Fluid Mechanics Course Code CHPE 207 and CIVL

Fluid Mechanics Course Code: CHPE 207 and CIVL 213 Lecturer: Dr Mustafa Saleh Nasser Office Number: 5 D-44 E-mail: mustafa. nasser@uniwa. edu. om Dr Mustafa Nasser 1

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Dimensions and Units In fluid mechanics we must describe various fluid characteristics in terms of certain basic quantities such as length, time and mass • A dimension is the measure by which a physical variable is expressed qualitatively, i. e. length is a dimension associated with distance, width, height, displacement. Ø Basic dimensions: Length, L (or primary quantities) Time, T Mass, M Temperature, Q Ø We can derive any secondary quantity from the primary quantities i. e. Force = (mass) x (acceleration) : F = M L T-2 • A unit is a particular way of attaching a number to the qualitative dimension: Systems of units can vary from country to country, but dimensions do not Dr Mustafa Nasser 9

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Units of Force – E system To make Newton’s law dimensionally consistent we must include a dimensional proportionality constant: where Dr Mustafa Nasser 11

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Density for Gasses – For gases: strong function of T and P from ideal gas law: r = P M/R T (can you drive this) where R = universal gas constant, M=mol. weight R= 8. 314 J/(g-mole K)=0. 08314 (liter bar)/(g-mole K)= 0. 08206 (liter atm)/(g-mole K)=1. 987 (cal)/(g-mole K)= 10. 73 (psia ft 3)/(lb-mole °R)=0. 7302 (atm ft 3)/(lb-mole °R) Dr Mustafa Nasser 15

Pressure is defined as the amount of force exerted on a unit area of a substance: P=F/A Pascal’s Laws Pascals’ laws: Pressure acts uniformly in all directions on a small volume (point) of a fluid In a fluid confined by solid boundaries, pressure acts perpendicular to the boundary – it is a normal force. Dr Mustafa Nasser 16

Direction of fluid pressure on boundaries Furnace duct Pipe or tube Heat exchanger Pressure is due to a Normal Force (acting perpendicular to the surface) It is also called a Surface Force Dam Dr Mustafa Nasser 17

Absolute and Gauge Pressure • Gauge pressure: Pressure expressed as the difference between the pressure of the fluid and that of the surrounding atmosphere. Dr Mustafa Nasser

Units for Pressure Unit Definition or Relationship 1 pascal (Pa) 1 kg m-1 s-2 1 bar 1 x 105 Pa 1 atmosphere (atm) 101, 325 Pa 1 torr 1 / 760 atm 760 mm Hg 1 atm 14. 696 pounds per sq. in. (psi) 1 atm Dr Mustafa Nasser 19

Pressure distribution for a fluid at rest We will determine the pressure distribution in a fluid at rest in which the only body force acting is due to gravity The sum of the forces acting on the fluid must equal zero Ø Consider an infinitesimal rectangular fluid element of dimensions Dx, Dy, Dz z y x Dr Mustafa Nasser 20

Pressure distribution for a fluid at rest Let Pz and Pz+Dz denote the pressures at the base and top of the cube, where the elevations are z and z+Dz respectively. -Force at base of cube: Pz A=Pz (Dx Dy) -Force at top of cube: Pz+Dz A= Pz+Dz (Dx Dy) -Force due to gravity: m g=r V g = r (Dx Dy Dz) g A force balance in the z direction gives: For an infinitesimal element (Dz 0) Fg Dr Mustafa Nasser 21

Variation of pressure with elevation

General variation of pressure in a static fluid due to gravity

General variation of pressure in a static fluid due to gravity for vertical direction: q = 0

Equality of pressure at the same level in a static fluid

Incompressible fluid Liquids are incompressible i. e. their density is assumed to be constant. When we have a liquid with a free surface the pressure P at any depth below the free surface is: where Po is the pressure at the free surface (Po=Patm) and h = zfree surface - z By using gauge pressures we can simply write: Dr Mustafa Nasser 26

Equality of pressure at the same level in a continuous fluid

Examples SG= 13. 6 Dr Mustafa Nasser 28

Solution: 1. 1 1. 2 Dr Mustafa Nasser 29

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Calculate the weight of a reservoir of oil if it has a mass of 825 kg. We have

If the reservoir from previous Example has a volume of 0. 917 m 3, compute the density, the specific weight, and the specific gravity of the oil.

3. 7 Pressure expressed as the Height of a Column of Liquid • Then we can use this as a conversion factor,

Pressure Gages and Transducers • • • For those situations where only a visual indication is needed at the site where the pressure is being measured, a pressure gage is used. In other cases there is a need to measure pressure at one point and display the value at another. The general term for such a device is pressure transducer, meaning that the sensed pressure causes an electrical signal to be generated that can be transmitted to a remote location such as a central control station where it is displayed digitally.

Pressure Gages • Below Figs shows the Bourdon tube pressure gage.

Pressure Gages • • • The pressure to be measured is applied to the inside of a flattened tube, which is normally shaped as a segment of a circle or a spiral. The increased pressure inside the tube causes it to be straightened somewhat. The movement of the end of the tube is transmitted through a linkage that causes a pointer to rotate.

Pressure Gages • Figure below shows a pressure gage using an actuation means called Magnehelic pressure gage.

3. 8. 2 Strain Gage Pressure Transducer • • Fig 3. 17 shows the strain gage pressure transducer and indicator. The pressure to be measured is introduced through the pressure port and acts on a diaphragm to which foil strain gages are bonded. As the strain gages sense the deformation of the diaphragm, their resistance changes. The readout device is typically a digital voltmeter, calibrated in pressure units.

Strain Gage Pressure Transducer

LVST-Type Pressure Transducer • • A linear variable differential transformer (LVDT) is composed of a cylindrical electric coil with a movable rod-shaped core. As the core moves along the axis of the coil, a voltage change occurs in relation to the position of the core. This type of transducer is applied to pressure measurement by attaching the core rod to a flexible diaphragm. Fig shows the Linear variable differential transformer (LVDT)-type pressure transducer.

LVST-Type Pressure Transducer

Piezoelectric Pressure Transducer • • • Certain crystals, such as quartz and barium titanate, exhibit a piezoelectric effect, in which the electrical charge across the crystal varies with stress in the crystal. Causing a pressure to exert a force, either directly or indirectly, on the crystal leads to a voltage change related to the pressure change. Fig shows the digital pressure gage.

Piezoelectric Pressure Transducer

Buoyancy Archimedes Principle Laws of buoyancy discovered by Archimedes: – A body immersed in a fluid experiences a vertical buoyant force equal to the weight of the fluid it displaces – A floating body displaces its own weight in the fluid in which it floats F 1 h 1 Free liquid surface The upper surface of the body is subjected to a smaller force than the lower surface A net force is acting upwards H h 2 F 2 Dr Mustafa Nasser 44

Buoyancy The net force due to pressure in the vertical direction is: FB = F 2 - F 1 = (Pbottom - Ptop) (Dx. Dy) The pressure difference is: Pbottom – Ptop = r g (h 2 -h 1) = r g H FB = r g H (Dx. Dy) Thus the buoyant force is: FB = r g V where r = the fluid density If the fluid density is greater than the average density of the object, the object floats. If less, the object sinks Dr Mustafa Nasser 45

Example Consider a solid cube of dimensions 1 ft x 1 ft (=0. 305 m x 0. 305 m). Its top surface is 10 ft (=3. 05 m) below the surface of the water. The density of water is rf=1000 kg/m 3. Consider two cases: a) The cube is made of cork (r. B=160. 2 kg/m 3) b) The cube is made of steel (r. B=7849 kg/m 3) In what direction does the body tend to move? Dr Mustafa Nasser 46

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Shear in Different Fluids • Shear-stress relations for different types of fluids • Newtonian fluids: linear relationship • Slope of line (coefficient of proportionality) is “viscosity”

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Viscosity v The constant of proportionality is designated by the Greek symbol (mu) and is called the absolute viscosity, dynamic viscosity, or simply the viscosity of the fluid. v The viscosity depends on the particular fluid, and for a particular fluid the viscosity is also dependent on temperature.

Viscosity and Temperature 1/3 v For fluids, the viscosity decreases with an increase in temperature. v For gases, an increase in temperature causes an increase in viscosity. ð WHY? molecular structure.

Fluid Properties Example 1 Explain why the viscosity of a liquid decreases while that of a gas increases with a temperature rise. The following is a table of measurement for a fluid at constant temperature. Determine the dynamic viscosity of the fluid. du/dy (rad s-1) t (N m-2) 0. 00 0. 20 0. 01 0. 40 1. 90 0. 60 3. 10 Dr Mustafa Nasser 0. 80 4. 00 52

Using Newton's law of viscocity where m is the viscosity. So viscosity is the gradient of a graph of shear stress against vellocity gradient of the above data, or Plot the data as a graph: Calculate the gradient for each section of the line du/dy (s-1) 0. 0 0. 2 0. 4 0. 6 0. 8 t (N m-2) 0. 0 1. 9 3. 1 4. 0 Gradient - 5. 0 4. 75 5. 17 5. 0 Thus the mean gradient = viscosity = 4. 98 N s / m 2 Dr Mustafa Nasser 53

Example 2: 5. 6 m 3 of oil weighs 46 800 N. Find its mass density, and relative density. Solution 2: Weight 46 800 = mg Mass m = 46 800 / 9. 81 = 4770. 6 kg Mass density r = Mass / volume = 4770. 6 / 5. 6 = 852 kg/m 3 Relative density Dr Mustafa Nasser 54

Example 3 The velocity distribution of a viscous liquid (dynamic viscosity m = 0. 9 Ns/m 2) flowing over a fixed plate is given by u = 0. 68 y - y 2 (u is velocity in m/s and y is the distance from the plate in m). What are the shear stresses at the plate surface and at y=0. 34 m? Dr Mustafa Nasser 55

At the plate face y = 0 m, Calculate the shear stress at the plate face At y = 0. 34 m, As the velocity gradient is zero at y=0. 34 then the shear stress must also be zero. Dr Mustafa Nasser 56

Example 4 In a fluid the velocity measured at a distance of 75 mm from the boundary is 1. 125 m/s. The fluid has absolute viscosity 0. 048 Pa s and relative density 0. 913. What is the velocity gradient and shear stress at the boundary assuming a linear velocity distribution. m = 0. 048 Pa s g = 0. 913 Dr Mustafa Nasser 57

Newtonian and Non-Newtonian Fluid v Fluids for which the shearing stress is linearly related to the rate of shearing strain are designated as Newtonian fluids v Most common fluids such as water, air, and gasoline are Newtonian fluid under normal conditions. v Fluids for which the shearing stress is not linearly related to the rate of shearing strain are designated as non-Newtonian fluids.

Non-Newtonian Fluids

Non-Newtonian Fluids Newtonian Fluid Non-Newtonian Fluid η is the apparent viscosity and is not constant for non-Newtonian fluids.

η - Apparent Viscosity The shear rate dependence of η categorizes non-Newtonian fluids into several types. Power Law Fluids: Ø Pseudoplastic – η (viscosity) decreases as shear rate increases (shear rate thinning) Ø Dilatant – η (viscosity) increases as shear rate increases (shear rate thickening) Bingham Plastics: Ø η depends on a critical shear stress (t 0) and then becomes constant

Non-Newtonian Fluids Bingham Plastic: sludge, paint, blood, ketchup Pseudoplastic: latex, paper pulp, clay solns. Newtonian Dilatant: quicksand

non-Newtonian Fluids 1/2 v Shear thinning fluids: The viscosity decreases with increasing shear rate – the harder the fluid is sheared, the less viscous it becomes. Many colloidal suspensions and polymer solutions are shear thinning. Latex paint is example.

non-Newtonian Fluids 2/2 v. Shear thickening fluids: The viscosity increases with increasing shear rate – the harder the fluid is sheared, the more viscous it becomes. Water-corn starch mixture watersand mixture are examples. Bingham plastic: neither a fluid nor a solid. Such material can withstand a finite shear stress without motion, but once the yield stress is exceeded it flows like a fluid. Toothpaste and mayonnaise are common examples.