HEAT TRANSFER ON MAGNETOHYDRODYNAMIC STAGNATION POINT NANOFLUID FLOW PAST A STRETCHING/SHRINKING SHEET
Nor Ain Azeany Mohd Nasir1*, Anuar Ishak2, Ioan Pop3, Fatin Amirah Ahmad Shukri1and Nurulhuda A. Manaf1
1Department of Mathematics, Centre for Defence Foundation Studies, Universiti Pertahanan Nasional Malaysia, Kem Sungai Besi
57000 Kuala Lumpur, Malaysia
2School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor, Malaysia
3Department of Mathematics, Babeş-Bolyai University, 400084 Cluj-Napoca, Romania
*Corresponding author: [email protected]
ABSTRACT
A steady flow of a nanofluid stagnation point magnetohydrodynamic (MHD) over a permeable shrinking sheet with convective boundary conditions and radiation effects were explored. The governing partial differential equations were converted by using similarity transformation into a system of non-linear ordinary differential equations (ODEs). The boundary value problem solver in MATLAB software was selected to solve the ODEs. The roles of physical parameters on the skin friction coefficient and local Nusselt number as well as velocity, temperature, and nanoparticle volume fraction profiles were described in graphs and addressed in detail. Findings showed that the suction strength improved the skin friction coefficient and heat transfer rate as the sheet was shrunk. Meanwhile, the study showed that the nanoparticle volume fraction and thermal boundary layer thickness were increased in the presence of radiation.
Keywords: Heat transfer; magnetohydrodynamic; nanofluid; stagnation-point flow INTRODUCTION
In the present world of fast technology, nanotechnology is considered as a significant factor that affects industrial applications like nuclear reactors, transportation, electronics, as well as food and biomedicine. Nanofluid is known as a catalyst that can boost thermal conductivity of fluid up to approximately two times with small amount of nanoparticles [1]. Many researchers have shown interest in nanofluid, such as [2,3,4].
Xuan and Li [5], reported that the thermal conductivity with low concentrations of nanoparticles increased more than 20% of the conventional fluid. The common nanoparticles that have been used are aluminium, copper, and silver, while water, ethylene glycol, and oil are the common base fluids. Recently, Aly [6] investigated graphene-water nanofluid with suction and heat source/sink effects. The nanofluid
could act as a coolant with increase in magnetic field, suction, and heat source/sink.
A boundary condition should be considered because when the surface and fluid are at different temperatures, the convective heat effects will occur. This type of boundary condition is called a convective boundary condition. Some relevant studies unfolding the convective boundary condition characteristics can be found in [7-10]. Das et al. [11]
investigated nanofluid flow with the effect of convective boundary condition. Further, Das et al. [12] explored the effect of heat source/sink towards nanofluid flow with convective boundary condition. They found that silver (Ag) had higher temperature as compared to copper (Cu) because Ag has higher thermal conductivity than Cu. Though Ag has higher thermal conductivity, Cu also achieved high temperature, but slightly less than Ag. Recently, Nasir et al. [13] discussed the MHD stagnation point flow over a stretching sheet in nanofluid. The velocity and temperature of the fluid were significantly affected in the presence of a magnetic parameter.
Radiation effects have major applications in electronics and physics. It can play a significant role in managing heat transfer in a polymer processing industry, whereby the final product quality depends to some degree on the heat managing factors. Several important applications of radiative heat transfer from a wall to conductive gray fluids are high-temperature plasmas, cooling of nuclear reactors, liquid metal fluids, and power generation systems. Among those found in literature, Mukhopadhyay [14] stated that the radiation would reduce the heat transfer rate when fluid flow over a symmetric wedge. Babu and Narayana [15] investigated the effect of radiation on Jeffry fluid flow over a continuously moving surface. They reported that the radiation increased the thermal boundary layer thickness; hence, increased the fluid temperature. Recently, the effect of radiation on ferrofluid flow over a nonlinearly moving surface was investigated by Jamaluddin et al. [16]. They found that the radiation caused the heat transfer rate to rapidly decrease. Other research studies on radiation effects were done by Kho et al., Hussain et al., and Yasmin et al. [17,18,19].
Motivated by the work of [13], this study closely followed the work done by adding shrinking effect. This study is important in solving applied problems, such as in paper production industry, polymer processing industry, glass fibre production industry, and packing of bulk products. The similarity transformation was used to transform the governing equations (in PDEs) into ordinary differential equations (ODEs). Then the ODEs were solved by using a boundary value problem solver, which is known as bvp4c built-in MATLAB software.
MATHEMATICAL FORMULATION
Consider a two-dimensional MHD stagnation boundary layer flow of nanofluid over a permeable shrinking sheet in the presence of convective boundary condition and radiation, as shown in Figure 1, where and are the Cartesian coordinate measure along the surface and normal to the surface, respectively. The flow starts at and its velocity is given as , where is the constant stretching/shrinking
parameter with for stretching surface, for shrinking surface, and for static surface. It is assumed that the inviscid flow is with is a positive constant. The constant mass velocity is assumed as with for suction and for injection. It is also assumed that the bottom surface is heated by a convection from a hot fluid at a constant temperature , which provides a constant heat transfer coefficient . The constant surface temperature and concentration of the surface are given as and ,respectively, whereas temperature for inviscid fluid is denoted as and concentration for inviscid fluid is . The flow is subjected to a constant transverse magnetic field of constant strength , which is assumed to be applied in positive direction, whereas the induced magnetic field is neglected since it is assumed to be smaller than applied magnetic field.
Basic fluids and nanoparticles are in thermal balance and there is no slippage between them. Therefore, by using nanofluid model by Buongiorno [20], the two dimensional governing equations are:
(1)
Figure 1: Physical model and coordinate system for the shrinking sheet
(2)
(3)
(4) along with the boundary conditions
(5) where and are the velocity components along axes and , is the nanofluid temperature, is the nanoparticle volume fraction, is the thermal diffusivity of the nanofluid, is the kinematic viscosity of the fluid, is the density, is the thermal conductivity, is the electrical conductivity, is the Brownian diffusion coefficient, is the thermophoretic diffusion coefficient, is the radiation term, is defined as , where is the effective heat capacity of the nanoparticle and is the heat capacity of the base fluid. The boundary condition in Equation 5 is a statement that, with thermophoresis taken into account, the normal flux of nanoparticles is zero at the boundary [21].
Based on Magyari and Pantokratoras [22], by applying the Rosseland approximation for radiation will produce
(6) where is the mean absorption coefficient and is the Stefan Boltzmann constant. As
a result, Equation 3 could be written as
(7)
With is the radiation parameter. Following Nasir et al. [13], the similarity variables are given as
(8)
By applying the above similarity variables (8) into Equation 2, Equation 4, and Equation 7 will easily obtain the similarity equations
(9) (10) (11) With the boundary conditions
(12) Where is the surface mass transfer parameter with for injection, for
suction, and for impermeable sheet. The parameter are the Prandtl number, Lewis number, Biot number, magnetic parameter, Brownian motion and thermophoresis, respectively, and are defined as
(13)
For practical purpose, the physical quantities are the skin friction coefficient and local Nusselt number , are defined as
(14) where is the shear stress and is the heat flux which is given by
(15) Replacing Equation 13 into Equation 14 and Equation 15 will produce
(16)
Where is the local Reynolds number.
Stability Analysis
Since the similarity equation produces multiple solutions, the stability analysis for the solution should be carried out. Following the work done by [23-26], the new dimensionless time variable was introduced as and the new similarity variables obtained as
(17) Substituting Equation 17 into Equation 2, Equation 4 and Equation 7, will obtain
(18) (19)
(20) and subject to boundary conditions
(21)
To determine the stability of the steady flow solution and satisfying the boundary value problem, disturbance equations are introduced given by,
(22)
with is an unknown eigenvalue parameter, and are small relative to and respectively. Using Equation 22, Equation 18 – Equation 21 can be written as
(23) (24)
(25) With boundary conditions
(26) This stability analysis is carried out by finding the eigenvalue in Equation 23 –
Equation 25. To get the values of , the Equation 23 – Equation 26 are solved by setting . If the smallest eigenvalue is positive, the flow is said to be stable or otherwise unstable.
RESULT AND DISCUSSION
The numerical solution for Equation 9 – Equation 12 was obtained by using a solver named bvp4c in MATLAB software. The solutions were compared with Nasir et al.
[13] to confirm the validity of the present work. Based on Table 1, the computed solutions are in very good agreement which ensured the justification of the present work.
Table 1: Comparison of ( ) for stretching sheet
Present study Nasir et al. [13] Nandy and Pop [27]
0.1 1.232587628 1.232588 1.232588
0.5 0.713294940 0.713295 0.713296
2.0 -1.887306655 -1.887307 -1.887307 5.0 -10.264749305 -10.264749 -10.264751
The variation of and for several values of suction parameter with and are presented in Figure 2. Note that, this figure represents the increasing in value for and
as increases. The solution range also increases as increases. The dual solutions exist when However, there is no solution for . It is observed that when
the critical values for were ,
respectively, and it can predict for larger values of , the critical values for will get smaller. This implied that the increment of can delay the boundary layer separation.
Figure 3 shows the effects of towards the velocity and temperature profiles with , and . These two figures illustrate the existence of dual solution profiles for a particular value of . It was found that the increased values of led to an increase in the fluid velocity for upper branch solution. Meanwhile, for the lower branch solution, increment of decreased the fluid velocity . The temperature profiles showed a reduction as the values
of increased. This happened because the velocity of the fluid was getting faster and more heat could be carried away from the surface; hence, it increased the heat transfer rate, causing the temperature to decrease. For lower branch solution, the temperature increased as the value of increased.
The effects of Brownian motion parameter towards temperature and nanoparticle fraction profiles can be observed in Figure 4. The temperature profile slightly increased as the value of increased while the value of gave great impact in reducing the nanoparticle fraction profiles. As for lower branch solution, the value of increased the temperature and nanoparticle fraction profiles. The same trend can be found in Figure 5 which demonstrates the effects of thermophoresis parameter towards temperature and nanoparticle profiles. The increment of had slightly increased the temperature profile. However, gave impact by increasing the nanoparticle fraction profile. Meanwhile, for lower branch solution the increment of can increase the temperature profile and decrease the nanoparticle fraction profile.
Figure 6 displays the effects of Biot number towards temperature and nanoparticle fraction profiles. It was clearly seen that the increment of increased the temperature and nanoparticle fraction profiles. Meanwhile, the value of decreased the nanoparticle fraction profile and increased temperature profile for lower branch solutions. The temperature profile with several values of radiation parameter is shown in Figure 7. The value of increased the temperature profile, and the thermal boundary layer thickness was also increases for both upper and lower branch solutions.
This happened because the value of Lewis number was large, and thus the effect of was small as compared to the effect of convection. Table 2 depicts the values of the smallest eigenvalues for different values of . This table indicated that the positive values of gave an initial decay of disturbance and the negatives values of showed an initial growth of disturbance. From this table, it was obvious that the upper branch solution produced positive and lower branch solution produced negative . Therefore, it can be concluded that the upper branch solution is stable and lower branch solution is unstable.
Figure 2(a): Variation of with several values of
Figure 2(b): Variation of with several values of
Figure 3(a): Effects of magnetic towards velocity profile
Figure 3(b): Effects of magnetic towards temperature profile
Figure 4(a): Effects of Brownian motion parameter towards temperature profile
Figure 4(b): Effects of towards nanoparticle fraction profile
Figure 5(a): Effects of thermophoresis parameter towards temperature profile
Figure 5(b): Effects of thermophoresis parameter towards nanoparticle fraction profiles
Figure 6(a): Effects of Biot number towards temperature profile
Figure 6(b): Effects of Biot number towards nanoparticle fraction profile
Figure 7: Effects of radiation parameter towards temperature profile Table 2. Smallest eigenvalues for various when
Upper branch solution Lower branch solution
2
-2.8 2.1335 -1.7731
-3.0 1.4740 -1.3016
-3.1 1.0019 -0.9212
-3.15 0.6577 -0.6224
-3.18 0.3141 -0.3059
-3.184 0.2348 -0.2302
-3.189 0.0371 -0.0370
CONCLUSION
The problem of magnetohydrodynamics (MHD) stagnation point past over a shrinking sheet in nanofluid including the presence of convective boundary condition and radiation effects were investigated. The effects of suction , magnetic , Brownian motion , thermophoresis , Biot number , and radiation parameter were presented and discussed. The results demonstrated that suction could delay the boundary layer separation, magnetic parameter enhances the heat transfer rate, while thermophoresis and Biot number increase the fraction of nanoparticle of the nanofluid. The stability analysis confirmed that the upper branch solution was stable
and was physically realisable, while the lower branch solution was unstable and was not physically realisable.
ACKNOWLEDGEMENT
The financial supports were received from the Universiti Kebangsaan Malaysia (Project code: DIP-2015-010) and Universiti Pertahanan Nasional Malaysia (Project code:
SF0059-UPNM/2019/SF/SG/1).
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