Fluid Statics and Its Applications

NATURE OF FLUIDS. A fluid is a substance that does not permanently resist distortion. An attempt to change the shape of a mass of fluid results in layers of fluid sliding over one another until a new shape is attained. During the change in shape, shear stresses exist, the magnitudes of which depend upon the viscosity of the fluid and the rate of sliding, but when a final shape has been reached, all shear stresses will have disappeared, A fluid in equilibrium is free from shear stresses. 

PRESSURE CONCEPT. The basic property of a static fiuid is pressure. Pressure is familiar as a surface force exerted by a fluid against the walls of its container. Pressure also exists at every point within a volume of fluid. A fundamental question is What kind of quantity is pressure Is pressure independent of direction, or does it vary with direction For a static fluid, as shown by the following analysis, pressure turns out to be independent of the orientation of any internal surface on which the pressure is assumed to act. 

The area of face COB is Ax Ay/2. Let the average pressure on this face be Pj. Then the upward force on the face is p. Ax Ay/2. Let p be the average pressure on face ABC. The area of this face is Ax Ay/ 2 cos 0), and the total force on the face is p Ax Ay (2 cos 6). The angle between the force vector of pressure p with the z axis is also 6, so the vertical component of this force acting downward is 

The volume of the tetrahedron is Ax Ay Az/6. If the fluid density is p, the force of gravity acting on the fluid in the tetrahedron is p Ax Ay Az p/6p ,. This component acts downward. The force balance in the z direction becomes 

keeping angle 0 constant, let plane ABC move toward the origin 0. As the distance between ABC and 0 approaches zero as a limit, Az approaches zero also and the gravity term vanishes. Also, the average pressures p and p, approach p and p, the loeal pressures at point 0, and Eq. (2.1) shows that p becomes equal to p. 

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