Before we identify forces acting on fluids it is important to
recall that we are interested (ultimately) in performing a momentum
balance. Recall that momentum is mass X velocity; therefore if we
are performing a differential balance on momentum, we will look at
*rates* at which momentum flows in or out of our
system/control volume (i.e., a momentum per unit time). Clearly
momentum per unit time is equivalent to mass X acceleration, and
therefore any differential momentum balance will necessarily
balances forces on the fluid!

### Stresses at a Point

Stresses in fluids
can be of two types: normal and shear.

##### DEFINITION:

A **normal stress** is force per unit area acting normal
(perpendicular) to a fluid area element $\Delta$A, and is usually
denoted as $\sigma_{ij}$.

$\displaystyle{\sigma_{ij} = \frac{dF_n}{dA} \equiv
\lim_{\Delta A\to \delta A} \frac{\Delta F_n}{\Delta
A}}$

##### DEFINITION:

A **shear stress** is force per unit area acting tangential
(parallel) to a fluid element $\Delta$A, and is usually denoted as
$\tau_{ij}$.

$\displaystyle{\tau_{ij} = \frac{dF_s}{dA} \equiv
\lim_{\Delta A\to \delta A} \frac{\Delta F_s}{\Delta
A}}$

### Fluid Pressure at a Point

The fluid pressure is one
of the components of the normal stress. In fact, if the fluid is at
rest, the pressure is *identically* the magnitude of the
compressive normal stress.

It is important to note that the pressure always acts
*compressively* on a fluid element so that while the
*force magnitude* due to the pressure is given as:

$dF_P = PdA$

the *direction* of the force (i.e., the force vector)
will always be in the direction opposite of the fluid element's
normal vector ( vector which points "outward" from a surface):

$d\vec{F_P} = - P\vec{n}dA$

##### NOTE:

Even in moving fluids the primary normal stress will be the
pressure. Other normal stresses will be ignored except in
Non-Newtonian fluids (we will define this term soon).

### Forces Acting on Fluid in Control Volume

In and open system (i.e., one with flow into and out of the
system), the forces acting on the fluid comprise three types:

- Pressure force at inlets and outlets (a surface force):
$\displaystyle{\vec{F_P} = - \int_{inlet}^{outlet}
P\vec{n}dA}$
- Pressure and shear force exerted by wall surfaces on fluid in
CV (surface forces): $\displaystyle{\vec{R}\equiv -
\int_{walls}P\vec{n}dA + \int_{walls}\vec{\tau}dA}$
- Gravitational force exerted on all fluid within CV (a body or
volumetric force): $\displaystyle{\vec{W} \equiv
\int_{CV}\rho\vec{g}dV = \rho V\vec{g}}$

##### OUTCOME:

Name and explain the origin of forces acting on fluids in a
control volume.