Chapter 3: Fluid Mechanics 3.1: Introduction • Fluid: refers to both liquids and gases • Density: mass per unit volume, and it’s typically denoted by the letter ρ o Note: this equation immediately implies that m is equal to ρV when ρ is a constant. ➢ Example: if 10−3 m3 of oil has a mass of 0.8 kg, then the density of this oil is ➢ Solution: • Pressure: force that is distributed over any small area of the object’s surface o SI unit: 1 pascal = 1 Pa = ➢ Example: A vertical column made of cement has a base area of 0.5 m2 . If its height is 2m, and the density of cement is 3000 kg/m3, how much pressure does this column exert on the ground? ➢ Solution: • Hydrostatic Pressure: liquid is at rest ➢ o Note: depends only on the density of the liquid and the depth below the surface 3.2: Buoyancy • Buoyant Force: net upward force, denoted Fbuoy ➢ ➢ Example: An object with a mass of 150 kg and a volume of 0.75 m3 is floating in ethyl alcohol, whose density is 800 kg/m3 . What fraction of the object’s volume is above the surface of the fluid? ➢ Solution: The density of the object is The ratio of the object’s density to the fluid’s density is This means that 1/4 of the object’s volume is below the surface of the fluid; therefore, the fraction above the surface is 1 − (1/4) = 3/4. 3.2.1: Buoyancy Diagrams Diagram of an object that is floating in a liquid Diagram of an object with a density greater than the fluid density as it sinks to the bottom of the tank 3.3: Flow Rate and The Continuity Equation • Volume Flow Rate: f, is the volume of fluid that passes a particular point per unit time. o In SI units expressed in m3/s o • Continuity Equation: the flow rate through a pipe (area times velocity) is constant ➢ Example: A circular pipe of non-uniform diameter carries water. At one point in the pipe, the radius is 2 cm and the flow speed is 6m/s. (a) What is the volume flow rate? (b) What is the flow speed at a point where the pipe constricts to a radius of 1 cm? ➢ Solution: (a) At any point, the volume flow rate, f, is equal to the cross-sectional area of the pipe multiplied by the flow speed. (b) By the Continuity Equation, we know that the volume flow rate must be the same at all points, so the volume flow rate must be the same as in part (a). 3.4: Bernoulli’s Equation • The most important equation in fluid mechanics which is the statement of conservation of energy for ideal fluid flow. • Three conditions that make fluid flow ideal: o The fluid is incompressible. o The fluid’s viscosity is negligible. o The flow is streamline. • Applied to any pair of points along a streamline within the flow o Let ρ be the density of the fluid that’s flowing. o Label the points we want to compare as Point 1 and Point 2. o Choose a horizontal reference level, and let y1 and y2 be the heights of these points above this level. 3.5: Torricelli’s Theorem • Bernoulli’s Equation becomes ➢ ➢ ➢ Example: The side of an aboveground pool is punctured, and water gushes out through the hole. If the total depth of the pool is 2.5 m, and the puncture is 1 m above ground level, what is the efflux speed of the water? ➢ Solution: Torricelli’s Theorem is v = , where h is the distance from the surface of the pool down to the hole. If the puncture is 1 m above ground level, then it’s 2.5 − 1 = 1.5 m below the surface of the water (because the pool is 2.5 m deep). Therefore, the efflux speed will be 3.6: The Bernoulli Effect • At comparable heights, the pressure is lower where the flow speed is greater. Illustration of Bernoulli (or Venturi) Effect o Air flow: Illustration of Air flow