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Fluid Statics

Objectives

To use the pressure-elevation relationship to find forces on submerged surfaces.

Fluid mechanics is broadly divided into two categories, fluid statics and fluid dynamics. Fluid statics, the subject of this section, is the branch of fluid mechanics that deals with the behavior of fluids at rest. In fluid statics, the fluid is at rest with respect to a frame of reference. This means that the fluid does not move with respect to a body or surface with which the fluid is in physical contact.Because the fluid is at rest, the fluid is in a state of equilibrium where the vector sum of the external forces acting on the fluid is zero. As a subject, fluid statics encompasses several areas for study including forces on submerged surfaces, pressure measurement and manometry, buoyancy, stability, and fluid masses subjected to acceleration. Our treatment of fluid statics will focus on the most fundamental of these subjects: forces on submerged surfaces.

1 Pressure-Elevation Relationship

Common experience tells us that the pressure increases with depth in a fluid. For example, a scuba diver experiences higher pressures as he descends below the water's surface. If we are to know how to analyze the effect of forces exerted on submerged surfaces, we must first understand how pressure changes with elevation (vertical distance) in a static fluid. To obtain a relationship between pressure and elevation in a static fluid, refer to the configuration shown in Figure 6. In Figure 6, we consider a static body of fluid with density, . Because the entire body of fluid is in equilibrium, every particle of fluid must therefore be in equilibrium. Thus, we can isolate an infinitesimally small fluid element for analysis. We choose as our fluid element a cylinder of height, dz, whose top and bottom surface area is A. Treating the fluid element as a free body in equilibrium, we observe that there are three external forces acting on the element in the z direction. Two of the forces are pressure forces acting on the top and bottom surfaces of the element. The pressure force acting on the top surface is PA, the product of the pressure at a given z coordinate and the surface area. The pressure force acting on the bottom surface is (P+dP)A, the product of the pressure at z+dz and the surface area. The pressure acting on the bottom surface is (P+dP) because the pressure has increased a differential amount corresponding to an elevation change of dz. Note that both pressure forces are compressive forces. (There are also pressure forces acting around the perimeter of the cylinder on its curved surface, but these forces cancel one another and are not a function of elevation.) The third force acting on the fluid element is the weight of the fluid element, W.

. Differential fluid element used to derive the pressure-elevation relation, =h.
Writing a force balance on the fluid element in the z direction, we obtain

The weight of the fluid element is

where the volume of the element is V=A dz. Substituting Equation 7-11 into Equation 7-10 and simplifying, we obtain

Equation 7-12 can now be integrated. Pressure is integrated from P₁ to P₂, and elevation is integrated from z₁ to z₂. Thus,

which yields

In many instances, P₁ is taken as the pressure at the origin, z=z₁=0. The pressure P₂ then becomes the pressure at a depth, z₂, below the free surface of the fluid. We are usually not concerned with the force exerted by atmospheric pressure, so the pressure, P₁, at the free surface of the fluid is zero; i.e., the gauge pressure at the free surface is zero, and P₂ is the gauge pressure at z₂. Equation 7-14 may be expressed in a simplified form by letting P=P₂P₁ and h=z₂ z₁. Noting that =g, Equation 7-14 reduces to

where is the specific weight of the fluid and h is the elevation change as referenced from the free surface. As h increases, pressure increases in accordance with our experience. We may draw some general conclusions from the relationship between pressure and elevation given by Equation 7-15:

Equation 7-15 is valid only for a homogenous static liquid. It does not apply to gases because is not constant for compressible fluids.
The change in pressure is directly proportional to the specific weight of the liquid.
Pressure varies linearly with depth, the specific weight of the liquid being the slope of the linear function.
Pressure increases with increasing depth and vice versa.
Points on the same horizontal plane have the same pressure.

Another important conclusion that may be drawn from Equation 7-15 is that, for a given liquid, the pressure change is a function of elevation change, h, only. Pressure is independent of any other geometrical parameter. The containers illustrated in Figure 7 are filled to a depth, h, with the same liquid, so the pressure at the bottom of these containers is the same. Each container has a different size and shape, and therefore contains different amounts of liquid, but the pressure is a function of depth only.

. For the same liquid, the pressures in these containers at a given depth, h, are equal, being independent of the shape or size of the container.
2 Forces on Submerged Surfaces

Now that the relationship between pressure and elevation in static liquids has been established, let us apply the relationship to the analysis of forces on submerged surfaces. We will examine two fundamental cases. The first case involves forces exerted by static liquids on horizontal submerged surfaces. The second case involves forces exerted by static liquids on partially submerged vertical surfaces. In both cases, we will restrict our analysis to plane surfaces.

In the first case, we find that the force exerted by a static liquid on a horizontal submerged surface is determined by a direct application of Equation 7-15 Consider a container with a plane horizontal surface filled with a liquid to a depth, h, as shown in Figure 8. The pressure at the bottom of the container is given by P=h. Because the bottom surface is horizontal, the pressure is uniform across the surface. The force exerted on the bottom surface is simply the product of the pressure and the surface area. Thus, the force exerted on a horizontal submerged surface is

where P=h and A is the surface area. Equation 7-16 is valid regardless of the shape of the horizontal surface. The force exerted on a horizontal submerged surface is equivalent to the weight, W, of the liquid above the surface. This fact is evident by writing Equation 7-16 as F=(hA)=V=W.

. The pressure is uniform on a horizontal submerged surface.
In the second case, we examine forces exerted on partially submerged vertical surfaces. One of the conclusions we gleaned from Equation 7-15 is that pressure varies linearly with depth in a static liquid. Consider the partially submerged vertical plane surface in Figure 9. The pressure (gauge pressure) is zero at the free surface of the liquid and increases linearly with depth. At a depth, h, below the free surface of the liquid, the gauge pressure is P=h. Because the pressure varies linearly from 0 to P over the range 0 to h, the average pressure, P_avg, is simply P/2. Thus,

The average pressure is a constant pressure that, when applied across the entire surface, is equivalent to the actual linearly varying pressure. Like pressure, the force exerted by the static liquid on the vertical surface increases linearly with depth. For purposes of structural design and analysis, we are generally interested in the total force or resultant force that acts on the vertical surface. The resultant force, F_R, is the product of the average pressure, P_avg, and the area, A, of the surface that is submerged. Hence,

The resultant force is a concentrated force (a force applied at a point) that is equivalent to the linear force distribution on the vertical surface. In order to make use of the resultant force, the point of application of F_R must be known. From principles of statics, it can be shown that for a linearly varying force distribution, the point of application of the equivalent resultant force is two-thirds the distance from the end with the zero force. Hence, the resultant force acts at a point 2h/3 from the free surface of the liquid or h/3 from the bottom of the vertical surface, as shown in Figure 9. The point at which the resultant force is applied is called the center of pressure. The resultant force, applied at the center of pressure, has the same structural effect on the surface as the actual linear force distribution. The reduction of a distributed force to a concentrated force simplifies the design and analysis of submerged surfaces such as dams, ship hulls, and storage tanks.

. Pressure variation and resultant force on a partially submerged vertical surface.

EXAMPLE

A small dam consists of a vertical plane wall with a height and width of 5 m and 30 m, respectively. The depth of the water (=9.81 kN/m³) is 4 m. Find the resultant force on the wall and the center of pressure.

Solution
Using Equation 7-18, and noting that only 4 m of the dam wall is submerged, the resultant force is

The center of pressure is located two-thirds from the free surface of the water. Thus, the center of pressure, which we denote by z_cp, is

Practice!

A barrel of motor oil (=8.61 kN/m³) is filled to a depth of 1.15 m. Neglecting atmospheric pressure, what is the pressure on the bottom of the barrel? If the radius of the barrel's bottom is 20 cm, what is the force exerted by the motor oil on the bottom?
The bottom portion of the hull of a barge is submerged 12 ft in seawater (=64.2 lb_f/ft³). The hull is horizontal and measures 30 ft×70 ft. Find the total force exerted by the seawater on the hull.
The gauge pressure at the bottom of a tank containing ethyl alcohol (=7.87 kN/m³) is 11 kPa. What is the depth of the alcohol?
A vertical gate in an irrigation canal holds back 2.2 m of water. Find the total force on the gate if its width is 3.6 m.
A simple dam is constructed by erecting a vertical concrete wall whose base is secured firmly to the ground. The width of the wall is 16 m, and 5 m of the wall is submerged in water. Find the moment of force about the base of the wall. (Hint: The moment of force is the product of the resultant force and the perpendicular distance from the center of pressure to the base of the wall.)