The motion of a rising disk in a rotating axially bounded fluid for large Taylor number

1995 ◽  
Vol 291 ◽  
pp. 1-32 ◽  
Author(s):  
Marius Ungarish ◽  
Dmitry Vedensky
Keyword(s):  
Exact Solution ◽  
Boundary Value Problem ◽  
Equations Of Motion ◽  
Taylor Number ◽  
Rossby Number ◽  
Rotating Fluid ◽  
Asymptotic Results ◽  
Dual Integral Equations ◽  
Wall Distance ◽  
Taylor Column

The motion of a disk rising steadily along the axis in a rotating fluid between two infinite plates is considered. In the limit of zero Rossby number and with the disk in the middle position, the boundary value problem based on the linear, viscous equations of motion is reduced to a system of dual-integral equations which renders ‘exact’ solutions for arbitrary values of the Taylor number, Ta, and disk-to-wall distance, H (scaled by the radius of the disk). The investigation is focused on the drag and on the flow field when Ta is large (but finite) for various H. Comparisons with previous asymptotic results for ‘short’ and ‘long’ containers, and with the preceding unbounded-configuration ‘exact’ solution, provide both confirmation and novel insights.In particular, it is shown that the ‘free’ Taylor column on the particle appears for H > 0.08 Ta and attains its fully developed features when H > 0.25 Ta (approximately). The present drag calculations improve the compatibility of the linear theory with Maxworthy's (1968) experiments in short containers, but for the long container the claimed discrepancy with experiments remains unexplained.

The motion generated by a slowly rising disk in an unbounded rotating fluid for arbitrary Taylor number

1994 ◽  
Vol 262 ◽  
pp. 1-26 ◽  
Author(s):  
D. Vedensky ◽  
M. Ungarish
Keyword(s):  
Equations Of Motion ◽  
Ekman Layer ◽  
The Body ◽  
Taylor Number ◽  
Present Formulation ◽  
Rotating Fluid ◽  
Asymptotic Results ◽  
Dual Integral Equations ◽  
Axis Of Rotation ◽  
Insight Into

The motion of a disk rising steadily parallel to the axis of rotation in a uniformly rotating unbounded liquid is considered. In the limit of zero Rossby number the linear viscous equations of motion are reduced to a system of dual integral equations which renders an ‘exact’ solution for arbitrary values of the Taylor number, Ta. The investigation is focused on the drag and the flow field. In the limits of small and large Ta the asymptotic results of the present formulation agree with – and extend – previous investigations by different approaches.A particular novel feature, for large Ta, is the contribution of the Ekman-layer flux to the outer motion. New insight into the structure of the Taylor column is gained; in particular, it is shown that the main part of the column is a ‘bubble’ of recirculating fluid, detached from the body and not communicating with the Ekman layer. However, it turns out that the essential discrepancy in drag between experiments (Maxworthy 1970) and previous theories cannot be attributed to the Ekman-layer suction effect.

Difference Schemes for Nonlinear BVPS on the Semiaxis

2007 ◽  
Vol 7 (1) ◽  
pp. 25-47 ◽  
Author(s):  
I.P. Gavrilyuk ◽  
M. Hermann ◽  
M.V. Kutniv ◽  
V.L. Makarov
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Boundary Value Problem ◽  
Difference Scheme ◽  
Adaptive Algorithm ◽  
Order Differential Equation ◽  
Constructive Method ◽  
Numerical Examples ◽  
Exact Difference ◽  
The Given

Abstract The scalar boundary value problem (BVP) for a nonlinear second order differential equation on the semiaxis is considered. Under some natural assumptions it is shown that on an arbitrary finite grid there exists a unique three-point exact difference scheme (EDS), i.e., a difference scheme whose solution coincides with the projection of the exact solution of the given differential equation onto the underlying grid. A constructive method is proposed to derive from the EDS a so-called truncated difference scheme (n-TDS) of rank n, where n is a freely selectable natural number. The n-TDS is the basis for a new adaptive algorithm which has all the advantages known from the modern IVP-solvers. Numerical examples are given which illustrate the theorems presented in the paper and demonstrate the reliability of the new algorithm.

scholarly journals Symmetry analysis of rotating fluid

2005 ◽  
Vol 47 (1) ◽  
pp. 65-74 ◽  
Author(s):  
K. Fakhar ◽  
Zu-Chi Chen ◽  
Xiaoda Ji
Keyword(s):  
Steady State ◽  
Infinite Number ◽  
Special Function ◽  
Equations Of Motion ◽  
Time Dependent ◽  
Lie Theory ◽  
Rotating Fluid ◽  
Type Solution ◽  
Function Type ◽  
Basic Equations

AbstractThe machinery of Lie theory (groups and algebras) is applied to the unsteady equations of motion of rotating fluid. A special-function type solution for the steady state is derived. It is then shown how the solution generates an infinite number of time-dependent solutions via three arbitrary functions of time. This algebraic structure also provides the mechanism to search for other solutions since its character is inferred from the basic equations.

Instability of quasi-geostrophic vortices in a two-layer ocean with a thin upper layer

2003 ◽  
Vol 475 ◽  
pp. 303-331 ◽  
Author(s):  
E. S. BENILOV
Keyword(s):  
Boundary Value Problem ◽  
Lower Layer ◽  
Boundary Value ◽  
Stability Boundary ◽  
Normal Modes ◽  
Layer Depth ◽  
Asymptotic Results ◽  
Gaussian Profile ◽  
Swirl Velocity ◽  
The Stability

We examine the stability of a quasi-geostrophic vortex in a two-layer ocean with a thin upper layer on the f-plane. It is assumed that the vortex has a sign-definite swirl velocity and is localized in the upper layer, whereas the disturbance is present in both layers. The stability boundary-value problem admits three types of normal modes: fast (upper-layer-dominated) modes, responsible for equivalent-barotropic instability, and two slow baroclinic types (mixed- and lower-layer-dominated modes). Fast modes exist only for unrealistically small vortices (with a radius smaller than half of the deformation radius), and this paper is mainly focused on the slow modes. They are examined by expanding the stability boundary-value problem in powers of the ratio of the upper-layer depth to the lower-layer depth. It is demonstrated that the instability of slow modes, if any, is associated with critical levels, which are located at the periphery of the vortex. The complete (sufficient and necessary) stability criterion with respect to slow modes is derived: the vortex is stable if and only if the potential-vorticity gradient at the critical level and swirl velocity are of the same sign. Several vortex profiles are examined, and it is shown that vortices with a slowly decaying periphery are more unstable baroclinically and less barotropically than those with a fast-decaying periphery, with the Gaussian profile being the most stable overall. The asymptotic results are verified by numerical integration of the exact boundary-value problem, and interpreted using oceanic observations.

A Mass-Conserving Algorithm for Dynamical Lubrication Problems With Cavitation

2009 ◽  
Vol 131 (3) ◽  
Author(s):  
Roberto F. Ausas ◽  
Mohammed Jai ◽  
Gustavo C. Buscaglia
Keyword(s):  
Exact Solution ◽  
Reynolds Equation ◽  
Journal Bearing ◽  
Equations Of Motion ◽  
Source Code ◽  
Squeeze Flow ◽  
Complementarity Conditions ◽  
Relaxation Scheme ◽  
Fluid Fraction ◽  
Oscillatory Squeeze Flow

A numerical algorithm for fully dynamical lubrication problems based on the Elrod–Adams formulation of the Reynolds equation with mass-conserving boundary conditions is described. A simple but effective relaxation scheme is used to update the solution maintaining the complementarity conditions on the variables that represent the pressure and fluid fraction. The equations of motion are discretized in time using Newmark’s scheme, and the dynamical variables are updated within the same relaxation process just mentioned. The good behavior of the proposed algorithm is illustrated in two examples: an oscillatory squeeze flow (for which the exact solution is available) and a dynamically loaded journal bearing. This article is accompanied by the ready-to-compile source code with the implementation of the proposed algorithm.

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