ARTICLE
J. Cent. South Univ. (2019) 26: 1283-1293
DOI: https://doi.org/10.1007/s11771-019-4087-6
Multiple solutions of Cu-C6H9NaO7 and Ag-C6H9NaO7 nanofluids flow over nonlinear shrinking surface
Liaquat Ali LUND1, Zurni OMAR1, Ilyas KHAN2, Sumera DERO1
1. School of Quantitative Sciences, Universiti Utara Malaysia, 06010 Sintok, Kedah, Malaysia;
2. Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam
Central South University Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract: Model of Casson nanofluid flow over a nonlinear shrinking surface is considered. Model of Tiwari and Das is applied to nanofluid comprising of sodium alginate with copper and silver. The governing nonlinear equations incorporating the effects of the viscous dissipation are transformed into boundary value problems (BVPs) of ordinary differential equations (ODEs) by using appropriate similarity transformations. The resulting equations are converted into initial value problems (IVPs) using the shooting method which are then solved by Runge-Kutta method of fourth order. In order to determine the stability of the dual solutions obtained, stability analysis is performed and discovered that the first (second) solution is stable (unstable) and physically realizable (unrealizable). Both the thickness of the thermal boundary layer as well as temperature increase when the Casson parameter (β) is increased in the second solution.
Key words: Cu-C6H9NaO7; Ag-C6H9NaO7; shrinking surface; dual solution; stability analysis; nanofluid
Cite this article as: Liaquat Ali LUND, Zurni OMAR, Ilyas KHAN, Sumera DERO. Multiple solutions of Cu-C6H9NaO7 and Ag-C6H9NaO7 nanofluids flow over nonlinear shrinking surface [J]. Journal of Central South University, 2019, 26(5): 1283–1293. DOI: https://doi.org/10.1007/s11771-019-4087-6.
1 Introduction
There are two approaches that can be used to study and understand the behavior of fluid dynamics: experimental and mathematical analysis [1]. The experimentation part of fluid dynamics comes under disciplines like aeronautical engineering, mechanical engineering, and hydraulics engineering etc. Meanwhile, the mathematical analysis approach is contained under the field of computational fluid dynamics (CFD). CFD can be defined as “a branch of fluid dynamics that uses numerical analysis and data structures to solve and analyzes the problems that involve fluid flows rather than model experiments”.
Initially, most of the systems used in industrial purpose were only dependent on the conventional heat transfer fluids like ethylene glycol, water, and oil. However, these fluids could not produce better performance in the cooling systems. To overcome these hindrances, CHOI et al [2] presented an alternative way by introducing a modern class of the fluid by suspending some quantity of the nanosized solid particles into the common base fluids, known as nanofluids, in order to enhance the thermal conductivity (MAHIAN et al [3, 4]). LAAD et al [5] observed that when nanoparticles of TiO2 were mixed to engine oil, the friction coefficient was decreased by 86% with 0.3% concentration. ROSTAMI et al [6] studied the combined convective stagnation point flow of the aqueous silica-alumina hybrid nanofluid and found dual solutions. NAKHCHI et al [7] have shown that the nanofluids enhance the thermo-physical properties and the heat transfer behavior as compared to common base fluids. Many researchers considered nanofluid in their studies and used different models (see Refs. [8–16]). The efficiency of heat conductivity in the nanofluids amongst the models is calculated, for the reason, to compare theoretical results with experimental results. Tiwari and Das’s model is a one-component (single-phase) model which reflects viscosity model. Many researchers such as CAPONE et al [17], BACHOK et al [18], RAHMAN et al [19], AHMAD et al [20] and NIELD [21] applied this model because it can explore the influence of the volumetric fraction of solid nanoparticles in base fluid.
However, most of the previous studies only focused on Newtonian based fluids. Hence, this study attempts to implicate the Tiwari and Das’s model in the non-Newtonian fluid due to its vast limit of the applications in many industrial and engineering sectors. In addition, this study also investigates the possibility of obtaining multiple solutions. It is worth mentioning here that there are still many questions to be answered. For instance, what kinds of the nanoparticles can be mixed in what kind of the base fluid to obtain the maximum rate of heat transfer? What can be the structure, shape, size etc. of each nanoparticle [22, 23]? So, many kinds of nanoparticles are used by researchers in theoretical or experimental studies.
The non-Newtonian fluids possess great importance in the research due to having the variety of applications in the engineering, petroleum and the chemical industrial sectors [24]. Casson fluid has also a large number of applications in the metallurgy, food processing, chemical production, nuclear reactors and microelectronics etc. Casson fluid is a pseudoplastic (shear thinning) liquid possessing zero viscosity at the infinite shear rate and an infinite viscosity at the zero-shear-rate to forecast flow behaviors of the pigment oil suspension as stated by CASSON [25] in 1959 in his introductory work. The boundary layer flow of the Casson fluid, as well as Casson nanofluid on various surfaces, was studied by many researchers (see Refs. [26–29]). MAHMOOD et al [30] found that the temperature profiles are increasing the function of the Hartman number, Casson parameter, nanoparticle volumetric fraction, thermal radiation, velocity slip, and sheet convective parameters. ALI et al [31] considered the effect of the magnetic field in the blood flow of the Casson fluid in the axisymmetrical cylindrical tube. KHALID et al [32] inspected the unsteady magneto hydrodynamics natural convective flow of the blood with carbon nanotubes. Recently, RAZA et al [33] found multiple solutions of Casson fluid flow between slowly expanding and contracting walls. It is worth mentioning that the multiple solutions exist due to non-linearity in governing differential equations of fluid flow. Furthermore, it is still a challenge to find all possible multiple solutions to any problems.
The main purpose of this work is to investigate the possibility of obtaining multiple solutions of the Cu-sodium alginate and Ag-sodium alginate based nanofluid on the nonlinear shrinking permeable sheet. To best of author’s knowledge, such type of work is not reported yet in the scientific research previously.
2 Problem formulation
Steady incompressible Casson nanofluid flow on a shrinking permeable flat sheet that is coincided by the y-axis, in the presence of viscous dissipation has been considered. The base fluid is sodium alginate nanofluid, which contains copper and silver nanoparticles. The geometry and physical model of the problem is illustrated in Figure 1. We also assume that the equation of incompressible Casson fluid flow and rheological state equation for an isotropic can be expressed as follows (see NAKAMURA et al [34]):
where Py and μB are the non-Newtonian fluid yield stress and plastic dynamic viscosity, respectively; critical value of π based on non-Newtonian model is πc and π is the product of the component of deformation rate with itself, namely, π=eijeij is the (i, j)th component of the deformation rate. Under the above assumptions, boundary layer equations of motion and the energy equation have the following form:
(1)
(2)
(3)
subject to boundary conditions:
as y→0, u→0,
T→T∞ as y→∞ (4)
In Eqs. (1)–(4), u and v indicate velocity components along directions x and y respectively; u∞ indicates free stream velocity; T stands for temperature; kf stands for thermal conductivity;
is the kinematic viscosity of nanofluid. The effective dynamic viscosity of nanofluid μnf is given by the Brinkmann as 
here f indicates the volumetric fraction of solid nanoparticles. Effective density of nanofluid, ρmf, heat capacity, (ρcp)nf, and thermal diffusivity, αnf, are taken as follows (ZAIB et al [35]).
(5)
(6)
(7)
The effective nanofluid thermal conductivity knf defined by the Maxwell Garnett model is written as
(8)
where knf denotes the nanofluid thermophilic properties, and subscripts f and s represent the base fluid and nanosized solid particles, respectively. It is worth mentioning that the expression for knf is restricted to spherical nanoparticles where it does not account for other shapes of nanoparticles [18]. For similarity solutions, the following similarity transformations are used:
,
(9)
Equation (1) is automatically satisfied. On the other side, when Eq. (9) is substituted in Eqs. (2) and (3), the following dimensionless equations are obtained:
(10)
(11)
here, prime denotes differentiation with respect to η; Pr and Ec represent the Prandtl and Eckert numbers, respectively, which are defined as
(12)
Now, the boundary conditions (4) become
;
(13)
It is worth mentioning that at f=0 the obtained governing equations will indicate for the viscous fluid flow. Also,
.
The coefficient of local skin friction Cf and Nusselt number Nux are explained as
where knf and μnf are thermal conductivity and dynamic viscosity of nanofluids, respectively. Applying similarity variables (9), we get:
(14)
where 
is local Reynold number.
Figure 1 Schematic representation of fluid flow
3 Stability analysis
ROSCA et al [36] and WEIDMAN et al [37] found dual solutions of convective flow through the vertical flat plate and the boundary layer flow with forced convective over permeable flat plate respectively. They discovered that the upper branch solutions are physically realizable, while the lower branch solutions are not. Many researchers [33, 38, 39] found multiple solutions but did not perform the stability analysis. The first step to find the stability of the solutions is to change the equation of momentum and heat into unsteady form by considering new variable τ. In our case, we have τ=cxm–1t. The unsteady form of momentum and heat equations can be written as
(15)
(16)
The new similarity transfer variables in the presence of τ and η can be expressed as
(17)
Applying Eq. (17) in Eqs. (15)–(16) yields
(18)
(19)
subject to boundary conditions:
(20)
In order to indicate the solution stability of f(η)=f0(η), and θ(η)=θ0(η) which satisfy the equation of boundary value problems (10–11) with boundary condition (13), we follow the suggestions [36, 37, 40]:
,
(21)
where γ denotes the unknown eigenvalue, both F(η, τ) and G(η, τ) are the small relative with F0(η), θ0(η). Equations (18)–(19) provide an infinite set of the eigenvalues γ1<γ2<γ3…, where, in the case of the smallest negative (positive) eigenvalue, there is an initial growth (decay) of the disturbance and flow becomes unstable (stable). Equations (18)–(20) can be written as:
(22)
(23)
subject to boundary conditions:
(24)
As proposed by WEIDMAN et al [37], there will be checked the stability of steady fluid flow as well as heat transfer solutions f0(0) and θ0(0) of Eqs. (10) and (11) subject to the boundary conditions (12) taking τ=0, and F=F0 and G=G0 in Eqs. (22) and (23) to detect initial decay or the growth of solution concerned to Eq. (21).
(25)
(26)
subject to boundary conditions:
,
(27)
It would be remembered that at the certain values of the m, Pr, Ec and f the concerned steady fluid flow solutions f0(η) and θ0(η) (see Eqs. (10) and (11)), the stability related to steady fluid flow solution is obtained by the smallest eigenvalue (γ). It is stated in the study of HARRIS et al [41] that eigenvalues can be determined if and only if boundary condition of any one function of the following functions F0(η) and G0(η) should be relaxed into the initial condition by differentiating that function one more time. To find the values of γ in this study, we relaxed G0(η)→0 as η→∞ and then solved the system of Eqs. (25)–(26) subject to Eq. (27) along with new relaxed boundary condition G′0(0)=1.
4 Result and discussion
The nonlinear coupled ordinary differential Eqs. (10)–(11) along with the boundary conditions given in Eq. (13) are solved with the help of the shooting method alongside the fourth-order Runge-Kutta algorithm in Maple software with aid of Shootlib function. Shooting method is widely used by researchers [39, 42]. Furthermore, dual solutions are achieved by setting various initial guesses for values of the skin friction coefficient f″(0) as well as local Nusselt number–θ′(0), where far-field boundary conditions (13) are satisfied by all the profiles simultaneously with the different shapes because of the unbounded physical domain. Whereas computational domain is finite, there has been applied far-field boundary conditions for similarity variable η for the finite value indicated here with ηmax=6 and ηmax=12. Most of the computations here are performed at the value ηmax=12, which was appropriate to get far-field boundary conditions for the values of considered parameters. The comprehensive explanations of that method are present by JALURIA et al [43]. In the present study, sodium alginate-Cu and sodium alginate-Ag nanofluid are taken into account. The thermo-physical properties are shown in Table 1. Additionally, the obtained results are compared with earlier published results. In order to verify and validate our numerical schemes, the values of the coefficient of skin friction f″(0) in the case of the Newtonian fluid (β=∞) at the various values of the suction and injection parameter (b) have been compared with the values obtained by ROHNI et al [43] and CORTELL [44] as shown in Table 2. The results are found good agreement. Table 3 indicates different turning points of Cu-sodium alginate nanofluid and Ag-sodium alginate nanofluid. Finally, we have checked the linear stability of steady flow solutions.
According to ROSCA et al [36], stability is determined by the sign of smallest eigenvalue. The positive minimum eigenvalue indicates the stable flow. Based on this argument, we have converted the problem to eigenvalue problem and the results are displayed in Table 4. For the first solutions, the eigenvalues are always positive, while negative for second solutions. Thus, it can be deduced that the first solution is stable, while the second solution is not for these values of particular parameters m=1.5, β=1.5, Ec=0.5 and Pr=10. Figure 2 shows that the thickness of velocity boundary layer for Ag Casson nanofluid is smaller than that of Cu Casson nanofluid. As a result, Ag and Cu Casson nanofluids tend to flow nearer to convectively non-linear shrinking surface and serve as a better heat transfer. It is observed in Figure 3 that the increasing nanoparticle volumetric fraction f pushes the fluid to the shrinking surface which causes both the fluid velocity and the thickness of momentum (velocity) boundary layer to decrease in the first solution. However, opposite behavior is seen in the second solution. The impact of volumetric fraction f at velocity profiles of Ag-sodium alginate-based type of Casson nanofluid is demonstrated in Figure 4. The increase in volumetric fraction f has contributed to the decrease in velocity and the thickness of the boundary layer in the first solutions. Furthermore, it is increased in the second solution. Figure 5 depicts the variation of velocity distribution that in the first solution velocity decreases by the increase in Casson parameter (β) as expected. Because of an increase in Casson parameter β, yield stress drops; therefore, the thickness of the velocity boundary layer is decreasing. Moreover, an opposite trend has been noticed in the second solution. It is worth mentioning here that the thickness of the velocity boundary layer of Ag- sodium alginate is less than Cu, which implies that Ag-sodium alginate can produce more heat transfer. The effect of a similarity constant (m) is demonstrated in Figure 6. The increasing values of m are caused by existing dual solutions. Velocity profiles increase (decrease) as m is increased in the first (second) solution for both Cu and Ag Casson nanofluids. The hydrodynamic boundary layer thickness is significantly enhanced as m increases in Cu Casson nanofluid as compared to Ag Casson nanofluid in the first solution. On the other hand, it seems that the velocity decreases as m is increased in the second solution. Figures 7 and 8 represent the influence of the volume fraction of solid nano- particles denoted by f at temperature distribution profiles for the two Cu and Ag Casson nanofluids. The increasing quantity of volume fraction of the solid nanoparticles increase the rate of the thermal conductivity of the nanofluids. Therefore, the thermal boundary layer thickness enhances in the second solutions of all considered nanofluid particles as expected. On the other hand, the dual behavior of temperature is noticed in the first solutions of both Figures 7 and 8. Figures 9 and 10 show the effect of Eckert number on the non- dimensional temperature profiles of Cu and Ag Casson nanofluids. There can be examined from the plots that the Ec significantly influences the temperature of nanofluids. It is also examined from the figures that, by the increase in the Eckert number the temperature is increasing in both solutions of both profiles because the Eckert number increases the flow’s kinetic energy. It is noted from Figure 11 that temperature distribution profile is increasing monotonically in the first solutions and temperature profiles of Ag Casson nanofluid close to the shrinking sheet as compared to Cu. On the other hand, the opposite trend has been noticed in the second solution. The influence of Casson parameter β on temperature fields is depicted in Figure 12. It is important to note that the temperature decreases at increasing value of β in the first solution of both Cu and Ag profiles. Whereas, the thickness of the velocity boundary layer enhances the second solution. The variations of the f″(0) and –θ(0) at suction parameter are presented in Figures 13 and 14 at the different values of the volume fraction of solid nanoparticle f for Cu– sodium alginate operational fluid. In these figures, it is shown that there is a region of no solutions for b>bc1, b>bc2 and b>bc3 and, dual (first and second solution) solutions for bc1>b, bc2>b and bc3>b. When suction is increased, the skin friction enhances in the second solution, but the opposite trend is noticed in the first solution (see Figure 3). On the other hand, heat transfer rate decreases as nanoparticle volume fractions f increases in both solutions. Figures 15 and 16 represent the variation of the skin friction or the shear stress and the rate of heat transfer with f. The same behavior of skin friction and the heat transfer have been noticed as we already discussed in Figures 13 and 14.
Table 1 Thermo-physical properties of water and nanoparticles [23, 45]
Table 2 Comparison of f″(0) for stretching sheet by keeping fixed values of following parameters as f=0 and β=∞
Table 3 Value of turning points of f″(0) and –θ′(0) for different nanoparticles
Table 4 Smallest eigen values γ for several values of suction parameter b at m=1.5, Pr=10, Ec=0.5 and f=0.1
Figure 2 Velocity plot for increasing Cu and Ag nanoparticles concentration
Figure 3 Velocity plot for increasing Cu nanoparticle concentration
Figure 4 Velocity plot for increasing Ag nanoparticle concentration
Figure 5 Velocity plot for increasing values of Casson parameter (β)
Figure 6 Velocity plot for increasing values of m
Figure 7 Temperature plot for increasing Cu nanoparticle concentration
Figure 8 Temperature plot for increasing Ag nanoparticle concentration
Figure 9 Temperature plot for increasing Eckert number with Cu
Figure 10 Temperature plot for increasing Eckert number with Ag
Figure 11 Temperature plot for increasing values of m
Figure 12 Temperature plot for increasing values of β
Figure 13 Graph of skin frictions with different values of suction (b) for different values of f with Cu
Figure 14 Graph of Nusselt number with different values of suction (b) for f with Cu
Figure 15 Graph of skin frictions with different values of suction (b) for different values of f with Ag
Figure 16 Graph of Nusselt number with different values of suction (b) for f with Cu
5 Conclusions
In the present paper, we have investigated steady 2-D flow and the heat transfer phenomena of the nanofluid using Tiwari and Dass’s model in two different nanofluids, namely Cu-sodium alginate and Ag-sodium alginate based Casson type nanofluids, over a non-linear permeable shrinking sheet in the presence of the viscous dissipation. Numerical computation is performed to investigate the occurrence of dual solutions. The critical parameters are identified as being dual and no solutions. From the present investigation, the following significant findings are summarized.
1) Velocity decreases (increases) for different nanoparticle volume fractions and Casson parameter for the first (second) solution.
2) Dual solution exists under the range of bc1, bc2, bc3.
3) As m increases, the thickness of the thermal boundary layer increases in the first solution and decreases in the second solution.
4) The skin friction and the heat transfer are higher for Ag-sodium alginate based Casson type nanofluids compared with Cu.
5) Eckert number enhances the thermal boundary layer.
6) The first solution is stable and physically possible.
Nomenclature
u, v
Velocity components
Cf
Skin friction coefficient
knf
Thermal conductivity of the nanofluid
T
Temperature
T0
A constant
Tw
Variable temperature at the sheet
T∞
Ambient temperature
m
Positive constant
b
b<0 for suction parameter and b>0 for blowing parameter
Nu
Nusselt number
Ec
Eckert number
uw
Velocity of shrinking surface
Py
Fluid’s yield stress
c
Positive constant
Re
Reynolds number
Pr
Prandtl number
vw
Suction/injection velocity
Greek letters
β
Casson parameter
Kinematic viscosity of nanofluid
γ1
Smallest eigen value
τ
Stability transformed variable
γ
Unknown eigen value
ψ
Stream function
μB
Plastic dynamic viscosity
αnf
Effective thermal diffusivity of the nanofluid
(ρcp)nf
Heat capacitance of the nanofluid
ρnf
Effective density of nanofluid
μnf
Effective dynamic viscosity of nanofluid
η
Transformed variable
f
Nanoparticle volume fraction
α
Thermal diffusivity
π
Product of the component of deformation rate with itself
Subscripts
s
Solid
nf
Nanofluid
f
Fluid
Acknowledgement
The authors would like to thank Universiti Utara Malaysia (UUM) for the moral and financial support in conducting this research.
References
[1] SHAH N A. Viscous fluid dynamics, for scientists and engineers [M]. Lahore: A-One Publishers, 2012.
[2] CHOI S U S, ESTMAN J A. Enhancing thermal conductivity of fluids with nanoparticles [J].ASME-Publications,1995, 231: 99–106.
[3] MAHIAN O, KOLSI L, AMANI M, ESTELL
P, AHMADI G, KLEINSTREUER C, MARSHALL J S, SIAVASHI M, TAYLOR R A, NIAZMAND H, WONGWISES S. Recent advances in modeling and simulation of nanofluid flows-Part I: Fundamental and theory [J]. Physics Reports, 2018, 790: 1–48.
[4] MAHIAN O, KOLSI L, AMANI M, ESTELL P, AHMADI G, KLEINSTREUER C, MARSHALL J S, TAYLOR R A, ABU-NADA E, RASHIDI S, NIAZMAND H. Recent advances in modeling and simulation of nanofluid flows-part II: Applications [J]. Physics Reports, 2018, 791: 1–59.
[5] LAAD M, JATTI V K S. Titanium oxide nanoparticles as additives in engine oil [J].Journal of King Saud University-Engineering Sciences, 2016, 30: 116–122.
[6] ROSTAMI M N, DINARVAND S, POP I. Dual solutions for mixed convective stagnation-point flow of an aqueous silica-alumina hybrid nanofluid [J].Chinese Journal of Physics, 2018, 56: 2465–2478.
[7] SHEIKHOLESLAMI M, HAYAT T, ALSAEDI A. On simulation of nanofluid radiation and natural convection in an enclosure with elliptical cylinders [J].International Journal of Heat and Mass Transfer, 2017,115: 981–991.
[8] BHATTI M M, RASHIDI M M. Effects of thermo-diffusion and thermal radiation on Williamson nanofluid over a porous shrinking/stretching sheet [J]. Journal of Molecular Liquids, 2016, 221: 567–573.
[9] MISHRA S R, BHATTI M M. Simultaneous effects of chemical reaction and Ohmic heating with heat and mass transfer over a stretching surface: A numerical study [J]. Chinese Journal of Chemical Engineering, 2017, 25(9): 1137–1142.
[10] SIAVASHI M, RASAM H, IZADI A. Similarity solution of air and nanofluid impingement cooling of a cylindrical porous heat sink [J]. Journal of Thermal Analysis and Calorimetry, 2018: 1–7.
[11] NAKHCHI M E, ESFAHANI J A. Cu-water nanofluid flow and heat transfer in a heat exchanger tube equipped with cross-cut twisted tape [J].Powder Technology, 2018, 339: 1399–1415.
[12] HAYAT T, RASHID M, ALSAEDI A, AHMAD B. Flow of nanofluid by nonlinear stretching velocity [J].Results in Physics, 2018,8: 1104–1109.
[13] BUONGIORNO J. Convective transport in nanofluids [J].Journal of Heat Transfer, 2006,128(3): 240–250.
[14] KHANAFER K, VAFAI K, LIGHTSTONE M. Buoyancy- driven heat transfer enhancement in a two-dimensional enclosure utilizing nanofluids [J].International Journal of Heat and Mass Transfer, 2003,46(19): 3639–3653.
[15] NIELD D A, KUZNETSOV A V. The Cheng–Minkowycz problem for natural convective boundary-layer flow in a porous medium saturated by a nanofluid [J].International Journal of Heat and Mass Transfer, 2009,52(25, 26): 5792–5795.
[16] TIWARI R K, DAS M K. Heat transfer augmentation in a two-sided lid-driven differentially heated square cavity utilizing nanofluids [J].International Journal of Heat and Mass Transfer, 2007, 50(9, 10): 2002–2018.
[17] CAPONE F, RIONERO S. Brinkmann viscosity action in porous MHD convection [J].International Journal of Non- Linear Mechanics,2016, 85: 109–117.
[18] BACHOK N, ISHAK A, POP I. Boundary layer stagnation- point flow and heat transfer over an exponentially stretching/shrinking sheet in a nanofluid [J].International Journal of Heat and Mass Transfer, 2012,55: 8122–8128.
[19] RAHMAN M M, RO
CA A V, POP I. Boundary layer flow of a nanofluid past a permeable exponentially shrinking/ stretching surface with second order slip using Buongiorno’s model [J].International Journal of Heat and Mass Transfer,2014, 77: 1133–1143.
[20] AHMAD R, MUSTAFA M, HAYAT T, ALSAEDI A. Numerical study of MHD nanofluid flow and heat transfer past a bidirectional exponentially stretching sheet [J].Journal of Magnetism and Magnetic Materials, 2016,407: 69–74.
[21] NIELD D A, KUZNETSOV A V. The Cheng–Minkowycz problem for the double-diffusive natural convective boundary layer flow in a porous medium saturated by a nanofluid [J].International Journal of Heat and Mass Transfer,2011, 54(1–3): 374–378.
[22] MATIN M H, POP I. Forced convection heat and mass transfer flow of a nanofluid through a porous channel with a first order chemical reaction on the wall [J].International Communications in Heat and Mass Transfer,2013, 46: 134–141.
[23] AHMED T N, KHAN, I. Mixed convection flow of sodium alginate (SA-NaAlg) based molybdenum disulphide (MoS2) nanofluids: Maxwell Garnetts and Brinkman models [J].Results in Physics, 2018,8: 752–757.
[24] SHAHID A, BHATTI M M, BG O A, KADIR A. Numerical study of radiative Maxwell viscoelastic magnetized flow from a stretching permeable sheet with the Cattaneo–Christov heat flux model [J]. Neural Computing and Applications, 2017, 30: 3467–3478.
[25] CASSON N. A flow equation for pigment-oil suspensions of the printing ink type [J].Rheology of Disperse Systems, 1959, 4: 227–233.
[26] SHEHZAD S A, HAYAT T, ALHUTHALI M, ASGHAR S. MHD three-dimensional flow of Jeffrey fluid with Newtonian heating [J]. Journal of Central South University,2014, 21(4): 1428–1433.
[27] RAMZAN M, FAROOQ M, HAYAT T, ALSAEDI A, CAO J. MHD stagnation point flow by a permeable stretching cylinder with Soret-Dufour effects [J]. Journal of Central South University,2015, 22(2): 707–716.
[28] KUMAR M S, SANDEEP N, KUMAR B R, SALEEM S. A comparative study of chemically reacting 2D flow of Casson and Maxwell fluids [J].Alexandria Engineering Journal, 2017, 57: 2027–2034.
[29] ABBAS T, BHATTI M M, AYUB M. Aiding and opposing of mixed convection Casson nanofluid flow with chemical reactions through a porous Riga plate [J]. Proceedings of the Institution of Mechanical Engineers, Part E: Journal of Process Mechanical Engineering, 2018, 232(5): 519–527.
[30] MAHMOOD A, JAMSHED W, AZIZ A. Entropy and heat transfer analysis using Cattaneo-Christov heat flux model for a boundary layer flow of Casson nanofluid [J].Results in Physics, 2018,10: 640–649.
[31] ALI F, SHEIKH N A, KHAN I, SAQIB M. Magnetic field effect on blood flow of Casson fluid in axisymmetric cylindrical tube: A fractional model [J]. Journal of Magnetism and Magnetic Materials, 2017,423: 327–336.
[32] KHALID A, KHAN I, KHAN A, SHAFIE S, TLILI I. Case study of MHD blood flow in a porous medium with CNTS and thermal analysis [J]. Case Studies in Thermal Engineering, 2018,12: 374–380.
[33] RAZA J, ROHNI A M, OMAR Z. Triple solutions of Casson fluid flow between slowly expanding and contracting walls [J].AIP Conference Proceedings, 2017,1905(1): 030029.
[34] NAKAMURA M, SAWADA T. Numerical study on the flow of a non-Newtonian fluid through an axisymmetric stenosis [J]. Journal of Biomechanical Engineering, 1988, 110(2): 137–143.
[35] ZAIB A, BHATTACHARYYA K, SHAFIE S. Unsteady boundary layer flow and heat transfer over an exponentially shrinking sheet with suction in a copper-water nanofluid [J]. Journal of Central South University, 2015, 22: 4856–4863.
[36] ROCA A V, POP I. Flow and heat transfer over a vertical permeable stretching/shrinking sheet with a second order slip [J].International Journal of Heat and Mass Transfer,2013, 60: 355–364.
[37] WEIDMAN P D, KUBITSCHEK D G, DAVIS A M J. The effect of transpiration on self-similar boundary layer flow over moving surfaces [J]. International Journal of Engineering Science,2006, 44(11, 12): 730–737.
[38] ABBAS Z, MASOOD T, OLANREWAJU P O. Dual solutions of MHD stagnation point flow and heat transfer over a stretching/shrinking sheet with generalized slip condition [J]. Journal of Central South University,2015, 22(6): 2376–2384.
[39] MEGAHED A M. Slip flow and variable properties of viscoelastic fluid past a stretching surface embedded in a porous medium with heat generation [J]. Journal of Central South University, 2016,23(4): 991–999.
[40] POSTELNICU A, POP I. Falkner–Skan boundary layer flow of a power-law fluid past a stretching wedge [J].Applied Mathematics and Computation,2011, 217(9): 4359–4368.
[41] HARRIS S D, INGHAM D B, POP I. Mixed convection boundary-layer flow near the stagnation point on a vertical surface in a porous medium: Brinkman model with slip [J].Transport in Porous Media,2009, 77(2): 267–285.
[42] ZHENG Lian-cun, LIANG Chen, ZHANG Xin-xin. Multiple values of skin friction for boundary layer separation flow in power law pseudoplastic fluids [J]. Journal of Central South University of Technology,2007, 14(1): 218–220.
[43] ROHNI A M, AHMAD S, POP I. Note on Cortell’s non- linearly stretching permeable sheet [J].International Journal of Heat and Mass Transfer,2012, 55(21, 22): 5846–5852.
[44] CORTELL R. Combined effect of viscous dissipation and thermal radiation on fluid flows over a non-linearly stretched permeable wall [J].Meccanica, 2012,47(3): 769–781.
[45] HATAMI M, GANJI D D. Natural convection of sodium alginate (SA) non-Newtonian nanofluid flow between two vertical flat plates by analytical and numerical methods [J].Case Studies in Thermal Engineering, 2014,2: 14–22.
(Edited by YANG Hua)
中文导读
Cu-C6H9NaO7 和 Ag-C6H9NaO7纳米流体在非线性收缩表面流动的多重解
摘要：本文研究了Casson纳米流体在一类非线性收缩表面的流动模型。将Tiwari和Das模型应用于含铜、银的海藻酸钠纳米流体中，通过适当的相似变换，将考虑黏性耗散影响的控制非线性方程组转化为常微分方程的边界值问题(BVPS)。将所得方程用打靶法转化为初值问题，再用四阶Runge-Kutta法求解。为了确定所得到的对偶解的稳定性，对第一(第二)解进行了稳定性分析，发现第一(第二)解是稳定的(不稳定的)和物理可实现的(不可实现的)。在第二解中，随着Casson参数(β)的增加，热边界层的厚度增加，温度也会升高。
关键词：Cu-C6H9NaO7；Ag-C6H9NaO7；收缩曲面；对偶解；稳定性分析；纳米流体
Received date: 2018-11-10; Accepted date: 2019-01-29
Corresponding author: Ilyas KHAN; E-mail: ilyaskhan@tdtu.edu.vn; ORCID: 0000-0002-2056-9371
