Analytical Solutions for Two-Dimensional Transport Equation with Time-Dependent Dispersion Coefficients

1996 ◽  
Vol 1 (1) ◽  
pp. 20-32 ◽  
Author(s):  
Mustafa M. Aral ◽  
Boshu Liao
Keyword(s):  
Transport Equation ◽  
Analytical Solutions ◽  
Time Dependent ◽  
Two Dimensional ◽  
Dispersion Coefficients

scholarly journals Semi-analytical solutions for two-dimensional elastic capsules in Stokes flow

2012 ◽  
Vol 468 (2146) ◽  
pp. 2915-2938 ◽  
Author(s):  
Michael Higley ◽  
Michael Siegel ◽  
Michael R. Booty
Keyword(s):  
Analytical Solutions ◽  
Stokes Flow ◽  
Finite Time ◽  
Capillary Number ◽  
Time Dependent ◽  
Two Dimensional ◽  
Elastic Interfaces ◽  
Critical Capillary Number ◽  
Field Velocity ◽  
Elastic Capsules

Elastic capsules occur in nature in the form of cells and vesicles and are manufactured for biomedical applications. They are widely modelled, but there are few analytical results. In this paper, complex variable techniques are used to derive semi-analytical solutions for the steady-state response and time-dependent evolution of two-dimensional elastic capsules with an inviscid interior in Stokes flow. This provides a complete picture of the steady response of initially circular capsules in linear strain and shear flows as a function of the capillary number Ca . The analysis is complemented by spectrally accurate numerical computations of the time-dependent evolution. An imposed nonlinear strain that models the far-field velocity in Taylor's four-roller mill is found to lead to cusped steady shapes at a critical capillary number Ca c for Hookean capsules. Numerical simulation of the time-dependent evolution for Ca > Ca c shows the development of finite-time cusp singularities. The dynamics immediately prior to cusp formation are asymptotically self-similar, and the similarity exponents are predicted analytically and confirmed numerically. This is compelling evidence of finite-time singularity formation in fluid flow with elastic interfaces.

scholarly journals Integrable unsteady motion with an application to ocean eddies

1998 ◽  
Vol 5 (3) ◽  
pp. 145-151
Author(s):  
A. D. Kirwan, Jr. ◽  
B. L. Lipphardt, Jr.
Keyword(s):  
Potential Vorticity ◽  
Particle Motion ◽  
Gulf Stream ◽  
Incompressible Flows ◽  
Unsteady Motion ◽  
Ring System ◽  
Time Dependent ◽  
Two Dimensional ◽  
Ocean Eddies ◽  
Multilayered Systems

Abstract. Application of the Brown-Samelson theorem, which shows that particle motion is integrable in a class of vorticity-conserving, two-dimensional incompressible flows, is extended here to a class of explicit time dependent dynamically balanced flows in multilayered systems. Particle motion for nonsteady two-dimensional flows with discontinuities in the vorticity or potential vorticity fields (modon solutions) is shown to be integrable. An example of a two-layer modon solution constrained by observations of a Gulf Stream ring system is discussed.

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