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Chapter - 11 Fluids

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**Fluids 11.1 Mass Density 11.2 Pressure**

11.3 Pressure and Depth in a Static Fluid 11.4 Pressure Guages 11.5 Pascal’e Principle 11.6 Archimedes’ Principle 11.7 Fluid in Motion 11.8 The Equation of Continuity 11.9 Bernoulli’s Equation Applications of Bernoulli’s Equation Viscous Flow

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**Definition of Mass Density **

The mass density of a substance is the mass of a substance per unit volume: SI Unit of Mass Density: kg / m3

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11.1 Mass Density

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**Example-1: Blood as a Fraction of Body Weight **

11.1 Mass Density Example-1: Blood as a Fraction of Body Weight The body of a man whose weight is 690 N contains about 5.2x10-3 m3 of blood. (a) Find the blood’s weight and (b) express it as a percentage of the body weight.

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11.1 Mass Density (a) (b)

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**Definition of Weight Density **

11.1 Mass Density Definition of Weight Density The Weight density of a substance is the Weight of a substance per unit volume: SI Unit of Weight Density: N / m3

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**Specific Gravity Is relative density.**

It is a ratio of the densities of two materials. Dimensionless. Using water as a reference can be convenient, since density of water is (approximately) 1000 kg/m³ or 1 g/cm³ at 4C. Specific gravity of Mercury is 13.6, means it is 13.6 times massive or heavier than water.

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**SI Unit of Pressure: 1 N/m2 = 1Pa**

Pascal

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**Pressure Pressure is force per unit area. P = F/A**

Pressure is not a vector quantity. To calculate the pressure we only consider the magnitude of the force. The force generated by the pressure of the static fluid is always perpendicular to the surface of the contact. Pa = N/m2 = x 10-4 lb/in2 (very small quantity) bar = 105 Pa = 100 kPa Atmospheric pressure is kPa = bar = lb/in2 = 760 torr lb/in2 or psi = x 103 Pa

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**Pressure causes Perpendicular Force**

The forces of a liquid pressing against a surface add up to a net force that is perpendicular to the surface. Components parallel to the surface cancels out Components vertical to the surface adds up

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**Example-2: The Force on a Swimmer **

11.2 Pressure Example-2: The Force on a Swimmer Suppose the pressure acting on the back of a swimmer’s hand is 1.2x105 Pa. The surface area of the back of the hand is 8.4x10-3 m2. Determine the magnitude of the force that acts on it. (b) Discuss the direction of the force.

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**Since the water pushes perpendicularly **

11.2 Pressure Since the water pushes perpendicularly against the back of the hand, the force is directed downward in the drawing.

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**Shape of the Dam Dam-B Dam-A Would you prefer Dam-A or Dam-B?**

Justify your answer. Dam-B Dam-A

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**Atmospheric Pressure at Sea Level: 1.013x105 Pa = 1 atmosphere**

Weight of the column of the air of 1m2 = x105 N Mass of the column of the air of 1m2 = x105 N/g = kg

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**11.3 Pressure and Depth in a Static Fluid**

Relation between Pressure and Depth Since the column is in equilibrium, The sum of the vertical forces equal to zero

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**11.3 Pressure and Depth in a Static Fluid**

The pressure increment = gh = weight density x height Pressure at a depth, h = P1 + gh = Atmospheric Pressure + Liquid Pressure So, Liquid Pressure at a depth, h = gh = weight density x depth

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Pressure Water pressure acts perpendicular to the sides of a container, and increases with increasing depth.

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**11.3 Pressure and Depth in a Static Fluid**

Conceptual Example-3: The Hoover Dam Lake Mead is the largest wholly artificial reservoir in the United States. The water in the reservoir backs up behind the dam for a considerable distance (120 miles). Suppose that all the water in Lake Mead were removed except a relatively narrow vertical column. Would the Hoover Dam still be needed to contain the water, or could a much less massive structure do the job?

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**11.3 Pressure and Depth in a Static Fluid**

Example-4: The Swimming Hole Points A and B are located a distance of 5.50 m beneath the surface of the water. Find the pressure at each of these two locations.

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**11.3 Pressure and Depth in a Static Fluid**

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**Where to put the pump? Pump-X Pump-Y Pump-X pushed the water up.**

Pump-Y removes the air from the pipe and air pressure outside pushes the water up. Pump-Y has a limit, what is that? Pump-X Pump-Y

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Pressure Gauges One of the simplest pressure gauges is the mercury barometer used for measuring atmospheric pressure. 760 mm of mercury

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**Point A and B are at the same depth, so, PB = PA Patm = P1 + gh **

11.4 Pressure Gauges Point A and B are at the same depth, so, PB = PA Patm = P1 + gh = 0 + gh

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Open-tube manometer The phrase “open-tube” refers to the fact that one side of the U-tube is open to atmospheric pressure. The tube contains mercury Its other side is connected to the container whose pressure P2 is to be measured. PB = PA P2 = Patm + gh P2 - Patm = gh P2 – Patm is called the guage pressure P2 is called the absolute pressure

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11.4 Pressure Gauges absolute pressure

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sphygmomanometer blood-pressure instrument: an instrument used to measure blood pressure in an artery that consists of a pressure gauge, an inflatable cuff placed around the upper arm, and an inflator bulb or pressure pump. Systolic pressure is 120 mm of mercury Diastolic pressure is 80 mm of mercury

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11.4 Pressure Gauges

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**Any change in the pressure applied **

11.5 Pascal’s Principle Pascal’s Principle Any change in the pressure applied to a completely enclosed fluid is transmitted undiminished to all parts of the fluid and enclosing walls.

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11.5 Pascal’s Principle

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**The input piston has a radius of 0.0120 m **

11.5 Pascal’s Principle Example-7: A Car Lift The input piston has a radius of m and the output plunger has a radius of 0.150 m. The combined weight of the car and the plunger is N. Suppose that the input piston has a negligible weight and the bottom surfaces of the piston and plunger are at the same level. What is the required input force?

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11.5 Pascal’s Principle

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**11.6 Archimedes’ Principle**

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**11.6 Archimedes’ Principle**

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:

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**11.6 Archimedes’ Principle**

If the object is floating then the magnitude of the buoyant force is equal to the magnitude of its weight.

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**11.6 Archimedes’ Principle**

Example-9: A Swimming Raft The raft is made of solid square pinewood. Determine whether the raft floats in water and if so, how much of the raft is beneath the surface.

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**11.6 Archimedes’ Principle**

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**11.6 Archimedes’ Principle**

The raft floats!

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**11.6 Archimedes’ Principle**

If the raft is floating:

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**11.6 Archimedes’ Principle**

Conceptual Example-10: How Much Water is Needed to Float a Ship? A ship floating in the ocean is a familiar sight. But is all that water really necessary? Can an ocean vessel float in the amount of water than a swimming pool contains?

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**11.6 Archimedes’ Principle**

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**Unsteady flow exists whenever the velocity of the fluid particles at a **

11.7 Fluids in Motion In steady flow the velocity of the fluid particles at any point is constant as time passes. Unsteady flow exists whenever the velocity of the fluid particles at a point changes as time passes. Turbulent flow is an extreme kind of unsteady flow in which the velocity of the fluid particles at a point change erratically in both magnitude and direction.

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**Fluid flow can be compressible or incompressible. Most liquids are **

11.7 Fluids in Motion Fluid flow can be compressible or incompressible. Most liquids are nearly incompressible. Fluid flow can be viscous or nonviscous. An incompressible, nonviscous fluid is called an ideal fluid.

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**When the flow is steady, streamlines are often used to represent **

11.7 Fluids in Motion When the flow is steady, streamlines are often used to represent the trajectories of the fluid particles.

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**Making streamlines with dye and smoke.**

11.7 Fluids in Motion Making streamlines with dye and smoke.

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**11.8 The Equation of Continuity**

The mass of fluid per second that flows through a tube is called the mass flow rate.

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**11.8 The Equation of Continuity**

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**11.8 The Equation of Continuity**

The mass flow rate has the same value at every position along a tube that has a single entry and a single exit for fluid flow. SI Unit of Mass Flow Rate: kg/s

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**11.8 The Equation of Continuity**

Incompressible fluid: Volume flow rate Q:

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**11.8 The Equation of Continuity**

Example-12: A Garden Hose A garden hose has an unobstructed opening with a cross sectional area of 2.85x10-4m2. It fills a bucket with a volume of 8.00x10-3m3 in 30 seconds. Find the speed of the water that leaves the hose through (a) the unobstructed opening and (b) an obstructed opening with half as much area.

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**11.8 The Equation of Continuity**

(b)

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**Bernoulli’s Equation For steady flow, The speed, Pressure, and**

Elevation of an incompressible and nonviscous fluid are related by Bernoulli’s equation.

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**The fluid accelerates toward the lower pressure regions. **

11.9 Bernoulli’s Equation The fluid accelerates toward the lower pressure regions. According to the pressure-depth relationship, the pressure is lower at higher levels, provided the area of the pipe does not change.

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**Bernoulli’s Equation Bernoulli’s Equation from Work-energy theorem:**

Work done on an object by external non-conservative forces is equal to the change in total mechanical energy. The pressure within a fluid is caused by collisional forces, which are non-conservative. Therefore, when a fluid is accelerated because of a difference in pressures, work is being done by non-conservative forces.

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11.9 Bernoulli’s Equation

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Bernoulli’s Equation Whenever a fluid is flowing in a horizontal pipe and encounters a region of reduced cross-sectional area, the pressure of the fluid drops. Can be explained by the Newton’s 2nd law. When moving from the wider region 2 to the narrower region 1, the fluid speeds up according to the equation of continuity. According to the 2nd law, th eaccelerating fluid must be subjected to an unbalanced force. There can be an unbalanced force only if the pressure in region 2 is higher than the pressure in region 1.

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**Bernoulli’s Equation Explanation:**

In figure-b on the top surface of the blue fluid element, the surrounding fluid exerts a pressure P. This pressure give rise to a force, F = P/A On the botom of the blue fluid element, the pressure is slightly higher, P+P. As a result, the force on the bottom surface has a magnitude of F+ F = (P+P)/A. The net force pushing the fluid element up the pipe is F=P/A

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Bernoulli’s Equation When the fluid element moves through its own length s, the work done is the W = F s = P A s = P V The total work done on the fluid element in moving it from region 2 to region 1is the sum of the small increments of work P V done as the element moves along the pipe. This sum amounts to Wnc = (P2 – P1)V Wnc = (P2 – P1)V = E1 – E2

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**fluid speed, and the elevation at two points are related by:**

11.9 Bernoulli’s Equation Bernoulli’s Equation In steady flow of a nonviscous, incompressible fluid, the pressure, the fluid speed, and the elevation at two points are related by:

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**Bernoulli’s Equation The term**

has a constant value at all positions in the flow.

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**Bernoulli’s Equation Bernoulli’s Equation**

reduces to the result for static fluids (v1=v2), as it is when the cross-sectional area remains constant.

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**11.10 Applications of Bernoulli’s Equation**

Conceptual Example-14: Tarpaulins and Bernoulli’s Equation When the truck is stationary, the tarpaulin lies flat, but it bulges outward when the truck is speeding down the highway. Account for this behavior.

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**Bernoulli’s Equation Bernoulli’s Equation reduces to**

When all parts have the same elevation (y1 = y2). The term P+ ½ remains constant throughout a horizontal pipe. If increases, P decreases and vice versa.

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**11.10 Applications of Bernoulli’s Equation**

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**11.10 Applications of Bernoulli’s Equation**

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**11.10 Applications of Bernoulli’s Equation**

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**11.10 Applications of Bernoulli’s Equation**

Example-16: Efflux Speed The tank is open to the atmosphere at the top. Find and expression for the speed of the liquid leaving the pipe at the bottom.

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**11.10 Applications of Bernoulli’s Equation**

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11.11 Viscous Flow Flow of an ideal fluid. Flow of a viscous fluid.

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**Force Needed to Move a Layer of Viscous Fluid with Constant Velocity **

11.11 Viscous Flow Force Needed to Move a Layer of Viscous Fluid with Constant Velocity The magnitude of the tangential force required to move a fluid layer at a constant speed is given by: coefficient of viscosity SI Unit of Viscosity: Pa·s Common Unit of Viscosity: poise (P) 1 poise (P) = 0.1 Pa·s

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**The volume flow rate is given by:**

11.11 Viscous Flow Poiseuille’s Law The volume flow rate is given by:

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**Example-17: Giving and Injection **

11.11 Viscous Flow Example-17: Giving and Injection A syringe is filled with a solution whose viscosity is 1.5x10-3 Pa·s. The internal radius of the needle is 4.0x10-4m. The gauge pressure in the vein is 1900 Pa. What force must be applied to the plunger, so that 1.0x10-6m3 of fluid can be injected in 3.0 s?

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11.11 Viscous Flow

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11.11 Viscous Flow

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Force A

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Force A

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Force A

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Force A

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