Bonni J. Dichone

Viscous Fluid Flows

Since including viscous effects in the fluid equations plays a fundamental role for the prototype problems of flow past bodies and natural convection to be treated in the next two chapters, respectively, this chapter considers viscosity, in some detail, as a prelude to those investigations. First, after discussing the behavior of the viscosity coefficients and restricting our attention to homogeneous fluids, the resulting Navier–Stokes governing equations of motion are presented in component form for both Cartesian and cylindrical coordinate systems.

Boundary Conditions for Fluid Mechanics

With this chapter, we complete the formulation of fluid mechanical model systems by considering the requisite boundary conditions for the governing partial differential equations developed in the last one. The no-penetration and no-slip fluid mechanical boundary conditions at rigid surfaces are catalogued for various configurations and the kinematic boundary condition at material surfaces is deduced from the relative normal speed of such a moving interface. The concept of a surface of discontinuity is introduced and careful application of the balance laws to that surface is shown to yield jump-type boundary conditions satisfied by the dependent variables across them. Then these are applied to the two-dimensional propagation of a shock front through an inviscid fluid to deduce the Rankine Hugoniot jump conditions for that situation. The problems deal with those specific jump-type conditions to be imposed for a variety of other continuum processes containing such surfaces when modeled in this manner, which involves its curvature and surface tension.

Finite Mathematical Models

Three finite mathematical models are examined. The first deals with the discrete-time rabbit reproduction population dynamics model posed by Leonardo of Pisa which gives rise to the finite difference equation the solution of which produces the Fibonacci sequence. In the chapter this is solved both directly in scalar form and by placing it in a system formulation that is then solved by a Jordan canonical form method. Two other methods of solution are introduced in the problems for the system formulation: Namely a Cayley-Hamilton Theorem approach and a direct eigenvalue-eigenvector expansion method. The second model deals with the minimum fraction of the popular vote that can elect the President of the United States posed by George Polya. The 1960 and 1996 Presidential elections are examined in the chapter while the 2008 election is considered in a problem. The third model deals with the financial mathematics problem of the repayment of a loan or mortgage. A loan shark example with 100% interest rate per pay period is examined in the chapter and a similar one with only a 50% interest rate per period is examined in the last problem.

Potential Flow Past a Circular Cylinder of a Homogeneous Inviscid Fluid

The steady-state two-dimensional potential flow of an inviscid fluid past a circular cylinder is considered. The resulting Laplace’s equation for the velocity potential is converted to cylindrical coordinates by the Calculus of Variations method of transformation of coordinates introduced in a pastoral interlude and that equation solved by a separation of variables technique. Then integration of the pressure determined by Bernoulli’s relation about the cylinder yields D’Alembert’s paradox for a two-dimensional situation that the drag on the cylinder is zero which is a consequence of the assumption that the small viscosity coefficient can be neglected. The problems fill in some details involving vector identities employing the alternating tensor introduced in Chap. 9, examine the properties of the orthogonal Hermite polynomials similar in behavior to the Legendre polynomials discussed below, and consider the corresponding companion situation of three-dimensional potential flow past a sphere. This requires that the resulting Laplace’s equation for the velocity potential be converted to spherical coordinates by the Calculus of Variations transformation method. Then the separation of variables technique of solution gives rise to Legendre polynomials the properties of which have been deduced by means of the two pastoral interludes that conclude the chapter. Integration of the pressure about the sphere yields D’Alembert’s paradox for a three-dimensional situation that the force on the sphere is zero.

Of Mites and Models

The nonlinear behavior of a particular Kolmogorov-type exploitation ordinary differential equation system assembled by May (Stability and complexity in model ecosystems. Princeton University Press, Princeton, 1973) [80] from predator and prey components developed by Leslie(Biometrica 35:213–245, 1948) [67] and Holling (Mem. Entomol. Soc. Can. 45:1–60, 1965) [49], respectively, is examined by means of the numerical bifurcation code AUTO with model parameters chosen appropriately for a temperature dependent mite interaction on fruit tree leaves. In particular the proper temperature-rate relationship for arthropods is developed by the knowledge of the results of singular perturbation theory applied to ordinary differential equations which is introduced as a pastoral interlude. The concepts of linear stability theory, phase-plane analysis, and limit cycle behavior are also introduced as pastoral interludes. The predictions of this model are then compared with general ecological field results and particular laboratory experimental data. The problems extend singular perturbation type analyses to the investigation of polynomials and the specific temperature-rate relationship for the predaceous mite and linear stability analyses to competing species models and predator–prey models employing a Holling-type I functional response as opposed to the type II response included in the May mite model.

Initiation of Cellular Slime Mold Aggregation Viewed as an Instability

The initiation of cellular slime mold aggregation is identified as the onset of a self-organized linear instability of a simplified reaction-diffusion model system for the slime mold amoeba density and the concentration of the extracellular chemical acrasin produced by them to which they are chemotactically attracted. To derive these governing equations a general balance law must be deduced employing the divergence, Stokes, and Green’s theorems which are the subject of a pastoral interlude. The initial conditions are satisfied by means of Fourier integrals, introduced by another pastoral interlude that deduces the relevant formula and in so doing also includes the concept of Laplace transforms. The factors that favor the initiation of such aggregation are predicted by examining the linear instability criterion. The problem considers the equivalent normal-mode linear stability analysis of a slightly more general four-component model system explicitly including the two other dependent variables: Namely, the enzyme acrasinase, a second chemical produced by the amoeba that degrades the acrasin to a product to which they are not chemotactically attracted, and an intermediate complex formed by the interaction of these two chemicals in a reversible equilibrium reaction.

Canonical Projectile Problem: Finding the Escape Velocity of the Earth

The escape velocity of the Earth is calculated using an idealized projectile model that allows for the determination of projectile velocity as a function of its altitude. In order to obtain an approximate solution for projectile altitude as a function of time which cannot be determined by an exact solution the concept of regular perturbation theory in ordinary differential equations is introduced as a pastoral interlude. Then a regular perturbation expansion is performed on the model to obtain the desired asymptotic solution of altitude as a function of time when the initial projectile velocity is much less than that of the escape velocity. An energy argument making use of the fact that gravity acts as a conservative force for this canonical model is also introduced to examine this phenomenon in more detail. The problems extend these analyses to the rest of the solar system planets and to two other canonical projectile problems that are nonconservative.

Nonlinear Optical Ring-Cavity Model Driven by a Gas Laser

The development of spontaneous stationary equilibrium patterns induced by the injection of a laser pump field into a purely absorptive two-level atomic sodium vapor ring cavity is investigated by means of a hexagonal planform nonlinear stability analysis applied to the appropriate governing evolution equation for this optical phenomenon. In the quasi-equilibrium limit for its atomic variables, the mathematical system modelling that phenomenon can be reduced to a single modified Swift-Hohenberg nonlinear partial differential time-evolution equation describing the intracavity field on an unbounded two-dimensional spatial domain. Diffraction of radiation can induce transverse patterns consisting of stripes and hexagonal arrays of bright spots or honeycombs in an initially uniform plane-wave configuration. Then, these theoretical predictions are compared with both relevant experimental evidence and existing numerical simulations from some recent nonlinear optical pattern formation studies. There are four problems: The first two fill in some details of this analysis while the last two examine bistability for a related nonlinear optical phenomenon and hexagonal pattern formation for the relevant amplitude-phase equations with a hypothetical growth rate and set of Landau constants.

Governing Equations of Fluid Mechanics

The basic equations of continuum mechanics for moving continua are derived from first principals employing the continuum hypothesis, substantial derivative, and material and spatial coordinates. After the continuity equation is deduced from conservation of mass using the Reynolds Transport Theorem, a general balance law is developed and applied to momentum (linear and angular) and energy for nonpolar continua. Then equations of state relevant to ideal and adiabatic gases and constitutive relations relevant to Newtonian fluid flow are introduced involving thermodynamics and Cartesian tensor notation, respectively. In addition the continuity equation in Cartesian coordinates is deduced by a fixed volume big box method as well and then that equation is converted to cylindrical coordinates by direct transformation. The problems examine various aspects of these concepts including incompressibility, stream functions, a big box derivation of the continuity equation in cylindrical coordinates, a transformation of the continuity equation in Cartesian coordinates to spherical coordinates, the constitutive relations for Newtonian fluids, the governing equation for chemical species conservation, the balance of angular momentum for polar continua, the terms in the energy equations for ideal gases, and the Clausius-Duhem inequality.

Chemical Turing Patterns and Diffusive Instabilities

The Brusselator activator-inhibitor reaction-diffusion model is considered and conditions deduced by a normal-mode linear stability analysis for the development of chemical Turing instabilities over a parameter range for which the dynamical system in the absence of diffusion would exhibit a stable homogeneous distribution. The effect the introduction of an immobilizer would have on such diffusive instabilities is also examined. The limitations of linear stability predictions of this sort are discussed and the results of a nonlinear stability analysis which will be treated in detail in later chapters are sketched for the Brusselator. In the problems similar normal-mode linear stability analyses of the Schnackenberg simplification of the Brusselator and a simplified version of the so-called CDIMA (Chlorine Dioxide Iodine Malonic Acid) chemical reaction-diffusion system are considered.