Thermocapillary flow of thin Cu-water nanoliquid film during spin coating process
Tóm tắt
Unsteady flow of thin Cu-water nanoliquid film over a horizontal rotating disk is studied numerically using finite difference technique under the assumption of planar interface. It is also assumed that the disk is cooling axisymmetrically from below. The effects of the nanolayer thickness and nanoparticle radius are considered for investigation. It is found that the film thinning rate decreases with increase of the nanoparticle volume fraction. It is also found that thickness of liquid decreases with increase of the thermocapillary parameter. The results show that the rate of film thinning is more for the thermal conductivity model of Yu and Choi [47] compared to the model of Maxwell [46]. It is observed that the film thinning rate increases with increase of nanolayer thickness but it decreases with the nanoparticle radius. A curve
$$R=R_c(z,t)$$
in
$$R-z$$
plane is delineated along which temperature gradient
$$T_z$$
is zero and positive or negative according to
$$RR_c$$
respectively. Furthermore, it is shown that the region for
$$T_z>0$$
enlarges with increase of the nanoparticle volume fraction and the nanolayer thickness.
Tài liệu tham khảo
Emslie, A.C., Bonner, F.D., Peck, L.G.: Flow of a viscous liquid on a rotating disk. J. Appl. Phys. 29, 858–862 (1958)
Acrivos, A., Shah, M.J., Petersen, E.E.: On the flow of a non-Newtonian liquid on a rotating disk. J. Appl. Phys. 31, 963–968 (1960)
Jenekhe, S.A., Schuldt, S.B.: Coating flow of non-Newtonian fuids on a flat rotating disk. Ind. Eng. Chem. Fundam. 23, 432–436 (1984)
Charpin, J.P.F., Lombe, M., Myers, T.G.: Spin coating of non-Newtonian fluids with a moving front. Phys. Rev. E 76, 016312 (2007)
Myerhofer, D.J.: Characteristics of resist films produced by spinning. J. Appl. Phys. 49, 3993–3997 (1978)
Mouhamad, Y., Mokarian-Tabari, P., Clarke, N., Jones, R.A.L., Geoghegan, M.: Dynamics of polymer film formation during spin coating. J. Appl. Phys. 116, 123513 (2014)
Middleman, S.: The effect of induced air flow on the spin coating of viscous liquids. J. Appl. Phys. 62, 2530–2532 (1987)
Ma, F., Hwang, J.H.: The effect of air shear on the flow of a thin liquid film over a rough rotating disk. J. Appl. Phys. 68, 1265–1271 (1990)
Higgins, B.G.: Film flow on a rotating disk. Phys. Fluids 29, 3522–3529 (1986)
Rehg, T.J., Higgins, B.G.: The effects of inertia and interfacial shear on film flow on a rotating disk. Phys. Fluids 31, 1360–1371 (1988)
Dandapat, B.S., Roy, P.C.: The effect of themocapillarity on the flow of a thin liquid film on a rotating disc. J. Phys. D Appl. Phys. 27, 2041–2045 (1994)
Dandapat, B.S., Roy, P.C.: Flow of a thin liquid film over a cold/hot rotating disk. Int. J. Nonlinear Mech. 28, 489–501 (1993)
Dandapat, B.S.: Unsteady flow of thin liquid film on a disk under nonuniform rotation. Phys. Fluids 13, 1860–1868 (2001)
Usha, R., Ravindran, R.: Numerical study of a film cooling on a disk. Int. J. Nonlinear Mech. 36, 147–154 (2001)
Wu, L.: Thermal effects on liquid film dynamics in spin coating. Sens. Actuators A 134, 140–145 (2007)
Cregan, V., O’Brien, S.B.G.: Extended asymptotic solutions to the spin-coating model with small evaporation. Appl. Math. Comput. 223, 76–87 (2013)
Matsumoto, Y., Ohara, T., Teruya, I., Ohashi, H.: Liquid film formation on a rotating disk. JSME Int. J. Ser. II(32), 52–56 (1989)
Kitamura, A.: Asymptotic solution for film flow on a rotating disk. Phys. Fluids 12, 2141–2144 (2000)
Dandapat, B.S., Santra, B., Kitamura, A.: Thermal effects on film development during spin coating. Phys. Fluids 17(1—-6), 062102 (2005)
Dandapat, B.S., Maity, S.: Effects of air-flow and evaporation on the development of thin liquid film over a rotating annular disk. Int. J. Nonlinear Mech. 44, 877–882 (2009)
Quinn, G., Cetegen, B.M.: Heat transfer in an evaporating liquid film flow over a rotating disk. Exp. Heat Transf. 24, 88–107 (2011)
Lin, M.C., Chen, C.K.: Finite amplitude long-wave instability of a thin viscoelastic fluid during spin coating. Appl. Math. Model. 36, 2536–2549 (2012)
Prieling, D., Steiner, H.: Unsteady thin film flow on spinning disks at large Ekman numbers using an integral boundary layer method. Int. J. Heat Mass Transf. 65, 10–22 (2013)
McIntyre, A., Brush, L.N.: Spin-coating of vertically stratified thin liquid films. J. Fluid Mech. 647, 265–285 (2010)
Dandapat, B.S., Singh, S.K.: Two-layer film flow over a rotating disk. Commun. Nonlinear Sci. Numer. Simul. 17, 2854–2863 (2012)
Dandapat, B.S., Singh, S.K.: Unsteady two-layer film flow on a non-uniform rotating disk in presence of uniform transverse magnetic field. Appl. Math. Comput. 258, 545–555 (2015)
Sahoo, S., Arora, A., Doshi, P.: Two-layer spin coating flow of Newtonian liquids: a computational study. Comput. Fluids 131, 180–189 (2016)
S. U. S. Choi, Enhancing thermal conductivity of fluids with nanoparticles. In: The proceedings of the 1995 ASME international mechanical engineering congress and exposition, San Francisco, USA, ASME, FED231/MD66, 99–105 (1995)
Choi, S.U.S., Zhang, Z.G., Yu, W., Lockwood, F.E., Grulke, E.A.: Anomalously thermal conductivity enhancement in nanotube suspensions. Appl. Phys. Lett. 79, 2252–2254 (2001)
Masuda, H., Ebata, A., Teramae, K., Hishinuma, N.: Alterlation of thermal conductivity and viscosity of liquid by dispersing ultra-fine particles. Netsu Bussei 7, 227–233 (1993). (dispersion of g-\(Al_2O_3\), \(SiO_2\), and \(TiO_2\) ultra-fine particles)
Xuan, Y., Li, Q.: Heat transfer enhancement of nanofluids. Int. J. Heat Fluid Flow 21, 58–64 (2000)
J. A. Eastman, S. U. S. Choi, S. Li, L. J. Thompson and S. Lee, Enhancemet thermal conductivity through the development of nanofluids. In: 1996 fall meeting of the materials research socity (MRS), Boston, USA (1997)
Das, S.K., Choi, S.U.S., Yu, W.: Nanofluids. Sciences and technology. Wiley, New Jersey (2008)
Buongiorno, J.: Convective transport in nanofluids. ASME J. Heat Transf. 128, 240–250 (2006)
Wang, X.Q., Majumdar, A.S.: Heat transfer characteristic of nanofluids: a review. Int. J. Therm. Sci. 46, 1–19 (2007)
Kakac, S., Pramuanjaroenkij, A.: Review of convective heat transfer enhancement with nanofluids. Int. J. Heat Mass Transf. 52, 3187–3196 (2009)
Ahmad, R., Mustafa, M., Hayat, T., Alsaedi, A.: Numerical study of MHD nanofluid flow and heat transfer past a bidirectional exponentially stretching sheet. J. Magn. Magn. Mater. 407, 69–74 (2016)
Ibanez, G., Lopez, A., Pantoja, J., Moreira, J.: Entropy generation analysis of a nanofluid flow in MHD porous microchannel with hydrodynamic slip and thermal radiation. Int. J. Heat Mass Transf. 100, 89–97 (2016)
Ferdows, M., Khan, M.S., Alam, M.M., Sun, S.: MHD mixed convective boundary layer flow of a nanofluid through a porous medium due to an exponentially stretching sheet. Math. Probl. Eng. 3(1–21), 408528 (2012)
Ferdows, M., Khan, M.S., Bg, O.A., Alam, M.M.: Numerical study of transient magnetohydrodynamic radiative free convection nanofluid flow from a stretching permeable surface. J. Process Mech. Eng. 25, 1–16 (2013)
Beg, O.A., Khan, M.S., Karim, I., Alam, M.M., Ferdows, M.: Explicit numerical study of unsteady hydromagnetic mixed convective nanofluid flow from an exponential stretching sheet in porous media. Appl. Nanosci. 4, 943–957 (2014)
Maity, S., Ghatani, Y., Dandapat, B.S.: Thermocapillary flow of a thin nanoliquid film over an unsteady stretching sheet. J. Heat Transf. 138(1–8), 041501 (2016)
Maity, S.: Unsteady flow of thin nanoliquid film over a stretching sheet in the presence of thermal radiation. Eur. Phys. J. Plus 131(1–16), 49 (2016)
Narayana, M., Sibanda, P.: Laminar flow of nanoliquid film over an unsteady stretching sheet. Int. J. Heat Mass Transf. 55, 7552–7560 (2012)
Oztop, H.F., Abu-Nada, E.: Numerical study of natural convection in partially heated rectangular enclosures filled with nanofluids. Int. J. Heat Fluid Flow 29, 1326–1336 (2008)
Maxwell, J.C.: A treatise on electricity and magnetism, 2nd edn. Oxford University Press, Cambridge (1904)
Yu, W., Choi, S.U.S.: The role of interfacial layers in the enhanced thermal conductivity of nanofluids: a renovated Maxwell model. J. Nanoparticle Res. 5, 167–171 (2003)
Robert, G.O.: Lecture notes in physics. Spinger, New York (1971)
Fletcher, C.A.J.: Computational technique for fluid dynamics. Springer, New York (1988)