Position and acceleration in a Lagrangian description of a 1D flow in fluid dynamics I have the following function of velocity of a one-dimensional flow inside a channel: $$u=k(1+x/L),$$ where $k$ is a constant, $L$ is a length of the channel and $x$ the distance from the enter. 
If I want to know the acceleration along the channel in an Eulerian description, I would need to do the substancial derivative of the function $u$, like the following:
$$a = u (du/dx) = ((k^2)/L)*(1+x/L)$$
But how would you do it if I want to obtain the acceleration as a function of time of a fluid particle located at $x=0$ at $t=0$?
 A: $$\frac{dx}{dt}=k(1+x/L)$$
$$\ln(1+x/L)=\frac{kt}{L}$$
$$(1+x/L)=\exp{(kt/L)}$$
$$u=k\exp{(kt/L)}$$
$$a=\frac{k^2}{L}\exp{(kt/L)}$$
In response to @Deep's comment, I am going to express the Lagrangian (embedded material coordinate system) quantities in the way I learned to do it and, (as an experienced non-Newtonian rheology person) have been doing it ever since.
Let $x(t,t_0,x_0)$ represent the position at time t of the material particle that was at position $x_0$ at time $t_0$.  Then similarly for $u(t,t_0,x_0)$ and $a(t,t_0,x_0)$.  So,
$$x(t,t_0,x_0)=L\left[(1+x_0/L)\exp{\left(\frac{k(t-t_0)}{L}\right)}-1\right]$$
$$u(t,t_0,x_0)=k(1+x_0/L)\exp{\left(\frac{k(t-t_0)}{L}\right)}$$
$$a(t,t_0,x_0)=\frac{k^2}{L}(1+x_0/L)\exp{\left(\frac{k(t-t_0)}{L}\right)}$$
A: So if you want this to be a steady flow then it is not quite "one-dimensional:" it is a steady flow for a channel which changes cross section linearly from $A \to A/2$ as $x$ goes from $0\to L.$
As for the underlying kinematics, prepare to be a little bored. As long as the flow is laminar, the particle simply goes from $x$ to $x + u~dt$ in a time $dt$, at which point its velocity has changed from $u(x)$ to $u(x + u~dt)$. Taylor-expanding we find:$$u(x + u~dt) \approx u(x) + \frac{\partial u}{\partial x}~u~dt,$$ which means that $$a = \lim_{dt\to0}\frac{u(x + u~dt) - u(x)}{dt} = u~\frac{\partial u}{\partial x}.$$ 
In other words the problem that you were solving is, in fact, the problem that you wanted to solve.
In a much more general context, we can speak of the continuity equation, which says "let me define a small box of volume $V$ around a point, it holds some conserved quantity $q = \rho~V.$ Then $q$ only increases due to either some current density $\vec J$ being imbalanced on the walls of the volume, or else by being teleported in/out of the system I am studying." In differential form the equation says, $$\frac{\partial \rho}{\partial t} = -\nabla \cdot \vec J + \Omega,$$where $\Omega$ describes the stuff coming into the system from outside, e.g. if you want to think about heat conduction in a plate that is being warmed by two lasers at specific points.
In fluid flow problems, we typically want to write that the particles of "stuff" are flowing downstream in some bigger fluid, with some flow field $\vec v.$ Of course it's boring if that's the only think they do, so we like to model that $\vec J = \rho \vec v + \vec j$ for some sort of "local deviation" $\vec j.$ A typical local deviation, for example, is diffusion according to Fick's law, $\vec j = -D\nabla \rho,$ expressing e.g. for thermal energy, "something twice as hot transfers heat to the things it's touching twice as fast."
Anyway plugging this in, we have the transport equation:$$\frac{\partial \rho}{\partial t} + \vec v\cdot\nabla \rho = -\rho \nabla \cdot \vec v - \nabla \cdot \vec j + \Omega.$$Note that the term on the left is $$\frac{D\rho}{Dt} = \lim_{dt\to0} \frac{\rho(\vec r + \vec v~dt, t + dt) - \rho(\vec r, \vec t)}{dt} = \frac{\partial \rho}{\partial t} + \vec v \cdot \nabla \rho,$$ and therefore describes roughly "the change in an amount of stuff in a little box flowing downstream." The first term on the right expresses the fact that the stuff inside that little box might come from many other boxes upstream of it, if the boxes are "shrinking" in volume according to the flow field, compressing all things that flow with the field.
So that might give you some guidance about how to approach this problem if there is a nontrivial $\vec j$ for a conserved quantity like, say, the $x$-momentum in some particles in the fluid.
