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C Tables and Diagrams for Compressible Flow. On the contrary, the behavior of a solid body is such that the deformation itself, not the velocity of deformation, goes to zero when the forces necessary to deform it tend to zero.

To illustrate this contrasting behavior, consider a material between two parallel plates and adhering to them acted on by a shearing force F Fig. If the extent of the material in the direction normal to the plane of Fig. At least one dimensional quantity typical for the material must enter this relation, and here this is the shear modulus G.

Shearing between two parallel plates 2 1 The Concept of the Continuum and Kinematics goes to zero. Often the relation for a solid body is of a more general form, e. This means there is no relationship between the displacement, or deformation, and the force. Again the force is inversely proportional to the distance between the plates.

We assume that the plate is being dragged at constant speed, so that the inertia of the material does not come into play. These include the glasslike materials which do not have a crystal structure and are structurally liquids.

Under short-term loads, they exhibit the behavior of a solid body. Under very short-term loads, e. Other materials behave like solids even in the long-term, provided they are kept below a certain shear stress, and then above this stress they will behave like liquids. A typical example of these substances Bingham materials is paint: it is this behavior which enables a coat of paint to stick to surfaces parallel to the force of gravity.

The essential difference between them lies in the greater compressibility of gases. When heated over the critical temperature Tc , liquid loses its ability to condense and it is then in the same thermodynamical state as a gas compressed to the same density. This graph shows that during dynamic processes where large changes of pressure and temperature occur, the change of volume has to be taken into account.

The behavior of solids, liquids and gases described up to now can be explained by the molecular structure, by the thermal motion of the molecules, and by the interactions between the molecules.

Microscopically the main Fig. With gases, the spacing at standard temperature and pressure Apart from occasional collisions, the molecules move along a straight path.

Only during the collision of, as a rule, two molecules, does an interaction take place. The mean free path is in general larger than the mean distance, and can occasionally be considerably larger.

In this case there is always an interaction between the molecules. Even gases have a resistance to change in volume, although at standard temperature and pressure it is much smaller and is proportional to the kinetic energy of the molecules.

When the gas is compressed so far that the spacing is comparable to that in a liquid, the resistance to volume change becomes large, for the same reason as referred to above. Real solids show a crystal structure: the molecules are arranged in a lattice and vibrate about their equilibrium position.

Above the melting point, this lattice disintegrates and the material becomes liquid. Now the molecules are still more or less ordered, and continue to carry out their oscillatory motions although they often exchange places.

The high mobility of the molecules explains why it is easy to deform liquids with shearing forces. It would appear obvious to describe the motion of the material by integrating the equations of motion for the molecules of which it consists: for computational reasons this procedure is impossible since in general the number of molecules in the material is very large.

In addition, detailed information about the molecular motion is not readily usable and therefore it would be necessary to average the molecular properties of the motion in some suitable way. It is therefore far more appropriate to consider the average properties of a cluster of molecules right from the start.

To justify this name, the volume which this cluster of molecules occupies must be small compared to 1. On the other hand, the number of molecules in the cluster must be large enough so that the averaging makes sense, i. Considering that the number of molecules in one cubic centimeter of gas at standard temperature and pressure is 2. On the other hand the linear measure of the volume must be small compared to the macroscopic length of interest.

This assumption forms the basis of the continuum hypothesis. Occasionally we will have to relax this assumption on certain curves or surfaces, since discontinuities in the density or temperature, say, may occur in the context of some idealizations. The number of molecules required to do any useful averaging then takes up such a large volume that it is comparable to the volume of the craft itself. Continuum theory is also inadequate to describe the structure of a shock see Chap.

Shocks have thicknesses of the same order of magnitude as the mean free path, so that the linear measures of the volumes required for averaging are comparable to the thickness of the shock. We have not yet considered the role the thermal motion of molecules plays in the continuum model.

Even if the macroscopic velocity given by 1. Obviously, molecules with other molecular properties e. Take as an example a gas 6 1 The Concept of the Continuum and Kinematics which consists of two types of molecule, say O2 and N2. The sign of this shear stress is such as to even out the velocity.

However nonuniformity of the velocity is maintained by the force on the upper plate, and thus the momentum transport is also maintained. From the point of view of continuum theory, this momentum transport is the source of the internal friction, i. The molecular transport of momentum accounts for internal friction only in the case of gases.

In liquids, where the molecules are packed as closely together as the repulsive forces will allow, each molecule is in the range of attraction of several others. The exchange of sites among molecules, responsible for the deformability, is impeded by the force of attraction from neighboring molecules.

Therefore the viscosity of liquids decreases with increasing temperature, since change of place among molecules is favored by more vigorous molecular motion. Yet the viscosity of gases, where the momentum transfer is basically its only source, increases with temperature, since increasing the temperature increases the thermal velocity of the molecules, and thus the momentum exchange is favored. In the most general sense, these relationships establish 1.

Continuum theory is however of a phenomenological nature: in order to look at the macroscopic behavior of the material, mathematical and therefore idealized models are developed. Yet this is necessary, since the real properties of matter can never be described exactly. But even if this possibility did exist, it would be wasteful to include all the material properties not relevant in a given technical problem.

The model of an ideal gas, for example, is evidently useful for many applications, although ideal gas is never encountered in reality. In principle, models could be constructed solely from experiments and experiences, without consideration for the molecular structure. Yet consideration of the microscopic structure gives us insight into the formulation and limitations of the constitutive equations. Mathematically, 1. Material description 1. Solving Eq. Using 1. For a given t 1. With the help of 1.

The mapping 1. In addition the experimental determination of the velocity as a function of the material coordinates 1. In particular the pressure distribution on the body can be found.

Just as the pathline is natural to the material description, so the streamline is natural to the Eulerian description.

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