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Combined Effect of variable viscosity and Thermal Conductivity on dissipative flow of oil-based nanofluid over a permeable vertical plate
Christian John Etwire1,a*, Ibrahim Yakubu Seini2,b, Rabiu Musah3,c
1Faculty of Mathematical Sciences, University for Development Studies, P.O. Box 24, Navrongo, UER, Ghana
2School of Engineering, University for Development Studies, Nyankpala Campus, Tamale, NR, Ghana
3Faculty of Applied Sciences, University for Development Studies, P.O. Box 24, Navrongo, UER, Ghana
jecpapa@yahoo.coma, yakubuseini@yahoo.comb, mrabiu@uds.edu.ghc
Abstract: The combined effect of variable viscosity and thermal conductivity on dissipative flow of oil-based nanofluid over a permeable vertical plate with suction has been studied. The governing partial differential equations were transformed into a coupled third order ordinary differential equations by similarity techniques. The third order ordinary differential equations were then reduced into a system of first order ordinary differential equations and solved numerically using the fourth order Runge-Kutta algorithm with a shooting method. The results were presented in tabular and graphically forms for various controlling parameters. It was revealed that increasing the viscosity parameter of CuO oil-based nanofluid increases the skin friction coefficient but slows the rate of heat transfer at the surface of the plate whilst increasing the thermal conductivity of CuO oil-based nanofluid depreciates both the skin friction coefficient and rate of heat transfer at the surface of the plate. Also, increasing the viscosity and thermal conductivity of CuO oil-based nanofluid, Prandtl number, suction parameter and Biot number weaken the thermal boundary layer.
Keywords: Cohesive, Collision, Lubricant, Piston, Viscosity
1 Introduction
Lubricant plays significant role in the automobile industry. It cushions the engine’s bearings from the shocks of cylinder firing, neutralizes the corrosive elements created during combustion, seals the engine’s metal surfaces from rust and cools internal engine parts that cannot be directly cooled by the engine’s water-cooling system. Cooling is critical for sustaining the desired performance and reliability of vehicle engines but the water cooling system of a vehicle cannot dissipate heat from the piston head of the engine due to detonation and pre-ignition. The piston, as a moving wall of the combustion chamber converts the heat generated as a result of the combustion of the fuel into mechanical work and drives the crankshaft through the connecting rod. Continuous heating of the piston without efficient coolant can lead to high fuel and oil consumption, harmful exhaust emissions, reduction in engine power output or permanent engine damage. To ensure durability, reliability and prolong lifespan of the engine, there is the need for oil with enhanced heat transfer characteristics. Nano-oil is the panacea and it exhibits improvement in thermophysical properties such as thermal conductivity, thermal diffusivity, viscosity, and convective heat transfer coefficients (Choi and Eastman, 1995). Pirhayati et al. (2012) examined the pressure drop of CuO-base oil nanofluid flow inside an inclined tube and found a reduction in pressure with the inclination of the tube at constant volume fraction concentrations and for Reynolds numbers less than 170. Pirhayati et al. (2014) further extended their work to include convective heat transfer characteristics. Their results showed that the heat transfer coefficient of nanofluid increased with increasing Reynolds number inside horizontal and inclined tubes. Bhaumik and Pathak (2015) analyzed the anti-wear properties of CuO nanoparticles in Mineral Oil using Pin-On-Disk Tribometer. Convective heat transfer and stability of Oil–based Nanofluid was investigated by Karamallah and Hussein (2016). Recently, Haq et al. (2017) discussed MHD pulsatile flow of engine oil based carbon nanotubes between two concentric cylinders.
Viscosity of oil is eminent in deciding its suitability for lubrication whether in the hydrodynamic or elasto-hydrodynamic regime. Oil’s viscosity is the measure of its thickness or resistance to flow. It is directly linked with how well oil lubricates and protects surfaces that move relative to each other. Viscous oil provides stronger oil film and the thicker the oil film, the more resistant it will be rubbed from lubricated surfaces. However, very thick oil offers excessive resistance to flow at low temperatures and as a result may not flow quickly enough to those parts requiring lubrication. It is therefore imperative that oil exhibits the right viscosity at both the highest and the lowest temperatures which is a requirement for proper functioning of the engine. Nanotechnology is paramount in developing oil with improved viscosity index. Viscosity index research has received remarkable admiration from scientists and engineers due to its industrial and engineering applications. Researchers such as; Makinde and Aziz (2010), Tshehla and Makinde (2011), Chinyoka and Makinde (2011), Moorthy and Senthilvadivu (2012), EL-Kabeir et al., (2013), Moorthy et al., (2013) , Eegunjobi and Makinde (2014) and Makinde et al., (2016) studied the effect of variable viscosity on conventional fluid.
However, Kuppalapalle et al., (2013) investigated the effect of variable viscosity on the flow and heat transfer of viscous Ag- water and Cu-water nanofluids. Shivakumara and Dhananjaya (2014) discussed the onset of convection in a Nanofluid saturated porous layer with temperature dependent viscosity. Uddin et al., (2014) analyzed g-Jitter mixed convective slip flow of Nanofluid past a permeable stretching sheet embedded in a Darcian porous media with variable viscosity. The effects of variable viscosity and thermal conductivity on natural-convection of Nanofluids past a vertical plate in porous media was reported by Noghrehabadi et al., (2014). James et al., (2015) discussed the effects of variable viscosity of nanofluid flow over a permeable wedge embedded in saturated porous medium with chemical reaction and thermal radiation. Nasrin and Alim (2015) investigated entropy generation by nanofluid with variable thermal conductivity and viscosity in a flat plate solar collector. Alvi et al., (2016) analyzed Peristalsis of non-constant viscosity Jeffrey fluid with nanoparticles. Ram et al., (2016) studied Variable Viscosity effects on time dependent magnetic Nanofluid flow past a stretchable rotating plate.
Most recently, Shahzadi et al., (2017) looked at the simultaneous effects of single wall carbon nanotube and effective variable viscosity on peristaltic flow through annulus having permeable walls. Chandra et al., (2017) studied effects of variable viscosity and thermal conductivity on MHD boundary layer flow of nanofluid with thermal radiation. Huda et al., (2017) analyzed the dynamics of variable-viscosity nanofluid flow with heat transfer in a flexible vertical tube under propagating waves. The effects of oil-based nanofluid on a stretching surface with variable suction and thermal conductivity was discussed by Etwire et al., (2017).
From the survey of literature, not much research work has been done on oil based nanofluid with variable viscosity which occurs in automobile industry since the engine of a vehicle operates at varying temperature and viscosity of oil depletes with temperature. The knowledge of the combined impact of viscosity and other thermophysical parameters on oil would help formulate lubricant which can maintain lubricating oil film at all operating temperature and ensures high temperature viscosity retention. Thus, this study sought to investigate the effects of variable viscosity and thermal conductivity on dissipative flow of oil-based nanofluid over a permeable vertical plate.
2 Mathematical Model
Consider a two-dimensional steady incompressible flow of dielectric and viscous CuO oil-based Nanofluid over a porous vertical plate with variable viscosity and thermal conductivity and suction. The x-axis is taken along the direction of the plate whilst the y-axis is taken normal to it as depicted in Figure 1. The left surface of the plate is heated by convection from a hot fluid at temperature , which provides a heat transfer coefficient while a stream of CuO oil-based Nanofluid at the free stream temperature moves over the right surface of the plate with a uniform free stream velocity .
Figure 1: Schematic diagram of flow problem
It is assumed that both the oil and CuO are in thermal equilibrium with no slip between them. The variation of density in the CuO oil-based nanofluid is taken into account using the Boussinesq approximation. The continuity, momentum and energy equations modeling the flow problem can be expressed as
(1) (2)
(3)
where and are and components of velocities respectively, is the temperature dependent dynamic viscosity of CuO oil-based nanofluid, is the temperature dependent thermal conductivity of CuO oil-based nanofluid, is the density of CuO oil-based nanofluid, is the thermal expansion coefficient of CuO oil-based nanofluid, is the permeability of the porous media, is the temperature of CuO oil-based nanofluid and is the heat capacitance of CuO oil-based nanofluid.
The boundary conditions on the right surface of the plate are;
at (4)
The boundary conditions of the CuO oil-based nanofluid at the far right surface of the plate are;
, as (5)
The properties of the nanofluid (Oztop and Abu-Nada, 2008) are defined as;
(6)
Where and are the densities of the oil and CuO respectively, and are the thermal expansion coefficients of the oil and CuO respectively, is the solid volume fraction of the CuO, and are the thermal conductivities of the oil and CuO nanoparticles respectively and is the dynamic viscosity of oil. The temperature dependent dynamic viscosity , and thermal conductivity , of CuO oil-based nanofluid are defined as;
(7)
(8)
Where is the dynamic viscosity of CuO oil-based nanofluid at ambient temperature , m is the viscosity parameter, is the thermal conductivity of CuO oil-based nanofluid at ambient temperature and q is the thermal conductivity parameter.
3 Similarity Procedure
Equation (1) is satisfied automatically by defining the stream function , in the usual way as:
and (9)
A similarity solution of equations (1) – (5) is achieved by defining an independent variable , a stream function in terms of a dependent variable and a dimensionless temperature , as;
(10)
Substituting equations (6) – (10) into equations (1) – (5), yield the desire coupled ordinary differential equations as;
(11)
(12)
Subject to the boundary conditions
at (13)
as (14)
where the prime symbol denotes differentiation with respect to , is the variable viscosity parameter of CuO oil-based nanofluid, is the thermal Grashof number, is the suction parameter, is the permeability parameter, is the variable thermal conductivity parameter of CuO oil-based nanofluid, is the Prandtl number, is the Biot number and is the Brinkman number. Parameters of engineering applications considered in this study are the skin-friction coefficient (Cf) and the Nusselt number (Nu) which are defined as;
(15)
where is the wall shear stress and is the wall heat flux which are given by
(16)
Substituting equation (16) into equation (15) yield
(19)
3 Numerical Procedure
The coupled third order nonlinear ordinary differential equations (11) and (12) are reduced into a system of first order ordinary differential equation by letting;
, (20)
Substituting equation (20) into equations (11) and (12) yield the required system of first order differential equations as;
(21)
Subject to the boundary conditions;
(22)
The Shooting technique is employed to guess the unknown b and c until the boundary conditions ( and ) is satisfied. The resulting differential equations are solved using the fourth order Runge Kutta integration scheme. Numerical computations are done using MAPLE 16 software package.
4 Results and Discussions
The thermophysical parameters of industrial and engineering applications considered in this study include; solid volume fraction of CuO ( ), Biot number (Bi), Brinkman number (Br), Prandtl number (Pr), Permeability parameter (K*), the viscosity (ᾳ) and the thermal conductivity (σ) of CuO oil-based nanofluid and Thermal Grashof number (GT). The effects of these parameters on the velocity profile, temperature profile, skin friction coefficient (Cf) and Nusselt number (Nu) were explored. The solid volume fraction of CuO was varied within the range . The permeability parameter and Grashof number are maintained at a constant value of 0.01. The thermophysical properties of oil and CuO are given in Table 1.
Table 1: Thermophysical properties of base fluid and nanoparticle
Physical property Cp (J/kgK) ρ (Kg/m3) k (W/mK) β x 10-5(K-1)
Oil 1670 920 0.138 64
CuO 540 6510 18 0.85
4.1 Numerical results
The numerical results of the present work for the plate surface temperature ( ) and the local Nusselt number represented by the rate of heat transfer ( ) were compared with the work of Aziz (2009) for varying values of the Biot number (Bi) and there was excellent agreement. The comparison is presented in Table 2.
Table 2: Computations showing comparison with Aziz (2009) for and
Aziz (2009) Present Work
Bi
0.05 0.1447 0.0428 0.1447 0.0428
0.10 0.2528 0.0747 0.2528 0.0747
0.20 0.4035 0.1193 0.4035 0.1193
0.40 0.5750 0.1700 0.5750 0.1700
0.60 0.6699 0.1981 0.6699 0.1981
0.80 0.7302 0.2159 0.7302 0.2159
1.00 0.7718 0.2282 0.7718 0.2282
The impact of the various thermo-physical parameters on the skin friction coefficient and the rate of heat transfer at the surface of the plate are presented in Table 3.
Table 3: Computation showing and for different parameter values
Br ⱷ Pr σ ᾳ S Bi
100 0.10 100 0.1 0.1 0.01 1 0.386727 1.902234
150 0.10 100 0.1 0.1 0.01 1 0.404480 2.881563
200 0.10 100 0.1 0.1 0.01 1 0.422451 3.745695
100 0.13 100 0.1 0.1 0.01 1 0.393888 2.176423
100 0.16 100 0.1 0.1 0.01 1 0.398046 2.463991
100 0.20 100 0.1 0.1 0.01 1 0.399411 2.873316
100 0.10 200 0.1 0.1 0.01 1 0.370614 1.127871
100 0.10 300 0.1 0.1 0.01 1 0.364551 0.727374
100 0.10 400 0.1 0.1 0.01 1 0.361333 0.469120
100 0.10 100 1.0 0.1 0.01 1 0.380246 1.092332
100 0.10 100 2.0 0.1 0.01 1 0.377653 0.806703
100 0.10 100 3.0 0.1 0.01 1 0.376132 0.650537
100 0.10 100 0.1 1.0 0.01 1 0.442927 0.553412
100 0.10 100 0.1 2.0 0.01 1 0.504076 0.154223
100 0.10 100 0.1 2.5 0.01 1 0.535795 0.042832
100 0.10 100 0.1 0.1 0.10 1 0.417849 1.177565
100 0.10 100 0.1 0.1 0.20 1 0.459698 0.721107
100 0.10 100 0.1 0.1 0.30 1 0.504852 0.464480
100 0.10 100 0.1 0.1 0.01 2 0.384661 2.976997
100 0.10 100 0.1 0.1 0.01 3 0.383394 3.672943
100 0.10 100 0.1 0.1 0.01 4 0.382537 4.161911
Table 3 reveals that increasing the Brinkman number increases the magnitude of both the skin friction coefficient and the rate of heat transfer at the surface of the plate. This is as a result of enhanced viscous dissipation over thermal conduction. Similar trend is observed with the solid volume fraction of CuO due to ballistic heat transfer occurring in the CuO. Conversely, increasing the intensity of the Prandtl number and thermal conductivity of CuO oil-based nanofluid decrease the magnitude of both the skin friction coefficient and the rate of heat transfer at the surface of the plate. However, increasing both the viscosity parameter of CuO oil-based nanofluid and suction parameter increase the intensity of the skin friction coefficient but slow the rate of heat transfer at the surface of the plate since the enhancement in the viscosity parameter strengthens the cohesive forces between the molecules of CuO oil-based nanofluid which enhances the viscous shear stresses in the nanofluid whilst the suction parameter delays the onset of the boundary layer. Also increasing the Biot number depreciates the skin friction coefficient but enhances the rate at which heat is transferred at the surface of the plate
4.2 Graphical Results
4.2.1 Effects of Parameter Variation on the Velocity Profiles
Figures 2 and 3, illustrate the velocity profiles for varying values of the thermophysical parameters. It is noted in Figure 2 that increasing the magnitude of the viscosity parameter of CuO oil-based nanofluid increases the velocity profile of the nanofluid. An increase in the viscosity parameter slows down the motion of the molecules of CuO oil-based nanofluid and increases the collision rate of the molecules which enhances the shearing stresses of the nanofluid. This thickens the momentum boundary layer of CuO oil-based nanofluid. Similar trend is observed in Figure 3 as the suction parameter is increased.
Figure 2: Velocity Profile for varying values of Viscosity of CuO oil-based Nanofluid for , and
Figure 3: Velocity Profile for varying values of Suction parameter for , and
4.2.2 Effects of Parameter Variation on Temperature Profiles
Figures 4–10, present the temperature profiles for varying values of the thermophysical parameters. It is evident in Figures 4 and 5 that increasing the solid volume fraction of CuO and Brinkman number enhance the temperature profile. The combined enhancement in solid volume fraction of CuO and Brinkman number intensify the shearing stresses of the nanofluid due to viscous diffusion. This increases the temperature of CuO oil-based nanofluid within the vicinity of the plate which intend thickens the thermal boundary layer. However, in Figure 6 increasing the viscosity of CuO oil-based nanofluid depreciates the temperature profile. As viscosity is increased, the cohesive intermolecular forces between the molecules of the nanofluid strengthen due to the drop in the energy level of the molecules. This deteriorates the thermal boundary layer thickness. Similar trend was observed in Figures 7-10 as the magnitudes of thermal conductivity of CuO oil-based nanofluid, Prandtl number, suction parameter and Biot number were increased due to enhanced viscous diffusion rate and the delay in the development of the boundary layer.
Figure 4: Temperature Profile for varying values of Solid Volume Fraction of CuO for , and
Figure 5: Temperature Profile for varying values of Brinkman number for , and
Figure 6: Temperature Profile for varying values of Viscosity of CuO oil-based Nanofluid for , and
Figure 7: Temperature Profile for varying values of Thermal Conductivity of CuO oil-based Nanofluid for , and
Figure 8: Temperature Profile for varying values of Prandtl number for , and
Figure 9: Temperature Profile for varying values of Suction parameter for , and
Figure 10: Temperature Profile for varying values of Biot number for , and
5 Conclusions
The combined effect of variable viscosity and thermal conductivity on dissipative flow of oil-based nanofluid over a permeable vertical plate with suction has been examined. The partial differential equations governing the flow were transformed into ordinary differential equations by similarity transformation and then solved numerically using the fourth order Runge-Kutta algorithm with a shooting method. Numerical results were presented whilst the velocity and temperature profiles illustrated graphically and analyzed. The study revealed that increasing the magnitudes of both the viscosity and thermal conductivity parameters of CuO oil-based nanofluid depreciate the thermal boundary layer.
Nomenclature
Cartesian coordinates
Velocity components
Ambient temperature
Free-stream temperature
Temperature of the sheet
Temperature of CuO oil-based nanofluid
Thermal conductivity of oil
Thermal conductivity of CuO
Specific heat at constant pressure
Permeability of the porous media
Permeability parameter
= Heat transfer coefficient
Prandtl number
Cf = skin-friction coefficient
Thermal Grashof number
Re = Reynolds number
Nu = Nusselt number
Wall heat flux
Biot number
Brinkman number
Greek Symbols
Wall shear stress
Dynamic viscosity of CuO oil-based nanofluid
Kinematic viscosity of oil
Density of CuO oil-based nanofluid
Thermal diffusivity of oil
Density of oil
Density of CuO
Solid volume fraction of CuO
Heat capacitance of CuO oil-based nanofluid
Stream function
Variable thermal conductivity parameter of CuO oil-based nanofluid
Variable viscosity parameter of CuO oil-based nanofluid,
= Thermal expansion coefficient of CuO oil-based nanofluid
= Thermal expansion coefficient of oil
= Thermal expansion coefficient of CuO
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Summary
The combined effect of variable viscosity and thermal conductivity on dissipative flow of oil-based nanofluid over a permeable vertical plate with suction has been studied. The governing partial differential equations were transformed into a coupled third order ordinary differential equations by similarity techniques. The third order ordinary differential equations were then reduced into a system of first order ordinary differential equations and solved numerically using the fourth order Runge-Kutta algorithm with a shooting method. The results were presented in tabular and graphically forms for various controlling parameters. It was revealed that increasing the viscosity parameter of CuO oil-based nanofluid increases the skin friction coefficient but slows the rate of heat transfer at the surface of the plate whilst increasing the thermal conductivity of CuO oil-based nanofluid depreciates both the skin friction coefficient and rate of heat transfer at the surface of the plate. Also, increasing the viscosity and thermal conductivity of CuO oil-based nanofluid, Prandtl number, suction parameter and Biot number weaken the thermal boundary layer.
