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Research Papers

A Thin Conducting Liquid Film on a Spinning Disk in the Presence of a Magnetic Field: Dynamics and Stability

[+] Author and Article Information
B. Uma

Center for Risk Studies and Safety, University of California, Santa Barbara, Goleta, CA 93117ubalakrishnan@engr.ucsb.edu

R. Usha

Department of Mathematics, Indian Institute of Technology Madras, Chennai 600 036, Indiaushar@iitm.ac.in

J. Appl. Mech 76(4), 041002 (Apr 21, 2009) (14 pages) doi:10.1115/1.3086589 History: Received February 06, 2008; Revised December 09, 2008; Published April 21, 2009

A theoretical analysis of the effects of a magnetic field on the dynamics of a thin nonuniform conducting film of an incompressible viscous fluid on a rotating disk has been considered. A nonlinear evolution equation describing the shape of the film interface has been derived as a function of space and time and has been solved numerically. The temporal evolution of the free surface of the fluid and the rate of retention of the liquid film on the spinning disk have been obtained for different values of Hartmann number M, evaporative mass flux parameter E, and Reynolds number Re. The results show that the relative volume of the fluid retained on the spinning disk is enhanced by the presence of the magnetic field. The stability characteristics of the evolution equation have been examined using linear theory. For both zero and nonzero values of the nondimensional parameter describing the magnetic field, the results show that (a) the infinitesimal disturbances decay for small wave numbers and are transiently stable for larger wave numbers when there is either no mass transfer or there is evaporation from the film surface, and although the magnitude of the disturbance amplitude is larger when the magnetic field is present, it decays to zero earlier than for the case when the magnetic field is absent, and (b) when absorption is present at the film surface, the film exhibits three different domains of stability: disturbances of small wave numbers decay, disturbances of intermediate wave numbers grow transiently, and those of large wave numbers grow exponentially. The range of stable wave numbers increases with increase in Hartmann number.

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Copyright © 2009 by American Society of Mechanical Engineers
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Figures

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Figure 1

Schematic representation of film flow on a rotating disk

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Figure 2

Successive free surface contours for an initially sinusoidal distribution for Re=6.2, E=0, and ϵ=0.01

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Figure 3

Successive free surface contours for an initially sinusoidal distribution for Re=6.2, E=0, and ϵ=0.01

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Figure 4

Relative volume of the lubricant retention for an initially sinusoidal distribution for Re=6.2 and ϵ=0.01

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Figure 5

Relative volume of the lubricant retention for an initially sinusoidal distribution for Re=0.0, E=0, and ϵ=0.01

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Figure 6

Successive free surface contours for an initially Gaussian plus and slowly falling distribution for Re=6.2, E=0, and ϵ=0.01

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Figure 7

Relative volume of the lubricant retention for an initially Gaussian plus and slowly falling distribution for Re=6.2, E=0, and ϵ=0.01

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Figure 8

Basic-state film thickness for principal values of the evaporation parameter and different values of Hartmann number

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Figure 9

Normal mode amplitude for a disturbance when Re=6.2, ϵ=0.01, and E=0: (a) stable and (b) transiently stable

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Figure 10

Normal mode amplitude for a disturbance when E=0.0012 and ϵ=0.01: (a) stable and (b) transiently stable

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Figure 11

Normal mode amplitude for a disturbance when E=−0.0012 and ϵ=0.01: (a) stable, (b) unstable, and (c) transiently stable

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Figure 12

Domains of instability for (a) M=0 and (b) M=2. S: stable; TS: transiently stable; U: unstable

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