Explorations in Complex Analysis

Brilleslyper, Michael; Dorff, Michael; McDougall, Jane; Rolf, James; Schaubroeck, Lisbeth; Stankewitz, Richard; Stephenson, Kenneth

doi:10.1090/clrm/040

Cited by 15 publications

(7 citation statements)

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“…As discussed, the prevertices ±δ are problem specific and, along with the constants, are often determined numerically [69]. We consider a symmetric set of prevertices because the polygon is rectangular [15]. We must also consider the branch points, and choices of branch cuts, that arise from square root and logarithm functions of complex variables.…”

Section: Mathematical Technicalitiesmentioning

confidence: 99%

“…The Schwarz-Christoffel mapping guarantees that angles are preserved, but it can be a challenge to prescribe lengths. One must solve for the appropriate prevertices, but this may not give an explicit result even when exploiting symmetry in the problem [15]. Here, we choose prevertices −1, −δ, δ, 1 and ∞, and define the height of the boulder via the value of δ (Figure 10).…”

Section: Mathematical Technicalitiesmentioning

confidence: 99%

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Potential flow over a submerged rectangular obstacle: Consequences for initiation of boulder motion

Herterich

Dias

2019

Eur. J. Appl. Math

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AbstractSteady two-dimensional fluid flow over an obstacle is solved using complex variable methods. We consider the cases of rectangular obstacles, such as large boulders, submerged in a potential flow. These may arise in geophysics, marine and civil engineering. Our models are applicable to initiation of motion that may result in subsequent transport. The local flow depends on the obstacle shape, slowing down in confining corners and speeding up in expanding corners. The flow generates hydrodynamic forces, drag and lift, and their associated moments, which differ around each face. Our model replaces the need for ill-defined drag and lift coefficients with geometry-dependent functions. We predict smaller flow velocities to initiate motion. We show how a joint-bound boulder can be transported against gravity, and analyse the influence of a wake region behind an isolated boulder.

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Section: Mathematical Technicalitiesmentioning

confidence: 99%

Section: Mathematical Technicalitiesmentioning

confidence: 99%

Potential flow over a submerged rectangular obstacle: Consequences for initiation of boulder motion

Herterich

Dias

2019

Eur. J. Appl. Math

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“…In particular, C 1/2 transforms to an origincentered nephroid, unsurprisingly with its two cusps at the critical values T 3 (± 1 2 ) = ∓1. Moreover, T 3 is univalent [6] (one-to-one) on the disk bounded by C 1/2 . However, its more relevant property is not yet apparent and will be revealed by the Joukowsky map.…”

Section: Distinct Critical Pointsmentioning

confidence: 99%

The Joukowsky Map Reveals the Cubic Equation

Sánchez-Reyes

2019

The American Mathematical Monthly

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Two canonical polynomials generate all cubics, via linear transformations of the polynomial map and the parameter: the cubic power function, with coincident critical points, and the third Chebyshev polynomial of the first kind, with two distinct critical points. Computing the roots of any cubic boils down to inverting these fundamental maps. In the more general case of distinct critical points, we show that the roots admit a startlingly simple expression in terms of a Joukowsky map and its inverse. Marden's theorem comes as a straightforward consequence, because the roots are the images, under a Joukowsky map, of the vertices of an equilateral triangle. The first time you come across the closed-form solution to the cubic equation [23], it looks intimidating, with square roots within cube roots and terms coming out of the blue, without apparent geometric meaning. Our goal is to change this perception, by reexamining known results and endowing them with an intuitive interpretation that leads to more compact expressions. To get this new insight, we analyze cubic polynomials as complex transformations, appealing to geometric intuition as exemplified by Needham in his beautifully illustrated book [18]. 1. CLASSIFYING CUBIC MAPS. The first question to answer is: How many essentially different cubic maps are there? We consider only true cubics, hence ruling out quadratic or linear maps. The answer arises naturally from a key feature of complex maps, namely their critical points z c , where the conformality of the otherwise conformal mapping breaks down. For a polynomial, critical points correspond to the points z c where the derivative vanishes. In particular, for a cubic c(z), the equation c (z c) = 0 is quadratic, furnishing only two cases: coincident or distinct critical points. Thus, we split the set of cubics into two subsets, depending on their number N = 1, 2 of distinct critical points. Formally speaking, we define two equivalence classes, whereby two cubics are equivalent if they share the same N. This case-by-case analysis will provide a deeper insight and prove simpler than the traditional approach of categorizing cubics according to the nature of their roots [2]. Next, we check that each class is generated by its respective canonical representative ζ(z), via linear transformations of ζ(z) and the parameter z. In the complex plane C, these transformations have a clear geometric meaning, since they amount to direct similarities [18], that is, combinations of uniform dilations, translations and rotations.

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“…For detailed definitions of analytical functions and line integrals, see Needham [, p. 197 and 383, respectively]. Numerous introductions and advanced texts on conformal mapping are readily available [ Churchill and Brown , ; Potter , ; Zill and Shanahan , ; Brilleslyper et al ., ]. Brief introductory articles on line integrals and complex analysis with a popular slant may provide useful entry‐level reading for students [ Braden , ; Gluchoff , ; Wegert and Semmler , ], which could be followed by studies discussing complex analysis of 3‐D flows requiring advanced mathematical skills [ Kelly et al ., ; Shaw , ].…”

Section: Previous Fundamental Workmentioning

confidence: 99%

Visualization of space competition and plume formation with complex potentials for multiple source flows: Some examples and novel application to Chao lava flow (Chile)

Weijermars

2014

JGR Solid Earth

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Fluid displacement in a continuum pressured by a variable constellation of source flows can be visualized as solutions of line integrals. The algorithms are based on complex potentials that provide exact solutions of the Navier-Stokes equation and allow users to specify both the location and flux strength of multiple sources. If relative strength and positioning of the sources are varied, a wide range of flow patterns and particle paths can be traced. Time-dependent variations in the strength of the sources can account for transient-flow problems. A further expansion is superposition of a background flow, which displaces the particle paths issued from the sources into concentrated plumes. The outlined modeling technique for visualization of multiple plume displacements is potentially relevant for a wide spectrum of practical situations. Detailed applications are possible, such as time tracking of groundwater-plume migration from a series of pollution sources, tracking of salt-feeder-stock flow and suture zone formation when salt issued from the stocks coalesces into a salt canopy, and designing of optimal spacing and relative pressure strength of multiple water injection wells in field-development plans for hydrocarbon production. Further applications are highlighted in the main text, including a simulation of geometrical features exposed in the Chao coulee (Chilean Andes).

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Explorations in Complex Analysis

Cited by 15 publications

References 0 publications

Potential flow over a submerged rectangular obstacle: Consequences for initiation of boulder motion

Potential flow over a submerged rectangular obstacle: Consequences for initiation of boulder motion

The Joukowsky Map Reveals the Cubic Equation

Visualization of space competition and plume formation with complex potentials for multiple source flows: Some examples and novel application to Chao lava flow (Chile)

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