scholarly journals EXACT SOLUTION FOR STEADY PAINT FILM FLOW OF A PSEUDO PLASTIC FLUID DOWN A VERTICAL WALL BY GRAVITY

2012 ◽  
Vol 16 (3) ◽  
pp. 181-192
Author(s):  
M.K. Alam ◽  
M.T. Rahim ◽  
S. Islam ◽  
A.M. Siddiqui
Keyword(s):  
Exact Solution ◽  
Film Flow ◽  
Paint Film ◽  
Vertical Wall ◽  
Plastic Fluid

Thin film flow of magnetohydrodynamic (MHD) pseudo-plastic fluid on vertical wall

2014 ◽  
Vol 245 ◽  
pp. 544-556 ◽  
Author(s):  
M.K. Alam ◽  
A.M. Siddiqui ◽  
M.T. Rahim ◽  
S. Islam ◽  
E.J. Avital ◽  
...  
Keyword(s):  
Thin Film ◽  
Film Flow ◽  
Vertical Wall ◽  
Thin Film Flow ◽  
Plastic Fluid

Falkner-Skan Solution for Gravity-Driven Film Flow

1985 ◽  
Vol 52 (4) ◽  
pp. 783-786 ◽  
Author(s):  
H. I. Andersson ◽  
T. Ytrehus
Keyword(s):  
Boundary Layer ◽  
Exact Solution ◽  
Film Flow ◽  
Vertical Wall ◽  
Boundary Layer Theory ◽  
Reference Solution ◽  
Viscous Boundary Layer ◽  
Viscous Film Flow ◽  
Gravity Driven ◽  
Viscous Boundary

It is demonstrated that the development of a viscous film flow down along a vertical wall can be described by the classical Falkner-Skan equation from aerodynamic boundary layer theory for the particular parameter-value m = 1/2. This leads to a well-known exact solution for the velocity field, as long as the viscous boundary layer can be considered not to interact with the free surface of the film. An exact reference solution for developing film flow is thus made available, against which approxiate solutions may be tested for accuracy.

scholarly journals Explicit Solutions of a Gravity-Induced Film Flow along a Convectively Heated Vertical Wall

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Ammarah Raees ◽  
Hang Xu
Keyword(s):  
Film Flow ◽  
Vertical Wall ◽  
Boussinesq Approximation ◽  
Convective Boundary Condition ◽  
Convective Boundary ◽  
Similarity Transformations ◽  
Reduced Equations ◽  
Homotopy Analysis ◽  
Gravity Driven ◽  
The Mathematical Model

The gravity-driven film flow has been analyzed along a vertical wall subjected to a convective boundary condition. The Boussinesq approximation is applied to simplify the buoyancy term, and similarity transformations are used on the mathematical model of the problem under consideration, to obtain a set of coupled ordinary differential equations. Then the reduced equations are solved explicitly by using homotopy analysis method (HAM). The resulting solutions are investigated for heat transfer effects on velocity and temperature profiles.

Unsteady non-linear waves in sloping channels

1958 ◽  
Vol 247 (1249) ◽  
pp. 186-198 ◽  
Keyword(s):  
Exact Solution ◽  
Water Wave ◽  
Legendre Function ◽  
Elliptic Integral ◽  
Vertical Wall ◽  
Auxiliary Function ◽  
Level Line ◽  
Wave Problem ◽  
Water Wave Problem ◽  
Non Linear

It is shown in general that the exact solution to every non-degenerate unsteady water-wave problem in a straight channel inclined at arbitrary slope, governed by the non-linear hydraulic equations, can be obtained in terms of the complete elliptic integral of the second kind, E . By means of a non-Newtonian reference frame, every such wave problem for a sloping channel can be replaced by an associated problem for a horizontal channel. For the latter, the partial differential equations become reducible and thus permit hodograph inversion. The Riemann integration method for the resulting Euler-Poisson equation yields an auxiliary function for these hydraulic problems which is transformable into a Legendre function and then into the elliptic integral. In particular, the procedure is applied to obtain the exact solution for the water wave in a sloping channel produced by sudden release of the triangular wedge of water (the reservoir) initially at rest behind a vertical wall. The behaviour of the solution is exhibited for convenience in two level-line charts, and representative wave profiles and velocity distributions are presented.

Thin-Film Flow at Moderate Reynolds Number

2000 ◽  
Vol 122 (4) ◽  
pp. 774-778 ◽  
Author(s):  
Kenneth J. Ruschak ◽  
Steven J. Weinstein
Keyword(s):  
Thin Film ◽  
Boundary Layer ◽  
Free Surface ◽  
Film Flow ◽  
Previous Analysis ◽  
Vertical Wall ◽  
Integral Approach ◽  
Internal Boundary ◽  
Thin Film Flow ◽  
Moderate Reynolds Number

Viscous, laminar, gravitationally-driven flow of a thin film over a round-crested weir is analyzed for moderate Reynolds numbers. A previous analysis of this flow utilized a momentum integral approach with a semiparabolic velocity profile to obtain an equation for the film thickness (Ruschak, K. J., and Weinstein, S. J., 1999, “Viscous Thin-Film Flow Over a Round-Crested Weir,” ASME J. Fluids Eng., 121, pp. 673–677). In this work, a viscous boundary layer is introduced in the manner of Haugen (Haugen, R., 1968, “Laminar Flow Around a Vertical Wall,” ASME J. Appl. Mech. 35, pp. 631–633). As in the previous analysis of Ruschak and Weinstein, the approximate equations have a critical point that provides an internal boundary condition for a bounded solution. The complication of a boundary layer is found to have little effect on the thickness profile while introducing a weak singularity at its beginning. The thickness of the boundary layer grows rapidly, and there is little cumulative effect of the increased wall friction. Regardless of whether a boundary layer is incorporated, the approximate free-surface profiles are close to profiles from finite-element solutions of the Navier-Stokes equation. Similar results are obtained for the related problem of developing flow on a vertical wall (Cerro, R. L., and Whitaker, S., 1971, “Entrance Region Flows With a Free Surface: the Falling Liquid Film,” Chem. Eng. Sci., 26, pp. 785–798). Less accurate results are obtained for decelerating flow on a horizontal wall (Watson, E. J., 1964, “The Radial Spread of a Liquid Jet Over a Horizontal Plane,” J. Fluid Mech. 20, pp. 481–499) where the flow is not gravitationally driven. [S0098-2202(00)01904-0]

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

2008 ◽  
Vol 130 (6) ◽  
Author(s):  
Roger E. Khayat
Keyword(s):  
Thin Film ◽  
Free Surface ◽  
Flow Field ◽  
Film Flow ◽  
Flow Direction ◽  
Vertical Wall ◽  
Galerkin Projection ◽  
Thin Film Flow ◽  
Thin Film Equations ◽  
Horizontal Wall

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.

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