UNIVERSITY OF GUYANA FACULTY OF NATURAL SCIENCES DEPART. OF MATH, PHYS & STATS PHY 110 – PHYSICS FOR ENGINEERS LECTURE 14 (THURSDAY, DECEMBER 8, 2011) 1

Lecture Notes: For this information, visit my website: In the event of any other issues to be resolved, 2

3.8 Variation of Pressure with Fluid Depth 3 Pressure in a Fluid: Given that the density and acceleration due to gravity are constant, A Level Physics by Stephen Pople, pg 68.

3.8 Variation of Pressure with Fluid Depth 4 Upthrust and Archimedes’ Principle: This principle states that for an object wholly or partially immersed in a fluid, the upthrust it experiences is equal the weight of the fluid it displaces. This upthrust is responsible for the apparent loss of weight of the object when submerged in a fluid. Consider an object completely submerged in a fluid of density ρ.

3.8 Variation of Pressure with Fluid Depth 5 A Level Physics by Stephen Pople, pg 69.

3.8 Variation of Pressure with Fluid Depth 6 A Level Physics by Stephen Pople, pg 69.

3.8 Variation of Pressure with Fluid Depth 7 Principle of Floatation: This principle states that a floating object displaces its own weight. In other words, the upthrust U acting on a partially immersed object must be equal to the weight W of the object.

3.8 Variation of Pressure with Fluid Depth 8 Principle of Floatation Cont’d: A Level Physics by Stephen Pople, pg 69.

3.9 Hydrodynamics. Viscosity 9 This is the study of fluids in motion. Definitions: 1. A flowline is a line along which a fluid element moves. 2. A streamline is a stable flowline, along which every element follows the same path. 3. Laminar flow (streamlined) is a stable flow of fluid elements i.e. stable flowlines established. 4. Turbulent flow is the chaotic (disorderly) flow of fluid elements i.e. flowlines do not exist. A Level Physics by J. Breithaupt and K. Dunn, pg 143.

3.9 Hydrodynamics. Viscosity 10 A Level Physics by Stephen Pople, pg 102.

3.9 Hydrodynamics. Viscosity 11 Viscosity: This is fluid friction that exists between layers of a fluid moving at different speeds. It is defined as the tangential stress in a fluid required to produce unit velocity gradient across that fluid. A Level Physics by J. Breithaupt and K. Dunn, pg 145.

3.9 Hydrodynamics. Viscosity 12 Viscosity: Units: Pascal - second (Pa s) 1 Pa s = 1Nsm -2 =1 kg m -1 s -1 A Level Physics by J. Breithaupt and K. Dunn, pg 145.

3.10 Critical Velocity 13 Critical Velocity: The flow of a fluid through a pipe is only streamline below a certain critical value. Above this value, the flow becomes turbulent. A Level Physics by Stephen Pople, pg 102.

3.10 Critical Velocity 14 Re - Reynold’s Number. ρ - Density of fluid. l - Characteristic length. For Laminar Flow: For Turbulent Flow: A Level Physics by Stephen Pople, pg 102.

3.10 Critical Velocity 15 Liquid Flow Through a Pipe: The viscosity of a liquid affects how it can flow through a pipe. Thus for viscous flow through a horizontal uniform pipe, assuming that it is streamlined, the rate of flow is: A Level Physics by J. Breithaupt and K. Dunn, pg 145.

3.10 Critical Velocity 16 Liquid Flow Through a Pipe: V – Volume of fluid t – Time r - Radius of pipe l - Length of pipe. A Level Physics by Stephen Pople, pg 103.

3.10 Critical Velocity 17 Stokes’ Law: When an object passes through a viscous fluid, the object experiences a frictional drag. For a uniform sphere moving at a constant velocity through this fluid, the drag force is: A Level Physics by J. Breithaupt and K. Dunn, pg 145.

3.10 Critical Velocity 18 Terminal Velocity: A falling sphere will reach its terminal velocity when the forces are balanced. But A Level Physics by Stephen Pople, pg 103.

3.10 Critical Velocity 19 Terminal Velocity Cont’d: But Rearranging gives

3.11 Bernoulli’s Theorem 20 Equation of Continuity: Fluid flow along a stream-tube will be fastest where the tube is narrowest. If the fluid is incompressible (density is constant), then the mass flow of the fluid per unit time along any segment of the tube is constant. This is the equation of continuity. A Level Physics by J. Breithaupt and K. Dunn, pg 144.

3.11 Bernoulli’s Theorem 21 Equation of Continuity: Given that the fluid is incompressible, A – cross-sectional area. v – velocity of fluid. A Level Physics by Stephen Pople, pg 103.

3.11 Bernoulli’s Theorem 22 Bernoulli’s Equation: For streamline motion of an incompressible non- viscous fluid, Bernoulli stated, “the sum of the pressure at any part plus the kinetic energy per unit volume plus the potential energy per unit volume there is always a constant.” This is the principle of the conservation of energy expressed for a fluid element. A Level Physics by Tom Duncan, pg.

3.11 Bernoulli’s Theorem 23 Bernoulli’s Equation : Three terms are: 1. Work done per unit volume. 2. Kinetic energy per unit volume. 3. Gravitational Potential energy. A Level Physics by J. Breithaupt and K. Dunn, pg 144.

3.12 Applications in Aerofoil 24 A Level Physics by Stephen Pople, pg 103. Aerofoil:

3.12 Applications in Aerofoil 25 A Level Physics by Stephen Pople, pg 103. Venturi meter:

3.12 Applications in Aerofoil 26 A Level Physics by Stephen Pople, pg 103. Spinning Ball:

3.13 Surface Tension. Capillarity. 27 Surface Tension: This is the ability of a liquid to wet a surface. This is due to the intermolecular attractions between the molecules of the liquid (Cohesive forces) as well as the attractions between the molecules of the surface and the liquid (Adhesive forces). Surface tension is responsible for a number of phenomena. For example, why does water wet glass but mercury does not? Why droplets of water are spherical? How does the water skater walk on water?

3.13 Surface Tension. Capillarity. 28 Surface Tension Cont’d It is defined as the surface tension force per unit length acting along the surface, perpendicular to a line in the surface. Units: 1 Nm -1 = 1 kg s -2 A Level Physics by J. Breithaupt and K. Dunn, pg 146.

3.13 Surface Tension. Capillarity. 29 Capillarity: This is the rise of water up the inside of a narrow vertical capillary tube with its lower end immersed in a beaker of water. This due to surface tension forces (adhesion) causing water to adhere to the glass. For mercury, it is a depression due to strong cohesive forces. Angle of Contact θ: This is the angle (in the liquid) that a liquid makes with a solid surface at the point of contact. A Level Physics by J. Breithaupt and K. Dunn, pg 147.

3.13 Surface Tension. Capillarity. 30 Capillary Rise: e.g. Water on Glass. Capillary Fall: e.g. Mercury on glass. A Level Physics by J. Breithaupt and K. Dunn, pg 147.

3.14 Molecular Forces. Implications. 31 Bubbles: The internal pressure of a bubble must be greater than the external pressure, so that the pressure difference is sufficient to withstand the surface tension forces that would burst the bubble. NB: For soap bubbles, it is doubled. A Level Physics by J. Breithaupt and K. Dunn, pg 147

Lecture Notes: For this information, visit my website: In the event of any other issues to be resolved, 32

33 END OF LECTURES.