# ITP: Forces Acting on Fluids

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.