# American Institute of Mathematical Sciences

• Previous Article
Numerical comparisons of smoothing functions for optimal correction of an infeasible system of absolute value equations
• NACO Home
• This Issue
• Next Article
Conformal deformations of a specific class of lorentzian manifolds with non-irreducible holonomy representation
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

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, doi: 10.3934/naco.2019040
##### References:
 [1] N. H. Z. Abidin, N. F. M. Mokhtar, I. K. Khalid, R. 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. Abidin, N. 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. Bergeon, D. Henry, H 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. Manga, D. 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. Nanjundappa, I. S. Shivakumara, R 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. Slavtchev, G. Simeonov, S 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. Stengel, D. 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. Abidin, N. F. M. Mokhtar, I. K. Khalid, R. 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. Abidin, N. 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. Bergeon, D. Henry, H 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. Manga, D. 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. Nanjundappa, I. S. Shivakumara, R 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. Slavtchev, G. Simeonov, S 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. Stengel, D. 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
Effect of $B$ to Rayleigh number
Effect of $Sr$ to Rayleigh number
Effect of $Df$ to Rayleigh number
Effect of $B$, $Sr$ and $Df$ to Marangoni number
Effect of $B$ on $Ra_c$ for various values of $Le$
Effect of $B$ on $Ra_c$ for various values of $Rs$
 [1] P. D. Howell, J. J. Wylie, Huaxiong Huang, Robert M. Miura. Stretching of heated threads with temperature-dependent viscosity: Asymptotic analysis. Discrete & Continuous Dynamical Systems - B, 2007, 7 (3) : 553-572. doi: 10.3934/dcdsb.2007.7.553 [2] Jishan Fan, Fucai Li, Gen Nakamura. Regularity criteria for the Boussinesq system with temperature-dependent viscosity and thermal diffusivity in a bounded domain. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 4915-4923. doi: 10.3934/dcds.2016012 [3] Tao Wang. One dimensional $p$-th power Newtonian fluid with temperature-dependent thermal conductivity. Communications on Pure & Applied Analysis, 2016, 15 (2) : 477-494. doi: 10.3934/cpaa.2016.15.477 [4] Isabel Mercader, Oriol Batiste, Arantxa Alonso, Edgar Knobloch. Dissipative solitons in binary fluid convection. Discrete & Continuous Dynamical Systems - S, 2011, 4 (5) : 1213-1225. doi: 10.3934/dcdss.2011.4.1213 [5] Chun-Hsiung Hsia, Tian Ma, Shouhong Wang. Bifurcation and stability of two-dimensional double-diffusive convection. Communications on Pure & Applied Analysis, 2008, 7 (1) : 23-48. doi: 10.3934/cpaa.2008.7.23 [6] Irena PawŃow, Wojciech M. Zajączkowski. Global regular solutions to three-dimensional thermo-visco-elasticity with nonlinear temperature-dependent specific heat. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1331-1372. doi: 10.3934/cpaa.2017065 [7] Zilai Li, Zhenhua Guo. On free boundary problem for compressible navier-stokes equations with temperature-dependent heat conductivity. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3903-3919. doi: 10.3934/dcdsb.2017201 [8] Adriana C. Briozzo, María F. Natale, Domingo A. Tarzia. The Stefan problem with temperature-dependent thermal conductivity and a convective term with a convective condition at the fixed face. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1209-1220. doi: 10.3934/cpaa.2010.9.1209 [9] Takeshi Fukao, Nobuyuki Kenmochi. Quasi-variational inequality approach to heat convection problems with temperature dependent velocity constraint. Discrete & Continuous Dynamical Systems - A, 2015, 35 (6) : 2523-2538. doi: 10.3934/dcds.2015.35.2523 [10] Francisco Guillén-González, Mamadou Sy. Iterative method for mass diffusion model with density dependent viscosity. Discrete & Continuous Dynamical Systems - B, 2008, 10 (4) : 823-841. doi: 10.3934/dcdsb.2008.10.823 [11] Liu Liu. Uniform spectral convergence of the stochastic Galerkin method for the linear semiconductor Boltzmann equation with random inputs and diffusive scaling. Kinetic & Related Models, 2018, 11 (5) : 1139-1156. doi: 10.3934/krm.2018044 [12] Takeshi Fukao, Nobuyuki Kenmochi. A thermohydraulics model with temperature dependent constraint on velocity fields. Discrete & Continuous Dynamical Systems - S, 2014, 7 (1) : 17-34. doi: 10.3934/dcdss.2014.7.17 [13] Mingxin Wang, Qianying Zhang. Dynamics for the diffusive Leslie-Gower model with double free boundaries. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2591-2607. doi: 10.3934/dcds.2018109 [14] W. E. Fitzgibbon, M.E. Parrott, Glenn Webb. Diffusive epidemic models with spatial and age dependent heterogeneity. Discrete & Continuous Dynamical Systems - A, 1995, 1 (1) : 35-57. doi: 10.3934/dcds.1995.1.35 [15] Bedr'Eddine Ainseba. Age-dependent population dynamics diffusive systems. Discrete & Continuous Dynamical Systems - B, 2004, 4 (4) : 1233-1247. doi: 10.3934/dcdsb.2004.4.1233 [16] Shuji Yoshikawa, Irena Pawłow, Wojciech M. Zajączkowski. A quasilinear thermoviscoelastic system for shape memory alloys with temperature dependent specific heat. Communications on Pure & Applied Analysis, 2009, 8 (3) : 1093-1115. doi: 10.3934/cpaa.2009.8.1093 [17] Yoshifumi Aimoto, Takayasu Matsuo, Yuto Miyatake. A local discontinuous Galerkin method based on variational structure. Discrete & Continuous Dynamical Systems - S, 2015, 8 (5) : 817-832. doi: 10.3934/dcdss.2015.8.817 [18] Chun-Hao Teng, I-Liang Chern, Ming-Chih Lai. Simulating binary fluid-surfactant dynamics by a phase field model. Discrete & Continuous Dynamical Systems - B, 2012, 17 (4) : 1289-1307. doi: 10.3934/dcdsb.2012.17.1289 [19] Tianwen Luo, Tao Tao, Liqun Zhang. Finite energy weak solutions of 2d Boussinesq equations with diffusive temperature. Discrete & Continuous Dynamical Systems - A, 2019, 0 (0) : 1-29. doi: 10.3934/dcds.2019230 [20] Ping Chen, Daoyuan Fang, Ting Zhang. Free boundary problem for compressible flows with density--dependent viscosity coefficients. Communications on Pure & Applied Analysis, 2011, 10 (2) : 459-478. doi: 10.3934/cpaa.2011.10.459

Impact Factor:

## Tools

Article outline

Figures and Tables