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\begin{center}{\large CHAPTER 1: BASICS}\end{center}
1-1-LagEul.tex\\
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\section{Methods of Describing Fluid Motion}
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To describe the fluid motion, one needs to know the variations of physical quantities such as , density, velocity, pressure, temperature, stresses, etc., as functions of time, everywhere within a certain spatial region. There are two ways to describe the fluid motion. One is called {\em Lagrangian}, where one follows all fluid particles and describes the variations around each fluid particle along its trajectory. The other is {\em Eulerian}, where the variations are described at all fixed stations as a function of time. In the second, different particles pass the same station at different times.
Consider first the Lagrangian description.
\subsection{Lagrangian description}
A particle is identified by its initial position at time $t_0$,
\be \vec{x}_0 = \lp x_0, y_0,z_0 \rp \ee
Let
\be \vec{x} =\vec{x} \lp \vec{x}_0,t \rp \ee
be the position of the same particle at the later time $
t$. By definition, at $t = t_0, \vec{x} \lp \vec{x}_0, t_0 \rp = \vec{x}_0$.
Instead of the initial position $\vec{x}_0$, one can use any three quantities $a,\,
b,\, c$, which are uniquely related to $\vec{x}_0$:
\be \vec{x}_0 = \vec{x}_0 \vec{(a)} \qquad \vec{a} \equiv (a,\, b, \, c) \ee to identify
the particle.
Thus the path of a particle identified by $\vec{a}$ is given by :
\be \vec{x} = \vec{x}(\vec{a},t) . \ee
Note that $\vec{a}, \, t$ are the independent variables,
and $\vec{x} = (x,y,z)$ are dependent variables.
From the particle position we can calculate the particle velocity:
\be \left. \vec{q} = \f{\p \vec{x}}{\p t} \right|_{\vec{a}} , \ee
as well as the
particle acceleration:
\be \left. \f{\p \vec{q}}{\p t} \right|_{\vec{a}} = \left. \f{\p^2 \vec{x}}{\p t^2}
\right|_{\vec{a}} . \ee
Other physical quantities such as density $\rho$ and pressure $p$ can also be expressed in terms of
$\vec{a}, \, t$, e.g.,
\be \rho = \rho \lp \vec{a}, t \rp, ~~~ p = p \lp \vec{a}, t \rp .
\ee
\subsection{Eulerian Description}
Here we specify the fluid properties (density, velocity, pressure...) at a fixed point
and a chosen time. Thus we are dealing with the time records of a property at a fixed measuring probe.
\be \vec{q} \lp \vec{x}, t \rp \, , \quad \rho \lp \vec{x} , t \rp \, , \quad p \lp \vec{x},
t \rp \, , \; \mbox{etc.} \ee
Once a physical property of all particles are known at all times, how do we get the property at all fixed points at all times? In other words, how are the Eulerian and Lagrangian systems related? Suppose first that
the Lagrangian information of the velocity of all particles are known :
\be \vec{q} = \vec{q} \lp \vec{a}, t \rp \; \label{Eq:1.1} \ee then we can integrate the definition
\be \f{\p \vec{x}}{\p t} = \vec{q} \lp \vec{a} , t \rp\ee
for to get the position of a particle,
\be \vec{x} = \vec{x} \lp \vec{a}, t
\rp\label{Eq:1.2}\ee
so that \[ \vec{x(\vec{a},0)}=\vec{a}.\]
The result can be inverted in principle to get
\be \vec{a} = \vec{a} \lp \vec{x}, t \rp \label{Eq:1.3}\ee
as long as the Jacobian of transformation
\be J = \f{\p(x,y,z)}{\p(a,b,c)}=\left| \begin{array}{lll}
\f{\p x}{\p a} & \f{\p x}{\p b} & \f{\p x}{\p c} \\
\f{\p y}{\p a} & \f{\p y}{\p b} & \f{\p y}{\p c} \\
\f{\p z}{\p a} & \f{\p z}{\p b} & \f{\p z}{\p c} \end{array} \right| \neq 0 . \ee
does not
vanish. Once the position of a particle at time $t$ is known, any physical property of a particle can be translated to the property at certain position in space. For example, by substituting (\ref{Eq:1.3}) into (\ref{Eq:1.1}), we get
\be \vec{q} = \vec{q} \lp \vec{x},t \rp , \ee which gives the Eulerian velocity, i.e., the velocity field at all fixed points. Similarly from the Lagrangian pressure $p((\vec{a},t)$ we get the Eulerian pressure
\[ p((\vec{a}(\vec{x},t),t)=p(\vec{x},t)\]
Consider next that
the Eulerian velocity field is known everywhere:
\be \vec{q} = \vec{q} \lp \vec{x},t \rp
\ee
Since at $t$ the particle $\vec{a}$ is at $\vec{x} \lp \vec{a},t \rp$; the
Lagrangian velocity of particle $\vec{a}$ is the same as the Eulerian velocity. Therefore,
\be \vec{q} \lp \vec{x},t \rp = \f{\p \vec{x}}{\p t} , \ee
This equation is a system of nonlinear ordinary differential equations. If they can be solved for $\vec{x} = \vec{x} \lp \vec{a},t \rp$
subjected to the initial condition $\vec{x} = \vec{x}_0 \lp \vec{a} \rp , t = t_0$, we
then have
\begin{eqnarray*}
\f{\p \vec{x}}{\p t} & = & \vec{q} \lb \vec{x} \lp \vec{a},t \rp, t \rb \\ & \equiv &
\vec{q}_L (a,t)\equiv \vec{q}_{\mbox{Lagrangian}} \end{eqnarray*}
For the rate of variations we need to calculate derivatives of some physical quantity $A \lp \vec{x},t \rp$ whose Eulerian information is given. The rate of
variation following a fluid particle is $\left. \f{\p A}{\p t} \right|_{\vec{a}}$,
\be \begin{array}{ll}\left. \f{\p A}{\p t} \right|_{\vec{a}} = \f{\p A}{\p t} + \f{\p A}{\p
x_i} \, \left. \f{\p x_i}{\p t} \right|_{\vec{a}} = & \f{\p A}{\p t} + q_i \, \f{\p A}{\p x_i} \\
\mbox{(Lagrangian)} & \mbox{(Eulerian)} \end{array} \ee
The operator \be \f{D}{Dt} =
\f{\p}{\p t} + q_i \, \f{\p}{\p x_i} \ee has different names: the substantial, or total, or material
{derivative}. The second term is referred to as the convective rate of variation since it arises because the fluid particle moves to new places. In particular, the acceleration of a fluid particle is
\be \f{\p^2 \vec{x}(\vec{a},t)}{\p t^2} = \f{\p \vec{q}}{\p t} + \vec{q} \cdot \nabla
\vec{q} = \f{D\vec{q}}{Dt} . \ee
A more physical interpretation of $Df/Dt$ can be
derived by using the Eulerian picture directly.
Consider $f \lp \vec{x},t \rp$. After $\delta t$, $f(\vec{x},t)$ is changed to
$f(\vec{x} + \vec{q} \delta t,t + \delta t)$. Therefore, the total change of
$f$ is
\begin{eqnarray*}
\delta f & = & f \lb \vec{x} + \vec{q} \delta t, \, t + \delta t \rb - f \lp \vec{x},t \rp \\
& = & f \lp \vec{x},t \rp + \lp \vec{q} \cdot \nabla f + \f{\p f}{\p t} \rp \delta t - f \lp
\vec{x},t \rp = \f{Df}{Dt} \, \delta t . \end{eqnarray*} It follows that
\be \left. \f{\delta f}{\delta t}\right|_{\vec{a}} = \f{Df}{Dt}= \f{\p f}{\p t} + \vec{q}
\cdot \nabla f \ee where $
\f{\p f}{\p t} $ is the instantaneous change, and $ \vec{q} \cdot \nabla f $ is the
convective change.
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