FLUID MECHANICS_101718.pptx

FLUID MECHANICS
BY ENGR. ALVIN M. ANTE

References:
Fluid Mechanics 3rd Ed. By Frank M. White
Fluid Mechanics and Hydraulics 4th Ed. By Gillesania
Other related books, PDF, etc.

What is Fluid Mechanics?
It is the science that deals with the behavior of fluids
at rest or in motion, and the interaction of fluids with
solids or other fluids at the boundaries.

Focus on rate of flow and the various
pressures that inhibit them

What is a Fluid?
A fluid is a substance that deforms when subjected to
force.
Liquids occupy definite volumes; gases expand to occupy
any containing vessel

The Concept of a Fluid
A solid can resist a shear stress by a static deformation; a fluid
cannot.
The fluid moves and deforms continuously as long as the shear
stress is applied.

LIQUIDS GASES
incompressible compressible
Compressibility is defined as the change of
specific weight with pressure

Dimensions and Units
A dimension is the measure by which a physical variable is
expressed quantitatively.
A unit is a particular way of attaching a number to the
quantitative dimension.

Primary Dimensions
In fluid mechanics there are only four primary dimensions from
which all other dimensions can be derived: mass, length, time, and
temperature.

FLUID PROPERTIES
The Velocity Field
Foremost among the properties of a flow is the velocity field
V(x, y, z, t).
In fact, determining the velocity is often tantamount to solving a
flow problem, since other properties follow directly from the velocity
field.

FLUID PROPERTIES
Kinematic Properties of the Velocity Field

FLUID PROPERTIES
Thermodynamic
Properties of
Fluids

FLUID PROPERTIES
Unit weight / Specific Weight
◦ Weight per unit volume of a fluid
γ =
𝒘
𝑽
Where:
γ = unit wt. (N/m3, lbf/ft3)
w = weight of fluid (N, lbf)
V = volume (m3, ft3)

FLUID PROPERTIES
Mass Density
◦ Mass per unit volume of a fluid
ρ =
𝒎
𝑽
; ρair = 1.205 kg/m3; ρwater = 1000 kg/m3
Where:
ρ = density (kg/m3, lb/ft3)
m = mass of fluid (kg, lb)
V = volume (m3, ft3)

FLUID PROPERTIES
Density of Gases
ρ =
𝒑
𝑹𝑻
Where:
ρ = density (kg/m3)
p = absolute pressure of gas (kPa)
R = gas constant (J/kg-K)
T = absolute temperature (K)

FLUID PROPERTIES
Specific Gravity
◦ Unit wt. of fluid divided by the unit wt. of water
s =
𝜸(𝒇)
𝜸(𝒘)
Where:
s = specific gravity (dmls)
𝜸(𝒇) = unit wt. of fluid (N/m3)
𝜸(𝒘) = unit wt. of water (N/m3)

FLUID PROPERTIES
Viscosity – resistance to deformation which causes flow
◦ Kinematic Viscosity
ν =
𝝁
𝝆
Where:
ν = (nu) k. viscosity (m2/s or stoke)
𝝁 = dynamic/absolute viscosity (Pa*s)
𝝆 = density (kg/m3) 1 stoke = 1 cm2/s

FLUID PROPERTIES
Viscosity
◦ Dynamic Viscosity
𝝁 =
𝝉
𝒅𝑽/𝒅𝒚
=
𝝉
𝜸
Where:
𝝁 = dynamic/absolute viscosity (Pa*s) or prop. const.
τ = shear stress (Pa)
dV/dy = γ = velocity gradient/shear rate (1/s)
1 centipoise (cP) = 10-3 Pa*s

FLUID PROPERTIES
Viscosity is a proportionality
constant suggesting a linear
relationship between shear
stress and velocity gradient.

FLUID PROPERTIES
Surface Tension
◦ The surface tension of a fluid is the work that must be done to
bring enough molecules from inside the liquid to the surface to
form a new unit area of that surface in ft-lb/ft2 or N-m/m2.

FLUID PROPERTIES
Capillarity
◦ Rise or fall of fluid in for example a tube which is caused by
surface tension and depends on the relative magnitudes of the
cohesion of the fluid and its adhesion to the walls.
Rise – cohesion < adhesion
Fall – cohesion > adhesion

Sample Problems: Fluid
Properties

Ideal Fluids
Assumed to have no viscosity (no resistance to shear)
Incompressible
Have no uniform velocity when flowing
No friction b/w moving layers of fluid
No eddy currents or turbulence

Real Fluids
Exhibit infinite viscosities
Non-uniform velocity distribution when flowing
Compressible
Experience friction and turbulence in flow

Newtonian Fluids
◦Newtonian fluids show linear relationship
between shear stress (τ) and velocity gradient
(du/dy) – that is the viscosity remains constant
irrespective of changes to shear stress and
velocity gradient

Non-Newtonian Fluids
◦The ratio between shear stress (τ) and velocity
gradient (du/dy) or shear rate is not constant but
depends on the shear force exerted on the fluid.
◦Non-Newtonian fluids doesn’t show linearity
between shear stress and shearing rate (velocity
gradient).

Non-Newtonian Fluids
Bingham Fluids - has true shear rate (damp clay, concentrated slurries,
cheese)
Pseudoplastics – viscosity decreases as shear rate increases (paint,
mayonnaise, blood, heavy slurries)
Dilatent fluids – viscosity increases as shear rate increases (most honeys,
sand)
Thixiotropic – become more fluid (viscosity decreases) when being more
stirred with time
Rheopetic – become less fluid with time, opposite of thixiotropic

Most fluid problems assume real fluids with Newtonian
characteristics for convenience. This assumption is appropriate
for:
 water
air
gases
steam
simple fluids: alcohol, gasoline, acid solutions

Unit Pressure or Pressure (p)
Pressure - is the force per unit area exerted by a liquid or
gad on a body or surface, with the force acting at right
angles to the surface uniformly in all directions.
Where:
p = pressure, N/m2 or Pa (psf, psi)
F = force, N or lbf
A = area, m2 or ft2 or in2
𝑝 =
𝐹
𝐴

Pascal’s Law
States that the pressure entity at a point in a fluid at rest is
the same in all directions
P1 = P2 =P3

Fluid-Static Law
States that the pressure in a fluid increases with increasing
depth
P = ρgh

Hydrostatic Pressure Distribution
Pressure in a continuously distributed uniform
static fluid varies only with vertical distance and
is independent of the shape of the container.
The pressure is the same at all points on a
given horizontal plane in the fluid. The pressure
increases with depth in the fluid

Absolute and Gage Pressures
Absolute pressure is the pressure above absolute zero (vacuum - no
air). It is the lowest possible pressure attainable. It can never be
negative value.
Gage pressure is pressure above or below the atmosphere and can
be measured by pressure gauges or manometers. The smallest gage
pressure is equal to the negative of ambient atmospheric pressure.
Atmospheric pressure is the pressure at any one point on the earth’s
surface from the weight of the air above it.

Absolute and Gage Pressures
𝑝𝑎𝑏𝑠 = 𝑝𝑔𝑎𝑔𝑒 + 𝑝𝑎𝑡𝑚

Steps in Solving Manometer Problems
1. Decide on the fluid in feet or meter, of which the heads are to be
expressed.
2. Starting from an end point, number in order, the interface of
different fluids.
3. Identify points of equal pressure. Label these points with the same
number.
4. Proceed from level to level, adding (if going down) or subtracting
(if going up) pressure heads as the elevation decreases or increases,
respectively with due regard for the specific gravity of the fluids.

Buoyancy
Principle which determines whether the body will
sink, rise or float

Buoyancy
Upthrust (F) or Buoyant force
F = Wair – Wliquid = ρ g V
V = volume of immersed object (or immersed part if it floats)

Hydrostatic Forces on
Surfaces

Hydrostatic Forces on Surfaces
1. Hydrostatic forces on plane surfaces
2. Hydrostatic forces on curved surfaces
3. Hydrostatic forces in layered surfaces

Hydrostatic Forces on Plane Surfaces
If the pressure over a plane area is uniform, as in
the case of a horizontal surface submerged in a
liquid, the total hydrostatic force/pressure is given by:
F= 𝑝𝐴

For an inclined or vertical plane submerged in a
liquid, the total pressure is given by:
Lifted from Fluid Mech. &
Hydraulics 4th Ed. By Gillesania

Lifted from Fluid Mech. & Hydraulics 4th Ed. By
Gillesania

The hydrostatic problem is reduced to simple formulas
involving centroid and moments of inertia of the plate cross-
sectional area.
Lifted from Fluid Mechanics 3rd Ed. by
White

Hydrostatic Forces on Curved Surfaces
The resultant force on a curved surface is most easily computed
by separating it into horizontal and vertical components.
Lifted from Fluid Mechanics 3rd
Ed. by White

Hydrostatic Forces on Curved Surfaces
The horizontal component of force on a curved surface equals the
force on the plane area formed by the projection of the curved surface
onto a vertical plane normal to the component.
The vertical component of pressure force on a curved surface equals
in magnitude and direction the weight of the entire column of fluid,
both liquid and atmosphere, above the curved surface.

Hydrostatic Forces on Layered Surfaces
If the fluid is layered with different
densities, a single formula cannot solve
the problem because the slope of the
linear pressure distribution changes
between layers.
However, the formulas apply separately
to each layer, and thus the appropriate
remedy is to compute and sum the
separate layer forces and moments.
Lifted from Fluid Mechanics 3rd
Ed. by White

Governing Eqns. Of Flow
BASIC EQUATIONS
1. continuity equation
2. energy equation
3. momentum equation

CONTINUITY EQUATON
2
1 m
m



2
2
2
1
1
1
t
t


 

2
2
1
1 Q
Q 
  2
1 Q
Q 
2
2
1
1 V
A
V
A 

ENERGY EQUATION (BERNOULLI’S EQUATION)
2
1 E
E 
g
v
z
p
g
v
z
p
2
2
2
2
2
2
2
1
1
1








Sample Problems:
Conservation Laws
involving Fluids

Open Channel Flow
The flow of liquids in a pipe is partially filled
with the liquid and there is a free surface

PRISMATIC VS. NON-PRISMATIC
A prismatic channel is a channel built with
constant cross-section and constant bottom
slope
Otherwise, it is a non-prismatic channel,
characterized by non-uniform cross section and
slope.

Artificial channels such as rectangular, trapezoid,
parabola and circle are the most commonly used
prismatic channels.
Natural channels are usually non-prismatic.

FLOW CLASSIFICATION
1. TIME AS THE CRITERION
2. SPACE AS THE CRITERION
3. BASED ON FLOW REGIMES
STEADY FLOW/UNSTEADY FLOW
UNIFORM FLOW/NON-UNIFORM FLOW
INERTIAL FORCES
VISCOUS FORCES
GRAVITATIONAL FORCES

Incompressible vs. Compressible
•Incompressible fluids maintain a nearly constant
density at given temperatures and pressures. Density
changes should be under 5%.
•Compressible fluids have significantly varying
densities during the flow

While all flows are compressible, flows are usually
treated as being incompressible when the Mach
number (the ratio of the speed of the flow to the speed
of sound) is less than 0.3 (30%)
Compressibility is NOT a fluid property but a flow
property.

Viscous vs. Inviscid Flow
Flows in which the effects of viscosity are significant
are called viscous flows.
Idealized flows of zero-viscosity fluids are called
frictionless or inviscid flows. The effects of viscosity is
very small.

Laminar vs. Turbulent Flow
The highly ordered fluid motion characterized by
smooth streamlines is called laminar.
The highly disordered fluid motion that typically
occurs at high velocities characterized by velocity
fluctuations is called turbulent.

Laminar vs. Turbulent Flow
Laminar: high-viscosity fluid such as oil at low velocity
Turbulent: low viscosity fluid such as air at high
velocity

Internal vs. External Flow
◦A fluid flow is classified as being internal and external,
depending on whether the fluid is forced to flow in a
confined channel or over a surface.

Internal Flow
The flow in a pipe or duct in which the fluid is
completely bounded by solid surfaces.

External Flow – The Boundary Layer
is the flow of an unbounded fluid
over a surface

Velocity Boundary Layer
The x-coordinate is measured along the plate surface from the leading
edge of the plate in the direction of the flow, and y is measured from
the surface in the normal direction.
The velocity of the particles in the first fluid layer adjacent to the plate
becomes zero because of the no-slip condition.
Thus, the presence of the plate is felt up to some normal distance from
the plate beyond which the free-stream velocity u remains essentially
unchanged.

Natural vs. Forced Flow
A fluid flow is said to be natural or forced, depending on how the fluid motion
is initiated.
In forced flow, a fluid is forced to flow over a surface or in a pipe by external
means such as a pump or a fan.
In natural flows, any fluid motion is due to a natural means such as the
buoyancy effect, which manifests itself as the rise of the warmer (and thus
lighter) fluid and the fall of cooler (and thus denser) fluid.

Steady vs. Unsteady (Transient) Flow
The term steady implies no change with time. The opposite
of steady is unsteady, or transient.
Many devices such as turbines, compressors, boilers,
condensers, and heat exchangers operate for long periods of
time under the same conditions, and they are classified as
steady-flow devices.

INERTIAL FORCES
VISCOUS FORCES
GRAVITATIONAL FORCES

Reynold’s Number
The transition from laminar to turbulent flow depends on
the surface geometry, surface roughness, free-stream
velocity, surface temperature, and type of fluid, among
other things.
After exhaustive experiments in the 1880s, Osborn Reynolds
discovered that the flow regime depends mainly on the
ratio of the inertia forces to viscous forces in the fluid.

REYNOLDS NO.
LAMINAR FLOW
◦Re < 500
TURBULENT FLOW
◦Re > 1,000
TRANSITION FLOW
◦500 < Re < 1,000

VL

Re
English Units

LAMINAR FLOW
Re < 2130
TURBULENT FLOW
Re > 4000
TRANSITION FLOW
2130 < Re < 4000
S.I. Units

Reynold’s Number
At large Reynolds numbers, the inertia forces, which are
proportional to the density and the velocity of the fluid, are large
relative to the viscous forces, and thus the viscous forces cannot
prevent the random and rapid fluctuations of the fluid.
At small Reynolds numbers, however, the viscous forces are large
enough to overcome the inertia forces and to keep the fluid “in line.”
The Reynolds number at which the flow becomes turbulent is called
the critical Reynolds number.

Reynold’s Number
For flow over flat plate:
where xcr is the distance from the leading edge of the
plate at which transition from laminar to turbulent flow
occurs

Streamlined vs. Turbulent Flow
Streamlined flow is when fluid moves in parallel elements,
depicted by streamlines. The velocity of any element is
constant but not necessarily the same as that of an adjacent
element.
Turbulent flow is when the fluid moves in elemental swirls or
eddies. Both velocity and direction of each element change
over time, thus a violent mixing results.

FROUDE NO.
CRITICAL FLOW
◦ Fr = 1
SUBCRITICAL FLOW
◦ Fr < 1
SUPERCRITICAL FLOW
◦ Fr > 1
gD
V
Fr 

CHEZY’S EQUATION
n
R
c
6
/
1

RS
c
V 
Manning

Darcy-Weisbach Equation
relates the head loss, or pressure loss, due to friction along a given
length of pipe to the average velocity of the fluid flow for an
incompressible fluid
hf =
𝟒𝒇𝑳
𝑫
𝒙 (𝑽𝟐)
𝟐𝒈

Cavitation
the vaporization that may occur at locations
where the pressure drops below the vapor
pressure
Cavities are known as “bubbles” or “voids”

FLUID MECHANICS_101718.pptx

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