### 物理代写|流体力学代写Fluid Mechanics代考|Equations of Motion for Particular

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## 物理代写|流体力学代写Fluid Mechanics代考|The Navier-Stokes Equations

We start with a Newtonian fluid which is defined by the constitutive relation (3.1) and, by setting (3.1) and (1.29) into (2.38), we obtain the Navier-Stokes equations
$$\varrho \frac{\mathrm{D} u_{i}}{\mathrm{D} t}=\varrho k_{i}+\frac{\partial}{\partial x_{i}}\left{-p+\lambda^{*} \frac{\partial u_{k}}{\partial x_{k}}\right}+\frac{\partial}{\partial x_{j}}\left{\eta\left[\frac{\partial u_{i}}{\partial x_{j}}+\frac{\partial u_{j}}{\partial x_{i}}\right]\right}$$
where we have used the exchange property of the Kronecker delta $\delta_{i j}$.
With the linear law for the friction stresses (3.2) and the linear law for the heat flux vector (3.8), we specialize the energy equation to the case of Newtonian fluids
$$\varrho \frac{\mathrm{D} e}{\mathrm{D} t}-\frac{p}{\varrho} \frac{\mathrm{D} \varrho}{\mathrm{D} t}=\Phi+\frac{\partial}{\partial x_{i}}\left[\lambda \frac{\partial T}{\partial x_{i}}\right]$$
where the dissipation function $\Phi$ is given hy (3.6). In the same way we deal with the forms (2.116) and (2.118) of the energy equation, which are often more appropriate. Another useful form of the energy equation arises by inserting the enthalpy $h=e+p / \varrho$ into (4.2). Because of

$$\varrho \frac{\mathrm{D} h}{\mathrm{D} t}=\varrho \frac{\mathrm{D} e}{\mathrm{D} t}-\frac{p}{\varrho} \frac{\mathrm{D} \varrho}{\mathrm{D} t}+\frac{\mathrm{D} p}{\mathrm{D} t}$$
(4.2) can also be written as
$$\varrho \frac{\mathrm{D} h}{\mathrm{D} t}-\frac{\mathrm{D} p}{\mathrm{D} t}=\Phi+\frac{\partial}{\partial x_{i}}\left[\lambda \frac{\partial T}{\partial x_{i}}\right]$$
As a consequence of Gibbs’ relation (2.133), the entropy equation for Newtonian fluids can also appear in place of (4.2)
$$\varrho T \frac{\mathrm{D} s}{\mathrm{D} t}=\Phi+\frac{\partial}{\partial x_{i}}\left[\lambda \frac{\partial T}{\partial x_{i}}\right]$$
If we choose the energy equation (4.2), together with the continuity equation and the Navier-Stokes equations we have five partial differential equations with seven unknown functions. But both the thermal equation of state $p=p(\varrho, T)$ and the caloric equation of state $e=e(\varrho, T)$ appear also. This set of equations forms the starting point for the calculation of frictional compressible flow.

## 物理代写|流体力学代写Fluid Mechanics代考|Vorticity Equation

Since a viscous incompressible fluid behaves like an inviscid fluid in regions where $\vec{\omega}=0$, the question arises of what the differential equation for the distribution of $\vec{\omega}$ is. Of course this question does not arise if we consider the velocity field as given, because then $\vec{\omega}$ can be calculated directly from the velocity field using Eq. (1.49). To obtain the desired relation, we take the curl of the Eg. (4.9b). For reasons of clarity, we shall use symbolic notation here. We assume further that $\vec{k}$ has a potential $(\vec{k}=-\nabla \psi)$, and use the identity (4.11) in Eq. (4.9b). In addition, we make use of (1.78) to obtain the Navier-Stokes equations in the form
$$\frac{1}{2} \frac{\partial \vec{u}}{\partial t}-\vec{u} \times \vec{\omega}=-\frac{1}{2} \nabla\left[\psi+\frac{p}{\varrho}+\frac{\vec{u} \cdot \vec{u}}{2}\right]-\nu \nabla \times \vec{\omega}$$
The operation $\nabla \times$ applied to (4.12), along with the identity (easily verified in index notation)
$$\nabla \times(\vec{u} \times \vec{\omega})=\vec{\omega} \cdot \nabla \vec{u}-\vec{u} \cdot \nabla \vec{\omega}-\vec{\omega} \nabla \cdot \vec{u}+\vec{u} \nabla \cdot \vec{\omega}$$
furnishes the new left-hand side $\partial \vec{\omega} / \partial t-\vec{\omega} \cdot \nabla \vec{u}+\vec{u} \cdot \nabla \vec{\omega}$, where we have already noted that the flow is incompressible $(\nabla \cdot \vec{u}=0)$ and that the divergence of the curl always vanishes
$$2 \nabla \cdot \vec{\omega}=\nabla \cdot(\nabla \times \vec{u})=0$$
This can be shown in index notation or simply explained by the fact that the symbolic vector $\nabla$ is orthogonal to $\nabla \times \vec{u}$. On the right-hand side of (4.12), the term in parantheses vanishes, since the symbolic vector $\nabla$ is parallel to the gradient. The remaining term on the right-hand side $-\nu \nabla \times(\nabla \times \vec{\omega})$ is recast using the identity (4.10), and because $\nabla \cdot \vec{\omega}=0$ from (4.14) we extract the new right-hand side $\nu \Delta \vec{\omega}$. In this manner we arrive at the vorticity equation

$$\frac{\partial \vec{\omega}}{\partial t}+\vec{u} \cdot \nabla \vec{\omega}=\vec{\omega} \cdot \nabla \vec{u}+\nu \Delta \vec{\omega}$$
Because $\partial / \partial t+\vec{u} \cdot \nabla=\mathrm{D} / \mathrm{D} t$ we can shorten this to
$$\frac{\mathrm{D} \vec{\omega}}{\mathrm{D} t}=\vec{\omega} \cdot \nabla \vec{u}+\nu \Delta \vec{\omega} .$$

## 物理代写|流体力学代写Fluid Mechanics代考|Effect of Reynolds’ Number

In viscous flow, the term, $\nu \Delta \vec{\omega}$ in (4.16) represents the change in the angular velocity of a material particle which is due to its neighboring particles. Clearly, the particle is set into rotation by its neighbors via viscous torques, and it itself exerts torques on other neighboring particles, thus setting these into rotation. The particle only passes on the vector of angular velocity $\vec{\omega}$ on to the next one, just as temperature is passed on by heat conduction, or concentration by diffusion. Thus we speak of the “diffusion” of the angular velocity vector $\vec{\omega}$ or of the vorticity vector curl $\vec{u}=\nabla \times \vec{u}=2 \vec{\omega}$. From what we have said before, we conclude that angular velocity cannot be produced within the interior of an incompressible fluid, but gets there by diffusion from the boundaries of the fluid region. Flow regions where the diffusion of the vorticity vector is negligible can be treated according to the rules of inviscid and irrotational fluids.

As we know, equations which express physical relationships and which are dimensionally homogeneous (only these are of interest in engineering) must be reducible to relations hetween dimensionless quantities. Ising the typical velocity $U$ of the problem, the typical length $L$ and the density $\varrho$, constant in incompressible flow, we introduce the dimensionless dependent variables

\begin{aligned} u_{i}^{+} &=\frac{u_{i}}{U} \ p^{+} &=\frac{p}{\varrho U^{2}} \end{aligned}
and the independent variables
\begin{aligned} x_{i}^{+} &=\frac{x_{i}}{L} \ t^{+} &=t \frac{U}{L} \end{aligned}
into the Navier-Stokes equations, and obtain (neglecting body forces)
$$\frac{\partial u_{i}^{+}}{\partial t^{+}}+u_{j}^{+} \frac{\partial u_{i}^{+}}{\partial x_{j}^{+}}=-\frac{\partial p^{+}}{\partial x_{i}^{+}}+R e^{-1} \frac{\partial^{2} u_{i}^{+}}{\partial x_{j}^{+} \partial x_{j}^{+}}$$
where $R e$ is the already known Reynolds’ number
$$R e=\frac{U L}{\nu}$$

## 物理代写|流体力学代写Fluid Mechanics代考|The Navier-Stokes Equations

\varrho \frac{\mathrm{D} u_{i}}{\mathrm{D} t}=\varrho k_{i}+\frac{\partial}{\partial x_{i}}\left{-p+ \lambda^{*} \frac{\partial u_{k}}{\partial x_{k}}\right}+\frac{\partial}{\partial x_{j}}\left{\eta\left[ \frac{\partial u_{i}}{\partial x_{j}}+\frac{\partial u_{j}}{\partial x_{i}}\right]\right}\varrho \frac{\mathrm{D} u_{i}}{\mathrm{D} t}=\varrho k_{i}+\frac{\partial}{\partial x_{i}}\left{-p+ \lambda^{*} \frac{\partial u_{k}}{\partial x_{k}}\right}+\frac{\partial}{\partial x_{j}}\left{\eta\left[ \frac{\partial u_{i}}{\partial x_{j}}+\frac{\partial u_{j}}{\partial x_{i}}\right]\right}

ϱD和D吨−pϱDϱD吨=披+∂∂X一世[λ∂吨∂X一世]

ϱDHD吨=ϱD和D吨−pϱDϱD吨+DpD吨
(4.2) 也可以写成

ϱDHD吨−DpD吨=披+∂∂X一世[λ∂吨∂X一世]

ϱ吨DsD吨=披+∂∂X一世[λ∂吨∂X一世]

## 物理代写|流体力学代写Fluid Mechanics代考|Vorticity Equation

12∂在→∂吨−在→×ω→=−12∇[ψ+pϱ+在→⋅在→2]−ν∇×ω→

∇×(在→×ω→)=ω→⋅∇在→−在→⋅∇ω→−ω→∇⋅在→+在→∇⋅ω→

2∇⋅ω→=∇⋅(∇×在→)=0

∂ω→∂吨+在→⋅∇ω→=ω→⋅∇在→+νΔω→

Dω→D吨=ω→⋅∇在→+νΔω→.

## 物理代写|流体力学代写Fluid Mechanics代考|Effect of Reynolds’ Number

X一世+=X一世大号 吨+=吨在大号

∂在一世+∂吨++在j+∂在一世+∂Xj+=−∂p+∂X一世++R和−1∂2在一世+∂Xj+∂Xj+

R和=在大号ν

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