Dr. Kamel Mohamed Guedri Umm Al-Qura University, Room H1091 ME804306-2 Fluid Mechanics Chapter 2 Fluid Statics Dr. Kamel Mohamed Guedri Mechanical Engineering Department, The College of Engineering and Islamic Architecture, Umm Al-Qura University, Room H1091 Website: https://uqu.edu.sa/kmguedri Email: kmguedri@uq.edu.sa

Objectives Determine the variation of pressure in a fluid at rest Calculate pressure using various kinds of manometers Calculate the forces exerted by a fluid at rest on plane or curved submerged surfaces. Analyze the stability of floating and submerged bodies. Analyze the rigid-body motion of fluids in containers during linear acceleration or rotation.

OUTLINE 1. Absolute and gauge pressure 2. Basic Equation of Fluid Statics 3. Pressure - Depth Relationships a) Constant density fluids b) Ideal gases 4. Pressure measurement devices a) The barometre b) The manometers b) Other pressure measurement devices 5. Hydrostatic forces on submerged plane surfaces 6. Hydrostatic forces on submerged curved surfaces 7. Buoyancy and stability 8. Fluid in rigid-body in motion

2.1 Absolute and Gauge Pressure Pressure measurements are generally indicated as being either absolute or gauge pressure. Gauge pressure is the pressure measured above or below the atmospheric pressure (i.e. taking the atmospheric as datum). can be positive or negative. A negative gauge pressure is also known as vacuum pressure. Absolute pressure uses absolute zero, which is the lowest possible pressure. Therefore, an absolute pressure will always be positive. A simple equation relating the two pressure measuring system can be written as: Pabs = Pgauge + Patm (2.2)

Atmospheric pressure refers to the prevailing pressure in the air around us. It varies somewhat with changing weather conditions, and it decreases with increasing altitude. At sea level, average atmospheric pressure is 101.3 kPa (abs), 14.7 psi (abs), or 1 atmosphere (1 bar = 1x105 Pa). This is commonly referred to as ‘standard atmospheric pressure’.

Example 2.1 Express a pressure of 155 kPa (gauge) as an absolute pressure. Express a pressure of –31 kPa (gauge) as an absolute pressure. The local atmospheric pressure is 101 kPa (abs). Solution: Pabs = Pgauge + Patm Pabs = 155 + 101 = 256 kPa Pabs = -31 + 101 = 70 kPa

2.2 Basic Equation of Fluid Statics The pressure at a point in a static fluid is the same in all directions. What this means that the pressure on the small cube in Figure 2.1 is the numerically the same on each face as the cube shrinks to zero volume. At the surface, the pressure is zero gage. Since depth increases in the downward z-direction, the sign on the specific weight in equation 2.1 is negative.

2.3 Pressure - Depth Relationships 2.3.1 Constant Density Fluids To establish the relationship between pressure and depth for a constant density fluid, the pressure - position relationship must be separated and integrated Equation 2.3 is one of the most important and useful equations in fluid statics.

Example 2.2 Figure below shows a tank with one side open to the atmosphere and the other side sealed with air above the oil (SG=0.90). Calculate the gauge pressure at points A,B,C,D,E.

Solution: At point A, the oil is exposed to the atmosphere Thus PA=Patm = 0 (gauge) Point B is 3 m below point A, Thus PB = PA + oilgh = 0 + 0.9x1000x9.81x3 = 26.5 kPa (gauge) Point C is 5 m below point A, Thus PC = PA + oilgh = 0 + 0.9x1000x9.81x5 = 44.15 kPa (gauge) Point D is at the same level of point B, Thus PD = PB Point E is higher by 1 m from point A, Thus PE = PA - oilgh = 0 - 0.9x1000x9.81x1 = -8.83 kPa (gauge).

2.3.2 Variable Density Fluids Most fluids are relatively incompressible, but gases are not. This means that the density increases with depth so we cannot use the specific weight as a constant in determining the pressure. If we assume an ideal gas, then the density is given by where M is the molecular weight, T is the absolute temperature, and R is the gas constant in appropriate units. Replacing in the differential equation (2.1) we get

2.4. Pressure measurement devices

Example 2.3

Example 2.4

Example 2.5

2.5. Hydrostatic forces on submerged plane surfaces

Example 2.6

2.6. Hydrostatic forces on submerged curved surfaces

Example 2.7 below. 2.7

2.7. Buoyancy and stability

Example 2.8

2.8. Fluid in rigid-body motion

Example 2.9