Rigid Body Motion

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[1]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.

Contents

Textbook Readings

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

Equation of Linear Motion

[2]The general equation of motion of a fluid is given by:

GneralEquationofMotion.png
Where
DelP.png

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

GeneralEquationofMotion1.png

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

GeneralEquationofMotion2.1.png

Where ax, ay and az 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

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

GeneralEquationofMotion-Rest.png

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

[4]When a fluid is falling at a rate equal to the acceleration due to gravity (i.e. az = -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

GeneralEquationofMotion-FreeFall.png

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

[5]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. ax and az while ay=0 since the motion is only in one direction. The equations therefore reduce to:

GeneralEquationofMotion-LinearAcceleration.png

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:

GeneralEquationofMotion-LinearAcceleration1.png

and by substitution:

GeneralEquationofMotion-LinearAcceleration2.png

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

GeneralEquationofMotion-LinearAcceleration3.png

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

VerticalSurfaceRise.png

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

IsobarSlope.png


References

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

  1. Textbook p.102
  2. Textbook p.103
  3. Textbook p.104
  4. Textbook p.104
  5. Textbook p.105
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