物理代写|流体力学代写Fluid Mechanics代考|The Concept of the Continuum and Kinematics

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• (Generalized) Linear Models 广义线性模型
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• Foundations of Data Science 数据科学基础

物理代写|流体力学代写Fluid Mechanics代考|Properties of Fluids, Continuum Hypothesis

Fluid mechanics is concerned with the behavior of materials which deform without limit under the influence of shearing forces. Even a very small shearing force will deform a fluid body, but the velocity of the deformation will be correspondingly small. This property serves as the definition of a fluid: the shearing forces necessary to deform a fluid body go to zero as the velocity of deformation tends to zero. 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. 1.1).

If the extent of the material in the direction normal to the plane of Fig. 1.1 and in the $x$-direction is much larger than that in the $y$-direction, experience shows that for many solids (Hooke’s solids), the force per unit area $\tau=F / A$ is proportional to the displacement $a$ and inversely proportional to the distance between the plates $h$. At least one dimensional quantity typical for the material must enter this relation, and here this is the shear modulus $G$. The relationship
$$\tau=G \gamma(\gamma \ll 1)$$
between the shearing angle $\gamma=a / h$ and $\tau$ satisfies the definition of a solid: the force per unit area $\tau$ tends to zero only when the deformation $\gamma$ itself goes to zero. Often the relation for a solid body is of a more general form, e.g., $\tau=f(\gamma)$, with $f(0)=0$.
If the material is a fluid, the displacement of the plate increases continually with time under a constant shearing force. This means there is no relationship between the displacement, or deformation, and the force. Experience shows here that with many fluids the force is proportional to the rate of change of the displacement, that is, to the velocity of the deformation. 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.) The dimensional quantity required is the shear viscosity $\eta$, and the relationship with $U=\mathrm{d} a / \mathrm{d} t$ now reads
$$\tau=\eta \frac{U}{h}=\eta \dot{\gamma},$$
or, if the shear rate $\dot{\gamma}$ is set equal to $\mathrm{d} u / \mathrm{d} y$.
$$\tau(y)=\eta \frac{\mathrm{d} u}{\mathrm{~d} y} .$$
$\tau(y)$ is the shêar streess on a surfacee eelement parallèl to the plates at point $y$. In so-called simple shearing flow (rectilinear shearing flow) only the $x$-component of the velocity is nonzero, and is a linear function of $y$.

The above relationship was known to Newton, and it is sometimes incorrectly used as the definition of a Newtonian fluid: there are also non-Newtonian fluids which show a linear relationship between the shear stress $\tau$ and the shear rate $\dot{\gamma}$ in this simple state of stress. In general, the relationship for a fluid reads $\tau=f(\dot{\gamma})$, with $f(0)=0$.

While there are many substances for which this classification criterion suffices, there are some which show dual character. These include the glasslike materials which do not have a crystal structure and are structurally liquids. Under prolonged loads these substances begin to flow, that is to deform without limit. Under short-term loads, they exhibit the behavior of a solid body. Asphalt is an oftquoted example: you can walk on asphalt without leaving footprints (short-term load), but if you remain standing on it for a long time, you will finally sink in. Under very short-term loads, e.g., a blow with a hammer, asphalt splinters, revealing its structural relationship to glass. 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.

物理代写|流体力学代写Fluid Mechanics代考|The behavior of solids

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 difference between gases on the one hand, and liquids and solids on the other is the mean distance between the molecules.

With gases, the spacing at standard temperature and pressure $(273.2 \mathrm{~K}$; $1.013$ bar) is about ten effective molecular diameters. 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 molecules first attract each other weakly, and then as the interval between them becomes noticeably smaller than the effective diameter, they repel strongly. The mean free path is in general larger than the mean distance, and can occasionally be considerably larger.

With liquids and solids the mean distance is about one effective molecular diameter. In this case there is always an interaction between the molecules. The large resistance which liquids and solids show to volume changes is explained by the repulsive force between molecules when the spacing becomes noticeably smaller than their effective diameter. 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. But it is impossible in principle anyway, since the position and momentum of a molecule cannot be simultaneously known (Heisenberg’s Uncertainty Principle) and thus the initial conditions for the integration do not exist. $\mathrm{~ I n ~ a ̊ d d i t i o ̄ n , ~ d e t a ̄ i l e ̉ ~ i n f o r m a ̄ t i o n ̃ ~ a b o o u t ~ t h e ~ m o ̄ l e ́ c u l a}$ 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.

物理代写|流体力学代写Fluid Mechanics代考|On the other hand the linear measure of the volume

On the other hand the linear measure of the volume must be small compared to the macroscopic length of interest. It is appropriate to assume that the volume of the fluid particle is infinitely small compared to the whole volume occupied by the fluid. This assumption forms the basis of the continuum hypothesis. Under this hypothesis we consider the fluid particle to be a material point and the density (or other properties) of the fluid to be continuous functions of place and time. 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 part of the fluid under observation consists then of infinitely many material points, and we expect that the motion of this continuum will be described by partial differential equations. However the assumptions which have led us from the material to the idealized model of the continuum are not always fulfilled. One example is the flow past a space craft at very high altitudes, where the air density is very low. 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. 9), a frequent occurrence in compressible flow. 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. This thermal motion is reflected in the macroscopic properties of the material and is the single source of viscosity in gases. Even if the macroscopic velocity given by (1.4) is zero, the molecular velocities $\vec{c}{i}$ are clearly not necessarily zero. The consequence of this is that the molecules migrate out of the fluid particle and are replaced by molecules drifting in. This exchange process gives rise to the macroscopic fluid properties called transport properties. Obviously, molecules with other molecular properties (e.g. mass) are brought into the fluid particle. Take as an example a gas which consists of two types of molecule, say $\mathrm{O}{2}$ and $\mathrm{N}{2}$. Let the number of $\mathrm{O}{2}$ molecules per unit volume in the fluid particle be larger than that of the surroundings. The number of $\mathrm{O}_{2}$ molecules which migrate out is proportional to the number density inside the fluid particle, while the number which drift in is proportional to that of the surroundings. The net effect is that more $\mathrm{O}{2}$ molecules drift in than drift out and so the $\mathrm{O}{2}$ number density adjusts itself to the surroundings. From the standpoint of continuum theory the process described above represents the diffusion.

物理代写|流体力学代写Fluid Mechanics代考|Properties of Fluids, Continuum Hypothesis

τ=GC(C≪1)

τ=这在H=这C˙,

τ(是)=这d在 d是.
τ(是)是表面元素上的剪切应力，平行于点的板是. 在所谓的简单剪切流（直线剪切流）中，只有X-速度的分量是非零的，并且是一个线性函数是.

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