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Similarity Solution (similarity + solution)

Distribution by Scientific Domains
Engineering75%

Selected Abstracts

Heat transfer for Marangoni-driven boundary layer flow

HEAT TRANSFER - ASIAN RESEARCH (FORMERLY HEAT TRANSFER-JAPANESE RESEARCH), Issue 2 2002
David M. Christopher
Abstract Marangoni convection induced by variation of the surface tension with temperature along a surface influences crystal growth melts and other processes with liquid,vapor interfaces, such as boiling in both microgravity and normal gravity in some cases. This paper presents the Nusselt number for Marangoni flow over a flat surface calculated using a similarity solution for both the momentum equations and the energy equation assuming developing boundary layer flow along a surface. Solutions are presented for the surface velocity, the total flow rate, and the Nusselt number for various temperature profiles, Marangoni numbers, and Prandtl numbers. For large bubbles, the predicted boundary layer thickness would be less than the bubble diameter, so the curvature effects could be neglected and this analysis could be used as a first estimate of the effect of Marangoni flow around a vapor bubble. © 2002 Scripta Technica, Heat Trans Asian Res, 31(2): 105,116, 2002; DOI 10.1002/htj.10019 [source]

Influence of inertia, topography and gravity on transient axisymmetric thin-film flow

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 4 2004
Roger E. Khayat
Abstract This study examines theoretically the development of early transients for axisymmetric flow of a thin film over a stationary cylindrical substrate of arbitrary shape. The fluid is assumed to emerge from an annular tube as it is driven by a pressure gradient maintained inside the annulus, and/or by gravity in the axial direction. The interplay between inertia, annulus aspect ratio, substrate topography and gravity is particularly emphasized. Initial conditions are found to have a drastic effect on the ensuing flow. The flow is governed by the thin-film equations of the ,boundary-layer' type, which are solved by expanding the flow field in terms of orthonormal modes in the radial direction. The formulation is validated upon comparison with the similarity solution of Watson (J. Fluid Mech 1964; 20:481) leading to an excellent agreement when only 2,3 modes are included. The wave and flow structure are examined for high and low inertia. It is found that low-inertia fluids tend to accumulate near the annulus exit, exhibiting a standing wave that grows with time. This behaviour clearly illustrates the difficulty faced with coating high-viscosity fluids. The annulus aspect is found to be influential only when inertia is significant; there is less flow resistance for a film over a cylinder of smaller diameter. For high inertia, the free surface evolves similarly to two-dimensional flow. The substrate topography is found to have a significant effect on transient behaviour, but this effect depends strongly on inertia. It is observed that the flow of a high-inertia fluid over a step-down exhibits the formation of a secondary wave that moves upstream of the primary wave. Gravity is found to help the film (coating) flow by halting or prohibiting the wave growth. The initial film profile and velocity distribution dictate whether the fluid will flow downstream or accumulate near the annulus exit. Copyright © 2004 John Wiley & Sons, Ltd. [source]

Drained cavity expansion in sands exhibiting particle crushing

INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS, Issue 4 2002
A. R. Russell
The expansion of cylindrical and spherical cavities in sands is modelled using similarity solutions. The conventional Mohr,Coulomb failure criterion and the state parameter sand behaviour model, which enables hardening,softening, are used in the analysis. The sand state is defined in terms of a new critical state line, designed to account for the three different modes of compressive deformation observed in sands across a wide range of stresses including particle rearrangement, particle crushing and pseudoelastic deformation. Solutions are generated for cavities expanded from zero and finite radii and are compared to those solutions where a conventional critical state line has been used. It is shown that for initial states typical of real quartz sand deposits, pseudoelastic deformation does not occur around an expanding cavity. Particle crushing does occur at these states and causes a reduction in the stress surrounding the cavity. This has major implications when using cavity expansion theory to interpret the cone penetration test and pressuremeter test. Copyright © 2002 John Wiley & Sons, Ltd. [source]

The Poisson equation with local nonregular similarities

NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 4 2001
Alexander Yakhot
Abstract Moffatt and Duffy [1] have shown that the solution to the Poisson equation, defined on rectangular domains, includes a local similarity term of the form: r2log(r)cos(2,). The latter means that the second (and higher) derivative of the solution with respect to r is singular at r = 0. Standard high-order numerical schemes require the existence of high-order derivatives of the solution. Thus, for the case considered by Moffatt and Duffy, the high-order finite-difference schemes loose their high-order convergence due to the nonregularity at r = 0. In this article, a simple method is outlined to regain the high-order accuracy of these schemes, without the need of any modification in the scheme's algorithm. This is a significant consideration when one wants to use a given finite-difference computer code for problems with local nonregular similarity solutions. Numerical examples using the modified scheme in conjunction with a sixth-order finite difference approximation are provided. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17:336,346, 2001 [source]