0
Research Papers: Fundamental Issues and Canonical Flows

Influence of Upstream Conditions and Gravity on Highly Inertial Thin-Film Flow

[+] Author and Article Information
Roger E. Khayat

Department of Mechanical and Materials Engineering, The University of Western Ontario, London, ON, N6A 5B9, Canadarkhayat@uwo.ca

J. Fluids Eng 130(6), 061202 (May 19, 2008) (13 pages) doi:10.1115/1.2928387 History: Received February 19, 2007; Revised December 03, 2007; Published May 19, 2008

Steady two-dimensional thin-film flow of a Newtonian fluid is examined in this theoretical study. The influence of exit conditions and gravity is examined in detail. The considered flow is of moderately high inertia. The flow is dictated by the thin-film equations of boundary layer type, which are solved by expanding the flow field in orthonormal modes in the transverse direction and using Galerkin projection method, combined with integration along the flow direction. Three types of exit conditions are investigated, namely, parabolic, semiparabolic, and uniform flow. It is found that the type of exit conditions has a significant effect on the development of the free surface and flow field near the exit. While for the parabolic velocity profile at the exit, the free surface exhibits a local depression, for semiparabolic and uniform velocity profiles, the height of the film increases monotonically with streamwise position. In order to examine the influence of gravity, the flow is studied down a vertical wall as well as over a horizontal wall. The role of gravity is different for the two types of wall orientation. It is found that for the horizontal wall, a hydraulic-jump-like structure is formed and the flow further downstream exhibits a shock. The influence of exit conditions on shock formation is examined in detail.

Copyright © 2008 by American Society of Mechanical Engineers
Your Session has timed out. Please sign back in to continue.

References

Figures

Grahic Jump Location
Figure 1

Schematic of two-dimensional thin-film flow emerging from a channel. The figure also shows the dimensionless notations used in the formulation.

Grahic Jump Location
Figure 2

Influence of the number of modes on the streamwise flow. The figure shows u profiles across the thickness of the film at x=10. Exact solution obtained by MATLAB is also included.

Grahic Jump Location
Figure 4

Influence of inertia on the film height of the flow at Re∊[25,100] subject to the parabolic exit condition (a). Flow field with contours of the velocity magnitude (b) and shear-stress distribution (c) at Re=25. Gravity is neglected.

Grahic Jump Location
Figure 5

Influence of inertia on the film height of the flow at Re∊[25,100] subject to the semiparabolic exit condition (a). Flow field with contours of the velocity magnitude (b) and shear-stress distribution (c) at Re=25. Gravity is neglected.

Grahic Jump Location
Figure 6

Influence of inertia on the film height of the flow at Re∊[25,100] subject to the uniform exit condition (a). Flow field with contours of the velocity magnitude (b) and shear-stress distribution (c) at Re=25. Gravity is neglected.

Grahic Jump Location
Figure 14

Influence of gravity on the film height for Fr∊[0.3,∞), Re=100, α=0, and ε=0.01. The exit conditions are (a) parabolic, (b) semiparabolic, and (c) uniform. The solution is based on six modes.

Grahic Jump Location
Figure 3

Convergence assessment based on the energy ratio criterion. The figure shows the relationship between velocity u convergence criterion IuM and the mode number M.

Grahic Jump Location
Figure 7

Convergence assessment based on the energy ratio criterion. The figure shows the relationship between velocity u convergence criterion IuM and the mode number M for the parabolic, semiparabolic, and uniform exit conditions (a). Error assessment based on the conservation of mass (b).

Grahic Jump Location
Figure 8

Influence of gravity on the film height for Fr∊[3,∞), Re=100, and α=π∕2 (a). Flow field with contours of the velocity magnitude for Fr=3 (b). The exit condition is parabolic.

Grahic Jump Location
Figure 9

Influence of gravity on the film height for Fr∊[3,∞), Re=100, and α=π∕2 (a). Flow field with contours of the velocity magnitude for Fr=3 (b). The exit condition is semiparabolic.

Grahic Jump Location
Figure 15

Influence of gravity on position of (a) maximum height hmax and (b) shock xshock for Re=100, α=0, and ε=0.01. The exit conditions are parabolic, semiparabolic, and uniform. The solution is based on six modes.

Grahic Jump Location
Figure 16

Flow field and contours of velocity magnitude (a) and shear-stress distribution and streamlines (b) of flow at Re=100, Fr=0.4, α=0, and ε=0.01. The exit condition is parabolic.

Grahic Jump Location
Figure 17

Influence of inertia on the film height for Re∊[25,100], Fr=0.4, α=0, and ε=0.01. The exit condition is parabolic. The solution is based on six modes.

Grahic Jump Location
Figure 18

Influence of inertia on position of xshock for Re=100, α=0, and ε=0.01. The exit conditions are parabolic, semiparabolic, and uniform. The solution is based on six modes.

Grahic Jump Location
Figure 10

Influence of gravity on the film height for Fr∊[3,∞), Re=100, and α=π∕2 (a). Flow field with contours of the velocity magnitude for Fr=3 (b). The exit condition is uniform.

Grahic Jump Location
Figure 11

Influence of gravity on the film height for Fr∊[0.2,∞), Re=100, α=0, and ε=0.01. The exit condition is parabolic. The solution is based on one mode.

Grahic Jump Location
Figure 12

Influence of gravity on position of a turning point for Re=100, α=0, and ε=0.01. The exit conditions are parabolic, semiparabolic, and uniform. The solution is based on one mode.

Grahic Jump Location
Figure 13

Influence of inertia on the film height for Re∊[25,100]Fr=0.4, α=0, and ε=0.01. The exit condition is parabolic. The solution is based on one mode.

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In