Download - Basic Fluid Mechanics - ase.uc.edupdisimil/classnotes/Bas Fluid Mech Sum2018/Stude… · Fluids: Substances (a liquid or gas) when at rest, cannot sustain ... •Table A.2 Properties

1

Basic Fluid Mechanics

Chapter 1C:

Basic Concepts and Definitions

Chapter 1C Basic Concepts and Definitions 14/16/2018

4/16/2018 Chapter 1C Basic Concepts and Definitions 2

States of MatterSolids: A substance, which has a definite shape regardless of whether small to moderate shear forces are applied to its surface. (Molecules are closely positioned and therefore have large intermolecular forces.)

Fluids: Substances (a liquid or gas) when at rest, cannot sustaina shear force (or tangential force).

solid liquid Gas or vapor

2


• Liquid- takes on the shape of the container it resides in and forms a free surface in the presence of gravity.

• Gas- expands until it encounters the walls of the container and fills the entire available space, and cannot form a free surface.

Classes of a Fluid

• Liquids – A state of matter in which molecules are relativelyfree to change their position with respect to each other butrestricted by intermolecular (cohesive) forces, so as tomaintain a relatively fixed volume.

• Gases – A state of matter in which the molecules arepractically unrestricted by intermolecular forces (moleculesspaced relatively far apart). Hence, a gas has neither adefinite shape nor volume.

• Density () - Mass per unit volume; in general = f (T,P)(typical units are kg/m3 or slugs/ft3 or lbm/ft3)

Chapter 1C Basic Concepts and Definitions 44/16/2018

Classes of Fluids

3


•Table A.2 Properties of the U.S. Standard Atmosphere (EE & BG Units)

Geometric Gravitational KinematicAltitude Temp Pressure Density Density Acceleration Viscosity Viscosityz (ft) T (°R) p (psia) (lbm/ft3) (slug/ft3) g (ft/s2) (lb s/ft2) (ft2/s)

-15000 572.2 24.626 1.162E-1 3.610E-3 32.220 4.031E-7 1.116E-4-10000 554.3 20.847 1.015E-1 3.155E-3 32.205 3.935E-7 1.247E-4

-5000 536.5 17.554 8.831E-2 2.745E-3 32.189 3.835E-7 1.398E-40 518.7 14.696 7.647E-2 2.377E-3 32.174 3.736E-7 1.572E-4

5000 500.8 12.054 6.590E-2 2.048E-3 32.159 3.636E-7 1.776E-410000 483.0 10.108 5.648E-2 1.756E-3 32.143 3.534E-7 2.013E-420000 447.4 6.759 4.077E-2 1.267E-3 32.112 3.326E-7 2.623E-430000 411.8 4.373 2.866E-2 8.907E-4 32.082 3.107E-7 3.488E-440000 390.0 2.730 1.890E-2 5.873E-4 32.051 2.969E-7 5.057E-450000 390.0 1.692 1.171E-2 3.639E-4 32.020 2.969E-7 8.159E-460000 390.0 1.049 7.259E-3 2.256E-4 31.990 2.969E-7 1.316E-370000 392.2 0.651 4.479E-3 1.392E-4 31.959 2.983E-7 2.143E-380000 397.7 0.406 2.758E-3 8.571E-5 31.929 3.018E-7 3.521E-390000 403.1 0.255 1.710E-3 5.315E-5 31.898 3.052E-7 5.743E-3

100000 408.6 0.162 1.068E-3 3.318E-5 31.868 3.087E-7 9.302E-3150000 479.1 0.020 1.112E-4 3.456E-6 31.716 3.512E-7 1.016E-1200000 457.0 0.003 1.696E-5 5.270E-7 31.566 3.382E-7 6.416E-1250000 351.8 0.000 2.263E-6 7.034E-8 31.42 2.721E-7 3.868E0300000 332.9 0.000 1.488E-7 4.625E-9 31.27 2.593E-7 5.608E1

Note:

i) ρ is a f(temperature, pressure)ii) The density of air at STD is ρ = 1.23 kg/m3 (or 0.00238 slug/ft3) at

P = 101.33 kPa (14.696 psia) and T = 15C (or 59F).

Note: Typical satellite in LEO is approx 160 to 2,000 kilometers

Properties of Air


• Continuum – When the properties of a fluid are considered tobe continuously distributed throughout the region of interest,or when the dimensions of the problem are large w.r.t. thespacing between the molecules.

Note:i) As the pressure is significantly reduced, the average

distance between molecules becomes large compared tothe dimensions of the object over which the fluid is flowing.Under these conditions the fluid is now considered a rarifiedgas & no longer fulfills the continuum assumption.

ii) There are 2.7x1016 molecules contained in a cubicmillimeter of air at standard conditions (STD).

iii) To determine if the continuum assumption is valid, comparethe characteristic length l, of the object under study to themolecular mean free path (), which is the average distancea molecule travels before it collides with another molecule.If l >> , the continuum model is acceptable.

iv) For air at STD conditions, is 6x10-6 cm = 60 nm.

4


• Gravitational Constant (gc) - used to correlate between two measurement systems with different units for mass.

British Gravitational System: 1 slug = 1 lbf·sec2/ft

English Engineering System: 1 lbm = 1/32.2 slug

gc=32.2 ft·lbm/lbf·sec2

S.I. units;presented in the absolute scale,

1 N/m2 = 1 pascal (Pa) or

English units;presented in either the absolute or gauge scale ;

1 lbf/in2 = 1 psi (psig or psia)

• Pressure - the force/unit area, exerted normal to a surface.


Solution: Examine a small volume of fluid in the form of a prism. The volume is small enough such that the properties (i.e., pressure) over each face can be assumed constant.

Consider the fluid is at rest and cannot sustain a shear stress (or force).

Include a body force (fx) due to gravity; x comp force/unit mass

Px – average pressure acting on the “x” surface lying in the y-z planeAx – surface area in y-z planePe - average pressure on inclined surfaceAe - area of the inclined surface - angle between the unit normals to Ax and Ae

xo - intercept of the prism face on the x-axis

Problem 1.2: Prove that for a fluid at rest, the pressure is independent of direction or orientation (i.e., it's isotropic)

Since fluid is at rest, net force acting on the fluid element must = 0.

5


Summing the forces in the x-direction:


This exercise can be repeated for the y and z directions, therefore, it can be concluded that for a fluid at rest the pressure is equal in all directions.

====== The End ======

• Ideal Gas Law - The equation of state for an ideal gas follows, where R is the gas constant;

RTp

6


• Incompressible fluid – a fluid in which the is assumed constant

Note: Air @ T=const only needs a P change from 101kPa to 119kPa to achieve a 17% change in density

Note:For water at constant T to experience a 17% change in density, the P would have to be increased by 5000 times


• For water:

Note: H2O density changes by approximately 4%, while air over the same T range (273 to 373 K) changes by 37%

Note: Hg density changes by < 2% as compared to 37% for air over the same T’s

7


• Body Forces – are those forces which involve action from adistance, and are proportional to either the volume or mass of abody.

Examples of body forces are those arising from:• gravity• magnetic fields• electrodynamics

• Surface Forces - those forces which are exerted at the controlsurface by the material outside the control volume on the materialinside the control volume.

Examples of these surface forces are those arising from:• normal stresses (or pressure),• shear stresses (viscous or turbulent),• surface tension (when interfaces between phasesexist).


The surface of a liquid can be compared to membrane under tension, which resists being stretched.

Surface tension allows the insect shown to walk on water.

8



Viscosity – a thermophysical property which represents theresistance to the sliding motion of one fluid layer over another.

• A fluid undergoes a continuous deformation (or strain) whensubjected to a shear stress, .

• Relating the to the rate of deformation is accomplished usingthe absolute viscosity, , which is a property of the fluid.

• Deformation rate (i.e., strain rate) is the velocity gradient, and in1D is dU/dy.

• Therefore, the 1D shear stress relation.

(1.2)dy

du

• Absolute or Dynamic viscosity () lb sec / ft2

• Kinematic viscosity () ft2 / sec or m2/sec

9


• If the fluid is compressible and if significant changes in volumeoccur, an additional viscous stress coefficient will be required.This coefficient is called the second or bulk viscosity.

• If is independent of the velocity gradient (i.e., the rate ofstrain),

that is if varies linearly with the fluid is called Newtonian.

• Examples of Newtonian fluids are water, air, alcohol, gases and most petroleum products (where is practically independent of the velocity gradient).

• Note: The absolute viscosity () is in general a f(P,T);- although changes in with P are usually small, - changes due to T may be very large. - (see Figures. 3A & 4A and Table 1A.)

dydu


Substance Pressure in Atmospheres100 300 500 750 1000 2000

Water at 30 0C 1.0 1.01 1.02 1.04 1.05 1.13Water at 10 0C 1.0 0.98 0.97 0.96 0.95 0.965RepresentativeLubricating Oil

(~ SAE 30) at 55 0C

1.45 2.50 4.7 9.4 19 ~ 150

TABLE 1-A, Displays the effect of pressure on the absolute viscosities of water and a typical lubricating oil (Similar to SAE 30).

The values in the table represent the absolute viscosity () at the a specified pressure divided by the () at 1 atmosphere.

Data adapted from International Critical Tables, McGraw-Hill Book Company, New York, 1926 (courtesy of the National Academy of Sciences, National Research Council, Washington, D.C.); H.A. Everett, High Pressure Viscosity as an Explanation of Apparent Oiliness, Soc. Aut. Eng., Trans., vol. 41, 5, p. 531, 1937; R. B. Dow, The Effect of Temperature and Pressure on the Viscosity of Lubricating Oils, Rheology Bulletin, Am. Inst. of Physics, 1937.

10


FIG. 4-A Kinematic viscosity of various fluids. S has the samemeaning as in Fig. 3-A. Prepared from data in R.L. Daughertyand A. C. Ingersoll, Fluid Mechanics, McGraw-Hill BookCompany, New York, 1954.

FIG. 3-A Absolute viscosity of various fluids. S refers to thedensity of the substance relative to water @ 60 deg F.Prepared from data in R.L. Daugherty and A. C. Ingersoll, FluidMechanics, McGraw-Hill Book Company, New York, 1954.

Note: 1- Air viscosity increased by 20% as T increased from 18 ºC to 100 ºC, however the viscosity of H2O decreases by almost a factor of 4 over the same T’s.2- In general gas ’s increase with T and the viscosity of liquids decrease with increasing T.


• When considering real fluids and viscosity, it has been shown that the fluid in contact with a solid surface has the same velocity as that surface. This is called the no-slip condition.

11


Problem 1.3: A 60 cm wide belt moves a 2 mm layer of water at 10 m/sec and 10 °C. Assume the flow has a linear velocity profile (i.e., Couette flow) at, calculate the required horsepower.

U = f(y)Solution:


====== The End ======

12


Newtonian vs. Non-Newtonian Fluids

• Air and water are considered to be Newtonian, however, not all fluids are considered Newtonian.

• Viscosity of non-Newtonian fluids are a function of strain rate. In general, solutions containing long chain polymers, as well as blood, slurries and suspensions are usually considered Non-Newtonian.

Non-Newtonian Fluid Examples


Newtonian vs. Non-Newtonian Fluids

Newtonian (high μ)

c

Shear-thinning

Shear-thickening

Strain rate (1/s)

Newtonian (low μ)

Shear Thickening – fluids that flow easy with low viscosity at low strain rates, but become more solid-like as the strain rate is increased.Examples of shear thickening fluids are medium greases, sludge's, and corn starch/water mixtures.

Shear Thinning – fluids in which viscous effects decrease as the strain rate increases.Examples of shear thinning fluids are ketchup and most salad dressings (i.e., they come out of the bottle all at once).

• Non-Newton fluids can be divided into two major sub groups: