- Original Article
- Open Access
MHD squeeze flow and heat transfer of a nanofluid between parallel disks with variable fluid properties and transpiration
- K. Vajravelu^{1}Email author,
- K. V. Prasad^{2},
- Chiu-On Ng^{3} and
- Hanumesh Vaidya^{2}
https://doi.org/10.1186/s40712-017-0076-4
© The Author(s). 2017
- Received: 5 December 2016
- Accepted: 15 February 2017
- Published: 9 March 2017
Abstract
Background
The purpose of the study is to investigate the effects of variable fluid properties, the velocity slip and the temperature slip on the time-dependent MHD squeezing flow of nanofluids between two parallel disks with transpiration.
Methods
The boundary layer approximation and the small magnetic Reynolds number assumptions are used. The non-linear governing equations with appropriate boundary conditions are initially cast into dimensionless form by using similarity transformations and then the resulting equations are solved analytically via Optimal Homotopy Analysis Method (OHAM). A detailed parametric analysis is carried out through plots and tables to explore the effects of various physical parameters on the velocity temperature and nanoparticles concentration fields.
Results
The velocity distribution profiles for transpiration (suction/blowing) are parabolic in nature. In general, at the central region, these profiles exhibit the cross-flow behavior and also exhibit the dual behavior with the increase in the pertinent parameters. The temperature distribution reduces in the case of suction whereas the reverse trend is observed in the case of injection.
Conclusion
The effects of temperature dependent thermophysical properties are significant on the flow field. For higher values of the fluid viscosity parameter, the velocity field increases near the walls. However, the transpiration effects are dominant and exhibit the cross-flow behavior as well as the dual behavior. The temperature and the concentration fields are respectively the increasing functions of the variable thermal conductivity and the variable species diffusivity parameters.
Keywords
- Squeeze flow
- Nanofluid
- Wall slip
- OHAM
- Transpiration
Background
Squeeze flow has promising applications in engineering and industrial processes such as bio-mechanics, flow through arteries, food processing, polymer processing, compression, injection modeling, and mechanical, industrial, and chemical engineering. Generally, squeeze flows are generated in many hydro-dynamical machines and tools where vertical velocities or normal stresses are applied due to moving boundary. Stefan (1874) initiated the study by considering the squeeze flow of a Newtonian fluid with lubrication approximation. Leider and Bird (1974) and Hamza (1988) extended the pioneering work of Stefan (1874) for flow between two parallel disks. Domairry and Aziz (2009) performed a comprehensive analysis of an electrically conducting fluid between two parallel disks of which the upper disk is impermeable and the lower disk is permeable. Joneidi et al. (2010) analyzed the effects of suction/injection and squeeze Reynolds number on the magnetohydrodynamic flow between two parallel disks. Hayat et al. (2012) extended the work of Domairry and Aziz (2009) to second-grade fluid using HAM homotopy analysis method. Further, Hussain et al. (2012) examined the unsteady MHD flow and heat transfer of a viscous fluid squeezed between two parallel disks. Furthermore, Shaban et al. (2013) reported the time-dependent squeeze MHD flow between two parallel disks via a new hybrid method based on the Tau method and the homotopy analysis method.
In recent years, manufacturing industries have begun to choose the fluids based on the heat transmission property: This has considerable implications in the performance of many devices such as in air-conditioning, electronic, chemical, and power. The traditional fluids such as water, ethylene glycol, mineral oils have limited heat transfer ability. Choi (1995) coined the word “nanofluid” for the fluids with suspended nanometer sized (10^{−9} nm) particles called nanoparticles, which were dispersed in traditional fluids. The distinctive nature of the nanofluid, higher thermal conductivity at low nanoparticle aggregation, strong temperature-dependent thermal conductivity, and non-linear increase in thermal conductivities are more useful in industrial and engineering applications which include nuclear reaction cooling, geothermal power extraction, automobile fuels, radiator cooling, cooling of electronic devices, smart fluids, and in bio- and pharmaceutical industry. Motivated by the aforementioned applications, numerous researchers have engaged in the discussion of flows of nanofluids through different geometries (see Vajravelu et al. 2011; Makinde and Aziz 2011; Bachok et al. 2012; Safaei et al. 2014; Safaei et al. 2014; Goodarzi et al. 2014; Togun et al. 2014; Prasad et al. 2016). The MHD squeezing flow of nanofluid between parallel disks is studied by Hashmi et al. (2012). Recently, Das et al. (2016) presented a numerical study on the squeeze flow of a nanofluid between two parallel disks in the presence of a magnetic field considering slip effects. Mohyud-Din et al. (2016) considered both velocity and temperature slip effects on squeeze MHD flow of a nanofluid between parallel disks.
All the above studies analyzed the characteristics of nanofluid squeeze flows assuming constant thermo-physical properties. However, several researchers (see Lai and Kulacki (1990), Vajravelu et al. (2013), Prasad et al. (2016)) have shown that the thermo-physical properties of the ambient fluid may change with temperature, especially the fluid viscosity and the fluid thermal conductivity. For lubricating fluids, heat generated by internal friction and the corresponding rise in the temperature affects the physical properties of the nanofluid, and hence the properties of the fluid are no longer assumed to be constant. The increase in temperature leads to an increase in the transport phenomena and so the heat transfer at the wall is also affected. The experimental results show that even a very low volume fraction of nanoparticles can significantly affect the thermo-physical properties of a nanofluid (see Vajjha and Das (2012)). Therefore, to predict the flow, heat, and mass transfer rates, it is necessary to take the variable fluid properties into account. Thus, the purpose here is to investigate the effects of variable fluid properties, the velocity slip, and the temperature slip on the time-dependent MHD squeezing flow of nanofluids between two parallel disks with transpiration. The governing partial differential equations are transformed into a set of ordinary differential equations: The non-linear coupled systems of equations have been solved for various values of sundry parameters via an efficient analytical method, optimal homotopy analysis method (OHAM) (for details see Liao (2003), Fan and You (2013)). The analysis revels that the fluid flow is appreciably influenced by the physical parameters. It is expected that the results presented here will not only provide useful information for industrial applications but also complement the existing literature.
Mathematical formulation
Similarity equations for nanofluid model
Methods
Semi-analytical solution: optimal homotopy analysis method (OHAM)
The governing equations are highly non-linear, coupled ODEs with variable coefficients. We use optimal homotopy analysis method (OHAM) to obtain appropriate analytic solutions for Eqs. (10)–(12) with associated boundary conditions (13). The OHAM is based on the homotopy concept from topology. In this regard, a non-linear problem is transformed into an infinite number of linear sub-problems. In the framework of OHAM, we have great freedom to choose the auxiliary linear operators and initial approximations. This is advantageous over other iterative techniques, where convergence is largely tied to a good initial approximation of the solution. The OHAM differs from other analytic approximation methods as it does not depend on small or large physical parameters. This is achieved by inclusion of an artificial “convergence control parameter,” which guarantees convergence of the solution series. The OHAM has been successfully applied to a wide variety of non-linear problems (for details see Liao (2003) Fan and You (2013)). Optimal homotopy analysis method is employed to solve the system of non-linear Eqs. (10)–(12) with boundary conditions (13).
\( {N}_f\left[\widehat{f}\left(\eta, 1\right),\widehat{\theta}\left(\eta, 1\right)\right]=0,{N}_{\theta}\left[\widehat{\theta}\left(\eta, 1\right),\widehat{f}\left(\eta, 1\right),\widehat{\phi}\left(\eta, 1\right)\right]=0,\ \mathrm{and}\ {N}_{\phi}\left[\widehat{\phi}\left(\eta, 1\right),\widehat{f}\left(\eta, 1\right),\widehat{\theta}\left(\eta, 1\right)\right]=0 \) which implies that \( \widehat{f}\left(\eta, 1\right)= f\left(\eta \right) \), \( \widehat{\theta}\left(\eta, 1\right)=\theta \left(\eta \right) \), and \( \widehat{\phi}\left(\eta, 1\right)=\phi \left(\eta \right) \) respectively.
Optimal convergence control parameter
Error analysis
Comparison of f″(1) and CPU time (s) incurred to evaluate the mth order approximation by exact residual error and average residual error when M = Pr = 1, S = Nb = Nt = 0.5, Le = 2, A = 0.01, γ = ε _{1} = ε _{2} = 0.1, θ _{ r } = − 5.0, andβ = 0.01.
Order m | Using exact residual error | Using average residual error | ||
---|---|---|---|---|
− f″(1) | CPU time (s) | − f″(1) | CPU time (s) | |
1 | 2.87099 | 9.83 | 2.87063 | 0.61 |
2 | 2.86492 | 26.86 | 2.86492 | 2.14 |
3 | 2.84464 | 73.93 | 2.84498 | 5.94 |
4 | 2.84315 | 1234.25 | 2.84332 | 14.43 |
Values of convergence control parameters \( {\overset{\frown }{h}}_f,{\overset{\frown }{h}}_{\theta},\mathrm{and}\;{\overset{\frown }{h}}_{\phi} \) and the corresponding total residual errors \( {E}_m^t \) for different orders of approximation m with S = M = 0.1, Pr = 0.72, γ = 0.1, Nb = Nt = 0.1, Le = 1, A = 0.01, ε _{1} = ε _{2} = 0.1, θ _{ r } = − 10.0, β = 0.1.
m | \( -{\overset{\frown }{h}}_f \) | \( -{\overset{\frown }{h}}_{\theta} \) | \( -{\overset{\frown }{h}}_{\phi} \) | \( {E}_m^t \) | CPU time (s) |
---|---|---|---|---|---|
1 | 0.855974 | 0.913900 | 0.213443 | 2.51 × 10^{−1} | 1.14 |
2 | 0.914521 | 0.894988 | 0.120164 | 1.08 × 10^{−2} | 5.34 |
3 | 0.905082 | 0.873693 | 0.917802 | 2.98 × 10^{−4} | 14.92 |
4 | 0.886267 | 0.835341 | 0.896250 | 5.54 × 10^{−6} | 38.95 |
5 | 0.858155 | 0.913425 | 0.876598 | 2.40 × 10^{−7} | 72.10 |
6 | 0.821699 | 0.919399 | 0.918880 | 3.42 × 10^{−8} | 74.52 |
Individual average residual error with S = M = 0.1, Pr = 0.72, γ = 0.1, Nb = Nt = 0.1, Le = 1, A = 0.01, ε _{1} = ε _{2} = 0.1, θ _{ r } = − 10.0, β = 0.1.
m | \( {E}_m^f \) | \( {E}_m^{\theta} \) | \( {E}_m^{\phi} \) | CPU time (s) |
---|---|---|---|---|
1 | 1.85 × 10^{−1} | 4.24 × 10^{−3} | 1.09 × 10^{−1} | 0.17 |
2 | 1.02 × 10^{−2} | 1.15 × 10^{−4} | 7.63 × 10^{−3} | 1.33 |
3 | 5.44 × 10^{−4} | 4.05 × 10^{−6} | 2.06 × 10^{−4} | 4.84 |
4 | 1.94 × 10^{−5} | 1.82 × 10^{−7} | 8.31 × 10^{−6} | 13.63 |
5 | 3.64 × 10^{−7} | 7.83 × 10^{−9} | 4.67 × 10^{−7} | 33.22 |
6 | 4.13 × 10^{−9} | 2.58 × 10^{−10} | 2.68 × 10^{−8} | 71.22 |
7 | 2.55 × 10^{−9} | 7.23 × 10^{−12} | 1.01 × 10^{−9} | 139.98 |
8 | 5.32 × 10^{−10} | 1.62 × 10^{−13} | 3.31 × 10^{−11} | 280.03 |
9 | 6.70 × 10^{−11} | 5.94 × 10^{−15} | 8.03 × 10^{−13} | 454.01 |
10 | 6.44 × 10^{−12} | 3.05 × 10^{−16} | 2.73 × 10^{−14} | 743.12 |
Results and discussion
Comparison of the values of the of skin friction coefficient for various values of Mn and S with A = 2, Pr = Le = Nb = 1, θ _{ r } → ∞, ε _{1} = ε _{2} = β = γ = Nt = 0.
Mn | S | Hashmi et al. (Hashmi et al. 2012) | Present Study | CPU time in seconds | ||
---|---|---|---|---|---|---|
− f″(1) | \( -{\overset{\frown }{h}}_f \) | \( {E}_{10}^f \) | ||||
0 | 1 | 7.53316579 | 7.533166134027173 | 0.9677134841537185 | 1.734689146770528 × 10^{− 7} | 12.0019918 |
2 | 8.26387231 | 8.263872005406636 | 0.9704415447020044 | 1.803753433567724 × 10^{− 7} | 11.4906528 | |
3 | 9.09732572 | 9.097325884047699 | 0.9714084422981315 | 2.017660259031556 × 10^{− 7} | 11.3115338 | |
5 | 11.3492890 | 11.349079068376334 | 0.9671141942979257 | 3.832529114797568 × 10^{− 7} | 12.2261407 | |
1 | 0.1 | 8.97552394 | 8.975523936918258 | 0.6586335149025293 | 3.901356871103989 × 10^{− 6} | 11.6837806 |
0.5 | 8.34924578 | 8.34924579267211 | 0.8848994230900776 | 5.175581204948535 × 10^{− 7} | 11.4095983 | |
1 | 7.72194601 | 7.721945958968551 | 0.9686906697036372 | 1.753316413825429 × 10^{− 7} | 11.1624343 | |
2 | 6.94077326 | 6.940264738765628 | 0.7930138508564271 | 1.820417866243763 × 10^{− 3} | 11.1854494 |
Comparison of the values of the local Nusselt number for various values of Nb and Nt with A = 2, Mn = Pr = S = Le = 1, θ _{ r } → ∞, ε _{1} = ε _{2} = β = γ = 0
Nb | Nt | Hashmi et al. (2012) | Das et al.(2016) | Present Study | CPU time of the system in seconds | ||
---|---|---|---|---|---|---|---|
− θ′(1) | \( -{\overset{\frown }{h}}_{\theta} \) | \( {E}_{15}^{\theta} \) | |||||
0.1 | 0.1 | 0.52628540 | 0.526285397692707 | 0.5262854795861431 | 0.908516 | 1.412748795457583 × 10^{−11} | 54.439 |
0.5 | 0.63433253 | 0.526285397692707 | 0.6343325523545158 | 0.914084 | 1.52497172481667 × 10^{−13} | 54.525 | |
1.0 | 0.78636385 | 0.634332530012476 | 0.7863638216341705 | 0.928943 | 9.805935149902197 × 10^{−13} | 54.009 | |
1.5 | 0.95569955 | 0.955699547716439 | 0.9556995220292246 | 0.918411 | 7.022421405151564 × 10^{−12} | 54.625 | |
1.5 | 0.5 | 1.17682119 | 1.176821184883260 | 1.1768208205076784 | 0.956179 | 2.358155300119884 × 10^{−10} | 53.940 |
1 | 1.48581207 | 1.485811936635642 | 1.4858197859422315 | 1.017842 | 1.217973748574313 × 10^{−8} | 53.585 | |
1.5 | 1.82305276 | 1.823053529110100 | 1.8231897567384212 | 0.953137 | 1.724206185011289 × 10^{− 6} | 54.439 | |
2 | 2.17915991 | 2.179227931099257 | 2.1772753199939645 | 0.817733 | 7.893665960448466 × 10^{− 4} | 57.834 |
Comparison of the values of the local Sherwood number for various values of Nb and Nt with A = 2, Mn = Pr = S = Le = 1, θ _{ r } → ∞, ε _{1} = ε _{2} = β = γ = 0.
Nb | Nt | Hashmi et al. (2012) | Das et al.(2016) | Present study | CPU time of the system in seconds | ||
---|---|---|---|---|---|---|---|
− ϕ′(1) | \( -{\overset{\frown }{h}}_{\phi} \) | \( {E}_{15}^{\phi} \) | |||||
0.1 | 0.1 | 0.86604666 | 0.866046665141329 | 0.8660467842530996 | 0.936748 | 2.426680292036448 × 10^{− 10} | 54.439 |
0.5 | 0.53012814 | 0.530128143896286 | 0.5301281821145534 | 0.928890 | 9.992908438598528 × 10^{− 13} | 54.525 | |
1.0 | 0.48603919 | 0.486039186120291 | 0.48603915346137855 | 0.891607 | 4.403963236292989 × 10^{− 13} | 54.009 | |
1.5 | 0.46986157 | 0.469861566526085 | 0.4698615495700593 | 0.884253 | 3.657227369041722 × 10^{− 13} | 54.625 | |
1.5 | 0.5 | 0.40180718 | 0.401807177532398 | 0.40180541220583815 | 0.908420 | 2.420566500000295 × 10^{− 10} | 53.940 |
1 | 0.12619334 | 0.126193335885242 | 0.12617433001495504 | 0.949699 | 2.832474925670632 × 10^{− 8} | 53.585 | |
1.5 | 0.39083080 | 0.390839865635371 | 0.3900777099943474 | 0.865141 | 6.281273402797455 × 10^{− 6} | 54.439 | |
2 | 1.16777723 | 1.16800852390237 | 1.1587904533044613 | 0.912246 | 3.5954596655078387 × 10^{− 4} | 57.834 |
Values of skin friction, Nusselt number, and Sherwood number for different physical parameters with γ = β = 0.1, Nt = Nb = 0.1, S = 0.1, A = 0.01.
Pr | Le | ε _{2} | ε _{1} | θ _{ r } | Mn | − f″(1) | \( -{\overset{\frown }{h}}_f \) | \( {E}_{10}^f \) | − θ′(1) | \( -{\overset{\frown }{h}}_{\theta} \) | \( {E}_{10}^{\theta} \) | − ϕ′(1) | \( -{\overset{\frown }{h}}_{\phi} \) | \( {E}_{10}^{\phi} \) | CPU time |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
6.2 | 1 | 0.1 | 0.1 | −10 | 0.5 | 6.93724 | 0.521876 | 1.85 × 10^{−5} | 2.56836 | 0.795575 | 3.93 × 10^{−6} | −0.20702 | 0.816672 | 6.56 × 10^{−5} | 452.31 |
1 | 7.16163 | 0.529837 | 1.90 × 10^{−5} | 2.56728 | 0.795370 | 3.87 × 10^{−6} | −0.20606 | 0.816681 | 6.49 × 10^{−5} | 417.61 | |||||
1.5 | 7.54951 | 0.540578 | 2.20 × 10^{−5} | 2.56541 | 0.794987 | 3.77 × 10^{−6} | −0.20439 | 0.816682 | 6.37 × 10^{−5} | 466.91 | |||||
2 | 8.12409 | 0.551879 | 2.94 × 10^{−5} | 2.56262 | 0.794530 | 3.16 × 10^{−6} | −0.20191 | 0.816656 | 6.21 × 10^{−5} | 415.70 | |||||
2.5 | 8.92326 | 0.561811 | 4.11 × 10^{−5} | 2.55874 | 0.793334 | 3.42 × 10^{−6} | −0.19845 | 0.816579 | 5.99 × 10^{−5} | 417.18 | |||||
6.2 | 1 | 0.1 | 0.1 | −10 | 0.1 | 3.23547 | 0.539759 | 3.88 × 10^{−6} | 2.61368 | 0.800566 | 4.60 × 10^{−6} | −0.24769 | 0.820601 | 7.36 × 10^{−5} | 445.34 |
−5 | 6.37108 | 0.796477 | 1.40 × 10^{−4} | 2.57160 | 0.796836 | 4.08 × 10^{−6} | −0.20998 | 0.817371 | 6.64 × 10^{−5} | 404.28 | |||||
−1 | 3.66322 | 0.344926 | 4.32 × 10^{−2} | 2.59901 | 0.400879 | 1.80 × 10^{−4} | −0.23434 | 0.538266 | 3.87 × 10^{−4} | 377.96 | |||||
6.2 | 1 | 0.1 | 0 | −10 | 0.1 | 6.88538 | 0.83426 | 2.12 × 10^{−6} | 4.37052 | 0.753998 | 2.89 × 10^{−6} | −0.05358 | 0.838724 | 5.27 × 10^{−6} | 413.24 |
0.2 | 6.84778 | 0.563282 | 1.20 × 10^{−5} | 3.29095 | 0.747359 | 7.96 × 10^{−5} | −0.38818 | 0.772511 | 2.83 × 10^{−4} | 416.86 | |||||
0.4 | 6.80107 | 0.690673 | 6.36 × 10^{−4} | 2.76736 | 0.689505 | 4.11 × 10^{−4} | −0.84408 | 0.694220 | 7.80 × 10^{−4} | 416.80 | |||||
0.8 | 6.66595 | 0.605364 | 0.602742 | 2.40018 | 0.386541 | 4.20245 | 1.00077 | 0.003287 | 1.63 × 10^{−4} | 536.38 | |||||
6.2 | 1 | 0 | 0.1 | −10 | 0.1 | 6.87282 | 0.817156 | 2.97 × 10^{−5} | 2.53070 | 0.787063 | 1.65 × 10^{−6} | −0.22122 | 0.827749 | 4.33 × 10^{−5} | 407.73 |
0.2 | 6.86217 | 0.528059 | 1.83 × 10^{−5} | 2.60736 | 0.795988 | 8.99 × 10^{−6} | −0.19379 | 0.805701 | 1.08 × 10^{−4} | 461.73 | |||||
0.4 | 6.85327 | 0.780916 | 1.10 × 10^{−4} | 2.68649 | 0.785656 | 3.00 × 10^{−5} | −0.16792 | 0.782738 | 3.30 × 10^{−4} | 384.13 | |||||
0.8 | 6.83347 | 0.580338 | 2.35 × 10^{−5} | 2.85208 | 0.739950 | 1.48 × 10^{−4} | −0.12115 | 0.716450 | 3.26 × 10^{−3} | 416.60 | |||||
6.2 | 1 | 0.1 | 0.1 | −10 | 0.1 | 6.86664 | 0.519571 | 1.84 × 10^{−5} | 2.56923 | 0.795649 | 3.95 × 10^{−6} | −0.20845 | 0.816398 | 6.51 × 10^{−5} | 429.74 |
5.0 | 6.86634 | 0.522037 | 1.71 × 10^{−5} | 2.57158 | 0.795701 | 3.95 × 10^{−6} | −0.21346 | 0.815193 | 6.23 × 10^{−5} | 444.22 | |||||
10.0 | 6.86703 | 0.806771 | 4.15 × 10^{−5} | 2.57445 | 0.795678 | 3.96 × 10^{−6} | −0.21969 | 0.813673 | 5.88 × 10^{−5} | 435.78 | |||||
25.0 | 6.86487 | 0.535357 | 1.18 × 10^{−5} | 2.58273 | 0.795091 | 4.05 × 10^{−6} | −0.23816 | 0.809001 | 4.96 × 10^{−5} | 436.47 | |||||
0.72 | 1 | 0.1 | 0.1 | −10 | 0.1 | 7.02711 | 0.910736 | 4.87 × 10^{−15} | 1.43830 | 0.913630 | 2.82 × 10^{−16} | 0.86420 | 0.918855 | 2.80 × 10^{−14} | 380.47 |
1.09 | 7.01781 | 0.896013 | 8.61 × 10^{−14} | 1.49661 | 0.908669 | 2.18 × 10^{−14} | 0.80733 | 0.905835 | 6.45 × 10^{−13} | 391.07 | |||||
2.0 | 6.99425 | 0.869605 | 2.11 × 10^{−11} | 1.64904 | 0.830756 | 2.05 × 10^{−11} | 0.65997 | 0.877078 | 1.20 × 10^{−10} | 848.27 | |||||
5.09 | 6.90493 | 0.826361 | 2.89 × 10^{−6} | 2.28346 | 0.796751 | 2.40 × 10^{−7} | 0.05966 | 0.832669 | 4.16 × 10^{−6} | 552.53 | |||||
6.2 | 6.86798 | 0.809634 | 4.61 × 10^{−5} | 2.56870 | 0.795627 | 3.95 × 10^{−6} | −0.20732 | 0.816668 | 6.58 × 10^{−5} | 534.81 |
Velocity field
The effect of θ _{ r }, Mn, S, and β on the velocity distribution f′(η) for transpiration (suction/blowing) is elucidated in Fig. 2a–d. Here, the profiles are parabolic in nature. In general, at the central region, these profiles exhibit the cross-flow behavior and also exhibit the dual behavior with the increase in the pertinent parameters. From Fig. 2a it is clear that, for higher values of θ _{ r }, the velocity profile increases near the walls where the transpiration effects are dominant when η ≤ 0.44, η ≤ 0.46, andη ≤ 0.49. However, for η > 0.44, η > 0.46, andη > 0.49, the velocity profiles decreases as θ _{ r } → 0. This may be attributed to the fact that, for a given base fluid (air or water), when ζ is fixed, lesser θ _{ r } implies higher temperature difference between the wall and the ambient fluid. Hence, the results explicitly manifest that θ _{ r } is the indicator of the variation of fluid viscosity with temperature which has a substantial effect on f′(η) and hence on f″(1) (see Table 7 for details). The effect of Mn on f′(η) is demonstrated in Fig. 2b. It is observed that, for the higher values of Mn, the fluid velocity increase and thereby decrease the thickness of the momentum boundary layer near the wall region, for absence of suction/injection parameter and 0.23 ≥ η ≥ 0.67 for suction) However, the reverse trend is observed in the central region. This dual behavior of the flow is due to the mass conservation constraint (see Lawal and Kalyon (1998)). Figure 2c depicts the behavior of S on f′(η) It is seen that f′(η) decreases with an increase in S at the central region for suction (0.32 ≤ η ≤ 0.74) and the reverse trend is recorded in the case of injection. Physically, in the case of suction, there will be an increase in h(t) which will cause it to decrease in velocity and the increase in squeezing parameter. Further, when η ≤ 0.32 or η ≤ 0.74, that is the neighborhood of 0.32 and 0.74, there will be a corresponding decrease in h(t) which in turn will increase in both fluid velocity and squeezing parameter. The effect of β on f′(η) is shown in Fig. 2d. The behavior of f′(η) at the surface of the disks is zero when β = 0 (no slip at the wall). For the suction flow, an increase in β strengthens the f′(η) near the wall, that is for η ≥ 0.16 and η ≥ 0.82, while at the central region, rising β results in weakening of f′(η). However, for the case of injection, the effect of f′(η) is opposite to that accounted for suction flow. Physically, in the case of injection, an increase in f′(η) near the wall region with increasing \( \beta \) gives rise to a decrease in velocity gradient at there. However, when the mass flow rate is kept constant, an increase in the fluid velocity near the wall region will be remunerated by an analogous fall in the fluid velocity near the mid region so that mass conservation limitation will not be dishonored (see ref. (Lawal and Kalyon 1998)).
Temperature distribution and nanoparticle volume distribution
In Table 7, we present the results of f″(1), θ′(1), and ϕ′(1) for several sets of values of the physical parameters. Increase in the magnetic parameter reduces the skin friction but increases the Nusselt number as well as the Sherwood number. Furthermore, an increase in the variable thermal conductivity parameter and the variable species diffusivity parameter results in an increase in the Nusselt number and the Sherwood number.
Conclusions
In this paper, an analytical technique is used to solve the mathematical model of MHD squeeze flow of nanofluid between parallel disks. This analysis gives unified results for the parameters Mn, S, Nt, and Nb, from which one can obtain the results for the special cases of Hashmi et al. (2012) and Das et al. (2016). The effects of temperature-dependent thermo-physical properties are significant on the flow field. For higher values of the fluid viscosity parameter, the velocity field increases near the walls. However, the transpiration effects are dominant and exhibit the cross-flow behavior as well as the dual behavior. The temperature and the concentration fields are respectively the increasing functions of the variable thermal conductivity and the variable species diffusivity parameters.
Nomenclature
A suction/injection parameter
a constant in Eq. (6)
C nanoparticle volumetric fraction
C _{ w } nanoparticle concentration at the lower disk
T temperature (K)
T _{ r } constant in Eq. (6)
T _{ w } temperature at the lower disk (K)
T _{∞} ambient temperature (K)
T _{ m } mean fluid temperature (K)
ΔT temperature difference (K)
D _{ T } thermophoretic diffusion coefficient (kg/ms K)
D _{ B ∞} Brownian diffusion coefficient (kg/ms)
B _{0} uniform magnetic field (Tesla)
C _{ p } specific heat at constant pressure (J/kg K)
C _{ fr } skin friction
h(t) variable distance = H(1 − αt)^{1/2}
K(T) temperature dependent thermal conductivity (W/m K)
K _{∞} thermal conductivity of the fluid far away from the sheet (W/m K)
Le Lewis number
Mn magnetic parameter
Nb Brownian motion parameter
Nt thermophoresis parameter
N _{ ur } reduced Nusselt number
Pr Prandtl number
Re_{x} local squeeze Reynolds number
S _{ hr } reduced Sherwood number
S squeezing parameter
r, u , and w radial and axial velocities (m/s)
Greek symbols
α characteristic parameter
ζ constant defined in equation (6)
ν _{∞} kinematic viscosity away from the sheet (kg/m^{3})
ρ _{∞} constant fluid density (kg/m^{3})
ρ _{ f } density of the fluid (kg/m^{3})
σ electric conductivity
ε _{1} variable thermal conductivity parameter
ε _{2} variable species diffusivity parameter (m^{2}/s)
η similarity variables
θ dimensionless temperature
ϕ dimensionless concentration
θ _{ r } fluid viscosity parameter
μ dynamic viscosity (Pa s)
μ _{ h } fluid viscosity of the fluid at the upper disk (Pa s)
k _{ h } thermal conductivity of the fluid at the upper disk
D _{Bh} species diffusivity/Brownian diffusion coefficient of the fluid at the upper disk
μ _{∞} constant value of dynamic viscosity (Pa s)
τ ratio between the effective heat capacity of the nanoparticle material and heat capacity of the fluid
τ _{ w } wall shear stress
ψ stream function
β velocity slip parameter
γ thermal slip parameter
Subscripts
∞ condition at infinity
w condition at the wall
′ differentiation with respect to η
Declarations
Authors’ contributions
All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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