Impacts of viscous dissipation and Joule heating on hydromagnetic boundary layer flow of nanofluids over a flat surface subjected to Newtonian heating
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Abstract
Main concern in this analysis is to study the two-dimensional, steady boundary layer flow of viscous, incompressible, electrically conducting nanofluids past a flat plate in the presence of magnetic field with heat being transferred by Newtonian heating way. Influences of viscous dissipation and Joule heating are considered also. Nonlinear partial differential equations are reduced into nonlinear ordinary differential equations by formulating similarity transformations. Numerical solutions of transformed boundary layer equations are clarified by applying the Keller-Box method. Effects of several types of nanofluids and various specified parameters such as solid volume fraction, magnetic parameter, Brinkmann number and local Biot number on velocity and temperature fields have been plotted graphically, while values of surface shear stress and surface heat flux are presented via table. Further, comparison of obtained computational values has been made with earlier published results for non-magnetic case.
Keywords
Viscous dissipation Joule heating Hydromagnetic boundary layer flow Nanofluids Flat surface Newtonian heatingList of symbols
- \(B_0\)
Uniform magnetic field strength (N m\(^{-1}\) A\(^{-1}\))
- Bi
Local Biot number
- Br
Brinkmann number
- \(C_f\)
Local skin friction coefficient
- \(C_p\)
Specific heat at constant pressure (J Kg\(^{-1}\) K\(^{-1}\))
- f
Dimensionless stream function
- \(h_t\)
Heat transfer coefficient
- M
Magnetic parameter
- \(Nu_x\)
Local Nusselt number
- Pr
Prandtl number
- \(Re_x\)
Local Reynolds number
- T
Temperature of nanofluid (K)
- \(T_w\)
Hot fluid temperature at surface (K)
- \(T_\infty\)
Ambient fluid temperature (K)
- \(U_\infty\)
Velocity of ambient fluid (m s\(^{-1}\))
- u
Velocity component parallel to the x-axis (m s\(^{-1}\))
- v
Velocity component parallel to the y-axis (m s\(^{-1}\))
- x
Direction along to the plate (m)
- y
Direction perpendicular to the plate (m)
Greek symbols
- \(\alpha\)
Thermal diffusivity (m\(^{2}\) s\(^{-1}\))
- \(\eta\)
Similarity variable
- \(\theta\)
Dimensionless temperature
- \(\kappa\)
Thermal conductivity (W m\(^{-1}\) K\(^{-1}\))
- \(\mu\)
Coefficient of viscosity (Kg m\(^{-1}\) s\(^{-1}\))
- \(\nu\)
Kinematic viscosity (m\(^{2}\) s\(^{-1}\))
- \(\rho\)
Density (Kg m\(^{-3}\))
- \(\sigma _e\)
Electrical conductivity (S m\(^{-1}\))
- \(\phi\)
Solid volume fraction
- \(\psi\)
Stream function (m\(^{2}\) s\(^{-1}\))
Superscripts
- \(\prime\)
Differentiation with respect to \(\eta\)
Subscripts
- f
Base fluid
- nf
Nanofluid
- s
Nano solid particles
1 Introduction
Along the shear forces action, an effort done through the fluid on adjoining layers is converted into a heat, which is known as viscous dissipation. For higher velocity and viscous flows, the viscous dissipation impact on heat transfer is essential. Viscous dissipation appears in natural convection for different devices and also in powerful gravitational fields. Whereas, a process in which electric current energy is transformed into heat as it flows by resistance is called Joule heating or ohmic heating. Joule heating has a lot of applications in the area of technology and industrial processing. Some examples of applications are electric heaters and fuses, handling of food, electric stoves, electronic cigarette and incandescent light bulb. Initially, El-Amin [1] developed the combined influence of viscous dissipation and Joule heating and observed the convectional boundary layer flow situation towards the embedded porous medium. Moreover, numerous authors like as Abo-Eldahab and El-Aziz [2], Jat and Chaudhary [3], Yavari et al. [4] and Das et al. [5] have studied the fluid flow with viscous dissipation and Joule heating influences for some extended effects. In recent years, various explorations have been found by Hayat et al. [6], Hussain et al. [7] and Chaudhary and Choudhary [8] and analyzed the boundary layer flow in the presence of viscous dissipation and Joule heating.
To adjust the structure of boundary layer, an efficient technique as magnetohydrodynamic (MHD) principle has been used. MHD is an observation of electrically conducting fluids in a magnetic field, which depends on the induced magnetic field strength. Some examples of magneto fluids are liquid of metals, salt water, plasmas and electrolytes, and few applications of MHD effect such as boundary layer control in aerodynamics, geothermal energy, bearing, MHD generators and sensors, crystal growth, pumps, electromagnetic castings and plasma studies have significant scope in the areas of engineering and technologies. Alfven [9] pioneered the analysis of MHD field and analyzed the electromagnetic hydrodynamic waves existence. After that Ganesan and Palani [10] executed the study of unsteady MHD flow towards an inclined plate with natural convection. Moreover, Jat and Chaudhary [11], Butt and Ali [12], Imtiaz et al. [13], Chaudhary and Choudhary [14], Aydin and Selvitopi [15], Rehman et al. [16, 17] and Jha and Malgwi [18] have inscribed some articles on heat and mass transfer in MHD flow against to various situations.
The conventional ordinary fluids specifically oil, toluene, water and ethylene glycol have lower thermal conductivity and heat transfer rate capability. But increasing demand of advanced technology in electronic devices miniaturization have requirement of heat transfer medium behaving like a liquid, which has greater heat transfer efficiency to develop the thermal characteristics. This type of medium is known as nanofluid. Nanofluids have two types of materials namely base fluid and ultrafine nanoparticles with the diameter size 1–100 nm. Some frequent nanoparticles made by metallic are silicon, titanium, copper, aluminum and silver. Nanofluids have drawn researchers awareness in the fields of engineering technology and science as a wide range of industrial applications such as food, drinks, dyes and toners, airplane engine, bio-chemical dispensation, microchip technology, pharmacological, dynamism, aerospace, remedial apparatus and devices. Choi [19] essentially discussed the concept of nanofluids for the enlargement of the thermal conductivity of fluids. Further, Buoyancy-driven heat transfer increment in a two-dimensional enclosure by using nanofluids established by Khanafer et al. [20]. Latterly, Chein and Chuang [21], Yang and Lai [22], Mital [23], Ibrahim and Makinde [24], Makinde et al. [25], Chaudhary and Kanika [26], Rehman et al. [27], Kandasamy et al. [28], Ma et al. [29] and Sheikholeslami et al. [30] have inspected the numerous numerical and analytical explorations for the improvement of nanofluids heat transfer.
Newtonian heating is a way of heat transfer into the conventional fluid from the boundary surface with specific heat capacity. That type of composition appears in the setup of convection flows and heat introduces through solar radiation. The Newtonian heating impact is found if heat flux from the wall is proportional to the local surface temperature. This heating process utilized some practical base applications, particularly thermal energy storage, conjugate heat transport around fines, petroleum industry, nuclear turbines and heat exchanger. Merkin [31] illustrated the first analysis in the area of Newtonian heating and applied the effect of Newtonian heating on natural convection boundary layer flow via a vertical plate. Until, Lesnic et al. [32] established the free convection boundary-layer flow above a nearly horizontal surface in a porous medium with Newtonian heating. Further, an extensive literature on the behavior of Newtonian heating on boundary layer flow have been discussed by Makinde [33], Akbar and Khan [34], Hayat et al. [35], Chaudhary et al. [36] and Kamran and Wiwatanapataphee [37].
Keller-Box method is a very versatile modeling scheme. It is comparatively easy to understand and implemented. The Keller-Box technique works well for the two-dimensional model with simplified system geometries. This method is used directly to the differential form of the governing equations. The details of Keller-Box method can be found in the book by Vajravelu and Prasad [38].
Above mentioned literature pointed out that Newtonian heating effect on MHD boundary layer flow of electrically conducting fluid as water containing nanoparticles specifically silver (Ag), copper (Cu), titanium dioxide (\(\text {TiO}_2\)) and alumina (\(\text {Al}_2\text {O}_3\)) is not examined yet. So the main objective of the present study is to extend the investigation of Makinde [39] with the impacts of viscous dissipation and Joule heating. Subsequently, the behaviors of nanofluids and considering parameters on the velocity, temperature, local skin friction coefficient and the local Nusselt number are given by plotted or tabulated values and discussed in detail.
2 Flow analysis
Thermophysical resources of water and nanoparticles
Properties | Water | Ag | Cu | \(\text {TiO}_2\) | \(\text {Al}_2\text {O}_3\) |
---|---|---|---|---|---|
\(\kappa \;(\text {W}\,\text {m}^{-1}\,\text {K}^{-1})\) | 0.613 | 429 | 400 | 8.9538 | 40 |
\(\rho \;(\text {Kg\,m}^{-3})\) | 997.1 | 10500 | 8933 | 4250 | 3970 |
\(C_p\; (\text {J}\,\text {Kg}^{-1}\,\text {K}^{-1})\) | 4179 | 235 | 385 | 686.2 | 765 |
\(\sigma _e\;(\text {S\,m}^{-1})\) | 0.05 | \(6.3\times 10^7\) | \(5.96\times 10^7\) | \(0.24\times 10^7\) | \(3.69\times 10^7\) |
3 Transformed problem
4 Declaration of curiosity
5 Solution methodology
The system of the nonlinear ordinary differential Eqs. (11) and (12) along with the associated boundary conditions Eq. (13) is solved numerically with a finite difference scheme as Keller-Box method (Kumar and Sood [43]). For the computational procedure, the suitable finite value of the far field boundary condition as \(\eta \rightarrow \infty =6\) is assumed.
5.1 Scheme of implicit finite difference
5.2 Newton’s method
5.3 Method of block elimination
6 Verification of the numerical method
Comparison of \(f^{\prime \prime }(0)\) for various values of \(\phi\) corresponding to different types of nanofluids with \(M=0\)
\(\phi\) | Makinde [39] | Present results | ||||
---|---|---|---|---|---|---|
Cu-water | \(\text {TiO}_2\)-water | \(\text {Al}_2\text {O}_3\)-water | Cu-water | \(\text {TiO}_2\)-water | \(\text {Al}_2\text {O}_3\)-water | |
0.000 | 0.3321 | 0.3321 | 0.3321 | 0.33258 | 0.33258 | 0.33258 |
0.008 | 0.3459 | 0.3398 | 0.3394 | 0.33945 | 0.33350 | 0.33315 |
0.014 | 0.3563 | 0.3456 | 0.3449 | 0.34429 | 0.33410 | 0.33350 |
0.016 | 0.3597 | 0.3476 | 0.3468 | 0.34585 | 0.33430 | 0.33360 |
0.020 | 0.3667 | 0.3515 | 0.3506 | 0.34888 | 0.33466 | 0.33379 |
0.100 | 0.5076 | 0.4362 | 0.4316 | 0.39015 | 0.33569 | 0.33418 |
7 Discussion of physical outcomes
To discuss the numerical values of the considered problem, the effects of the different types of nanofluids and the effects of the pertinent parameters namely the solid volume fraction \(\phi\), the magnetic parameter M, the Brinkmann number Br and the local Biot number Bi along with the Cu-water nanofluid on the velocity \(f^{\prime }(\eta )\) and the temperature \(\theta (\eta )\) distributions are plotted graphically. Subsequently, the computational values of the wall shear stress \(f^{\prime \prime }(0)\) and the wall heat flux \(\theta ^{\prime }(0)\) for the impacts of the different nanofluids as well as the specified parameters are given via table. It is also noted that due to find the impact of anyone specified parameter, all remaining controlling parameters are taken constant.
Computed values of \(f^{\prime \prime }(0)\) and \(\theta ^{\prime }(0)\) for distinct types of nanofluids and several values of specified parameters when \(Pr=6.2\)
Nanofluids | \(\phi\) | M | Br | Bi | \(f^{\prime \prime }(0)\) | \(-\,\theta ^{\prime }(0)\) |
---|---|---|---|---|---|---|
Ag-water | 0.07 | 0.01 | 0.1 | 0.1 | 0.40348 | 0.0835793 |
Cu-water | 0.39077 | 0.0836163 | ||||
\(\text {TiO}_2\)-water | 0.35026 | 0.0834815 | ||||
\(\text {Al}_2\text {O}_3\)-water | 0.34770 | 0.0833913 | ||||
Cu-water | 0.01 | 0.35439 | 0.0843381 | |||
0.04 | 0.37516 | 0.0840001 | ||||
0.10 | 0.40207 | 0.0831959 | ||||
0.07 | 1.00 | 1.06816 | 0.0801331 | |||
2.00 | 1.46935 | 0.0775452 | ||||
3.00 | 1.78315 | 0.0754303 | ||||
0.01 | 1.0 | 0.39077 | 0.0636384 | |||
2.0 | 0.0414407 | |||||
3.0 | 0.0192430 | |||||
0.1 | 0.5 | 0.2668798 | ||||
1.0 | 0.3675850 | |||||
10.0 | 0.5566177 |
Finally, in the previous studies such as Chaudhary and Choudhary [8], and Chaudhary et al. [36], the magnetohydrodynamic flows of ordinary fluid have been examined and Galerkin finite element method applied to find the solution of the governing equations. Whereas the magnetohydrodynamic flow of various types of nanofluids have been explored by using the Keller-Box scheme in the present analysis. From these illustrations, it can be observed that nanofluids flow gives the comparatively better performance of the flow system than the ordinary fluids.
8 Conclusions
- (1)
The velocity profile and the surface gradient are higher for Ag-water nanofluid than remaining nanofluids like Cu-water, \(\text {TiO}_2\)-water and \(\text {Al}_2\text {O}_3\)-water respectively. Whereas, \(\text {Al}_2\text {O}_3\)-water nanofluid has higher improvement on the temperature profile and the surface heat flux than other nanofluids.
- (2)
The momentum boundary layer, the thermal boundary layer, the local skin friction coefficient and the local Nusselt number increase as the solid volume fraction and the magnetic parameter develop, while opposite phenomenon occurs in the thermal boundary layer for increment in the magnetic parameter when \(\eta >1\).
- (3)
In case of enhancement in the Brinkmann number and the local Biot number, the fluid temperature as well as the heat transfer rate step-up, while reverse trend is happened in the heat transfer rate for the booming value of the local Biot number.
Notes
Funding
This study was funded by Malaviya National Institute of Technology Jaipur (Grant Number F.4.R (Ph.D.) Acdm/MNIT/2016/5499).
Compliance with ethical standards
Conflict of interest
Authors declare that they have no conflict of interest.
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