## The Pressure Force

A fluid, either a gas or a liquid, is composed of molecules in motion. The collective motion of the molecules gives rise to the flow of the fluid—the winds of the atmosphere and the currents of the ocean—even when there is no such organized flow, the molecules are moving, but their motion is more random. In a gas at room temperature, the molecules are moving at typical speeds of 450-500 m s-1. These molecules naturally collide with each other (in fact, a typical distance a molecule travels before colliding with another is only about 7 X 10-8 m, and this distance is covered in about 0.14 nanoseconds) and with the walls of any enclosing container, and these collisions are the origin of the pressure force in a gas. In a liquid, the molecular motion is not nearly so random, but nevertheless there is a similar pressure force. The pressure force acts not only on the walls of the container but also on the fluid itself, and this force may cause the fluid to move.

If the pressure in a fluid is uniform everywhere, then there is no net force. To see this, imagine a small, neutrally buoyant slab floating within the fluid, as illustrated in figure 3.3. There is a pressure force on the left side pushing the slab to the right and a pressure force on the right side pushing it to the left. If the pressure is the same everywhere (that is, Sp = 0 in the figure), these two forces cancel out and the body remains stationary, but if there is a pressure gradient (Sp ^ 0), then there is a net force on the body and it will move. If we imagine removing the body and replacing it with the fluid itself (so that we may think of the new piece of fluid as floating in the rest of the fluid) then, again, if there is a pressure gradient, there is a net force on the fluid and, in the absence of opposing forces, the fluid accelerates.

Figure 3.3. A slab (dark shading) floating within a fluid, with x and z the horizontal and vertical directions, respectively. The force to the right is just the difference of the pressure forces between the right and left surfaces of the slab, and so proportional to Sp. Thus, the net force is proportional to the pressure gradient within the fluid.

Figure 3.3. A slab (dark shading) floating within a fluid, with x and z the horizontal and vertical directions, respectively. The force to the right is just the difference of the pressure forces between the right and left surfaces of the slab, and so proportional to Sp. Thus, the net force is proportional to the pressure gradient within the fluid.

We may express the above arguments in mathematical form. Suppose now that the slab in figure 3.3 is just the fluid itself, with a cross-sectional area of A and a thickness of Sx. The total pressure force on the left surface of the slab is the pressure times the area, equal to pA, and this force pushes the slab to the right (the positive x direction). The pressure on the right surface is equal to

(p + 8p), so the force pushing the slab to the left is equal to (p + 8p)A. Thus, the net force pushing the slab to the right is equal to —SpA. The volume of the slab is just A8x, so that the net pressure force per unit volume is just —SpA/ASx = —Sp/Sx. That is, if we denote the pressure gradient force per unit volume in the x direction as Fx then we have

where we have used the notation of a partial derivative (dp/dx) because pressure may be changing in other directions too. Thus, in words, the pressure gradient force per unit volume in a particular direction is equal to the rate of change of pressure with respect to distance in that same direction. Why is there a minus sign in equation 3.5? It is because the pressure force is directed from high pressure to low pressure. Thus, if the pressure is increasing in the x direction (i.e., to the right), so that dp/dx is positive, then the pressure force is directed in the negative x direction, and to the left.

In the vertical direction, precisely the same considerations apply, so that the pressure gradient force in the vertical direction is given by

where z measures distance upward. If the pressure decreases upward (as it does), then dp/dz is negative, and the pressure force Fz is postive and directed upward.

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