of the solid volume fraction of copper and silver nanoparticles in Casson fluid enhance rate of skin friction and decline rate of the heat transfer in both solutions.

The silver nanoparticles show more rate of heat transfer enhancement comparatively to copper nanoparticles in Casson fluid.

The Eckert number enhances the heat transfer rate whereas Prandtl number declines the rate of the heat transfer in both solutions.

Casson parameter declines the heat transfer rate in first solution and the reverse result is seen in second solution.

Declarations

Author contribution statement

Sumera Dero: Analyzed and interpreted the data; Wrote the paper.

Azizah Mohd Rohni: Conceived and designed the analysis.

Azizan Saaban: Contributed analysis tools or data; Wrote the paper.

Funding statement

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Competing interest statement

The authors declare no conflict of interest.

Additional information

No additional information is available for this paper.