1
Introduction to the Fluid Model

While dealing with a fluid, in reality, one deals with a system that has many particles which interact with one another. The main utility of fluid dynamics is the ability to develop a formalism which deals solely with a few macroscopic quantities like pressure, while ignoring the details of the particle interactions. Therefore, the techniques of fluid dynamics have often been found useful in modeling systems with complicated interactions (which are either not known or very difficult to describe) between the constituents. Thus, the first successful model of the nuclear fission of heavy elements was the liquid drop model of the nucleus, which treats the nucleus as a fluid. This replaces the many body problem of calculating the interactions of all the protons and neutrons with the much simpler problem of calculating the pressures and surface tension in this fluid.1 Of course, this treatment gives only a very rough approximation to reality, but it is nonetheless a very useful way of approaching the problem.

The primary purpose of fluid dynamics is to study the causes and effects of the motion of fluids. Fluid dynamics seeks to construct a mathematical theory of fluid motion based on the smallest number of dynamical principles, which are adequate to correlate the different types of fluid flow as far as their macroscopic features are concerned. In many circumstances, the incompressible, inviscid fluid model is sufficiently representative of real fluid properties to provide a satisfactory account of a great variety of fluid motions. It turns out that such a model makes accurate predictions for the airflow around streamlined bodies moving at low speeds. While dealing with streamlined bodies (which minimize flow-separation) in flows of fluids of small viscosities, one may divide the flow field into two parts. The first part, where the viscous effects are appreciable consists of a thin boundary layer adjacent to the body and a small wake behind the body. The second part is the rest of the flow field that behaves essentially like an inviscid fluid. Such a division greatly facilitates the mathematical analysis in that the inviscid flow field can first be determined independent of the boundary layer near the body. The pressure field obtained from the inviscid-flow calculation is then used to calculate the flow in the boundary layer.

An attractive feature of fluid dynamics is that it provides ample room for the subject to be expounded as a branch of applied mathematics and theoretical physics.

1.1 The Fluid State

A fluid is a material that offers resistance to attempts to produce relative motions of its different elements, but deforms continually upon the application of surface forces. A fluid does not have a preferred shape, and different elements of a homogeneous fluid may be rearranged freely without affecting the macroscopic properties of the fluid. Fluids, unlike solids, cannot support tension or negative pressure. Thus, the occurrence of negative pressures in a mathematical solution of a fluid flow is an indication that this solution does not correspond to a physically possible situation. However, a thin layer of fluid can support a large normal load while offering very little resistance to tangential motion - a property which finds practical use in lubricated bearings.

A fluid of course, is discrete on the microscopic level, and the fluid properties fluctuate violently, when viewed at this level. However, while considering problems in which the dimensions of interest are very large compared to molecular distances, one ignores the molecular structure and endows the fluid with a continuous distribution of matter. The fluid properties can then be taken to vary smoothly in space and time. The characteristics of a fluid, caused by molecular effects, such as viscosity, are incorporated into the equation of fluid flows as experimentally obtained empirical parameters.

Fluids can exist in either of two stable phases - liquids, and gases. In case of gases, under ordinary conditions, the molecules are so far apart that each molecule moves independently of its neighbors except when making an occasional "collision." In liquids, on the other hand, a molecule is continually within the strong cohesive force fields of several neighbors at all times. Gases can be compressed much more readily than liquids. Consequently, for a gas, any flow involving appreciable variations in pressure will be accompanied by much larger changes in density. However, in cases where the fluid flows are accompanied by only slight variations in pressure, gases and liquids behave similarly.

In formulations of fluid flows, it is useful to think that a fluid particle is small enough on a macroscopic level that it may be taken to have uniform macroscopic properties. However, it is large enough to contain sufficient number of molecules to diminish the molecular fluctuations. This allows one to associate with it a macroscopic property which is a statistical average of the corresponding molecular property over a large number of molecules. This is the continuum hypothesis.

As an illustration of the limiting process by which the local continuum properties are defined, consider the mass density ?. Imagine a small volume dV surrounding a point P, let dm be the total mass of material instantaneously in dV . The ratio , as dV reduces to , where is the volume of fluid particle, is taken to give the mass density ? at P.

1.2 Description of the Flow-Field

The continuum model affords a field description, in that the average properties in the volume surrounding the point P are assigned in the limit to the point P itself. If q represents a typical continuum property, then one has a fictitious continuum characterized by an aggregate of such local values of q, i.e., . This enables one to consider what happens at every fixed point in space as a function of time - the so-called Eulerian description. In an alternative approach, called the Lagrangian description, the dynamical quantities, as in particle mechanics, refer more fundamentally to identifiable pieces of matter, and one looks for the dynamical history of a selected fluid element.

Imagine a fluid moving in a region O (see Figure 1.1). Each particle of fluid follows a certain trajectory. Thus, for each point in O, there exists a path line given by

Figure 1.1 Motion of a fluid particle.

where the flow mapping , depends continuously on the parameter t. and its inverse are both continuous, so this mapping is one-to-one and onto.2

The velocity of the flow is given by

We then have the following results.

Theorem 1.1 (Existence) Assume that the flow velocity is a C1 function of x and t. Then, for each pair , there exists a unique integral curve - the path line , defined on some small interval in t about t0, such that

Theorem 1.2 (Boundedness) Consider a region O with a smooth boundary . If the flow velocity x is parallel to , then the integral curves of i.e., the path lines starting in O remain in O.

Streamlines are obtained by holding t fixed, say , and solving the differential equation

Streamlines coincide with pathlines if the flow is steady, i.e., if

The transformation from the Eulerian to the Lagrangian description is given by

Since the Lagrangian description makes the formalism cumbersome, we shall instead use the Eulerian description. However, the Lagrangian concepts of material volumes, material surfaces, and material lines which consist of the same fluid particles and move with them are still useful in developing the Eulerian description.

1.3 Volume Forces and Surface Forces

One may think of two distinct kinds of forces acting on a fluid continuum. Long-range forces such as gravity penetrate into the interior of the fluid, and act on all elements of the fluid. If such a force varies smoothly in space, then it acts equally on all the matter within a fluid particle of density ? and volume dV. The total force acting on the particle is proportional to its mass and is equal to . In this sense, long-range forces are called body forces.

The short-range forces (which have a molecular origin) between two fluid elements, on the other hand, are effective only if they interact through direct mechanical contact. Since the short-range forces on an element are determined by its surface area, one considers a plane surface element of area dA in the fluid. The local short-range force is then specified as the total force exerted on the fluid on one side of dA by the fluid on the other side and is equal to (Cauchy, 1827). The direction of this force is not known a priori for a viscous fluid (unlike in the case of an inviscid fluid). Here, is the unit normal to the surface element...