doi: 10.3934/dcdss.2020197

Marangoni convective flow of nanoliquid towards a riga surface

1. 

School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China

2. 

1/4B Ton Duc Thang street, My Binh ward, Long Xuyen city, An Giang province 881092, Vietnam

* Corresponding author: Anum Shafiq

Received  March 2019 Revised  May 2019 Published  December 2019

This manuscript describes the marangoni mixed convective flow over a Riga surface is investigated. Heat phenomenon is examined with thermal radiation impact. The governing PDEs are converted into a set of nonlinear ODEs using suitable transformations. The governing system is analytically solved invoking the HAM. Convergence of the obtained series solutions is explicitly discussed. Examination of different relevant parameters on the speed and temperature fields are investigated through charts. Friction coefficient and Nusselt number are arranged and talked about for dimensionless rising parameters.

Citation: Anum Shafiq, Loc Nguyen. Marangoni convective flow of nanoliquid towards a riga surface. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020197
References:
[1]

K. Arafune and A. Hirata, Thermal and solutal Marangoni convection in In-Ga-Sb system, J. Crystal Growth., 197 (1999), 811-817.   Google Scholar

[2]

K. Arafune and A. Hirata, Interactive solutal and thermal Marangoni convection in a rectangular open boat, Numer. Heat Trans. Part A, 34 (1998), 421-429.  doi: 10.1080/10407789808913995.  Google Scholar

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N. Biswas and N. K. Manna, Magneto-hydrodynamic Marangoni flow in bottom-heated lid-driven cavity, J. Molecular Liquids, 251 (2018), 249-266.  doi: 10.1016/j.molliq.2017.12.053.  Google Scholar

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H. Chen, T. L. Xiao, J.-Y. Chen and M. Shen, Effects of solid matrix and porosity of porous medium on heat transfer of Marangoni boundary layer flow saturated with power-law nanofluids, Chin. Phys. Lett., 33 (2016). doi: 0.1088/0256-307X/33/10/104401.  Google Scholar

[5]

D. Christopher and B. Wang, Marangoni convection around a bubble in microgravity, heat transfer, Proceedings of the 11$^{th}$ International Heat Transfer Conference, 3, Taylor Francis, Levittown, 1998, 489–494. Google Scholar

[6]

A. CrollW. Muller-Sebert and R. Nitsche, The critical Marangoni number for the onset of time-dependent convection in silicon, Mater. Res. Bull., 24 (1989), 995-1004.  doi: 10.1016/0025-5408(89)90184-0.  Google Scholar

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I. Da-Riva and L. G. Napolitano, Fluid physics under reduced gravity - An overview, ESA SP-191, (1983). Google Scholar

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R. EllahiA. Zeeshan and M. Hassan, Particle shape effects on Marangoni convection boundary layer flow of a nanofluid, Internat. J. Numer. Methods Heat Fluid Flow, 26 (2016), 2160-2174.  doi: 10.1108/HFF-11-2014-0348.  Google Scholar

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C. Golia and A. Viviani, Non isobaric boundary layers related to Marangoni flows, Meccanica, 21 (1986), 200-204.  doi: 10.1007/BF01556486.  Google Scholar

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T. Hayat, U. Shaheen, A. Shafiq, A. Alsaedi and S. Asghar, Marangoni mixed convection flow with Joule heating and nonlinear radiation, AIP Advances, 5 (2015). doi: 10.1063/1.4927209.  Google Scholar

[11]

T. HayatM. I. KhanM. FarooqA. Alsaedi and T. Yasmeen, Impact of Marangoni convection in the flow of carbon-water nanofluid with thermal radiation, Internat. J. Heat Mass Transfer, 106 (2017), 810-815.  doi: 10.1016/j.ijheatmasstransfer.2016.08.115.  Google Scholar

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T. HayatA. Shafiq and A. Alsaedi, Characteristics of magnetic field and melting heat transfer in stagnation point flow of Tangent-hyperbolic liquid, J. Magnetism Magnetic Materials, 405 (2016), 97-106.  doi: 10.1016/j.jmmm.2015.10.080.  Google Scholar

[13]

T. Hayat, S. Jabeen, A. Shafiq and A. Alsaedi, Radiative squeezing flow of second grade fluid with convective boundary conditions, PLoS One, 11 (2016). doi: 10.1371/journal.pone.0152555.  Google Scholar

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F. MaboodA. ShafiqT. Hayat and S. Abelman, Radiation effects on stagnation point flow with melting heat transfer and second order slip, Results Phys., 7 (2017), 31-42.  doi: 10.1016/j.rinp.2016.11.051.  Google Scholar

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L. G. Napolitano, Microgravity fluid dynamics, $2^{nd}$ Levitch Conference, Washington, 1978. Google Scholar

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L. G. Napolitano, Marangoni boundary layers, Proceedings of the 3rd European symposium on material science in space, Grenoble, 1979. Google Scholar

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A. NaseemA. ShafiqL. Zhaoa and M. U. Farooq, Analytical investigation of third grade nanofluidic flow over a riga plate using Cattaneo-Christov model, Results Phys., 9 (2018), 961-969.  doi: 10.1016/j.rinp.2018.01.013.  Google Scholar

[22]

F. Naseem, A. Shafiq, L. Zhao and A. Naseem, MHD biconvective flow of Powell Eyring nanofluid over stretched surface, AIP Advances, 7 (2017). doi: 10.1063/1.4983014.  Google Scholar

[23]

M. K. NayakS. ShawO. D. Makinde and A. J. Chamkha, Effects of homogenous-heterogeneous reactions on radiative NaCl-CNP nanofluid flow past a convectively heated vertical Riga plate, J. Nanofluids, 7 (2018), 657-667.   Google Scholar

[24]

M. K. NayakS. ShawO. D. Makinde and A. J. Chamkha, Investigation of partial slip and viscous dissipation effects on the radiative tangent hyperbolic nanofluid flow past a vertical permeable Riga plate with internal heating: Bungiorno model, J. Nanofluids, 8 (2019), 51-62.  doi: 10.1166/jon.2019.1576.  Google Scholar

[25]

D. R. V. S. R. Sastry, MHD thermosolutal Marangoni convection boundary layer nanofluid flow past a flat plate with radiation and chemical reaction, Indian J. Science Technology, 8 (2015), 1-8.  doi: 10.17485/ijst/2015/v8i13/55226.  Google Scholar

[26]

A. ShafiqZ. Hammouch and A. Turab, Impact of radiation in a stagnation point flow of Walters' B fluid towards a Riga plate, Thermal Sci. Engineering Progress, 6 (2018), 27-33.  doi: 10.1016/j.tsep.2017.11.005.  Google Scholar

[27]

A. ShafiqZ. Hammouch and T. N. Sindhu, Bioconvective MHD flow of tangent hyperbolic nanofluid with Newtonian heating, Internat. J. Mechanical Sci., 133 (2017), 759-766.  doi: 10.1016/j.ijmecsci.2017.07.048.  Google Scholar

[28]

A. Shafiq, S. Jabeen, T. Hayat and A. Alsaedi, Cattaneo-Christov heat flux model for squeezed flow of third grade fluid, Surface Rev. Lett., (2017). doi: 10.1142/S0218625X17500986.  Google Scholar

[29]

A. Shafiq and T. N. Sindhu, Statistical study of hydromagnetic boundary layer flow of Williamson fluid regarding a radiative surface, Results Phys., 7 (2017), 3059-3067.  doi: 10.1016/j.rinp.2017.07.077.  Google Scholar

[30]

S. Slavtchev and S. Miladinova, Thermocapillary flow in a liquid layer at minimum in surface tension, Acta Mechanica, 127 (1998), 209-224.  doi: 10.1007/BF01170374.  Google Scholar

[31]

J. Straub, The role of surface tension for two-phase heat and mass transfer in the absence of gravity, Experiment. Therm. Fluid Sci., 9 (1994), 253-273.  doi: 10.1016/0894-1777(94)90028-0.  Google Scholar

[32]

X. Xu and S. Chen, Cattaneo-Christov heat flux model for heat transfer of Marangoni boundary layer flow in a copper–water nanofluid, Heat Transfer-Asian Res., 46 (2017), 1-13.  doi: 10.1002/htj.21273.  Google Scholar

show all references

References:
[1]

K. Arafune and A. Hirata, Thermal and solutal Marangoni convection in In-Ga-Sb system, J. Crystal Growth., 197 (1999), 811-817.   Google Scholar

[2]

K. Arafune and A. Hirata, Interactive solutal and thermal Marangoni convection in a rectangular open boat, Numer. Heat Trans. Part A, 34 (1998), 421-429.  doi: 10.1080/10407789808913995.  Google Scholar

[3]

N. Biswas and N. K. Manna, Magneto-hydrodynamic Marangoni flow in bottom-heated lid-driven cavity, J. Molecular Liquids, 251 (2018), 249-266.  doi: 10.1016/j.molliq.2017.12.053.  Google Scholar

[4]

H. Chen, T. L. Xiao, J.-Y. Chen and M. Shen, Effects of solid matrix and porosity of porous medium on heat transfer of Marangoni boundary layer flow saturated with power-law nanofluids, Chin. Phys. Lett., 33 (2016). doi: 0.1088/0256-307X/33/10/104401.  Google Scholar

[5]

D. Christopher and B. Wang, Marangoni convection around a bubble in microgravity, heat transfer, Proceedings of the 11$^{th}$ International Heat Transfer Conference, 3, Taylor Francis, Levittown, 1998, 489–494. Google Scholar

[6]

A. CrollW. Muller-Sebert and R. Nitsche, The critical Marangoni number for the onset of time-dependent convection in silicon, Mater. Res. Bull., 24 (1989), 995-1004.  doi: 10.1016/0025-5408(89)90184-0.  Google Scholar

[7]

I. Da-Riva and L. G. Napolitano, Fluid physics under reduced gravity - An overview, ESA SP-191, (1983). Google Scholar

[8]

R. EllahiA. Zeeshan and M. Hassan, Particle shape effects on Marangoni convection boundary layer flow of a nanofluid, Internat. J. Numer. Methods Heat Fluid Flow, 26 (2016), 2160-2174.  doi: 10.1108/HFF-11-2014-0348.  Google Scholar

[9]

C. Golia and A. Viviani, Non isobaric boundary layers related to Marangoni flows, Meccanica, 21 (1986), 200-204.  doi: 10.1007/BF01556486.  Google Scholar

[10]

T. Hayat, U. Shaheen, A. Shafiq, A. Alsaedi and S. Asghar, Marangoni mixed convection flow with Joule heating and nonlinear radiation, AIP Advances, 5 (2015). doi: 10.1063/1.4927209.  Google Scholar

[11]

T. HayatM. I. KhanM. FarooqA. Alsaedi and T. Yasmeen, Impact of Marangoni convection in the flow of carbon-water nanofluid with thermal radiation, Internat. J. Heat Mass Transfer, 106 (2017), 810-815.  doi: 10.1016/j.ijheatmasstransfer.2016.08.115.  Google Scholar

[12]

T. HayatA. Shafiq and A. Alsaedi, Characteristics of magnetic field and melting heat transfer in stagnation point flow of Tangent-hyperbolic liquid, J. Magnetism Magnetic Materials, 405 (2016), 97-106.  doi: 10.1016/j.jmmm.2015.10.080.  Google Scholar

[13]

T. Hayat, S. Jabeen, A. Shafiq and A. Alsaedi, Radiative squeezing flow of second grade fluid with convective boundary conditions, PLoS One, 11 (2016). doi: 10.1371/journal.pone.0152555.  Google Scholar

[14]

S. J. Liao, Homotopy Analysis Method in Nonlinear Differential Equations, Springer, Berlin, Heidelberg, 2012. doi: 10.1007/978-3-642-25132-0.  Google Scholar

[15]

F. MaboodA. ShafiqT. Hayat and S. Abelman, Radiation effects on stagnation point flow with melting heat transfer and second order slip, Results Phys., 7 (2017), 31-42.  doi: 10.1016/j.rinp.2016.11.051.  Google Scholar

[16]

F. Mabood and W. A. Khan, Corrigendum to ``Homotopy analysis method for boundary layer flow and heat transfer over a permeable flat plate in a Darcian porous medium with radiation effects", J. Taiwan Inst. Chem. Engineers, 45 (2014). doi: 10.1016/j.jtice.2014.05.013.  Google Scholar

[17]

Y. A. Malmejac, Challenges and perspectives of microgravity research in space, ESA-BR-05, (1981). Google Scholar

[18]

R. MehmoodM. K. NayakN. S. Akbar and O. D. Makinde, Effects of thermal-diffusion and diffusion-thermo on oblique stagnation point flow of couple stress casson fluid over a stretched horizontal Riga plate with higher order chemical reaction, J. Nanofluids, 8 (2019), 94-102.   Google Scholar

[19]

L. G. Napolitano, Microgravity fluid dynamics, $2^{nd}$ Levitch Conference, Washington, 1978. Google Scholar

[20]

L. G. Napolitano, Marangoni boundary layers, Proceedings of the 3rd European symposium on material science in space, Grenoble, 1979. Google Scholar

[21]

A. NaseemA. ShafiqL. Zhaoa and M. U. Farooq, Analytical investigation of third grade nanofluidic flow over a riga plate using Cattaneo-Christov model, Results Phys., 9 (2018), 961-969.  doi: 10.1016/j.rinp.2018.01.013.  Google Scholar

[22]

F. Naseem, A. Shafiq, L. Zhao and A. Naseem, MHD biconvective flow of Powell Eyring nanofluid over stretched surface, AIP Advances, 7 (2017). doi: 10.1063/1.4983014.  Google Scholar

[23]

M. K. NayakS. ShawO. D. Makinde and A. J. Chamkha, Effects of homogenous-heterogeneous reactions on radiative NaCl-CNP nanofluid flow past a convectively heated vertical Riga plate, J. Nanofluids, 7 (2018), 657-667.   Google Scholar

[24]

M. K. NayakS. ShawO. D. Makinde and A. J. Chamkha, Investigation of partial slip and viscous dissipation effects on the radiative tangent hyperbolic nanofluid flow past a vertical permeable Riga plate with internal heating: Bungiorno model, J. Nanofluids, 8 (2019), 51-62.  doi: 10.1166/jon.2019.1576.  Google Scholar

[25]

D. R. V. S. R. Sastry, MHD thermosolutal Marangoni convection boundary layer nanofluid flow past a flat plate with radiation and chemical reaction, Indian J. Science Technology, 8 (2015), 1-8.  doi: 10.17485/ijst/2015/v8i13/55226.  Google Scholar

[26]

A. ShafiqZ. Hammouch and A. Turab, Impact of radiation in a stagnation point flow of Walters' B fluid towards a Riga plate, Thermal Sci. Engineering Progress, 6 (2018), 27-33.  doi: 10.1016/j.tsep.2017.11.005.  Google Scholar

[27]

A. ShafiqZ. Hammouch and T. N. Sindhu, Bioconvective MHD flow of tangent hyperbolic nanofluid with Newtonian heating, Internat. J. Mechanical Sci., 133 (2017), 759-766.  doi: 10.1016/j.ijmecsci.2017.07.048.  Google Scholar

[28]

A. Shafiq, S. Jabeen, T. Hayat and A. Alsaedi, Cattaneo-Christov heat flux model for squeezed flow of third grade fluid, Surface Rev. Lett., (2017). doi: 10.1142/S0218625X17500986.  Google Scholar

[29]

A. Shafiq and T. N. Sindhu, Statistical study of hydromagnetic boundary layer flow of Williamson fluid regarding a radiative surface, Results Phys., 7 (2017), 3059-3067.  doi: 10.1016/j.rinp.2017.07.077.  Google Scholar

[30]

S. Slavtchev and S. Miladinova, Thermocapillary flow in a liquid layer at minimum in surface tension, Acta Mechanica, 127 (1998), 209-224.  doi: 10.1007/BF01170374.  Google Scholar

[31]

J. Straub, The role of surface tension for two-phase heat and mass transfer in the absence of gravity, Experiment. Therm. Fluid Sci., 9 (1994), 253-273.  doi: 10.1016/0894-1777(94)90028-0.  Google Scholar

[32]

X. Xu and S. Chen, Cattaneo-Christov heat flux model for heat transfer of Marangoni boundary layer flow in a copper–water nanofluid, Heat Transfer-Asian Res., 46 (2017), 1-13.  doi: 10.1002/htj.21273.  Google Scholar

Figure 1.  Physical diagram
Figure 2.  $ \hbar $-curve of $ f^{\prime \prime \prime }\left( 0\right) $ at $ 12^{th} $ order of approximation
Figure 3.  $ \hbar $-curve of $ \theta ^{\prime }\left( 0\right) $ at $ 12^{th} $ order of approximation
Figure 4.  Influence of $ Q $ on $ f^{\prime }\left( \eta \right) $
Figure 5.  Influence of $ r $ on $ f^{\prime }\left( \eta \right) $
Figure 6.  Influence of $ \phi $ on $ f^{\prime }\left( \eta \right) $
Figure 7.  Influence of $ \gamma $ on $ f^{\prime }\left( \eta \right) $
Figure 8.  Influence of $ R_{d} $ on $ \theta \left( \eta \right) $
Figure 9.  Influence of $ \phi $ on $ \theta \left( \eta \right) $
Figure 10.  Influence of $ r $ on $ \theta \left( \eta \right) $
Figure 11.  Influence of $ Q $ on $ \theta \left( \eta \right) $
Table 1.  Numerical values of thermophysical properties of base fluid and nanoparticles [28]
Physical Properties Base fluid Nanoparticles
Water $ Cu $ $ Al_{2}O_{3} $
$ \rho \left( kg/m^{3}\right) $ 997 8933 3970
$ c_{p}\left( J/kgK\right) $ 4179 385 765
$ k\left( W/mK\right) $ 0.613 400 40
Physical Properties Base fluid Nanoparticles
Water $ Cu $ $ Al_{2}O_{3} $
$ \rho \left( kg/m^{3}\right) $ 997 8933 3970
$ c_{p}\left( J/kgK\right) $ 4179 385 765
$ k\left( W/mK\right) $ 0.613 400 40
Table 2.  Convergence of homotopy solutions when $ \alpha = 0.01, $ $ \beta = 0.1, $ $ \gamma = 0.1, $ $ \delta = 0.3, $ $ M = 0.1, $ $ {Re} = 2, $ $ \Pr = 6.2, $ $ Ec = 0.1 $ and $ \hbar _{f} = \hbar _{\theta } = -0.7. $
Order of approximations $ f^{\prime \prime \prime }\left( 0\right) $ $ -\theta ^{\prime }(1) $
$ Cu $ $ AI_{2}O_{3} $ $ Cu $ $ AI_{2}O_{3} $
1 0.19230 0.1306 1.186 0.7009
2 0.19040 0.1067 1.259 0.6466
5 0.16050 0.04472 1.367 0.5566
8 0.12480 0.02153 1.433 0.5130
10 0.10030 0.01245 1.472 0.4852
15 0.05241 0.01139 1.562 0.3787
20 0.07751 0.01139 1.892 0.3787
27 0.07751 0.01139 1.892 0.3787
30 0.07751 0.01139 1.892 0.3787
35 0.07751 0.01139 1.892 0.3787
40 0.07751 0.01139 1.892 0.3787
Order of approximations $ f^{\prime \prime \prime }\left( 0\right) $ $ -\theta ^{\prime }(1) $
$ Cu $ $ AI_{2}O_{3} $ $ Cu $ $ AI_{2}O_{3} $
1 0.19230 0.1306 1.186 0.7009
2 0.19040 0.1067 1.259 0.6466
5 0.16050 0.04472 1.367 0.5566
8 0.12480 0.02153 1.433 0.5130
10 0.10030 0.01245 1.472 0.4852
15 0.05241 0.01139 1.562 0.3787
20 0.07751 0.01139 1.892 0.3787
27 0.07751 0.01139 1.892 0.3787
30 0.07751 0.01139 1.892 0.3787
35 0.07751 0.01139 1.892 0.3787
40 0.07751 0.01139 1.892 0.3787
Table 3.  Numerical values of Nusselt number $ {Re}_{x}Nu_{x}.$
$r$ $\gamma $ $Q$ $\phi $ $R$ ${Re} _{x}Nu $
$Cu$ $AI_{2}O_{3}$
0.1 0.2 0.5 0.5 0.5 2.0910 4.8872
0.2 1.8690 2.7106
0.3 0.9587 1.2883
0.2 0.0 0.5 0.5 0.5 2.5600 2.0971
0.1 1.0830 1.8846
0.2 1.0490 0.9505
0.3 0.4 0.0 0.5 0.5 0.5418 0.6415
0.2 1.4470 0.8727
0.4 1.5960 0.9666
0.3 0.2 0.5 0.0 0.5 4.0860 3.6910
0.1 3.6601 1.7362
0.2 2.5121 1.5705
0.1 0.2 0.5 0.1 0.0 3.8691 1.1792
0.2 5.4701 1.4803
0.5 6.0772 1.5912
$r$ $\gamma $ $Q$ $\phi $ $R$ ${Re} _{x}Nu $
$Cu$ $AI_{2}O_{3}$
0.1 0.2 0.5 0.5 0.5 2.0910 4.8872
0.2 1.8690 2.7106
0.3 0.9587 1.2883
0.2 0.0 0.5 0.5 0.5 2.5600 2.0971
0.1 1.0830 1.8846
0.2 1.0490 0.9505
0.3 0.4 0.0 0.5 0.5 0.5418 0.6415
0.2 1.4470 0.8727
0.4 1.5960 0.9666
0.3 0.2 0.5 0.0 0.5 4.0860 3.6910
0.1 3.6601 1.7362
0.2 2.5121 1.5705
0.1 0.2 0.5 0.1 0.0 3.8691 1.1792
0.2 5.4701 1.4803
0.5 6.0772 1.5912
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