In the preceding chapters, a number of asymptotic methods were introduced for the approximate solution of nonlinear flow problems. In many of the cases considered so far, including all of the problems of the two preceding chapters, the asymptotic limiting process produced a simplification of the full nonlinear problem by restricting the geometry of the flow domain to one in which certain terms in the equations could be neglected because the length scales in some direction (or directions) become very large compared with the length scales in other directions. In retrospect, even the exact unidirectional flow problems of Chap. 3 can often be regarded as a first approximation of some more general problem in which the geometry reduces to a unidirectional form in the limit as a ratio of two length scales vanishes, e.g., the “Dean” problem of Chap. 4, which reduces to the unidirectional Poiseuille flow problem in the limit as the ratio of the tube radius to the radius of curvature of the tube in the axial direction goes to zero. In some cases, this disparate ratio of length scales was true everywhere in the flow domain, and then the asymptotic solution was found to be “regular”; e.g., the Dean problem or the eccentric cylinder problem of Chap. 5. In others, the region with a small length-scale ratio was restricted to a local part of the overall flow domain, and in these cases, the asymptotic approximation was of the “singular” type, in which the simplified form of the governing equations is valid only locally, and the resulting approximate solution must be “matched” to a solution of the unapproximated equations that are valid elsewhere.

Although the full Navier–Stokes equations are nonlinear, we have studied a number of problems in Chap. 3 in which the flow was either unidirectional so that the nonlinear terms u · ∇u were identically equal to zero or else appeared only in an equation for the cross-stream pressure gradient, which was decoupled from the primary linear flow equation, as in the 1D analog of circular Couette flow. This class of flow problems is unusual in the sense that exact solutions could be obtained by use of standard methods of analysis for linear PDEs. In virtually all circumstances besides the special class of flows described in Chap. 3, we must utilize the original, nonlinear Navier–Stokes equations. In such cases, the analytic methods of the preceding chapter do not apply because they rely explicitly on the so-called superposition principle, according to which a sum of solutions of linear equations is still a solution. In fact, no generally applicable analytic method exists for the exact solution of nonlinear PDEs.

The question then is whether methods exist to achieve approximate solutions for such problems. In fluid mechanics and in convective transport problems there are three possible approaches to obtaining approximate results from the nonlinear Navier–Stokes equations and boundary conditions.

First, we may discretize the DEs and boundary conditions, using such formalisms as finite-difference, finite-element, or related approximations, and thus convert them to a large but finite set of nonlinear algebraic equations that is suitable for attack by means of numerical (or computational) methods.

A BRIEF HISTORICAL PERSPECTIVE OF TRANSPORT PHENOMENA IN CHEMICAL ENGINEERING

“Transport phenomena” is the name used by chemical engineers to describe the subjects of fluid mechanics and heat and mass transfer. The earliest step toward the inclusion of specialized courses in fluid mechanics and heat or mass transfer processes within the chemical engineering curriculum probably occurred with the publication in 1923 of the pioneering text Principles of Chemical Engineering by Walker, Lewis, and McAdams. This was the first major departure from curricula that regarded the techniques involved in the production of specific products as largely unique, to a formal recognition of the fact that certain physical or chemical processes, and corresponding fundamental principles, are common to many widely differing industrial technologies.

A natural outgrowth of this radical new view was the gradual appearance of fluid mechanics and transport in both teaching and research as the underlying basis for many of the unit operations. Of course, many of the most important unit operations take place in equipment of complicated geometry, with strongly coupled combinations of heat and mass transfer, fluid mechanics, and chemical reaction, so that the exact equations could not be solved in a context of any direct relevance to the process of interest. Hence, insofar as the large-scale industrial processes of chemical technology were concerned, even at the unit operations level, the impact of fundamental studies of fluid mechanics or transport phenomena was certainly less important than a well-developed empirical approach (and this remains true today in many cases).

In Chap. 4 we explored the consequences of a weak departure from strict adherence to the conditions for unidirectional flow; namely, the effect of slight curvature in flow through a circular tube. For that case, the centripetal acceleration associated with the curved path of the primary flow was shown to produce a weak secondary motion in the plane orthogonal to the tube axis. In this chapter we consider another class of deviations from unidirectional flow that occur when the boundaries are slightly nonparallel.

If the boundaries of the flow domain are not parallel, the magnitude of the primary velocity component must vary as a function of distance in the flow direction. This not only introduces a number of new physical phenomena, as we shall see, but it also means that the Navier–Stokes equations cannot be simplified following the unidirectional flow assumptions of Chap. 3, and exact analytical solutions are no longer possible. In this chapter, we thus consider only a special limiting case, known as the “thin-gap” limit, in which the distance between the boundaries is small compared with the lateral gap width. In this case, we shall see that we can obtain approximate analytical solutions by using the asymptotic and scaling techniques that were introduced in the preceding chapter.

The resulting theory, at the leading order of approximation, is applicable to a number of important phenomena. There are two generic classes of problems.

Although the solution methods of the preceding chapter are very useful and allowed us to derive a number of interesting results, they were all based on transforming the creeping-flow and continuity equations into a single higher-order differential for the streamfunction so that the classical methods of eigenfunction expansions can be used to obtain a general representation of the solution. This approach will work whenever the geometry of the boundaries and the form of any imposed flow are consistent with either a 2D or axisymmetric velocity and pressure field. We applied it to some representative 2D problems, as well as to problems involving the motions of spherical or near-spherical particles, bubbles, and drops in axisymmetric applied flows. There are, of course, other problems involving axisymmetric bodies in an axisymmetric flow – for example, any body of rotation in which the rotation axis is parallel to the axis of symmetry of the undisturbed flow. These problems can all be formulated in the same way, and, provided the geometry of the body is coincident with a coordinate surface in some orthogonal coordinate system, the same procedures can be followed.

In spite of the fact that there are actually quite a large number of axisymmetric problems, however, there are many important and apparently simple-sounding problems that are not axisymmetric.

We are concerned in this book primarily with a description of the motion of fluids under the action of some applied force and with convective heat transfer in moving fluids that are not isothermal. We also consider a few analogous mass transfer problems involving the convective transport of a single solute in a solvent.

It is assumed that the reader is familiar with the basic principles and equations that describe these processes from a continuum mechanics point of view. Nevertheless, we begin our discussion with a review of these principles and the derivation of the governing differential equations (DEs). The aim is to provide a reasonably concise and unified point of view. It has been my experience that the lack of an adequate understanding of the basic foundations of the subject frequently leads to a feeling on the part of students that the whole subject is impossibly complex. However, the physical principles are actually quite simple and generally familiar to any student with a physics background in classical mechanics. Indeed, the main problems of fluid mechanics and of convective heat transfer are not in the complexity of the underlying physical principles, but rather in the attempt to understand and describe the fascinating and complicated phenomena that they allow. From a mathematical point of view, the main problem is not the derivation of the governing equations that is presented in this second chapter, but in their solution. The latter topic will occupy the remaining chapters of this book.

The atmospheric radiation field

The theory presented in this book applies to the lower 50 km of the Earth's atmosphere, that is to the troposphere and to the stratosphere. In this part of the atmosphere the so-called local thermodynamic equilibrium is observed.

In general, the condition of thermodynamic equilibrium is described by the state of matter and radiation inside a constant temperature enclosure. The radiation inside the enclosure is known as black body radiation. The conditions describing thermodynamic equilibrium were first formulated by Kirchhoff (1882). He stated that within the enclosure the radiation field is:

1. isotropic and unpolarized;

2. independent of the nature and shape of the cavity walls;

3. dependent only on the temperature.

The existence of local thermodynamic equilibrium in the atmosphere implies that a local temperature can be assigned everywhere. In this case the thermal radiation emitted by each atmospheric layer can be described by Planck's radiation law. This results in a relatively simple treatment of the thermal radiation transport in the lower sections of the atmosphere.

Kirchhoff's and Planck's laws, fundamental in radiative transfer theory, will be described in the following chapters. While the derivation of Planck's law requires a detailed microscopic picture, Kirchhoff's law may be obtained by using purely thermodynamic arguments.

Introduction

In this chapter we are going to discuss some of the more elementary ideas in connection with the absorption spectra of gases. The energy E of a molecule may be expressed as the sum of the rotational energy Erot, vibrational energy Evib and electronic energy Eel. Of these three types of energy, Erot is generally the smallest, typically a few hundredths of an electron Volt. Vibrational energies are of the order of a few tenths of one electron Volt, while the largest energies are of the electronic type which generally amount to a few electron Volts.

The absorption (emission) spectrum arising from the rotational and vibrational motion of a molecule which is not electronically excited will be located in the infrared region. In infrared absorption experiments light from a suitable source penetrates an absorption chamber containing the gas to be studied and then enters a spectrograph. If the instrument is of low resolving power, a series of wide bands is observed which correspond to the vibrational transitions. If an instrument of high resolving power is used, these bands are seen to consist of numerous spectral lines resulting from the energy levels of rotation.

In the next section we are going to discuss the vibrational motion of two relatively simple molecules (CO2 and H2O) which are particularly important in the study of radiative transfer in the atmosphere.

In the final chapter of this book we will briefly treat the radiative influence of clouds on the climate of the Earth. A brief introduction to this topic has already been presented in the first chapter where we have discussed the radiation budget of the Earth. In the chapters that followed we have studied the radiative transfer theory in some detail and have learned how to calculate the radiances and flux densities in the solar and long-wave spectrum.

We have omitted any discussion of measurement programs such as the satellite experiments ERB and ERBE which were specifically devised to study the global radiation budget. A brief description of some of the sophisticated instrumentation used to measure the reflected solar energy and the outgoing long-wave radiation is given by Lenoble (1993) where many references to this topic can be found.

Globally the planet Earth is in radiative equilibrium implying that the reflected solar radiation and the outgoing long-wave radiation are in balance with the incoming solar radiation. If this balance is disturbed by natural or by anthropogenic processes the global climate will be changed. A detailed study of the radiative impact of clouds on the climate is very difficult and can be carried out only with the help of sophisticated climate models.

Introduction

The evaluation of the radiative transfer equation requires a detailed knowledge of the extinction and scattering properties of atmospheric particles. In most cases we rely on theoretical calculations assuming that the scattering particles have a spherical shape. In order to carry out the calculations we must specify the particle size and the wavelength-dependent complex index of refraction. The required calculations are known as Mie calculations while the entire theory is called the Mie theory. This goes back to Gustav Mie who published the whole theory in 1908. Nowadays a large number of reliable computer programs are available to provide the required information without understanding the theory behind it. This might be sufficient in some cases, but it is more valuable to comprehend the theoretical background which will be presented in the following sections.

The Mie theory of light scattering by homogeneous spheres is based on the formal solution of Maxwell's equations using proper boundary conditions. In this text we will follow Stratton's discussion of the Electromagnetic Theory (1941). In van De Hulst's (1957) treatise Light Scattering by Small Particles many mathematical details of the Mie theory are omitted, but numerous helpful physical explanations are given. In particular he gives a number of approximate formulas for special cases.

Radiation in the Atmosphere is the third volume in the series A Course in Theoretical Meteorology. The first two volumes entitled Dynamics of the Atmosphere and Thermodynamics of the Atmosphere were first published in the years 2003 and 2004.

The present textbook is written for graduate students and researchers in the field of meteorology and related sciences. Radiative transfer theory has reached a high point of development and is still a vastly expanding subject. Kourganoff (1952) in the postscript of his well-known book on radiative transfer speaks of the three olympians named completeness, up-to-date-ness and clarity. We have not made any attempt to be complete, but we have tried to be reasonably up-to-date, if this is possible at all with the many articles on radiative transfer appearing in various monthly journals. Moreover, we have tried very hard to present a coherent and consistent development of radiative transfer theory as it applies to the atmosphere. We have given principle allegiance to the olympian clarity and sincerely hope that we have succeeded.

In the selection of topics we have resisted temptation to include various additional themes which traditionally belong to the fields of physical meteorology and physical climatology. Had we included these topics, our book, indeed, would be very bulky, and furthermore, we would not have been able to cover these subjects in the required depth.

The principle of invariance in the original form was stated by Ambartsumian (1942) expressing the invariance of the diffusely reflected radiation emerging from a semi-infinite atmosphere to the addition or subtraction of an infinitely thin atmospheric layer. Chandrasekhar (1960) advanced the original from and stated four general principles of invariance which apply to finite atmospheric layers. These principles are not based on the radiative transfer equation, but they are of equal physical validity. We accept Goody's (1964a) statement that the principles of invariance may be viewed as a series of common-sense relations between the scattering and transmission functions with the radiances emerging from the upper and lower boundaries of an atmospheric layer and at some intermediate variable level.

Definitions of the scattering and transmission functions

Let us consider a plane–parallel atmospheric layer of vertical optical thickness τ1 bounded on both sides by a vacuum, see Figure 3.1. The upper boundary of this layer is illuminated by a beam of parallel downward directed radiation S0, while at τ = τ1 no radiation is incident in the upward direction. For simplicity, only short-wave radiation will be considered. However, inclusion of infrared radiation causes no particular difficulties. We call this situation the restricted or standard problem.