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Unformatted text preview: I. FLUID MECHANICS
I.1 Basic Concepts & Definitions:
Fluid Mechanics  Study of fluids at rest, in motion, and the effects of fluids on
boundaries.
Note: This definition outlines the key topics in the study of fluids:
(1) fluid statics (fluids at rest), (2) momentum and energy analyses (fluids
in motion), and (3) viscous effects and all sections considering pressure
forces (effects of fluids on boundaries). Fluid  A substance which moves and deforms continuously as a result of an
applied shear stress.
The definition also clearly shows that viscous effects are not considered in the
study of fluid statics.
Two important properties in the study of fluid mechanics are:
Pressure and Velocity
These are defined as follows: Pressure  The normal stress on any plane through a fluid element at rest.
Key Point: The direction of pressure forces will always be perpendicular to
the surface of interest.
Velocity  The rate of change of position at a point in a flow field. It is used
not only to specify flow field characteristics but also to specify
flow rate, momentum, and viscous effects for a fluid in motion.  v v I1  eText Main Menu  Textbook Table of Contents  Study Guide I.4 Dimensions and Units
This text will use both the International System of Units (S.I.) and British
Gravitational System (B.G.).
A key feature of both is that neither system uses gc. Rather, in both systems
the combination of units for mass * acceleration yields the unit of force, i.e.
Newton’s second law yields
S.I. 1 Newton (N) = 1 kg m/s2 B.G. 1 lbf = 1 slug ft/s2 This will be particularly useful in the following:
Concept Expression Units
kg/s * m/s = kg m/s2 =N &
mV momentum slug/s * ft/s = slug ft/s2 = lbf
ρgh manometry kg/m3*m/s2*m = (kg m/s2)/ m2 =N/m2
slug/ft3*ft/s2*ft = (slug ft/s2)/ft2 = lbf/ft2 dynamic viscosity µ N s /m2 = (kg m/s2) s /m2 = kg/m s
lbf s /ft2 = (slug ft/s2) s /ft2 = slug/ft s Key Point: In the B.G. system of units, the mass unit is the slug
and not the lbm. and 1 slug = 32.174 lbm. Therefore, be careful not
to use conventional values for fluid density in English units without
appropriate conversions, e.g., ρw = 62.4 lb/ft3
For this case the manometer equation would be written as ∆P=ρ g
h
gc  v v I2  eText Main Menu  Textbook Table of Contents  Study Guide Example:
Given: Pump power requirements are given by
&
Wp = fluid density*volume flow rate*g*pump head = ρ Q g hp For ρ = 1.928 slug/ft3, Q = 500 gal/min, and hp = 70 ft,
Determine: The power required in kW.
3
&
Wp = 1.928 slug/ft3 * 500 gal/min*1 ft /s /448.8 gpm*32.2 ft/s2 * 70 ft &
Wp = 4841 ft–lbf/s * 1.3558*103 kW/ft–lbf/s = 6.564 kW Note: We used the following: 1 lbf = 1 slug ft/s2 to obtain the desired
units
Recommendation: Properties of the velocity Field
Two important properties in the study of fluid mechanics are
Pressure and Velocity The basic definition for velocity has been given previously, however, one of
its most important uses in fluid mechanics is to specify both the volume
and mass flow rate of a fluid.
I3  v v 1.5 In working with problems with complex or mixed system
units, at the start of the problem convert all parameters with
units to the base units being used in the problem, e.g. for
S.I. problems, convert all parameters to kg, m, & s; for BG
problems, convert all parameters to slug, ft, & s. Then
convert the final answer to the desired final units.  eText Main Menu  Textbook Table of Contents  Study Guide Volume flow rate: &
Q = ∫ V ⋅ n dA = ∫ Vn dA
cs cs where Vn is the normal component of
velocity at a point on the area across
which fluid flows.
Key Point: Note that only the normal
component of velocity contributes to
flow rate across a boundary. Mass flow rate: &
m = ∫ ρV ⋅n d A = ∫ ρV d A
n
cs cs NOTE: While not obvious in the basic
equation, Vn must also be measured
relative to any flow area boundary
motion, i.e., if the flow boundary is
moving, Vn is measured relative to the
moving boundary. This will be particularly important for problems involving moving control
volumes in Ch. III.  v v I4  eText Main Menu  Textbook Table of Contents  Study Guide 1.6 Thermodynamic Properties
All of the usual thermodynamic properties are important in fluid mechanics
P  Pressure (kPa, psi) T Temperature ( C, F) ρ ñ Density (kg/m3, slug/ft3) o o Alternatives for density
γ  specific weight = weight per unit volume (N/m3, lbf/ft3)
H2O: γ = 9790 N/m3 = 62.4 lbf/ft3 Air: γ=ρg γ = 11.8 N/m3 = 0.0752 lbf/ft3 S.G.  specific gravity = ρ / ρ (ref)
where: ρ (ref) = ρ (water at 1 atm, 20˚C) for liquids = 998 kg/m3
= ρ (air at 1 atm, 20˚C) for gases = 1.205 kg/m3
Example: Determine the static pressure difference indicated by an 18 cm
column of fluid (liquid) with a specific gravity of 0.85.
∆P = ρ g h = S.G. γ h = 0.85* 9790 N/m3 0.18 m = 1498 N/m2 = 1.5 kPa
I.7 Transport Properties
Certain transport properties are important as they relate to the diffusion of
momentum due to shear stresses. Specifically:
µ ≡ coefficient of viscosity (dynamic viscosity) {M / L t }
ν ≡ kinematic viscosity ( µ / ρ ) 2 {L /t}  v v I5  eText Main Menu  Textbook Table of Contents  Study Guide This gives rise to the definition of a Newtonian fluid.
Newtonian fluid: A fluid which
has a linear relationship between
shear stress and velocity gradient.
dU
dy
The linearity coefficient in the
equation is the coefficient of
viscosity µ . τ =µ Flows constrained by solid surfaces can typically be divided into two regimes:
a. Flow near a bounding surface with
1. significant velocity gradients
2. significant shear stresses
This flow region is referred to as a "boundary layer."
b. Flows far from bounding surface with
1. negligible velocity gradients
2. negligible shear stresses
3. significant inertia effects
This flow region is referred to as "free stream" or "inviscid flow region."
An important parameter in identifying the characteristics of these flows is the
Reynolds number = Re = ρV L
µ This physically represents the ratio of inertia forces in the flow to viscous
forces. For most flows of engineering significance, both the characteristics
of the flow and the important effects due to the flow, e.g., drag, pressure
drop, aerodynamic loads, etc., are dependent on this parameter.  v v I6  eText Main Menu  Textbook Table of Contents  Study Guide ...
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