Acceleration Field

To apply Newton’s second law, we want to be able to describe particle acceleration. This is analogous to the velocity field. Note that the acceleration field applies for the Eulerian flow description; for Lagrangian, we simply use $\mathbf{a}=\mathbf{a}(t)$ for each particle.

Material Derivative

Consider a fluid particle moving along its pathline as shown below.

The position is given by:

$$\mathbf{r}=x\mathbf{i}+y\mathbf{j}+z\mathbf{k}$$

The velocity is in turn given by:

$$\mathbf{V}=\frac{d\mathbf{r}}{dt}=\frac{dx}{dt}\mathbf{i}+\frac{dy}{dt}\mathbf{j}+\frac{dz}{dt}\mathbf{k}=u\mathbf{i}+v\mathbf{j}+w\mathbf{k}$$

where we can say $\mathbf{V}=\mathbf{V}(\mathbf{r},t)=\mathbf{V}(x(t),y(t),z(t),t)$.

Then the acceleration is given by

$$\begin{array}{rl}\mathbf{a}(t)& =\frac{d\mathbf{V}}{dt}=\frac{\partial \mathbf{V}}{\partial t}+\frac{\partial \mathbf{V}}{\partial x}\frac{dx}{dt}+\frac{\partial \mathbf{V}}{\partial y}\frac{dy}{dt}+\frac{\partial \mathbf{V}}{\partial z}\frac{dz}{dt}\\ & =\frac{\partial \mathbf{V}}{\partial t}+u\frac{\partial \mathbf{V}}{\partial x}+v\frac{\partial \mathbf{V}}{\partial y}+w\frac{\partial \mathbf{V}}{\partial z}\end{array}$$

• The first term is local derivative/acceleration
• The next 3 terms are convective derivative/acceleration

We can also write $\mathbf{a}=a_x\mathbf{i}+a_y\mathbf{j}+a_z\mathbf{k}$. Each scalar component can be written as:

$$\begin{array}{rl}{a}_x& =\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}+w\frac{\partial u}{\partial z}\\ a_y& =\frac{\partial v}{\partial t}+u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}+w\frac{\partial v}{\partial z}\\ a_z& =\frac{\partial w}{\partial t}+u\frac{\partial w}{\partial x}+v\frac{\partial w}{\partial y}+w\frac{\partial w}{\partial z}\end{array}$$

In shorthand notation, the above result is given as

$$\mathbf{a}=\frac{D\mathbf{V}}{Dt}$$

where the operator

$$\frac{D()}{dt}\equiv \frac{\partial ()}{\partial t}+u\frac{\partial ()}{\partial t}+v\frac{\partial ()}{\partial t}+w\frac{\partial ()}{\partial t}$$

is called the material derivative.

We can make a simplification by noting that

$$\mathbf{V}\cdot \nabla =(u\mathbf{i}+v\mathbf{j}+w\mathbf{k})\cdot \left(\frac{\partial }{\partial x}\mathbf{i}+\frac{\partial }{\partial y}\mathbf{j}+\frac{\partial }{\partial z}\mathbf{k}\right)$$

which then lets us write

$$\mathbf{a}=\frac{\partial \mathbf{V}}{\partial t}+(\mathbf{V}\cdot \nabla )\mathbf{V}$$

Note that if $\frac{\partial \mathbf{V}}{\partial t}\ne 0$, we have unsteady flow.

Streamline Coordinates

One of the major advantages of using the streamline coordinate system is that the velocity is always tangent to the $s$ direction:

• Cartesian – $\mathbf{V}=u\mathbf{i}+v\mathbf{j}+w\mathbf{k}$
• Streamline – $\mathbf{V}=V\mathbf{s}$

In streamline coordinates, the material derivative becomes:

$$\frac{D()}{Dt}=\frac{\partial ()}{\partial t}+\frac{\partial ()}{\partial s}\frac{ds}{dt}+\frac{\partial ()}{\partial n}\frac{dn}{dt}$$

Then, for acceleration we have:

$$\begin{array}{rl}\mathbf{a}& =\frac{D\mathbf{V}}{Dt}=\frac{D}{Dt}(V\mathbf{s})=\frac{DV}{Dt}\mathbf{s}+V\frac{D\mathbf{s}}{dt}\\ & =\left(\frac{\partial V}{\partial t}+\frac{\partial V}{\partial s}\frac{ds}{dt}+\frac{\partial V}{\partial n}\frac{dn}{dt}\right)\mathbf{s}+V\left(\frac{\partial \mathbf{s}}{\partial t}+\frac{\partial \mathbf{s}}{\partial s}\frac{ds}{dt}+\frac{\partial \mathbf{s}}{\partial n}\frac{dn}{dt}\right)\end{array}$$

For steady flows and along streamlines:

$$\begin{array}{rl}\mathbf{a}& =\left(\text{cancelto}0\frac{\partial V}{\partial t}+\frac{\partial V}{\partial s}\frac{ds}{dt}+\text{cancelto}0\frac{\partial V}{\partial n}\frac{dn}{dt}\right)\mathbf{s}+V\left(\text{cancelto}0\frac{\partial \mathbf{s}}{\partial t}+\frac{\partial \mathbf{s}}{\partial s}\frac{ds}{dt}+\text{cancelto}0\frac{\partial \mathbf{s}}{\partial n}\frac{dn}{dt}\right)\\ & =V\frac{\partial V}{\partial s}\mathbf{s}+V^2\frac{\partial \mathbf{s}}{\partial s}\end{array}$$

Note that

$$\frac{\partial \mathbf{s}}{\partial s}=\underset{\Delta s,\Delta t\to 0}{\lim }\frac{\mathbf{s}(t+\Delta t)-\mathbf{s}(t)}{s(t+\Delta t)-s(t)}=\frac{|\mathbf{s}|\delta \theta \mathbf{n}}{R\delta \theta }=\frac{\mathbf{n}}{R}$$

Then, we can write our previous expression as

$$\mathbf{a}=V\frac{\partial V}{\partial s}\mathbf{s}+\frac{V^2}{R}\mathbf{n}$$

or

$$\mathbf{a}=a_s\mathbf{s}+a_n\mathbf{n}$$

with $a_s=V\frac{\partial V}{\partial s}$, $a_n=\frac{V^2}{R}$.