#25Fundamentals_Navier-Stokes equation

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(It seems a little bit awkward for me to consider Navier-Stokes equation as fundamentals.....XD")

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Navier-Stokes equation is one of the most important equations in fluid mechanics. From the simulation of weather to the current in blood vessels, every complex problem related to fluid requires it to be solved or approximated. However, because the solution of Navier-Stokes equations are usually very complex or even unsolvable currently, most of the time we will rely on numerical simulation to finish the task. Today we will focus on the physical meaning of Navier-Stokes equation, which is basically the Newtonian second law used to describe the change in flow field.

Navier-Stokes equation可說是流體力學中最重要的方程式之一，舉凡天氣、洋流、飛機感受到的氣流、血管中的血流等，要對他們進行模擬與計算，最基本的統馭方程大概都是Navier-Stokes equation或他的化簡變形版本。但因為這個方程式太過複雜，只有非常化簡的情形可以人工計算，大部分的時候都是仰賴數值模擬。克雷數學研究所(Clay Mathematics Institute,CMI)於2000年公告的七個千禧年大獎難題中，就有一個是關於Navier-Stokes equation。今天我們要來和大家解釋Navier-Stokes equation中每個項所代表的物理意義，本質上他就是牛頓第二運動定律用來描述流場的變化而已。

Firstly, we shall consider the acceleration of a fluid element. Since the location of the fluid element would also change with time, we have to account both the changes with time and the changes with space to fully consider its acceleration. For simplicity, we should consider motion in 1 dimension first. Assume that the fluid element move Δy in time Δt, then its change in velocity could be written as: 首先我們要先考慮一個單位流體(fluid element)的加速度。由於組成流場的單位流體隨時間流逝，他的位置也有改變，因此我們在探討流場的加速度時，除了時間的變化之外也要考慮空間的變化。如果我們只考慮一維的情形，要算出加速度必須知道速度的變化量，所以對一單位流體而言，假設經過Δt他往前移動了Δy，那麼它的速度變化為：

$\Delta v(y,t) = v(y+\Delta y, t+\Delta t) - v(y,t) = \frac{\partial v}{\partial y} \Delta y + \frac{\partial v}{\partial t} \Delta t = \frac{\partial v}{\partial y} (v\Delta t) + \frac{\partial v}{\partial t} \Delta t$

So we can write down the acceleration as:
所以我們就可以寫出加速度為：

$a(y,t) = \frac{\Delta v}{\Delta t}= \frac{\partial v}{\partial y} v_{y} + \frac{\partial v}{\partial t}$

And the motion in 3 dimensions would then be:
擴展到三維的情形，我們就可以寫出：

$\Delta v(x,y,z,t) = \frac{\partial v}{\partial x}v_{x}\Delta t + \frac{\partial v}{\partial y}v_{y}\Delta t + \frac{\partial v}{\partial z}v_{z}\Delta t + \frac{\partial v}{\partial t}\Delta t$

$\overrightarrow{a}(x,y,z,t) = \frac{\partial \overrightarrow{v}}{\partial x}v_{x} + \frac{\partial \overrightarrow{v}}{\partial y}v_{y} + \frac{\partial \overrightarrow{v}}{\partial z}v_{z} + \frac{\partial \overrightarrow{v}}{\partial t}$

This could be simplified by some vector calculus:
也就可以寫成

$\overrightarrow{a}(x,y,z,t) = \frac{\partial \overrightarrow{v}}{\partial t} + (\overrightarrow{v}\cdot \bigtriangledown )\overrightarrow{v}$

The former term in the RHS is the change in flow field with respect to time while the latter term is the change in flow field with respect to space.
前項是大家比較熟悉的流場隨時間變化，後項則是場的空間變化。

Recall that the Newtonian second law told us "F=ma." If we could write down the acceleration of a mass, we could know how much forces the fluid element is enduring. Here we shall consider body force, pressure force and viscous force.
還記得牛頓第二運動定律告訴我們F=ma，一旦我們知道加速度應該如何表達之後，我們就要來看看單位流體到底受了哪些力。在這裡我們要考慮實體力(body force)、壓力(pressure force)和黏滯力(viscous force)。

The simplest term of the three is the body force, which means forces only related to the volume of the fluid element, rather than to the surface area of it. Classic examples include gravitational forces and electrostatic forces. Assume that the fluid has a density of ρ, we could write down the body force as:
首先最簡單的是body force，他是只和單位流體的體積有關的力，而和流體的表面無關。例如重力、靜電力等。假設流體的密度為ρ，我們可以簡單把body force表示成：

$\rho \overrightarrow{Y}$

Sometimes this term could be ignored all at once if we are discuss a system which limits the motion of fluid in a horizontal plane.
之後再看系統的假設來化簡就可以了。(例如，如果我們探討的系統是水平平面上的流體行為，重力的項就可以整個丟掉。)

Let's move on to the effect of pressure force. We shall focus on y direction as we just did previously. From the figure shown above, the net pressure force in y direction could be written as:
再來看壓力的作用。一樣我們只先看y方向的貢獻。從上面的圖我們知道，y方向來自壓力的淨力可以寫成：

$\delta f_{y}^{p} = p(x,y,z,t)(\Delta x\Delta z) - p(x,y+\Delta y,z,t)(\Delta x\Delta z) = - \frac{\partial p}{\partial y}(\Delta x\Delta y\Delta z)$

So this term would become something like this if we consider all 3 dimensions:
所以擴展到三維我們就可以寫成：

$\delta \vec{f^{p}} = - (\Delta x\Delta y\Delta z)(\frac{\partial p}{\partial x} ,\frac{\partial p}{\partial y}, \frac{\partial p}{\partial z}) = -\bigtriangledown p(\Delta x\Delta y\Delta z)$

The last one would be the viscous force. According to the hypothesis proposed by Newton, viscous force could be calculated as viscosity times the gradient of fluid speed. Therefore, let's first consider the viscous force generated from the change of Vz with respect to y direction:
最後我們來看黏滯力的作用。根據牛頓的假設，黏滯力等於黏滯係數η乘上流速變化的梯度。因此我們先只考慮z方向速度隨y方向變化所產生黏滯力的淨變化。

$f_{z,y=y}^{\nu} =-\eta \frac{\partial v_{z}(y)}{\partial y}(\Delta x\Delta z)$

If we consider the changes in both side of the fluid element, the net viscous force would be:
那如果同時考慮左右兩側面，那所受到z方向的黏滯力的淨力為：

$\delta f_{z,y=y}^{\nu} =-\eta \frac{\partial v_{z}(y)}{\partial y}(\Delta x\Delta z) + \eta \frac{\partial v_{z}(y+\Delta y)}{\partial y}(\Delta x\Delta z) = \eta \frac{\partial^2 v_{z}}{\partial y^2}(\Delta x\Delta y\Delta z)$

Then we further consider the changes of Vz with respect to 3 directions, x, y, and z, and we could write down the net viscous force as:
所以如果我們同時考慮z方向速度隨x、y、z三個方向的變化，我們就可以寫出z方向淨力為：

$\delta f_{z}^{\nu} = \eta (\frac{\partial^2 v_{z}}{\partial x^2}+\frac{\partial^2 v_{z}}{\partial y^2}+\frac{\partial^2 v_{z}}{\partial z^2})(\Delta x\Delta y\Delta z)$

Finally we consider the changes of Vx, Vy and Vz, then we will get:
最後同時考慮x, y, z三個方向的話，我們就可以寫出

$\delta \vec{f}^{\nu} =\eta \bigtriangledown ^2 \vec{v} (\Delta x \Delta y\Delta z)$

Add all these stuffs together and we will get our Navier-Stokes equation:
所以把所有東西兜在一起，我們就可以得到Navier-Stokes equation：

$\rho[\frac{\partial \vec{v}}{\partial t} + (\vec{v}\cdot\bigtriangledown)\vec{v}] = \rho\vec{Y}-\bigtriangledown p +\eta\bigtriangledown^2\vec{v}$

We may use these terms to perform dimensional analysis, or we may compare the effect of these terms to determine which one we could ignore. Please make sure you fully understand the meaning of each terms before you move on. Stay tuned!
未來我們在進行因次分析或數量級的比較時，常常用到這個方程式裡面的term之間的比值，所以請讀者再回顧一下，是不是已經清楚每個項代表的意義是甚麼了呢？今天的介紹就到這邊。

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