The Framework ofk-Harmonically Analytic Functions for Three-Dimensional Stokes Flow Problems, Part II

Written by a master of the subject, this text will be appreciated by students and experts for the way it develops the classical theory of functions of a complex variable in a clear and straightforward manner. In general, the approach taken here emphasises geometrical aspects of the theory in order to avoid some of the topological pitfalls associated with this subject. Thus, Cauchy's integral formula is first proved in a topologically simple case from which the author deduces the basic properties of holomorphic functions. Starting from the basics, students are led on to the study of conformal mappings, Riemann's mapping theorem, analytic functions on a Riemann surface, and ultimately the Riemann-Roch and Abel theorems. Profusely illustrated, and with plenty of examples, and problems (solutions to many of which are included), this book should be a stimulating text for advanced courses in complex analysis.

The history of complex function can be dated back to about 18 century when Euler came up with two equations derived from integrals of complex function. Over the age, mathematicians have been trying to explore and discuss more in this mysterious field, and this new branch gradually became prevalent in the 19 century mathematics like when calculus ruled the 18 century mathematics. In particular, the concept of residue was invented and became an important part in complex function, being applied in some special types of real integral. Also it is a generalization of Cauchy integral theorem and Cauchy integral formula. Nowadays, the area of complex function is filled with crystallization of the wisdom of countless scholars and researchers, and the precious mathematical treasure can also meet the needs of other academic areas like physics and biology. Summarizing the various approaches and examples of application in fields of mathematics and others can be a valuable topic. In this paper, the history of application and some examples for references of complex functions, and introduce the main concepts and formulas about them will be briefly discussed, including the area of complex numbers, analytic function, and residue. There are some relationships between some of the applications to be dig out and considerable real-world problems, and they are supposed to the generalization of those subjects can provide certain enlightenment to people interested in or during the study in fields of complex function.

Particle-in-cell models for Stokes flow through a relatively homogeneous swarm of particles are of substantial practical interest, because they provide a relatively simple platform for the analytical or semianalytical solution of heat and mass transport problems. Despite the fact that many practical applications involve relatively small particles (inorganic, organic, biological) with axisymmetric shapes, the general consideration consists of rigid particles of arbitrary shape. The present work is concerned with some interesting aspects of the theoretical analysis of creeping flow in ellipsoidal, hence nonaxisymmetric domains. More specifically, the low Reynolds number flow of a swarm of ellipsoidal particles in an otherwise quiescent Newtonian fluid, that move with constant uniform velocity in an arbitrary direction and rotate with an arbitrary constant angular velocity, is analyzed with an ellipsoid-in-cell model. The solid internal ellipsoid represents a particle of the swarm. The external ellipsoid contains the ellipsoidal particle and the amount of fluid required to match the fluid volume fraction of the swarm. The nonslip flow condition on the surface of the solid ellipsoid is supplemented by the boundary conditions on the external ellipsoidal surface which are similar to those of the sphere-in-cell model of Happel (self-sufficient in mechanical energy). This model requires zero normal velocity component and shear stress. The boundary value problem is solved with the aim of the potential representation theory. In particular, the Papkovich–Neuber complete differential representation of Stokes flow, valid for nonaxisymmetric geometries, is considered here, which provides the velocity and total pressure fields in terms of harmonic ellipsoidal eigenfunctions. The flexibility of the particular representation is demonstrated by imposing some conditions, which made the calculations possible. It turns out that the velocity of first degree, which represents the leading term of the series, is sufficient for most engineering applications, so long as the aspect ratios of the ellipsoids remains within moderate bounds. Analytical expressions for the leading terms of the velocity, the total pressure, the angular velocity, and the stress tensor fields are obtained. Corresponding results for the prolate and the oblate spheroid, the needle and the disk, as well as for the sphere are recovered as degenerate cases. Novel relations concerning the ellipsoidal harmonics are included in the Appendix.

Based on the England–Spencer plate theory, the elastic field in a transversely isotropic functionally graded plate with holes is analyzed by transforming the original three-dimensional (3D) problem into a two-dimensional (2D) one. The general solutions of the governing equations for the 2D problem are expressed by four analytic functions α(ζ), β(ζ), ϕ(ζ) and ψ(ζ) when there are no transverse forces acting on the plate surfaces. In a multiply-connected domain, the four analytic functions are expressed as the sum of logarithmic functions that are multi-valued and holomorphic functions, and should meet the conditions of single-valuedness of displacements. 3D elasticity solutions are then obtained for a transversely isotropic FGM plate with a circular hole subject to loads at infinity based on the power series method. Explicit expressions for concentration factors of the resultant forces are presented. For the degenerated case of homogeneous materials, the present elasticity solutions are exactly the same as those based on plane stress elasticity and classical plate theory. In addition, the elasticity solutions for an infinite FGM plate subject to in-plane point forces (X, Y) and moments (MX, MY) are derived, among which the solutions of point moments are reported for the first time. There are obvious differences between the present solutions of point forces and the existing solutions reported in the literature. In particular, the present solutions for homogeneous materials include a 3D correction term, which depends on the thickness-to-radius ratio, to those of plane stress elasticity. Finally, the elasticity solution for the point moment MZ is discussed.

It has long been customary in the theory of linear transformations-integral equations, linear equations in an infinite number of unknowns, etc.-to make use of certain notions from the theory of functions of a complex variable. In particular, the well known Neumann expansion of the resolvent is a formal power series whose coefficients, instead of being numbers, are iterates of a certain linear transformation.' These resemblances to the classical CauchyWeierstrass theory can be described accurately in terms of a theory of analytic functions whose values lie in a complex Banach space. The possibility of such a theory, derived in the usual manner from Cauchy's theorem and integral formula, was recognized by Wiener.2 Still further generalizations, in which the independent variable may also range over a complex Banach space, have been considered by a number of people.3 It therefore seems desirable to outline a general abstract theory of linear operations depending analytically on a parameter, which we shall here suppose to be a complex variable. DEFINITION 1.1 A function f(X) defined in an open set A of the complex plane, with values in a complex Banach space E, is called analytic in A if the limit

In this work, we investigate a nonconforming finite element (FE) approximation of phase‐field parameterized topology optimization governed by the Stokes flow. The phase field, the velocity field and the pressure field are approximated by conforming linear FEs, nonconforming linear FEs (Crouzeix‐Raviart elements) and piecewise constants, respectively. When compared with the standard conforming counterpart, the nonconforming FEM can provide an approximation with fewer degrees of freedom, leading to improved computational efficiency. We establish the convergence of the resulting numerical scheme in the sense that the sequences of phase‐field functions and discrete velocity fields contain subsequences that converge to a minimizing pair of the continuous problem in the ‐norm and a mesh‐dependent norm, respectively. We present extensive numerical results to illustrate the performance of the approach, including a comparison with the popular Taylor‐Hood elements.

A recently introduced fast algorithm for the computation of the first N terms in an expansion of an analytic function into ultraspherical polynomials consists of three steps: Firstly, each expansion coefficient is represented as a linear combination of derivatives; secondly, it is represented, using the Cauchy integral formula, as a contour integral of the function multiplied by a kernel; finally, the integrand is transformed to accelerate the convergence of the Taylor expansion of the kernel, allowing for rapid computation using Fast Fourier Transform. In the current paper we demonstrate that the first two steps remain valid in the general setting of orthogonal polynomials on the real line with finite support, orthogonal polynomials on the unit circle and Laurent orthogonal polynomials on the unit circle.

An experimental investigation of the influence of a leading edge separation region on the unsteady aerodynamic problem for thin, symmetric airfoils exposed to periodic gusting was performed in a subsonic indraft wind tunnel. Two leading edge shapes for the airfoil were used to examine the influence of flow separation both in the unsteady velocity and pressure fields. It is shown that flow separation near the leading edge causes a sharp increase in the unsteady RMS velocity and unsteady surface pressure variation associated with the gust frequency. This also causes a departure from the expectation that the gust information is convected downstream by the freestream in the velocity field. In the absence of such a separation region, velocity and surface pressure results closely resemble inviscid theoretical predictions and expectations. Accurate prediction of unsteady pressure is crucial in estimating the acoustic production of the system.

This research investigates a number of problems, related to Stokes flow and to heat flow. The Stokes flow is inspired by glass flow in the process of making bottles or jars. The heat flow is related to a heat conduction model problem, and a problem about hot-spots formation in the microwave heating. We will discuss the first problem. There are two phases during the industrial process of making glass, viz. the pressing phase and the blowing phase. We consider some mathematical aspects of the pressing phase. The motion of glass at temperatures above 6000C can be described by the Navier-Stokes equations. Since glass is a highly viscous fluid, those equations can be simplified to the Stokes equations. We use two different methods to solve these equations, viz. perturbation and operator methods. The perturbation method is based on the geometry being slowly varying. As a result, we obtain the velocity analytically. This result has a good agreement with numerical results based on finite element modelling. Using the velocity obtained we derive the formula for the force on the plunger. Next, we consider the operator method. Using this method, the Stokes equations can be transformed into an operator equation on the boundary ??? with a tangent vector field a on the boundary ??? as unknown. Solving this operator equation shows, that the solutions of the Stokes equations can be parameterized by aH, the harmonic extension of a to the interior of the domain ??. As an application, we present some full explicit solutions of the Stokes equations for several domains such as the interior and exterior of the unit ball and of the unit disk, an infinite strip, a half space, and a wedge. In the second problem, we consider the heat conduction problem inside two types of geometry, viz. slowly and slightly varying geometry. Using this problem, we show the difference between those geometries. An example that involves the boundary layers at the ends is presented. Finally, we consider a simplified model of the microwave heating of a one-dimensional unit slab. This slab consists of three layers that have different thermal conductivities. We consider only the steady state problem with Dirichlet boundary conditions and continuity of heat flux across the layers. Using a fundamental-mode approximation of eigenfunction expansion, we investigate the effect of thermal conductivity on the formation of hot-spots where the temperature increases catastrophically as a function of d, the amplitude of the applied electric field. First, we consider a unit slab geometry. In this geometry, we find the critical value dcr, for which slight changes in d yields a sudden jump to another stable solution, now with a much higher temperature. Next, we consider a unit slab consisting of three layers of material with different thermal conductivity (µ). We assume the inner layer has the smallest value of µ. We find the temperature in this layer is much higher than that in other layers. Then, we consider only the inner layer. For a given value of d and changing values of µ, we get a temperature jump near some values of µ. This jump shows that there is a critical value of µ and signifies the formation of a hot-spot.

The flow of a viscous fluid at small Reynolds numbers (Stokes flow) in a three-dimensional formulation is investigated. In this case, the inertial terms in the equations of motion can be neglected. Such flows can occur in nanotubes that can be considered as inclusions in representative volume elements of nanomaterials. By using the fundamental solution of Ossen, an integral representation of the velocity is proposed. This representation is used to receive an integral equation for an unknown density. The solution of the resulting equation makes it possible to calculate the fluid pressure on the walls of the shell. The case of axially symmetric flows is investigated. For this, an integral representation of the unknown velocity in cylindrical coordinates is obtained. By integrating over the circumferential coordinate, the two-dimensional singular integral equation is reduced to one-dimensional one. It has been proved that the components of the kernels in singular operators are expressed in terms of elliptic integrals of the first and second kind. It has been proved that the singularities of the kernels of one-dimensional singular integral equations have a logarithmic character. To calculate elliptic integrals, the Gaussian algorithm based on the use of the arithmetic-geometric mean value is proposed. This procedure allows us to obtain logarithmic singular components with high accuracy, which makes it possible to use special quadrature formulas to calculate such integrals. An algorithm with usage of the boundary element method for the numerical solution of the obtained singular integral equations is proposed. The method for solving one-dimensional singular equations, where the kernels contain elliptic integrals with logarithmic singularities (i.e logarithmic singularity is not expressed explicitly) has been tested. The obtained numerical results have been compared with the well-known analytical solutions. The data obtained indicate the high efficiency of the proposed numerical method.

The multigrid solution of coupled porous media and Stokes flow problems is considered. The Darcy equation as the saturated porous medium model is coupled to the Stokes equations by means of appropriate interface conditions. We focus on an efficient multigrid solution technique for the coupled problem, which is discretized by finite volumes on staggered grids, giving rise to a saddle point linear system. Special treatment is required regarding the discretization at the interface. An Uzawa smoother is employed in multigrid, which is a decoupled procedure based on symmetric Gauss-Seidel smoothing for velocity components and a simple Richardson iteration for the pressure field. Since a relaxation parameter is part of a Richardson iteration, Local Fourier Analysis (LFA) is applied to determine the optimal parameters. Highly satisfactory multigrid convergence is reported, and, moreover, the algorithm performs well for small values of the hydraulic conductivity and fluid viscosity, that are relevant for applications.

We study the evolution of drops of a very viscous and conducting fluid under the influence of an external electric field. The drops may be neutral or may be charged with some amount of electric charge. If both the external electric field and total drop charge are sufficiently small, then prolate spherical shapes develop according to Taylor’s observations. For sufficiently large charge and/or external field a self-similar conelike singularity develops in a mechanism different from Taylor’s prediction. The opening semiangle of the cones both for uncharged and charged drops in a constant electric field is typically around 30° with a very slight dependence on the viscosity ratio and independence from both total charge and external field. We also discuss the structure of electric and velocity fields near the tip.

Non-linear singular integral representation analysis for inviscid flowfields of unsteady airfoils

In this chapter we use the Cauchy theory to study functions that are holomorphic on an annulus, and analytic functions with isolated singularities. We describe a classification for isolated singularities. Functions that are holomorphic on an annulus have Laurent series expansions, an analogue of power series expansions for holomorphic functions on discs. Holomorphic functions with a finite number of isolated singularities in a domain can be integrated using the residue theorem, an analogue of the Cauchy integral formula. We discuss local properties of these functions. The study of zeros and poles of meromorphic functions leads to a theorem of Rouché that connects the number of zeros and poles to an integral. The theorem is not only aesthetically pleasing in its own right but also allows us to give alternate proofs of many important results. In the penultimate section of this chapter we illustrate the use of complex function theory in the evaluation of real definite integrals. The appendix discusses Cauchy principal values, a way to integrate functions with certain singularities.KeywordsCauchy TheoryHolomorphic FunctionReal Definite IntegralsLaurent Series ExpansionNonconstant Meromorphic FunctionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Acoustic vector sensors that measure pressure and orthogonal particle velocity are gaining widespread interest. Predicting their performance requires calculating the pressure field and the velocity fields, which require spatial gradients of the pressure field. In typical hydrophone applications, significant computational savings are realized by using reciprocity to generate the pressure field as a function of source position rather than receive position. However, the presence of the spatial gradients in the velocity fields means that reciprocity cannot be used to model the vector field for inverse problems. Instead, the inverse vector velocity field must be computed point by point, even for the simplest environments. Examples of this effect are demonstrated by the derivation of analytic expressions for pressure and particle velocity in a Pekeris waveguide. These simple waveguide results are extended to arbitrary, range-dependent, environment parameters using a parabolic equation model.