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December  2019, 9(4): 413-421. doi: 10.3934/naco.2019040

Onset of Benard-Marangoni instabilities in a double diffusive binary fluid layer with temperature-dependent viscosity

1. 

Laboratory of Computational Sciences and Mathematical Physics, Institute for Mathematical Research (INSPEM), Universiti Putra Malaysia, 43400 Serdang, Selangor, Malaysia

2. 

Faculty of Computer and Mathematical Sciences, Universiti Teknologi MARA, Perak Branch, Tapah Campus, 35400 Tapah Road, Perak, Malaysia

3. 

Department of Mathematics, Faculty of Science, Universiti Putra Malaysia, 43400 Serdang, Selangor, Malaysia

* Corresponding author: norfadzillah.mokhtar@gmail.com

The reviewing process of the paper is handled by Gafurjan Ibragimov, Siti Hasana Sapar and Siti Nur Iqmal Ibrahim

Received  January 2018 Revised  July 2018 Published  August 2019

Fund Project: he present research was partially supported by MOHE for FRGS Vote no 5524808.

The effect of temperature-dependent viscosity in a horizontal double diffusive binary fluid layer is investigated. When the layer is heated from below, the convection of Benard-Marangoni will start to exists. Linear stability analysis is performed and the eigenvalues from few cases of boundary conditions were obtained. Galerkin method were used to solve the numerical calculation and marginal stability curve is obtained. Results shows that an increase of temperature-dependent viscosity will destabilized the system. The impact of double diffusive coefficients are also revealed. It is found that the effect of Soret parameter exhibits destabilizing reaction on the system while an opposite response is noted with an increase of Dufour parameter.

Citation: Nurul Hafizah Zainal Abidin, Nor Fadzillah Mohd Mokhtar, Zanariah Abdul Majid. Onset of Benard-Marangoni instabilities in a double diffusive binary fluid layer with temperature-dependent viscosity. Numerical Algebra, Control & Optimization, 2019, 9 (4) : 413-421. doi: 10.3934/naco.2019040
References:
[1]

N. H. Z. AbidinN. F. M. MokhtarI. K. KhalidR. A. Rahim and S. S. A. Gani, Stability control in a binary fluid mixture subjected to cross diffusive coefficients, International Journal on Advanced Science, Engineering and Information Technology, 7 (2017), 322-328.  doi: 10.18517/ijaseit.7.1.1321.  Google Scholar

[2]

N. H. Z. AbidinN. M. Arifin and M. S. Noorani, Boundary effect on Marangoni convection in a variable viscosity fluid layer, Journal of Mathematics and Statistics, 4 (2008), 1-8.  doi: 10.3844/jmssp.2008.1.8.  Google Scholar

[3]

N. M. Arifin and N. H. Z. Abidin, Marangoni convection in a variable viscosity fluid layer with feedback control, Journal of Applied Computer Science & Mathematics, 3 (2009), 373-382.   Google Scholar

[4]

N. M. Arifin and N. H. Z. Abidin, Stability of Marangoni convection in a fluid layer with variable viscosity and deformable free surface under free-slip condition, Journal of Applied Computer Science & Mathematics, 3 (2009), 43-47. doi: https://www.ingentaconnect.com/content/doaj/20664273/2009/00000003/00000006/art00007.  Google Scholar

[5]

A. BergeonD. HenryH Benhadid and L. S. Tuckerman, Marangoni convection in binary mixtures with Soret effect, J. Fluid Mech., 375 (1998), 143-177.  doi: 10.1017/S0022112098002614.  Google Scholar

[6]

C. F. Chen and T. F. Su, Effect of surface tension on the onset of convection in a double-diffusive layer, Physics of Fluids A: Fluid Dynamics, 4 (1992), 2360-2367.  doi: 10.1063/1.858477.  Google Scholar

[7]

A. Cloot and G. Lebon, Marangoni instability in a fluid layer with variable viscosity and free interface, in microgravity, PhysicoChemical Hydrodynamics, 6 (1985), 453-462.   Google Scholar

[8]

F. Franchi and B. Straughan, Nonlinear stability for thermal convection in a micropolar fluid with temperature dependent viscosity, Int. J. Eng. Sci., 30 (1992), 1349-1360.  doi: 10.1016/0020-7225(92)90146-8.  Google Scholar

[9]

M. Hilt, M. Gl$\ddot{a}$ssl and W. Zimmermann, Effects of a temperature-dependent viscosity on thermal convection in binary mixtures, , Physical Review E, 89 (2014), 052312. doi: 10.1103/PhysRevE.89.052312.  Google Scholar

[10]

D. T. J. Hurle and E. Jakeman, Soret-driven thermosolutal convection, J. Fluid Mech., 47 (1971), 667-1360.  doi: 10.1017/S0022112071001319.  Google Scholar

[11]

Z. Kozhoukharova and C. Rozé, Influence of the surface deformability and variable viscosity on buoyant-thermocapillary instability in a liquid layer, Eur. Phys. J. B, 8 (1999), 125-135.  doi: 10.1007/s100510050674.  Google Scholar

[12]

J. W. Lu and F. Chen, Onset of double-diffusive convection of unidirectionally solidifying binary solution with variable viscosity, J. Cryst. Growth, 149 (1995), 131-140.  doi: 10.1016/0022-0248(94)01006-4.  Google Scholar

[13]

M. MangaD. Weeraratne and S. J. S. Morris, Boundary-layer thickness and instabilities in Bnard convection of a liquid with a temperature-dependent viscosity, Phys. Fluids, 13 (2001), 802-805.  doi: 10.1063/1.1345719.  Google Scholar

[14]

C. E. NanjundappaI. S. ShivakumaraR Arunkumar and Ar unkumar, Onset of Marangoni-B$\acute{e}$nard ferroconvection with temperature dependent viscosity, Microgravity Sci. Technol, 25 (2013), 103-1360.  doi: 10.1007/s12217-012-9330-9.  Google Scholar

[15]

D. A. Nield and A. V. Kuznetsov, The onset of double-diffusive convection in a nanofluid layer, Int. J. Heat Fluid Flow, 32 (1967), 771-776.  doi: 10.1016/j.ijheatfluidflow.2011.03.010.  Google Scholar

[16]

E. Palm, On the tendency towards hexagonal cells in steady convection, J. Fluid Mech, 8 (1960), 183–3011., doi: 10.1017/S0022112060000530.  Google Scholar

[17]

N. E. Ramirez and A. E. Saez, The effect of variable viscosity on boundary-layer heat transfer in a porous medium, Int. Commun. Heat Mass Transfer, 17 (1990), 477-488.  doi: 10.1016/0735-1933(90)90066-S.  Google Scholar

[18]

M. M. Rashidi, N. Kavyani, S. Abelman, M.J. Uddin and N. Freidoonimehr, Double diffusive magnetohydrodynamic (MHD) mixed convective slip flow along a radiating moving vertical flat plate with convective boundary condition, , PLoS ONEs, 9 (2014), e109404. doi: 10.1371/journal.pone.0109404.  Google Scholar

[19]

S. Saravanan and T. Sivakumar, Exact solution of Marangoni convection in a binary fluid with throughflow and Soret effect, Applied Mathematical Modelling, 33 (2009), 3674-3681.  doi: 10.1016/j.apm.2008.12.017.  Google Scholar

[20]

S. SlavtchevG. SimeonovS Van Vaerenbergh and J. C. Legros, Technical note Marangoni instability of a layer of binary liquid in the presence of nonlinear Soret effect, Int. J. Heat Mass Transfer, 42 (1999), 3007-3011.  doi: 10.1016/S0017-9310(98)00353-6.  Google Scholar

[21]

K. C. StengelD. S. Oliver and J. R. Booker, Onset of convection in a variable-viscosity fluid, J. Fluid Mech, 120 (1982), 411-431.  doi: 10.1017/S0022112082002821.  Google Scholar

[22]

K. E. Torrance and D. L. Turcotte, Thermal convection with large viscosity variations, J. Fluid Mech, 47 (1971), 113-125.  doi: 10.1017/S002211207100096X.  Google Scholar

[23]

D. B. White, The planforms and onset of convection with a temperature-dependent viscosity, J. Fluid Mech., 191 (1988), 247-286.  doi: 10.1017/S0022112088001582.  Google Scholar

show all references

References:
[1]

N. H. Z. AbidinN. F. M. MokhtarI. K. KhalidR. A. Rahim and S. S. A. Gani, Stability control in a binary fluid mixture subjected to cross diffusive coefficients, International Journal on Advanced Science, Engineering and Information Technology, 7 (2017), 322-328.  doi: 10.18517/ijaseit.7.1.1321.  Google Scholar

[2]

N. H. Z. AbidinN. M. Arifin and M. S. Noorani, Boundary effect on Marangoni convection in a variable viscosity fluid layer, Journal of Mathematics and Statistics, 4 (2008), 1-8.  doi: 10.3844/jmssp.2008.1.8.  Google Scholar

[3]

N. M. Arifin and N. H. Z. Abidin, Marangoni convection in a variable viscosity fluid layer with feedback control, Journal of Applied Computer Science & Mathematics, 3 (2009), 373-382.   Google Scholar

[4]

N. M. Arifin and N. H. Z. Abidin, Stability of Marangoni convection in a fluid layer with variable viscosity and deformable free surface under free-slip condition, Journal of Applied Computer Science & Mathematics, 3 (2009), 43-47. doi: https://www.ingentaconnect.com/content/doaj/20664273/2009/00000003/00000006/art00007.  Google Scholar

[5]

A. BergeonD. HenryH Benhadid and L. S. Tuckerman, Marangoni convection in binary mixtures with Soret effect, J. Fluid Mech., 375 (1998), 143-177.  doi: 10.1017/S0022112098002614.  Google Scholar

[6]

C. F. Chen and T. F. Su, Effect of surface tension on the onset of convection in a double-diffusive layer, Physics of Fluids A: Fluid Dynamics, 4 (1992), 2360-2367.  doi: 10.1063/1.858477.  Google Scholar

[7]

A. Cloot and G. Lebon, Marangoni instability in a fluid layer with variable viscosity and free interface, in microgravity, PhysicoChemical Hydrodynamics, 6 (1985), 453-462.   Google Scholar

[8]

F. Franchi and B. Straughan, Nonlinear stability for thermal convection in a micropolar fluid with temperature dependent viscosity, Int. J. Eng. Sci., 30 (1992), 1349-1360.  doi: 10.1016/0020-7225(92)90146-8.  Google Scholar

[9]

M. Hilt, M. Gl$\ddot{a}$ssl and W. Zimmermann, Effects of a temperature-dependent viscosity on thermal convection in binary mixtures, , Physical Review E, 89 (2014), 052312. doi: 10.1103/PhysRevE.89.052312.  Google Scholar

[10]

D. T. J. Hurle and E. Jakeman, Soret-driven thermosolutal convection, J. Fluid Mech., 47 (1971), 667-1360.  doi: 10.1017/S0022112071001319.  Google Scholar

[11]

Z. Kozhoukharova and C. Rozé, Influence of the surface deformability and variable viscosity on buoyant-thermocapillary instability in a liquid layer, Eur. Phys. J. B, 8 (1999), 125-135.  doi: 10.1007/s100510050674.  Google Scholar

[12]

J. W. Lu and F. Chen, Onset of double-diffusive convection of unidirectionally solidifying binary solution with variable viscosity, J. Cryst. Growth, 149 (1995), 131-140.  doi: 10.1016/0022-0248(94)01006-4.  Google Scholar

[13]

M. MangaD. Weeraratne and S. J. S. Morris, Boundary-layer thickness and instabilities in Bnard convection of a liquid with a temperature-dependent viscosity, Phys. Fluids, 13 (2001), 802-805.  doi: 10.1063/1.1345719.  Google Scholar

[14]

C. E. NanjundappaI. S. ShivakumaraR Arunkumar and Ar unkumar, Onset of Marangoni-B$\acute{e}$nard ferroconvection with temperature dependent viscosity, Microgravity Sci. Technol, 25 (2013), 103-1360.  doi: 10.1007/s12217-012-9330-9.  Google Scholar

[15]

D. A. Nield and A. V. Kuznetsov, The onset of double-diffusive convection in a nanofluid layer, Int. J. Heat Fluid Flow, 32 (1967), 771-776.  doi: 10.1016/j.ijheatfluidflow.2011.03.010.  Google Scholar

[16]

E. Palm, On the tendency towards hexagonal cells in steady convection, J. Fluid Mech, 8 (1960), 183–3011., doi: 10.1017/S0022112060000530.  Google Scholar

[17]

N. E. Ramirez and A. E. Saez, The effect of variable viscosity on boundary-layer heat transfer in a porous medium, Int. Commun. Heat Mass Transfer, 17 (1990), 477-488.  doi: 10.1016/0735-1933(90)90066-S.  Google Scholar

[18]

M. M. Rashidi, N. Kavyani, S. Abelman, M.J. Uddin and N. Freidoonimehr, Double diffusive magnetohydrodynamic (MHD) mixed convective slip flow along a radiating moving vertical flat plate with convective boundary condition, , PLoS ONEs, 9 (2014), e109404. doi: 10.1371/journal.pone.0109404.  Google Scholar

[19]

S. Saravanan and T. Sivakumar, Exact solution of Marangoni convection in a binary fluid with throughflow and Soret effect, Applied Mathematical Modelling, 33 (2009), 3674-3681.  doi: 10.1016/j.apm.2008.12.017.  Google Scholar

[20]

S. SlavtchevG. SimeonovS Van Vaerenbergh and J. C. Legros, Technical note Marangoni instability of a layer of binary liquid in the presence of nonlinear Soret effect, Int. J. Heat Mass Transfer, 42 (1999), 3007-3011.  doi: 10.1016/S0017-9310(98)00353-6.  Google Scholar

[21]

K. C. StengelD. S. Oliver and J. R. Booker, Onset of convection in a variable-viscosity fluid, J. Fluid Mech, 120 (1982), 411-431.  doi: 10.1017/S0022112082002821.  Google Scholar

[22]

K. E. Torrance and D. L. Turcotte, Thermal convection with large viscosity variations, J. Fluid Mech, 47 (1971), 113-125.  doi: 10.1017/S002211207100096X.  Google Scholar

[23]

D. B. White, The planforms and onset of convection with a temperature-dependent viscosity, J. Fluid Mech., 191 (1988), 247-286.  doi: 10.1017/S0022112088001582.  Google Scholar

Figure 1.  Effect of $ B $ to Rayleigh number
Figure 2.  Effect of $ Sr $ to Rayleigh number
Figure 3.  Effect of $ Df $ to Rayleigh number
Figure 4.  Effect of $ B $, $ Sr $ and $ Df $ to Marangoni number
Figure 5.  Effect of $ B $ on $ Ra_c $ for various values of $ Le $
Figure 6.  Effect of $ B $ on $ Ra_c $ for various values of $ Rs $
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