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Impinging jet flow and hydraulic jump on a rotating disk

Published online by Cambridge University Press:  02 February 2018

Yunpeng Wang  and
Roger E. Khayat Open the ORCID record for Roger E. Khayat [Opens in a new window]
Yunpeng Wang
Affiliation:
Department of Mechanical and Materials Engineering, University of Western Ontario, London, ON N6A 5B9, Canada
Roger E. Khayat*
Affiliation:
Department of Mechanical and Materials Engineering, University of Western Ontario, London, ON N6A 5B9, Canada
*
†Email address for correspondence: rkhayat@uwo.ca
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Abstract

The free-surface flow formed by a circular jet impinging on a rotating disk is analysed theoretically. The study explores the effects of rotation and inertia on the thin-film flow. Both boundary-layer height and film thickness are found to diminish with rotation speed. A maximum film thickness develops in the supercritical region, which reflects the competition between the convective and centrifugal effects. Unlike the flow on a stationary disk, an increase in the wall shear stress along the radial direction is predicted, at a rate that strengthens with rotating speed. Our results corroborate well existing measurements. The location and height of the hydraulic jump are determined subject to the value of the thickness at the edge of the disk, which is established first for a stationary disk based on the capillary length, and then for a rotating disk using existing analyses and measurements in spin coating. The case of a stationary is revisited in an effort to predict the location and height of the jump uniquely. The formulated value of the height at the edge of the disk seems to give excellent results for a jet at moderately high flow rate (or low viscosity) where the jump structure is well identifiable in reality.

JFM classification

Boundary Layers: Boundary layer structure Interfacial Flows (free surface): Capillary flows Interfacial Flows (free surface): Interfacial Flows (free surface)
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 839 , 25 March 2018 , pp. 525 - 560
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Copyright
© 2018 Cambridge University Press 

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References

Aoune, A. & Ramshaw, C. 1999 Process intensification: heat and mass transfer characteristics of liquid films on rotating discs. Intl J. Heat Mass Transfer 42, 2543–2556.Google Scholar
Avedisian, C. T. & Zhao, Z. 2000 The circular hydraulic jump in low gravity. Proc. R. Soc. Lond. A 456, 2127–2151.Google Scholar
Bakhmeteff, B. A. 1966 Hydraulics of Open Channels. McGraw-Hill.Google Scholar
Baonga, J. B., Gualous, H. L. & Imbert, M. 2006 Experimental study of hydrodynamic and heat transfer of free liquid jet impinging a flat circular heated disk. Appl. Therm. Engng 26, 1125–1138.Google Scholar
Basu, S. & Cetegen, B. M. 2007 Effect of hydraulic jump on hydrodynamics and heat transfer in a thin liquid film flowing over a rotating disk analyzed by integral method. Trans. ASME J. Heat Transfer 129, 657–663.Google Scholar
Bauer, J., Drescher, G., Silz, H., Frankenfeld, H. & Illig, M. 1997 Surface tension and adhesion of photo and electron-beam resists. In Proc. SPIE 3049, Advances in Resist Technology and Processing XIV, 640 (July 7).Google Scholar
Benilov, E. S. 2015 Hydraulic jumps in a shallow flow down a slightly inclined substrate. J. Fluid Mech. 782, 5–24.Google Scholar
Bohr, T., Dimon, P. & Putzkaradze, V. 1993 Shallow-water approach to the circular hydraulic jump. J. Fluid Mech. 254, 635–648.Google Scholar
Bohr, T., Ellegaard, C., Hansen, A. E. & Haaning, A. 1996 Hydraulic jumps, flow separation and wave breaking: an experimental study. Physica B 228, 1–10.Google Scholar
Bohr, T., Putkaradze, V. & Watanabe, S. 1997 Averaging theory for the structure of hydraulic jumps and separation in laminar free-surface flows. Phys. Rev. Lett. 79, 1038–1041.Google Scholar
Brechet, Y. & Neda, Z. 1999 On the circular hydraulic jump. Am. J. Phys. 67, 723–731.Google Scholar
Burns, J., Ramshaw, C. & Jachuck, R. 2003 Measurement of liquid film thickness and the determination of spin-up radius on a rotating disc using an electrical resistance technique. Chem. Engng Sci. 58, 2245–2253.Google Scholar
Bush, J. W. M. & Aristoff, J. M. 2003 The influence of surface tension on the circular hydraulic jump. J. Fluid Mech. 489, 229–238.Google Scholar
Bush, J. W. M., Aristoff, J. M. & Hosoi, A. E. 2006 An experimental investigation of the stability of the circular hydraulic jump. J. Fluid Mech. 558, 33–52.Google Scholar
Butuzov, A. I. & Pukhovoi, I. I. 1976 Liquid film flow regimes on a rotating surface. J. Engng Phys. 31, 886–891.CrossRefGoogle Scholar
Charwat, A., Kelly, R. & Gazley, C. 1972 The flow and stability of thin liquid films on a rotating disk. J. Fluid Mech. 53, 227–255.Google Scholar
Craik, A., Latham, R., Fawkes, M. & Gibbon, P. 1981 The circular hydraulic jump. J. Fluid Mech. 112, 347–362.Google Scholar
Crowe, C. T. 2009 Engineering Fluid Mechanics, 9th edn. Wiley.Google Scholar
Deng, H. & Ouyang, H. 2011 Vibration of spinning discs and powder formation in centrifugal atomization. Proc. R. Soc. Lond. A 467, 361–380.Google Scholar
Diversified Enterprises2009 Surface Energy Data for PDMS (Polydimethylsiloxane) http://www.accudynetest.com/polymer_surface_data/polydimethylsiloxane.pdf.Google Scholar
Dressaire, E., Courbin, L., Crest, J. & Stone, H. A. 2010 Inertia dominated thin-film flows over microdecorated surfaces. Phys. Fluids 22, 073602.Google Scholar
Duchesne, A., Lebon, L. & Limat, L. 2014 Constant Froude number in a circular hydraulic jump and its implication on the jump radius selection. Europhys. Lett. 107, 54002.Google Scholar
Ellegaard, C, Hansen, A. E., Haaning, A., Hansen, K., Marcussen, A., Bohr, T., Hansen, J. L. & Watanabe, S. 1999 Polygonal hydraulic jumps. Nonlinearity 12, 1–7.Google Scholar
Ellegaard, C, Hansen, A. E., Haaning, A., Marcussen, A., Bohr, T., Hansen, J. L. & Watanabe, S. 1998 Creating corners in kitchen sink flows. Nature 392, 767–768.Google Scholar
Godwin, R. P. 1993 The hydraulic jump (shocks and viscous flow in the kitchen sink). Am. J. Phys. 61, 829–832.Google Scholar
Hall, D. B., Underhill, P. & Torkelson, J. M. 1998 Spin coating of thin and ultrathin polymers. Polym. Engng Sci. 38, 2039–2045.CrossRefGoogle Scholar
Hansen, S. H., Horluck, S., Zauner, D., Dimon, P., Ellegaard, C. & Creagh, S. C. 1997 Geometric orbits of surface waves from a circular hydraulic jump. Phys. Rev. E 55, 7048–7061.Google Scholar
Henderson, F. M. 1966 Open Channel Flow. Macmillan.Google Scholar
Higuera, F. J. 1994 The hydraulic jump in a viscous laminar flow. J. Fluid Mech. 274, 69–92.Google Scholar
Kasimov, A. R. 2008 A stationary circular hydraulic jump, the limits of its existence and its gasdynamic analogue. J. Fluid Mech. 601, 189–198.Google Scholar
Kate, R. P., Das, P. K. & Chakraborty, S. 2007 Hydraulic jumps due to oblique impingement of circular liquid jets on a flat horizontal surface. J. Fluid Mech. 573, 247–263.Google Scholar
Khayat, R. E. 2016 Impinging planar jet flow and hydraulic jump on a horizontal surface with slip. J. Fluid Mech. 808, 258–289.Google Scholar
Khayat, R. E. & Kim, K. 2006 Thin-film flow of a viscoelastic fluid on an axisymmetric substrate of arbitrary shape. J. Fluid Mech. 552, 37–71.Google Scholar
Lawley, A. 1992 Atomization: The Production of Metal Powders. Metal Powder Industries Federation.Google Scholar
Lienhard, J. 2006 Heat transfer by impingement of circular free-surface liquid jets. In 18th National & 7th ISHMT-ASME Heat and Mass Transfer Conference, pp. 1–17.Google Scholar
Liu, X. & Lienhard, J. 1993 The hydraulic jump in circular jet impingement and in other thin liquid films. Exp. Fluids 15, 108–116.Google Scholar
Liu, X., Gabour, L. A. & Lienhard, J. 1993 Stagnation-point heat transfer during impingement of laminar liquid jets: analysis including surface tension. Trans. ASME J. Heat Transfer 115, 99–105.Google Scholar
Lubarda, V. & Talke, K. A. 2011 Analysis of the equilibrium droplet shape based on an ellipsoidal droplet Model. Langmuir 27, 10705–10713.Google Scholar
Majeed, M. H. 2014 Static contact angle and large water droplet thickness measurements with the change of water temperature. Nahrain Univ. College Eng. J. 17 (1), 114–128.Google Scholar
Martens, E. A., Watanabe, S. & Bohr, T. 2012 Model for polygonal hydraulic jumps. Phys. Rev. E 85, 036316.Google Scholar
Matar, O. K., Sisoev, G. M. & Lawrence, C. J. 2004 Evolution scales for wave regimes in liquid film flow over a spinning disk. Phys. Fluids 16, 1532–1545.Google Scholar
MicroChemicals2013 Spin coating of photoresists http://www.microchemicals.com/technical_information/spin_coating_photoresist.pdf.Google Scholar
Miyasaka, Y. 1974 On the flow of a viscous free boundary jet on a rotating disk. Bull J. Soc. Mech. Engng 17, 1469–1475.Google Scholar
Mohajer, B. & Li, R. 2015 Circular hydraulic jump on finite surfaces with capillary limit. Phys. Fluids 27, 117102.Google Scholar
Ozar, B., Cetegen, B. M. & Faghri, A. 2003 Experiments on the flow of a thin liquid film over a horizontal stationary and rotating disk surface. Exp. Fluids 34, 556–565.Google Scholar
Passandideh-Fard, M., Teymourtash, A. R. & Khavari, M. 2011 Numerical study of circular hydraulic jump using volume-of-fluid method. Trans. ASME J. Fluids Engng 133, 011401.Google Scholar
Prince, J. F., Maynes, D. & Crockett, J. 2012 Analysis of laminar jet impingement and hydraulic jump on a horizontal surface with slip. Phys. Fluids 24, 102103.Google Scholar
Rahman, M. & Faghri, A. 1992 Numerical simulation of fluid flow and heat transfer in a thin liquid film over a rotating disk. Intl J. Heat Mass Transfer 35, 1441–1453.Google Scholar
Rao, A. & Arakeri, J. H. 1998 Integral analysis applied to radial film flows. Intl J. Heat Mass Transfer 41, 2757–2767.Google Scholar
Rauscher, J., Kelly, R. & Cole, J. 1973 An asymptotic solution for the laminar flow of a thin film on a rotating disk. Trans. ASME J. Appl. Mech. 40, 43–47.Google Scholar
Rayleigh, O. M. 1914 On the theory of long waves and bores. Proc. R. Soc. Lond. A 90, 324–328.Google Scholar
Rice, J., Faghri, A. & Cetegen, B. 2005 Analysis of a free surface film from a controlled liquid impinging jet over a rotating disk including conjugate effects, with and without evaporation. Intl. J. Heat Mass Transfer 48, 5192–5204.Google Scholar
Rojas, N., Argentina, M. & Tirapegui, E. 2010 Inertial lubrication theory. Phys. Rev. Lett. 104, 187801.Google Scholar
Rojas, N., Argentina, M. & Tirapegui, E. 2013 A progressive correction to the circular hydraulic jump scaling. Phys. Fluids 25, 042105.Google Scholar
Rojas, N. & Tirapegui, E. 2015 Harmonic solutions for polygonal hydraulic jumps in thin fluid films. J. Fluid Mech. 780, 99–119.Google Scholar
Schlichtling, H. 2000 Boundary-Layer Theory, 8th edn. Springer.Google Scholar
Sisoev, G. M., Goldgof, D. B. & Korzhova, V. N. 2010 Stationary spiral waves in film flow over a spinning disk. Phys. Fluids. 22, 052106.Google Scholar
Sisoev, G. M., Matar, O. K. & Lawrence, C. J. 2003 Axisymmetric wave regimes in viscous liquid film flow over a spinning disk. J. Fluid Mech. 495, 385–411.Google Scholar
Stevens, J. & Webb, B. W. 1992 Measurements of the free surface flow structure under an impinging free liquid jet. Trans. ASME J. Heat Transfer 114, 79–83.Google Scholar
Tani, I. 1949 Water jump in the boundary layer. J. Phys. Soc. Japan 4, 212–215.Google Scholar
Teymourtash, A. R. & Mokhlesi, M. 2015 Experimental investigation of stationary and rotational structures in non-circular hydraulic jumps. J. Fluid Mech. 762, 344–360.Google Scholar
Thomas, S., Faghri, A. & Hankey, W. 1991 Experimental analysis and flow visualization of a thin liquid film on a stationary and rotating disk. Trans. ASME J. Fluids Engng 113, 73–80.Google Scholar
Thomas, S., Hankey, W., Faghri, A. & Swanson, T. 1990 One-dimensional analysis of the hydrodynamic and thermal characteristics of thin film flows including hydraulic jump and rotation. Trans. ASME J. Heat Transfer 112, 728–735.Google Scholar
Uma, B. & Usha, R. 2009 A thin conducting liquid film on a spinning disk in the presence of a magnetic field: dynamics and stability. Trans. ASME J. Appl. Mech. 76, 041002.Google Scholar
Vicente, C. M. S., Andre, P. S. & Ferreira, R. A. S. 2012 Simple measurement of surface free energy using a web cam. Rev. Brasil. Ens. Fisica 34, 3312.Google Scholar
Watson, E. 1964 The spread of a liquid jet over a horizontal plane. J. Fluid Mech. 20, 481–499.Google Scholar
Yang, S. & Chen, C. 1992 Laminar film condensation on a finite-size horizontal plate with suction at the wall. Appl. Math. Model. 16, 325–329.Google Scholar
Yang, Y., Chen, C. & Hsu, P. 1997 Laminar film condensation on a finite-size wavy disk. Appl. Math. Model. 21, 139–144.Google Scholar
Zhao, J. & Khayat, R. E. 2008 Spread of a non-Newtonian liquid jet over a horizontal plate. J. Fluid Mech. 613, 411–443.Google Scholar