## Tuesday, June 14, 2011

### MrsDrPoe: Rigid Body Motion

And now...back to fluids! Today we'll be delving into rigid body motion. For this case to exist, we must have an accelerating fluid with no shearing stresses present. If you recall, the presence of shear stresses causes a fluid to continually deform (or flow). So in a nutshell, rigid body motion involves a fluid that is accelerating but not flowing. We don't have to look far to find examples of this type of motion: liquid being transported in a tanker truck that is speeding up on the interstate, a thin layer of oil in the bed of a pickup truck slowing to a stop, etc.

Since the fluid is no longer static (motionless), we do not have a hydrostatic pressure distribution...which meant that pressure in a fluid varied with depth only. For this case we have pressure variation in at least two directions; consequently, lines of constant pressure are no longer horizontal!

Let's consider the example of a layer of oil in the back of a pickup truck that this slowing down (the level of the oil is exaggerated so we can clearly examine it):

To ensure that we have to proper orientation, note that the velocity (v) of the truck is in the -x-direction and the acceleration (a) (or deceleration) could have components in both the +x-direction (ax) and the +y-direction (ay).* The pressure at each point on the oil surface is equal to the pressure of the atmosphere (or zero gage pressure), so we know that the oil surface makes a line of constant pressure. Therefore, all other lines of constant pressure must be parallel to this line.

We can determine the slope of these lines of constant pressure from the following equation:

Slope = rise/run = dy/dx = -ax/(g+ay)

Here that g is the gravitational constant that always acts in the downward (here -y) direction, and dy or dx means a change in y or x, respectively. *From experience we know that in this case of the decelerating truck on a flat, horizontal road, the acceleration is actually in the +x-direction only, so the equation becomes dy/dx = -ax/g.

We can also determine how the pressure varies in the x- and y-directions by:

dp/dx = -rho(ax) and dp/dy = -rho(ay+g)

where rho is the density of the fluid, and dp is the change in pressure. Again since ay is zero for our truck case, dp/dy = -rho(g).

That's rigid body motion, folks!