scholarly journals NEARSHORE CIRCULATIONS ON A WAVY COAST

1988 ◽  
Vol 1 (21) ◽  
pp. 193
Author(s):  
Ming-Chung Lin ◽  
Sheng-Yeh Hwang
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Total Water ◽  
Radiation Stress ◽  
Circular Arc ◽  
Wave Characteristics ◽  
Circulation Patterns ◽  
The Mean ◽  
Set Up ◽  
Wave Induced

Nearshore circulations, produced by wave-induced radiation stress gradients, form different circulation patterns under different wave characteristics and topographical conditions. Although numerous studies of nearshore circulations, such as Bowen (1969), Miller and Barcilon (1978), Dalrymple & Lozano (1978) and so on, have been appeared in the literature, it seems that little attention is paid on the case of a non-straight shoreline. O'Rorske & Leblond (1972) have investigated the wave-induced longshore currents in a semicircular bay, while Oda (1982) has used the coordinate transformation to treat nearshore circulations on a circular-arc shaped coast. Lin and Lee (1982), introduced a small perturbation quantity of wave set-up and set-down, induced by the non-straight shoreline, into the mean total water depth to obtain a governing partial differential equation by which they investigated the nearshore circulations on a cuspate coast. In Lin and Liou (1986), a more general equation was deduced on the orthogonal curvlinear coordinate system to unify the diversities among the related theories, and moreover, to investigate the nearshore circulations on the arc-shaped coast.

The linear stability of a core–annular flow in an asymptotically corrugated tube

2002 ◽  
Vol 466 ◽  
pp. 113-147 ◽  
Author(s):  
HSIEN-HUNG WEI ◽  
DAVID S. RUMSCHITZKI
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Linear Stability ◽  
Annular Flow ◽  
Immiscible Fluids ◽  
Base Flow ◽  
Leading Order ◽  
Partial Differential ◽  
The Mean ◽  
Liquid Displacement

This paper examines the core–annular flow of two immiscible fluids in a straight circular tube with a small corrugation, in the limit where the ratio ε of the mean undisturbed annulus thickness to the mean core radius and the corrugation (characterized by the parameter σ) are both asymptotically small and where the surface tension is small. It is motivated by the problems of liquid–liquid displacement in irregular rock pores such as occur in secondary oil recovery and in the evolution of the liquid film lining the bronchii in the lungs whose diameters vary over different generations of branching. We investigate the asymptotic base flow in this limit and consider the linear stability of its leading order (in the corrugation parameter) solution. For the chosen scalings of the non-dimensional parameters the core's base flow slaves that of the annulus. The equation governing the leading-order interfacial position for a given wall corrugation function shows a competition between shear and capillarity. The former tends to align the interface shape with that of the wall and the latter tends to introduce a phase shift, which can be of either sign depending on whether the circumferential or the longitudinal component of capillarity dominates. The asymptotic linear stability of this leading-order base flow reduces to a single partial differential equation with non-constant coefficients deriving from the non-uniform base flow for the time evolution of an interfacial disturbance. Examination of a single mode k wall function allows the use of Floquet theory to analyse this equation. Direct numerical solutions of the above partial differential equation agree with the predictions of the Floquet analysis. The resulting spectrum is periodic in α- space, α being the disturbance wavenumber space. The presence of a small corrugation not only modifies (at order σ2) the primary eigenvalue of the system. In addition, short-wave order-one disturbances that would be stabilized flowing to capillarity in the absence of corrugation can, in the presence of corrugation and over time scales of order ln(1/σ), excite higher wall harmonics (α±nk) leading to the growth of unstable long waves. Similar results obtain for more complicated wall shape functions. The main result is that a small corrugation makes a core–annular flow unstable to far more disturbances than would destabilize the same uncorrugated flow system. A companion paper examines that competition between this added destabilization due to pore corrugation with the wave steepening and stabilization in the weakly nonlinear regime.

SYSTEM CHARACTERISTICS OF DISTRIBUTED PARAMETER SYSTEMS

2020 ◽  
Vol 70 (3) ◽  
pp. 34-44
Author(s):  
Kamen Perev
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Initial Conditions ◽  
Distributed Parameter Systems ◽  
Point Of View ◽  
Distributed Parameter ◽  
Partial Differential ◽  
Special Cases ◽  
The Green Function ◽  
Set Up

The paper considers the problem of distributed parameter systems modeling. The basic model types are presented, depending on the partial differential equation, which determines the physical processes dynamics. The similarities and the differences with the models described in terms of ordinary differential equations are discussed. A special attention is paid to the problem of heat flow in a rod. The problem set up is demonstrated and the methods of its solution are discussed. The main characteristics from a system point of view are presented, namely the Green function and the transfer function. Different special cases for these characteristics are discussed, depending on the specific partial differential equation, as well as the initial conditions and the boundary conditions.

scholarly journals On the conformal deformation of Riemannian structures

1990 ◽  
Vol 42 (2) ◽  
pp. 307-313
Author(s):  
Yoon-Tae Jung
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Positive Constant ◽  
Smooth Function ◽  
Compact Manifold ◽  
Nonlinear Partial Differential Equation ◽  
Necessary Condition ◽  
Conformal Deformation ◽  
The Mean ◽  
Image Position

In this paper, we study a nonlinear partial differential equation on a compact manifold;where a > 1 is a constant, r is a positive constant, and H is a prescribed smooth function.Kazdan and Warner showed that if λ1(g) < 0 and < 0, where is the mean of H, then there is a constant 0 < r0(H) ≤ ∞ such that one can solve this equation for 0 < r < r0(H), but not for r > r0(H). They also proved that if r0(H) = ∞, then H(x) ≤ 0 (≢0) for all x ∈ M. They conjectured that this necessary condition might be sufficient.I show that this conjecture is right; that is, if H(x) ≤ 0 (≠ 0) for all x ∈ M, then r0(H) = ∞.

scholarly journals Exploring Capabilities within ForTrilinos by Solving the 3D Burgers Equation

2012 ◽  
Vol 20 (3) ◽  
pp. 275-292 ◽  
Author(s):  
Karla Morris ◽  
Damian W.I. Rouson ◽  
M. Nicole Lemaster ◽  
Salvatore Filippone
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Message Passing ◽  
Message Passing Interface ◽  
Mathematical Formulation ◽  
Three Dimensional ◽  
Abstract Data Type ◽  
Partial Differential ◽  
Sparse Linear Solvers ◽  
Set Up

We present the first three-dimensional, partial differential equation solver to be built atop the recently released, open-source ForTrilinos package (http://trilinos.sandia.gov/packages/fortrilinos). ForTrilinos currently provides portable, object-oriented Fortran 2003 interfaces to the C++ packages Epetra, AztecOO and Pliris in the Trilinos library and framework [ACM Trans. Math. Softw.31(3) (2005), 397–423]. Epetra provides distributed matrix and vector storage and basic linear algebra calculations. Pliris provides direct solvers for dense linear systems. AztecOO provides iterative sparse linear solvers. We demonstrate how to build a parallel application that encapsulates the Message Passing Interface (MPI) without requiring the user to make direct calls to MPI except for startup and shutdown. The presented example demonstrates the level of effort required to set up a high-order, finite-difference solution on a Cartesian grid. The example employs an abstract data type (ADT) calculus [Sci. Program.16(4) (2008), 329–339] that empowers programmers to write serial code that lower-level abstractions resolve into distributed-memory, parallel implementations. The ADT calculus uses compilable Fortran constructs that resemble the mathematical formulation of the partial differential equation of interest.

A Coupled-Mode Technique for the Prediction of Wave-Induced Set-Up and Mean Flow in Variable Bathymetry Domains

2007 ◽  
Author(s):  
K. A. Belibassakis ◽  
Th. P. Gerostathis ◽  
G. A. Athanassoulis
Keyword(s):  
Shallow Water ◽  
Water Waves ◽  
Water Environment ◽  
Wave Model ◽  
Mean Flow ◽  
Flow Equations ◽  
Coupled Mode ◽  
The Mean ◽  
Set Up ◽  
Wave Induced

In the present work, a complete, phase-resolving wave model is coupled with an iterative solver of the mean-flow equations in intermediate and shallow water depth, permitting an accurate calculation of wave set-up and wave-induced current in intermediate and shallow water environment with possibly steep bathymetric variations. The wave model is based on the consistent coupled-mode system of equations, developed by Athanassoulis & Belibassakis (1999) for the propagation of water waves in variable bathymetry regions. This model improves the predictions of the mild-slope equation, permitting the treatment of wave propagation in regions with steep bottom slope and/or large curvature. In addition, it supports the consistent calculation of wave velocity up to and including the bottom boundary. The above wave model has been further extended to include the effects of bottom friction and wave breaking, which are important factors for the calculation of radiation stresses on decreasing depth. The latter have been used as forcing terms to the mean flow equations in order to predict wave-induced set up and mean flow in open and closed domains. Numerical results obtained by the present model are presented and compared with predictions obtained by the mild-slope approximation (Massel & Gourlay 2000), and experimental data (Gourlay 1996).

Mechanical Model and Analysis on Movement of Rock Strata between Coal Seams in Pillar Upward Mining of Left-Over Coal

2011 ◽  
Vol 58-60 ◽  
pp. 393-398
Author(s):  
Guo Rui Feng ◽  
Jing Zheng ◽  
Ya Feng Ren ◽  
Cong Ming Zhong ◽  
Li Xun Kang
Keyword(s):  
Differential Equation ◽  
Mechanical Properties ◽  
Partial Differential Equation ◽  
Mechanical Model ◽  
Theoretical Foundation ◽  
Control Measures ◽  
Coal Seams ◽  
Set Up ◽  
Rock Strata ◽  
Upward Mining

Work out control measures for pillar upward mining of left-over coal, in view of coal floor movement, this paper analyzes theoretical foundation and mechanical properties of mechanical model on coal floor movement of rock strata between coal seams in pillar upward mining of left-over coal and presents a simplified corresponding mechanical condition. Taking into the relation between layer movement and time consideration, the PTh model is used to show its rheological properties. By using a two-dimension model, mechanical model of coal floor movement is set up and both ends of cantilever beam are fixed. Finally, subsiding partial differential equation is induced at different districts in pillar upward mining of left-over coal. After solving the equation, prediction model about law of coal floor movement at different districts is gotten in pillar upward mining of left-over coal, which provides a theory basis for further study on upward mining.

Acoustic Field in Ducts With Sinusoidal Area Variation

2013 ◽  
Vol 136 (1) ◽  
Author(s):  
N. S. Vikramaditya ◽  
R. B. Kaligatla
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Acoustic Field ◽  
Second Order ◽  
Ordinary Linear Differential Equation ◽  
Order Partial Differential Equation ◽  
Mean Flow ◽  
Area Variation ◽  
The Mean ◽  
Linear Differential

The purpose of this article is to provide an analytical solution for the acoustic field in a duct with sinusoidal area variation along the length. The equation describing the acoustic field in a variable area duct is a second-order partial differential equation. It is converted into a second-order ordinary linear differential equation, whose solution is dependent on the choice of area variation. The solution for the differential equation is obtained in terms of the area and is obtained neglecting the mean flow. Therefore, it is applicable in the absence of mean flow or in cases where the effects of mean flow are insignificant.

scholarly journals Relationship between Three-Dimensional Radiation Stress and Vortex-Force Representations

2021 ◽  
Vol 9 (8) ◽  
pp. 791
Author(s):  
Duoc Tan Nguyen ◽  
Ad J. H. M. Reniers ◽  
Dano Roelvink
Keyword(s):  
Three Dimensional ◽  
Ocean Model ◽  
Radiation Stress ◽  
Mean Motion ◽  
Numerical Ocean Model ◽  
The Mean ◽  
Ocean Models ◽  
Wave Induced ◽  
Mean Current

In numerical ocean models, the effect of waves on currents is usually expressed by either vortex-force or radiation stress representations. In this paper, the differences and similarities between those two representations are investigated in detail in conditions of both conservative and nonconservative waves. In addition, comparisons between different sets of equations of mean motion that apply different representations of wave-induced forcing terms are included. The comparisons are useful for selecting a suitable numerical ocean model to simulate the mean current in conditions of waves combined with currents.

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