### 物理代写|流体力学代写Fluid Mechanics代考|Constitutive Relations for Fluids

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## 物理代写|流体力学代写Fluid Mechanics代考|already explained in the previous

As already explained in the previous chapter on the fundamental laws of continuum mechanics, bodies behave in such a way that the universal balances of mass, momentum, energy and entropy are satisfied. Yet only in very few cases, like, for example, the idealizations of a point mass or of a rigid body without heat conduction, are these laws enough to describe a body’s behavior. In these special cases, the characteristics of “mass” and “mass distribution” belonging to each body are the only important features. In order to describe a deformable medium, the material from which it is made must be characterized, because clearly, the deformation or the rate of deformation under a given load is dependent on the material. Because the balance laws yield more unknowns than independent equations, we can already conclude that a specification of the material through relationships describing the way in which the stress and heat flux vectors depend on the other field quantities is generally required. Thus the balance laws yield more unknowns than independent equations. The summarizing list of the balance laws of mass (2.2)
$$\frac{\partial \varrho}{\partial t}+\frac{\partial}{\partial x_{i}}\left(\varrho u_{i}\right)=0$$
of momentum (2.38)
$$\varrho \frac{D u_{i}}{D t}=\varrho k_{i}+\frac{\partial \tau_{j i}}{\partial x_{j}}$$
of angular momentum (2.53)
$$\tau_{i j}=\tau_{j i}$$

and of energy (2.119)
$$\varrho \frac{D e}{D t}=\tau_{i j} \frac{\partial u_{i}}{\partial x_{j}}-\frac{\partial q_{i}}{\partial x_{i}}$$
yield 17 unknown functions $\left(\varrho, u_{i}, \tau_{i j}, q_{i}, e\right)$ in only eight independently available equations. Instead of the energy balance, we could also use the entropy balance (2.134) here, which would introduce the unknown function $s$ instead of $e$, but by doing this the number of equations and unknown functions would not change. Of course we could solve this system of equations by specifying nine of the unknown functions arbitrarily, but the solution found is then not a solution to a particular technical problem.

## 物理代写|流体力学代写Fluid Mechanics代考|The final condition

The final condition is here of particular importance, since, as we know from Sect. $2.4$, the equations of motion (momentum balance) are not frame independent in this sense. In accelerating reference frames, the apparent forces are introduced, and only the axiom of objectivity ensures that this remains the only difference for the transition from an inertial system to a relative system. However, it is clear that an observer in an accelerating reference frame detects the same material properties as an observer in an inertial system. To illustrate this, for a given deflection of a massless spring, an observer in a rotating reference frame would detect exactly the same force as in an inertial frame.

In so-called simple fluids, the stress on a material point at time $t$ is determined by the history of the deformation involving only gradients of the first order or more exactly, by the relative deformation tensor (relative Cauchy-Green-tensor) as every fluid is isotropic. Essentially all non-Newtonian fluids belong to this group.

The most simple constitutive relation for the stress tensor of a viscous fluid is a linear relationship between the components of the stress tensor $\tau_{i j}$ and those of the rate of deformation tensor $e_{i j}$. Almost trivially, this constitutive relation satisfies all the above axioms. The material theory shows that the most gêneraal linéar rèlationship of this kind must be of the form
$$\tau_{i j}=-p \delta_{i j}+\lambda^{} e_{k k} \delta_{i j}+2 \eta e_{i j}$$ or, using the unit tensor $\mathbf{I}$ $$\mathbf{T}=\left(-p+\hat{\lambda}^{} \nabla \cdot \vec{u}\right) \mathbf{I}+2 \eta \mathbf{E}$$
(Cauchy-Poisson law), so that noting the decomposition (2.35), the tensor of the friction stresses is given by
$$P_{i j}=\lambda^{*} e_{k k} \delta_{i j}+2 \eta e_{i j}$$

or
$$\mathbf{P}=\lambda^{} \nabla \cdot \vec{u} \mathbf{I}+2 \eta \mathbf{E}$$ We next note that the friction stresses at the position $\vec{x}$ are given by the rate of deformation tensor $e_{i j}$ at $\vec{x}$, and are not explicitly dependent on $\vec{x}$ itself. Since the friction stress tensor $P_{i j}$ at $\vec{x}$ determines the stress acting on the material particle at $\vec{x}$, we conclude that the stress on the particle only depends on the instantaneous value of the rate of deformation tensor and is not influenced by the history of the deformation. We remind ourselves that for a fluid at rest or for a fluid undergoing rigid body motion, $e_{i j}=0$, and (3.1a) reduces to (2.33). The quantities $\lambda^{}$ and $\eta$ are scalar functions of the thermodynamic state, typical to the material. Thus (3.1a, $3.1 \mathrm{~b})$ is the generalization of $\tau=\eta \dot{\gamma}$, which we have already met in connection with simple shearing flow and defines the Newtonian fluid.

The extraordinary importance of the linear relationship ( $3.1 \mathrm{a}, 3.1 \mathrm{~b})$ lies in the fact that it describes the actual material behavior of most technically important fluids very well. This includes practically all gases, in particular air and steam, gas mixtures and all liquids of low molecular weight, like water, and also all mineral oils.

## 物理代写|流体力学代写Fluid Mechanics代考|Here k is a positive function of the thermodynamic

Here $\lambda$ is a positive function of the thermodynamic state, and is called the thermal conductivity. The minus sign here is in agreement with the inequality (2.141). Experiments show that this linear law describes the actual behavior of materials very well. The dependency of the thermal conductivity on $p$ and $T$ remains open in (3.8), and has to be determined experimentally. For gases the kinetic theory leads to the result $\lambda \sim \eta$, so that the thermal conductivity shows the same temperature dependence as the shear viscosiry. (For liquids, one discovers theorerically thar the thermal conductivity is proportional to the velocity of sound in the fluid.)

In the limiting case $\eta, \lambda^{}=0$, we extract from the Cauchy-Poisson law the constitutive relation for inviscid fluids $$\tau_{i j}=-p \delta_{i j}$$ Thus, as with a fluid at rest, the stress tensor is only determined by the pressure $p$. As far as the stress state is concerned, the limiting case $\eta, \lambda^{}=0$ leads to the sảmé result as $e_{i j}=0$. Also consistênt with $\eta, \lambda^{*}=0$ is the casse $\lambda=0$; ignoring the friction stresses implies that we should in general also ignore the heat conduction.
It would now appear that there is no technical importance attached to the condition $\eta, \lambda *, \lambda=0$. Yet the opposite is actually the case. Many technically important, real flows are described very well using this assumption. This has already been stressed in connection with the flow through turbomachines. Indeed the flow past a flying object can often be predicted using the assumption of inviscid flow. The reason for this can be clearly seen when we note that fluids which occur in applications (mostly air or water) only have “small” viscosities. However, the viscosity is a dimensional quantity, and the expression “small viscosity” is vague, since the numerical value of the physical quantity “viscosity” may be arbitrarily changed by suitable choice of the units in the dimensional formula. The question of whether the viscosity is small or not can only be settled in connection with the specific problem, however this is already possible using simple dimensional arguments. For incompressible fluids, or by using Stokes’ relation (3.5), only the shear viscosity appears in the constitutive relation (3.1a, 3.1b). If, in addition, the temperature field is homogeneous, no thermodynamic quantities enter the problem, and the incident flow is determined by the velocity $U$, the density $\varrho$ and the shear viscosity $\eta$. We characterize the body past which the fluid flows by its typical length $L$, and we form the dimensionless quantity
$$R e=\frac{U L \varrho}{\eta}=\frac{U L}{v}$$

## 物理代写|流体力学代写Fluid Mechanics代考|already explained in the previous

∂ϱ∂吨+∂∂X一世(ϱ在一世)=0

ϱD在一世D吨=ϱķ一世+∂τj一世∂Xj

τ一世j=τj一世

ϱD和D吨=τ一世j∂在一世∂Xj−∂q一世∂X一世

## 物理代写|流体力学代写Fluid Mechanics代考|The final condition

τ一世j=−pd一世j+λ和ķķd一世j+2这和一世j或者，使用单位张量我

（Cauchy-Poisson 定律），因此注意到分解 (2.35)，摩擦应力的张量由下式给出

## 物理代写|流体力学代写Fluid Mechanics代考|Here k is a positive function of the thermodynamic

τ一世j=−pd一世j因此，与静止的流体一样，应力张量仅由压力决定p. 就应力状态而言，极限情况这,λ=0导致 sảmé 结果为和一世j=0. 也符合这,λ∗=0是案例λ=0; 忽略摩擦应力意味着我们通常也应该忽略热传导。

R和=在大号ϱ这=在大号在

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