### 物理代写|流体力学代写Fluid Mechanics代考|Momentum and Angular Momentum

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## 物理代写|流体力学代写Fluid Mechanics代考|Accelerating Frame

The balance of momentum and angular momentum that we have discussed so far are only valid in inertial reference frames. An inertial reference frame in classical mechanics could be a Cartesian coordinate system whose axes are fixed in space (relative, for example, to the fixed stars), and which uses the average solar day as a unit of time, the basis of all our chronology. All reference frames which move uniformly, i.e.., not acceelerrating in this systêm, arre equivallênt añd thus aree inertiāl frames.

The above balances do not hold in frames which are accelerating relative to an inertial frame. But the forces of inertia which arise from nonuniform motion of the frame are often so small that reference frames can by regarded as being approximately inertial frames. On the other hand, we often have to use reference frames where such forces of inertia cannot be neglected.

To illustrate this we will look at a horizontal table which is rotating with angular velocity $\Omega$. On the table and rotating with it is an observer, who is holding a string at the end of which is a stone, lying a distance $R$ from the fulcrum of the table. The observer experiences a force (the centrifugal force) in the string. Since the stone is at rest in his frame, and therefore the acceleration in his reference frame is zero, the rate of change of momentum must also be zero, and thus, by the balance of momentum (2.9), the force in the string should vanish. The observer then correctly concludes that the balance of momentum does not hoold in his reference frame. The rotating table must be treated as an noninertial reference frame. The source of the force in the string is obvious to an observer who is standing beside the rotating table. He sees that the stone is moving on a circular path and so it experiences an acceleration toward the center of the circle, and that according to the balance of momentum, there must be an external force acting on the stone. The acceleration is the centripetal acceleration, which is given here by $\Omega^{2} \mathrm{R}$. The force acting inwards is the centripetal force which is exactly the same size as the centrifugal force experienced by the rotating observer.

In this example the reference frame of the observer at rest, that is the earth, can be taken as an inertial reference frame. Yet in other cases deviations from what is expected from the balance of momentum appear. This is because the earth is rotating and therefore the balance of momentum strictly does not hold in a reference frame moving with the earth. With respect to a frame fixed relative to the earth we observe, for example, the deflection of a free falling body to the east, or the way that the plane of oscillation of Foucault’s pendulum rotates. These examples, and many others, are not compatible with the validity of the balance of momentum in the reference frame chosen to be the earth. For most terrestrial events, however, a coordinate system whose origin is at the center of the earth, and whose axes are directed towards the fixed stars, is valid as an inertial reference frame. The easterly deflection mentioned above can then be explained by the fact that the body, in its initial position, has a somewhat higher circumferential speed because of the rotation of the earth than at the impact point nearer the center of the earth. To explain Foucault’s pendulum, we notice that, in agreement with (2.9), the pendulum maintains its plane of oscillation relative to the inertial frame. The reference frame attached to the earth rotates about this plane, and an observer in the laboratory experiences a rotation of the plane of oscillation relative to his system with a period of twenty-four hours.

## 物理代写|流体力学代写Fluid Mechanics代考|Applications to Turbomachines

Typical applications of the balances of momentum and of angular momentum can be found in the theory of turbomachines. The essential element present in all turbomachines is a rotor equipped with blades surrounding it, either in the axial or radial direction.

When the fluid exerts a force on the moving blades, the fluid does work. In this case we can also speak of turbo force machines (turbines, wind wheels, etc.). If the moving blades exert a force on the fluid, and thus do work on it, increasing its energy, we speak of turbo work machines (fans, compressors, pumps, propellers).
Often the rotor has an outer casing, called stator, which itself is lined with blades. Since these blades are fixed, no work is done on them. Their task is to direct the flow either towards or away from the moving blades attached to the rotor. These blades are called guide blades or guide vanes. A row of fixed blades together with a row of moving blades is called a stage. A turbomachine can be constructed with one or more of these stages. If the cylindrical surface of Fig. $2.6$ at radius $r$ through the stage is cut and straightened, the contours of the blade sections originally on the cylindrical surface form two straight cascades. The set up shown consists of a

turbine stage where the fixed cascade is placed before the moving cascade seen in the direction of the flow.

Obviously the cascades are used to turn the flow. If the turning is such that the magnitude of the velocity is not changed, the cascade is a pure turning or constant pressure cascade, since then no change of pressure occurs through the cascade (only in the case of frictionless flow). In general the magnitude of the velocity changes with the turning and therefore also the pressure. If the magnitude of the velocity is increased we have an acceleration cascade, typically found in turbines, and if it is decreased we have a deceleration cascade, typically found in compressors. We shall consider the cascade to be a strictly periodic ordering of blades, that is, an infinitely long row of blades with exactly the same spacing $s$ between blades along the cascade. Because of this the flow is also strictly periodic.

## 物理代写|流体力学代写Fluid Mechanics代考|Balance of Energy

The fact that mechanical energy can be changed into heat and heat can be changed into mechanical energy shows that the balance laws of mechanics we have discussed up to now are not enough for a complete description of the motion of a fluid. As well as the two laws we have already treated, therefore a third basic empirical law, the balance of energy, appears:
The rate of change of the total energy of a body is equal to the power of the external forces plus the rate at which heat is transferred to the body.
This law can be “deduced” from the well known first law of thermodynamics together with a mechanical energy equation which follows from Cauchy’s Eq. (2.38a, 2.38b). However here we prefer to postulate the balance of the total energy, and to infer from it the more restrictive statement of the first law of thermodynamics.

We shall assume the fundamentals of classical thermodynamics as known. Thermodynamics is concerned with processes where the material is at rest and where all quantities appearing are independent of position (homogeneous), and therefore are only dependent on time. An important step to the thermodynamics of irreversible processes as they appear in the motion of fluids, consists of simply applying the classical laws to a material particle. If $e$ is the internal energy per unit

mass, then the internal energy of a material particle is given by $e \mathrm{~d} m$, and we can calculate the internal energy $E$ of a body, that is, the energy of a bounded part of the fluid, as the integral over the region occupied by the body
$$E=\iiint_{(V(t))} e \varrho \mathrm{d} V$$
In order to obtain the total energy of the fluid body under consideration, the kinetic energy which does not appear in the classical theory must be added to (2.109). The kinetic energy of the material particle is $\left(u^{2} / 2\right) \mathrm{d} m$, and the kinetic energy $K$ of the body is correspondingly
$$K=\iiint_{(V(t))} \frac{u_{i} u_{i}}{2} \varrho \mathrm{d} V$$
The applied forces which appear are the surface and body forces which were discussed in the context of the balance of momentum. The power of the surface force $\vec{t} \mathrm{~d} S$ is $\vec{u} \cdot \vec{t} \mathrm{~d} S$, while that of the body force $\varrho \vec{k} \mathrm{~d} V$ is $\vec{u} \cdot \vec{k} \varrho \mathrm{d} V$. The power of the applied forces is then
$$P=\iiint_{(V(t))} \varrho u_{i} k_{i} \mathrm{~d} V+\iint_{(S(t))} u_{i} t_{i} \mathrm{~d} S$$
In analogy to the volume flow $\vec{u} \cdot \vec{n} \mathrm{~d} S$ through an element of the surfacee, we introduce the heat flux through an element of the surface with $-\vec{q} \cdot \vec{n} \mathrm{~d} S$ and denote $\vec{q}$ as the heat flux vector. The minus sign is chosen so that inflowing energy ( $\vec{q}$ and $\vec{n}$ forming an obtuse angle) is counted as positive. From now we shall limit ourselves to the transfer of heat by conduction, although $\vec{q}$ can also contain other kinds of heat transfer, for example, heat transfer by radiation, via Poynting’s vector.

## 物理代写|流体力学代写Fluid Mechanics代考|Balance of Energy

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