Rigid Body Motion

In many situations fluids act as if they are rigid bodies, that is the shape does not change (and there is no deformation). Such situations occur when fluids are at both at rest and under acceleration. When a fluid undergoes acceleration, it initially gets pushed backwards creating a splash but after a while a new free surface is formed where each particle assumes the same acceleration. Once this occurs, the fluid is said to act as a rigid body.

Textbook Readings

Cengel and Cimbala, Fluid Mechanics: Fundamentals and Applications, (2nd ed, Singapore, McGraw Hill Education, 2010), pp. 102 - 106.

Equation of Linear Motion

The general equation of motion of a fluid is given by:

Where

This equation can be resolved into the three principle components as:

In scalar form in the three orthogonal directions, the equation can be split and written as:

Where a_x, a_y and a_z are the accelerations in the x-, y- and z- directions. It should be noted that in the analysis, the z- direction is up rather than y-.

Application

Fluids at Rest

When fluids are at rest, or at constant velocity, all acceleration components become zero, yielding:

Which confirms that pressure in a fluid is related to the vertical height, and that at equal heights, the pressure is constant.

Fluids at Free Fall

When a fluid is falling at a rate equal to the acceleration due to gravity (i.e. a_z = -g), then the accelerations in the x- and y- directions are zero, as well as the acceleration in the horizontal direction. In such a case the fluid is said to be in "free fall" and the pressure is said to be constant, since

implies no change in pressure.

Fluids Accelerating in a Straight Line

Fluid undergoing linear acceleration. Picture taken from Cengel and Cimbala, Fluid Mechanics: Fundamentals and Applications, (2nd ed, Singapore, McGraw Hill Education, 2010), p. 105

Fluids accelerating in a straight line tend to get pushed backwards, causing the fluid to increase in height towards the back. This gives acceleration in both a horisontal (taken as x- direction) and the vertical direction (z- direction), i.e. a_x and a_z while a_y=0 since the motion is only in one direction. The equations therefore reduce to:

Implying that pressure does not vary with the y- direction.

Then the total differential of P is a function of x and z, that is:

and by substitution:

For constant density (ρ = constant), the pressure between two points 1 and 2 is given by:

By choosing P₁ and P₂ both on the surface, thats is P₁=P₂, the vertical surface rise is given by:

Since Δz and Δx are two sides of a right angle triangle, the slope of the surface of the liquid (also the slope of the isobars) is given by

References

"Textbook" refers to Cengel and Cimbala, Fluid Mechanics: Fundamentals and Applications, (2nd ed, Singapore, McGraw Hill Education, 2010).