Fluids Chapter 11

Why Does Something Float?

11.1 Mass Density In this chapter, we discuss the behavior of fluids. Fluids by definition, include both gases and liquids Mass density is the mass per unit volume of a liquid or gas Greek letter rho (ρ) is the symbol. kg/m3 is the unit 𝜌= 𝑚 𝑉

Remember Chemistry The relative closeness of particles of solids and liquids lead to greater densities than gases. In chemistry you learned that changes in temperature, volume, and pressure can change the density of a substance. That is TRUE! However, for the range of temperatures and pressures used in this course, the densities of solids and liquids will not differ much from those given on page 322, Table 11.1.

Mass vs Weight Fg=mg Mass, not weight, is part of the definition of mass density. When weight is needed, you must solve using mass density, volume, and g.

Specific Gravity Mass density can be used to help determine pressure. Specific Gravity can be used to compare the densities of substances Specific Gravity is density of a substance divided by the density of a standard reference material, usually water at 4⁰C. 𝑺𝒑𝒆𝒄𝒊𝒇𝒊𝒄 𝒈𝒓𝒂𝒗𝒊𝒕𝒚= 𝒅𝒆𝒏𝒔𝒊𝒕𝒚 𝒐𝒇 𝒔𝒖𝒃𝒔𝒕𝒂𝒏𝒄𝒆 𝒅𝒆𝒏𝒔𝒊𝒕𝒚 𝒐𝒇 𝑯 𝟐 𝑶 𝒂𝒕 𝟒°𝑪 Since specific gravity is a ratio, no unit is involved!

11.2 Pressure Pressure exerted by gases and liquids comes from collisions between molecules and container walls. Pressure, technically, is defined as magnitude F of the force acting perpendicular to a surface divided by the area over which the force acts. 𝑷= 𝑭 𝑨

Units of Pressure and Conversions SI unit: N/m2 referred to as a Pascal 105 Pa = one bar Pounds per square inch (psi) 1.010 x 105 Pa 1 atm 14.70 lbs/in2 760 mmHg Atmospheric Pressure Force is a vector, pressure is NOT! The FORCE generated by the pressure of a static fluid is always perpendicular to the surface that the fluid contacts.

11.3 Pressure and Depth in a Static Fluid As a diver descends to deeper water, more pressure will be exerted on his or her body. Since more water is supported by the same horizontal area of water at greater depths (namely the weight of the water above it) there is more force per unit area or pressure. Check the derivation on pg. 325 of your text. 𝑃 2 = 𝑃 1 +𝜌𝑔ℎ P1 is the “higher level” Equation assumes an incompressible fluid (reasonable for liquids) With gases, this equation can only be used for values of h where differences in ρ are negligible. Horizontal distances in fluids DO NOT affect pressure.

Click on the professor for a video lesson recap and then some! Time for a Dork Break! Click on the professor for a video lesson recap and then some!

More Pressure and Depth Problem Solving Insight: The pressure at any point in a fluid depends on the vertical distance (h) of the point beneath the surface. However, for a given vertical distance, the pressure is the same, no matter where the point is located horizontally on the fluid. The pressure on the surface of water at sea level is the atmospheric pressure of the air above it. Try this self-assessment test!

11.4 Pressure Gauges Mercury Barometer measures atmospheric pressure with respect to the height of mercury in an *evacuated column. Patm = 0 Pa + ρgh Open – tube manometer contains one open side of a U-tube. P2 – Patm = ρgh Changes in air pressure cause changes in the height of the mercury in both cases. The gauge pressure is the amount by which the container pressure differs from atmospheric pressure. Gauge pressure is proportional to height. The actual value for P2 is the absolute pressure.

Learn more by clicking the picture! SPHYGMOMANOMETER This word deserves it’s own title! Fancy word for device used to measure blood pressure. Learn more by clicking the picture!

11.5 Pascal’s Principle Deals with completely enclosed fluids and the pressure resulting from external forces. Any change in the pressure applied to a completely enclosed fluid is transmitted undiminished to all parts of the fluid and the enclosing walls. 𝐹 2 𝐴 2 = 𝐹 1 𝐹 2 because the pressure is equal This relationship only works when points 1 and 2 lie at the same depth (h = 0m) in the fluid.

Hydraulic Lifts Use Pascal’s Principle to apply small forces in order to move objects with large weight. The same amount of work is done by both the input and output forces (in this magical world of no friction!) The larger output force moves through a smaller distance while the smaller input force moves through a larger distance. http://www.explainthatstuff.com/hydraulics.html http://www.wfu.edu/physics/demolabs/demos/avimov/bychptr/chptr4_matter.htm#FluidPressure

11.6 Archimedes’ Principle The upward force supplied by a fluid is called the buoyant force. All fluids apply such a force to objects immersed in them (similar to normal force from solids). Exists because fluid pressure is larger at greater depths. Not a new type of force; simply the name given to the net upward force exerted by the fluid on the object. 𝐹 𝐵 =𝜌𝑔ℎ𝐴=𝜌ℎ𝑉 Any fluid applies a buoyant force to an object that is partially or completely immersed in it; the magnitude of the buoyant force equals the weight of the fluid that the object displaces.

What floats? Compare the weight of a floating object to the maximum possible buoyant force. Since the Fg =ρVg, and V and g are the same for both the floating object and the water, the comparison depends ONLY ON DENSITY. Any object that is solid throughout will float in a liquid of the density of the object is less than or equal to the density of the liquid.

Large Boats Floating is Really Cool Boats float because it is NOT a solid material. Large, steel boats contain A LOT of “empty space” and because of their shape, displace enough water to balance their own weight. Watch this! http://science.howstuffworks.com/science-vs-myth/everyday-myths/question254.htm Try this problem. http://bcs.wiley.com/he-bcs/Books?action=resource&bcsId=6796&itemId=0470879521&resourceId=26562&chapterId=73116

11.7 Fluids in Motion Steady Flow Unsteady Flow The velocity of the fluid particles at any point is constant as time passes. Within a stream, all particles passing through one point will that the same velocity. Other particles at other points may have a different velocity. The velocity at a point in the fluid changes as time passes. Turbulent flow occurs when the velocity at a point changes erratically from moment to moment, both in magnitude and direction.

Compressible vs Incompressible Gases are highly compressible. Most gases will change density very easily when they are subject to changes in pressure. Most liquids are nearly incompressible. Density remains almost constant as pressure changes. Liquids flow in an incompressible manner.

Viscosity High Viscosity No Viscosity Honey does not flow very readily Flow of a viscous fluid is said to be energy dissipating. Viscosity of one layer hinders the flow of nearby layers. Ideal fluids are incompressible, nonviscous fluids. Flow in an unhindered manner. No dissipation of energy.

Streamlines Used to represent the trajectories of the fluid particles. A line (like a vector) drawn in a fluid such that a tangent to the streamline at any point is parallel to the fluid velocity. Remember: velocity is a vector quantity. While fluid velocity can vary from point to point, the velocity is constant in time. Steady flow is often referred to as streamline flow. Liquids use colored dye. Gases use smoke streamers.

11.8 The Equation of Continuity If a fluid enters one end of a pipe at a certain rate, then the fluid must also leave at the same rate, assuming that there are no places between the entry and exit point to add or remove fluid. Mass flow rate (ρAv) is the mass of fluid per second that flows through a tube. SI unit kg/s. 𝜌 1 𝐴 1 𝑣 1 = 𝜌 2 𝐴 2 𝑣 2 ρ = fluid density (kg/m3) A = cross-sectional area of tube (m2) v = fluid speed (m/s)

Volume Flow Rate Since the density of a flowing fluid doesn’t change, it can be eliminated from previous equation. For an incompressible fluid 𝐴 1 𝑣 1 = 𝐴 2 𝑣 2 Av represents the volume flow rate (Q) passing through the tube at any second. Be careful: if the fluid density changes due to compression, you must use previous equation!

11.9 Bernoulli’s Equation The work-energy theorem leads us to Bernoulli’s equation (helps us relate concepts and derive equation) Wnc = ΔE is the work energy -theorem. Conditions necessary for equation to work: steady flow, incompressible, nonviscous fluid. Pressure on a fluid is caused by collision forces which are nonconservative. When a fluid is accelerated because of a difference in pressures, work is being done by nonconservative forces. Total mechanical energy is NOT conserved.

Bernoulli’s Observations of Moving FluΔids When a fluid is flowing in a horizontal pipe and meets a region of reduced cross-sectional area, the pressure drops. When moving from wider region to narrower region, the fluid speeds up (accelerates) consistent with equation of continuity. According to 2nd Law of Motion, if something accelerates, an unbalanced force exist. (ΔP) Also, if fluid moves to higher elevation, P2 > P1

Bernoulli’s Equation In the steady flow of a nonviscous, incompressible fluid of density ρ, the pressure P, the fluid speed v, and the elevation y at any two points (1 and 2) are related by 𝑃 1 + 1 2 𝜌 𝑣 1 2 +𝜌𝑔 𝑦 1 = 𝑃 2 + 1 2 𝜌 𝑣 2 2 +𝜌𝑔 𝑦 2 For a static fluid, this equation reduces when the speed of the fluid is uniform throughout. (A is constant) 𝑃 2 = 𝑃 1 +𝜌𝑔 𝑦 1 − 𝑦 2 = 𝑃 1 +𝜌𝑔ℎ

11.10 Applications of Bernoulli’s Equation If a moving fluid is in a horizontal pipe, all parts have the same elevation, so simplify Bernoulli’s equation to eliminate ρgy. DUH! Watch this video. It is basically a review of the nature of fluids in motion. Bernoulli’s equation is summarized here. Applications, including airplane flight are included. Physics of a curveball.