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Is any study more important
than Continuum / Fluid
Mechanics?
Imagine water: sea, rivers, lakes,
ponds, rain, snow ! Imagine air above water! Imagine space above air!
Imagine inside the earth!
Imagine people
/living objects!
Imagine oil and gasses
Upper Space
Other fluids
300km
Continuum-Fluid Mechanics Lectures’ Contents
The Equations of Fluid
Motion Lectures 3-5: 4/26; 5/10, 17
Introduction Lecture 1: 4/12
Background:
Basic Physics of Fluids Lecture 2: 4/19
Applications of the
Navier–Stokes
Equations Lectures 6-8: 5/24, 31; 6/14
The Continuum Hypothesis
Definition of a Fluid
Fluid Properties
Classification of Flow Phenomena
Flow Visualization
Continuum Mechanics (Solid & Fluid Mechanics)
Importance of Fluids (Science & Technology)
The Study of Fluids (Approaches)
Lagrangian & Eulerian Systems; the Substantial Derivative
Review of Pertinent Vector Calculus
Conservation of Mass- Continuity Equation
Momentum Balance- Navier-Stokes Equations
Analysis of Navier-Stokes Equations
Scaling and Dimension Analysis
Fluid Statics
Bernoulli’s Equation
Control-Volume Momentum Equation
Classical Exact Solutions to N-S. Equations
Pipe Flow (Brief)
Continuum mechanics: study of the physics of continuous materials
Solid Mechanics: physics of continuous
materials with a defined rest shape.
Fluid mechanics : physics of continuous materials
which deform when subjected to a force.
Elasticity describes materials that
return to their rest shape after applied
stresses are removed.
Plasticity describes materials that
permanently deform after a sufficient
applied stress
Rheology: study of
materials with both solid
and fluid characteristics
Fluid mechanics including liquids, gases, and plasmas. is the study of fluids and the forces on them.
Fluid statics: study of
fluid at rest, and has NO
share stress.
Fluid dynamics: study of fluid in
motion WITH an account of the
forces acting on it
Hydrodynamics: study of
water flow, and deals with
the properties of liquids in
motion.
Aerodynamics: study
of airflow and deals
with the properties of
gases in motion.
Fluid kinematics: study of fluid in
motion WITHOUT an account of the
forces acting on it
Newtonian fluids undergo
strain rates proportional to
the applied shear stress.
water, kerosene oil, air,
glycerine
Non-Newtonian fluids do not
undergo strain rates proportional to
the applied shear stress: blood,
paint, ketchup, toothpaste, coconut
oil, mud flows
Importance of Fluids: Fluids in the pure sciences
Oceanography
Geophysics
Astrophysics
Biological sciences
(b) mesoscale weather patterns: short-range
weather prediction; tornado and hurricane
warnings; pollutant transport
(a) global circulation: long-range
weather prediction; analysis of
climate change (global warming
Atmospheric sciences
(a) ocean circulation patterns: causes of El
Ni˜no, effects of ocean currents on weather
and climate
(b) effects of pollution on living organisms
(a) convection (thermally-driven fluid motion)
In the Earth’s mantle: understanding of plate
tectonics, earthquakes, volcanoes
(b) convection in Earth’s molten core:
production of the magnetic field
(a) galactic structure
and clustering
(b) stellar evolution—from formation by gravitational
collapse to death as a supernovae,
from which the basic elements are distributed
throughout the universe, all via fluid motion
(a) circulatory and respiratory systems in animals (b) cellular processes
Importance of Fluids: Fluids in technology
1. Internal combustion engines—all types of transportation systems
2. Turbojet, scramjet, rocket engines—aerospace propulsion systems
3. Waste disposal
(a) chemical treatment
(b) incineration
(c) sewage transport and treatment
4. Pollution dispersal—in the atmosphere (smog);
in rivers and oceans
5. Steam, gas and wind turbines,
and hydroelectric facilities for
electric power generation
6. Pipelines
(a) crude oil and natural gas transferral
(b) irrigation facilities
(c) office building and household plumbing 7. Fluid/structure interaction
(a) design of tall buildings
(b) continental shelf oil-drilling rigs
(c) dams, bridges, etc.
(d) aircraft and launch vehicle airframes
and control systems
8. Heating, ventilating and air-
conditioning (HVAC) systems
9. Cooling systems for high-density electronic devices—digital computers
from PCs to supercomputers
10. Solar and geothermal heat utilization
11. Artificial hearts, kidney dialysis machines, insulin pumps
12. Manufacturing processes
(a) spray painting automobiles, trucks, etc.
(b) filling of containers, e.g., cans of soup, cartons of milk, plastic bottles of soda
(c) operation of various hydraulic devices
(d) chemical vapor deposition, drawing of synthetic fibers, wires, rods, etc.
Approaches to study Fluid Mechanics
Analytical Method Using advanced mathematics, we can solve
governing equations of fluid motions and obtain
specific solutions for various flow problems.
Experiments This approach utilities facilities to measure
considered flow fields or uses various
visualization methods to visualize flow pattern.
Computations For most of flow problems, we cannot obtain an
analytical solution. The results are so-called
numerical solutions. On the other hands, costs
of experiments become very expensive.
Numerical solutions provides an alternative
approach to observe flow fields without built-up
a real flow field.
The Navier–Stokes equations
Pressure distribution and
qualitative nature of the velocity
field for flow over a race car.
Temperature field and a portion
of the velocity field in a PC
The Continuum Hypothesis
Basic Concept
Background: Basic Physics of Fluids
When dealing with fluids we ignore that
they actually comprise billions of
individual molecules in a small region.
Instead treat the properties of that
ENTIRE region as if it were a continuum.
Advantage
This way fluid property is possible to be treated as
varying continuously from one point to the next within
the fluid;
Difficulties at Molecular level:
Molecules move randomly in different directions
Widely separated (=mean free path)
Difficult to find a molecule at an observation point
Therefore, cannot measure a property of an
individual molecule (say v ).
Average velocity in
(a) volume possible
Average velocity in (b)
volume NOT possible
(not enough molecules)
Continuum Hypothesis: Formal Definition.
We can associate with a small -non-zero-
volume of fluid, those macroscopic properties
such as velocity, temperature, that we
associate with the bulk fluid.
Av distances between mean free
path = collisions of green
molecule with light blue
Sea-level air is a fluid satisfying the continuum hypothesis
in the majority of common engineering situation
Continuum hypothesis is invalid 300 Km
At sea-level the no of molecules in
a small volume ( ) =
Background: Basic Physics of Fluids
What is a fluid?
Substance that can readily flow from place to
place , and that take on the shape of a container
rather than retain a shape of their own.
A fluid is any substance that deforms
contineously when subjected to a shear stress-
tangential force per unit area), no matter how
small.
No deformation (in shape)
Share stress
Deformation (in shape)
No (contineous)
response to share
stress
Response to
share stress:
Continued flowing/
deforming
Response to share
stress = Take on the
shape of the
container/Deformation
F
Tangential
force
Normal force
Background: Basic Physics of Fluids
Fluid Properties
Physical properties
Density
It is the amount of mass per unit volume.
Pressure
Pressure is a normal force per unit area
in a fluid.
Surface Tension
The ability of liquids to support
weak tensile force.
A phenomenon in which the
surface of a liquid, where the liquid
is in contact with gas, acts like a
thin elastic sheet.
Transport properties
Viscosity
Through viscosity a fluid offers resistance
to shear stresses.
Mass Diffusivity
A “mixing” of two or more substances at
the molecular level. For example, mass
diffusion of salt into fresh water and
quantify the degree of mixing with the
concentration of salt
Thermal Conductivity
It mediates diffusion of heat through a
substance in a manner analogous to
viscosity.
Background: Basic Physics of Fluids
Density One of the best way is to quantify a fluid is it in terms of its density.
It is the mass M of a material (fluid or not) in a volume V.
The denser a material, the more mass it has in a given volume.
Maple syrup
Corn
Shampoo
Water
Dish detergent
Antifreeze
The equation implies that density is proportional to the mass
and inversely proportional to the volume the mass is
contained..
Units of density
Background: Basic Physics of Fluids
Pressure
In every day life, if you push a button or pressed a
key on a keyboard, you apply a force. That force
spreads out over an area, however.
The equation implies that pressure is
proportional to the applied force and inversely
proportional to the area over which it acts.
Pressure is increased if the force
applied to a given area increased or if a
given force is applied to a smaller area.
This bird exerts a small pressure on the lily
pad as its weight spreads out over a large
area by its long toes
Since the pressure is not enough to sink a
lily pad, the bird can “walk on water.”
Why a balloon burst with a needle?
Why the bird can walk on water? Small area, big pressure
Big area, small pressure
Background: Basic Physics of Fluids
Surface Tension Origin
Surface tension arises at liquid-solid and liquid-gas
interfaces.
At the interface what was a 3-D liquid molecular structure
is disrupted and becomes a 2-D one.
Thus, the molecular forces that are elsewhere distributed
over three directions become concentrated into two
directions at the interface, leading to an increase in
pressure.
Water (3-D)
Air
Interface
Water (2-D)
Surface tension in
spherical water droplet
Because of a force balance a droplet
of water in air, would lose its shape.
But in the presence of surface tension
the internal pressure at the surface is
increased, and the droplet maintains
its shape.
The change in pressure across the
interface to the surface tension:
Surface tension
Background: Basic Physics of Fluids
Effect of surface tension on Wet Sand
The pin, which has a higher
density than water, is floating on
the surface.
See the surface of the water is
depressed around the pin.
The vertical component of
surface tension is what supports
the weight of the pin.
Walking on water is difficult, . But walking on dry sand is also
difficult—think beach volleyball. However, when we wet sand
with water sufficiently to “activate” surface tension effects we
can easily walk on the mixture.
Effect of surface tension on a pin
Effect of surface tension on water
Background: Basic Physics of Fluids
Effect of surface tension in Capillary Tube
Surface tension acts in
a direction parallel to
the surface of the
interface.
The interaction of the
gas liquid interface is
altered at the solid wall
This is caused by a
combination of liquid
molecular structure
and details of the wall
surface structure
The oxygen exchange in
the lungs takes place
across the membranes
of small balloon-like
structures called alveoli
attached to the branches
of the bronchial
passages.
These alveoli inflate and
deflate with inhalation
and exhalation.
The behavior of the
alveoli is largely dictated
by LaPlace’s law and
surface tension.
Effect of surface tension in Alveoli
The surface tension of water provides the
necessary wall tension for the formation of bubbles
with water. The tendency to minimize that wall
tension pulls the bubbles into spherical shapes
(LaPlace’s law).
Effect of surface tension and Bubbles
Background: Basic Physics of Fluids
Viscosity Concept
Same speed with no friction & no
interaction between molecules Speed of the fluid zero on the walls
maximum at the center.
Frictional Force
Force opposite the flow
Adjacent portions of he fluid flow past one
another with different speeds, a force must
be exerted on the fluid to maintain the flow.
Experiment show that
coefficient of viscosity
Ideal Fluid Real Fluid
Block sliding across a rough
surface experiences a frictional
force opposing the motion.
The tendency to resists flow = Viscosity of a fluid.
The pulmonary artery-
connecting the hear to the
lungs, is 8.5 cm long, and has
pressure difference over this
length of 450 Pa. If the inside of
the artery is 2.4 mm, what is the
average speed of blood in the
pulmonary artery?
What’s the pressure difference
if v is 15 m/s?
Practical Example: Blood Speed
v = 1.4 m/s
P Difference = 480 Pa
Background: Basic Physics of Fluids
Newton’s Law of Viscosity
For a given rate of angular deformation of
a fluid, shear stress is directly
proportional to viscosity
Consider flow between two horizontal
parallel flat plates spaced a distance h
apart.
Apply a tangential force F to the upper
plate sufficient to move it at constant
velocity U in the x direction
At y = 0 the velocity is zero, and at y =
h it is U, the speed of the upper plate.
Experiments show that
F directly proportional to the plate area (A), speed (U),
and inversely proportional to the spacing (h).
Why F decreases with h?
The upper plate, through viscosity, attempt to “drag” the
lower plate; but this effect diminishes with distance h
between the plates, so the force needed to move the
upper plate decreases.
Background: Basic Physics of Fluids
Why is velocity zero at the stationary bottom plate,
and equal to the speed of the moving top plate?
Because of the no-slip condition for viscous fluids:
it is an experimental observation that such fluids
always take on the (tangential) velocity of the
surfaces to which they are adjacent.
We need to consider the nature of Real
Surfaces in contrast to Perfectly-smooth ideal
Surfaces.
Real surfaces are very jagged on microscopic
scales.
Their roughness permits parcels of fluid to be
trapped and temporarily immobilized.
Such a fluid parcel clearly has zero velocity with
respect to the surface,
It is in this trapped state only momentarily before another
fluid particle having sufficient momentum to dislodge it.
This constant exchange of fluid parcels at the solid surface
gives rise to the zero surface velocity in the tangential
direction characterizing the no-slip condition.
The effect of shear stress between the parcels on the
surface and the immediately adjacent ones impose a
tangential force on these elements causing them to attain
velocities nearly the same as those of the adjacent
surfaces.
Units of Viscosity
Background: Basic Physics of Fluids
At this point it is useful to consider the dimensions
and units of viscosity.
Background: Basic Physics of Fluids
Newtonian Fluids
Typically, liquids take on the shape of the container they are
poured into. We call these ‘normal liquids’ Newtonian fluids
Examples:
glycerine water Air/gasses
drilling mud toothpaste mayonnaise chocolate ketchup mustard modern paints blood
Mathematically
Non-Newtonian fluids
Fluids not following the Newtonian law. Non-Newtonian fluids.
Examples
Kerosene oil
Flow behavior index Consistency index
Background: Basic Physics of Fluids
Diffusion of Momentum
Mixing on molecular scales of a
high-momentum portion of flow
with a lower-momentum portion.
a general “smoothing” of
the velocity profiles.
Physical description
Plates have zero speed, fluid
motionless.
Top plate move with speed U, at
later instant, because of a
tangential force F. Initial velocity
profile
appears
More velocity profile
appears:
The high-momentum
adjacent fluid parcels
(upper plate) collided
with zero-momentum
parcels, and exchanged
momentum.
At a later time,
velocity field is
smooth and
nearly linear and
steady.
Background: Basic Physics of Fluids
Classification of Flow Phenomena
Steady and unsteady flows
Flow dimensionality
Uniform and non-uniform flows
Rotational and irrotational flows
Viscous and inviscid flows
Incompressible and compressible flows
Laminar and turbulent flows
Separated and unseparated flows
Background: Basic Physics of Fluids
Steady and unsteady flows If ALL properties are independent of time, the flow is steady; otherwise, unsteady.
Transient Doesn’t persist for ‘long times’
Stationary Qualitative behavior fixed
detail motion change with time.
Complete unsteady
Background: Basic Physics of Fluids
Flow dimensionality
The dimensionality = the number of coordinates
needed to describe ALL properties of the flow.
Infinite plates
v and w don’ t depend on any coordinates
= constant = zero.
u depends only on y.
assuming density and temperature constant
(say a gas), easy to see that p also constant.
The flow is 1D.
Varies with x and y.
Two planes of vectors (in z-
direction) exactly alike
The flow is 2D
Infinite z-direction
Two planes of velocity
vectors differ.
(i) z dependence of u and v,
(ii) the x and y dependence.
w, also varies with x, y and z.
Background: Basic Physics of Fluids
Uniform and Non-uniform flows
ALL velocity vectors are identical at
every point instant of time.
Otherwise non-uniform.
Uniform: All same length
and same direction.
Locally uniform: Different
magnitudes, same directions.
Away from real flow physics,
but useful.
Locally same
Different in flow direction
Non-uniform: All different
lengths, directions.
Common in actual flows
Mathematical definition:
U: velocity vector
S: direction of differentiation( e.g. x)
Background: Basic Physics of Fluids
Rotational and Irrotational flows
A flow field with velocity vector U is said to be
rotational if
otherwise, it is irrotational.
Curl U is called vorticity, denoted w.
if the corresponding flow field is rotational
Uniform flow
Any uniform flow is
automatically irrotational since
Case 2
Case 1
Background: Basic Physics of Fluids
1-D Shear Flows
Only nonzero component u and varies
only with y.
all derivatives are zero except of u.
1-D flows are rotational
2-D Shear Flows over a step
Red: high magnitude, but negative, vorticity
Blue: high magnitude, but positive, vorticity Black: Paths followed by fluid elements.
Flow from
left to right..
Green: (Large area) Zero
vorticity
Flow from left to right implies large u and small v.
Why vorticity is negative?
v is very small, so no contribution to
Along the top of the step u is increasing with y - from a zero on the step
(no-slip condition) out to the speed of the oncoming flow, so
A similar argument holds for the positive vorticity at the upper boundary.
Background: Basic Physics of Fluids
3-D Shear Flow in a Box
Blue: Large values of vorticity
Red: vorticity values near zero
Extreme variability in a three-dimensional
sense of the magnitude of vorticity
throughout the flow field.
Complicated structure of the lines indicating
motion of fluid parcels
The vortical shape near the top of the box
Specific features of the flow field
Evaluation: The behavior of the flow filed seem
to be a combined effects of the viscosity, no-slip
condition and diffusion of momentum.
Background: Basic Physics of Fluids
Viscous and inviscid flows
Consider a case in which viscosity is small (e.g. a gas flow at low temperature);
Hence, in view of the expression above, the shear stress will be reasonably small
And, in turn, the corresponding shear forces will be small.
Assume that pressure forces are large by comparison with the shear forces.
In this situation it might be appropriate to treat the flow as inviscid and ignore the effects of viscosity.
In situations where viscous effects are important, they must not be neglected, and the flow is said to be viscous.
Incompressible and compressible flows
Gases are, in general, quite compressible; yet flows of gases can often be treated as incompressible flows.
A simple, and quite important, example of this is flow of air in air-conditioning ducts.
A flow will be considered as incompressible if its density is constant.
However, there are some flows exhibiting variable density, and which can still be analyzed accurately as
incompressible.
Background: Basic Physics of Fluids
Laminar and Turbulent flows: Simplified Example
Laminar and turbulent flow of water from a faucet
Steady laminar Periodic, wavy laminar Turbulent
Low-speed flow in which the
trajectories followed by fluid parcels
are very regular and smooth.
No indication that these trajectories
might exhibit drastic changes in
direction.
The flow is steady.
Still laminar flow, but permitting a
higher flow speed if we open the
faucet more.
The surface of the stream of water
begins to exhibit waves changing
in time.
The flow has become time
dependent
Show a turbulent flow
corresponding to much higher
flow speed.
Paths followed by fluid parcels,
complicated and entangled
indicating a high degree of mixing.
3-D, unsteady, and very difficult to
predict in detail.
Background: Basic Physics of Fluids
Laminar and Turbulent flows: Experiment Example
Reynolds’ experiment using water in a pipe to study
transition to turbulence; (a) low-speed flow, (b) higher-speed
flow.
As long as the flow speed is low the flow
will be laminar,
As soon as the flow speed is fast enough
turbulent flow will occur.
The transition to turbulence as the flow
speed is increased
Transition to turbulence in spatially-evolving flow
As the flow moves from left to right
we see the path of the dye streak
become more complicated and
irregular as transition begins.
Only a little farther down stream the
flow is turbulent leading to complete
mixing of the dye and water if the
flow speed is sufficiently high.
Evaluations: Most flows in
engineering practice are turbulent,
and the main tool available for their
analysis is CFD.
Navier–Stokes equations are capable
of exhibiting turbulent solutions,
Background: Basic Physics of Fluids
Flow with no separation
The pancake syrup oozes along the top of the
step.
As the flow reaches the corner its momentum
is very low due to its low speed, and it exhibits
no tendency to “overshoot” the corner.
Flow with separation
Same experiment but with a less viscous
fluid higher-speed flow.
The flow momentum is high, and difficult to
turn the sharp corner without overshooting
The result is the primary recirculation region.
Common Features of Separated Flows
Dividing streamline; Reattachment point;
Secondary recirculation region
Separated flow can occur for flow over airfoils.
Evaluation: Understanding of separation mechanism is
very important since the behavior are often seen In
engineering device: interior cooling-air circuits of aircraft
engine turbine blades.
Background: Basic Physics of Fluids
Flow Visualization
Importance: Now CFD can produce
details of 3-D, time-dependent fluid
flows. So visualization is all the
more important than ever!
Streamlines (Fluid elements trajectories)
A continuous line such that the tangent
at each point is the direction of the
velocity vector at that point.
Pathlines (time-dependent)
The trajectory of an individual element of fluid.
Background: Basic Physics of Fluids
Streaklines (time-dependent)
The locus of all fluid elements that have previously passed through a given point.
Streamlines, Pathlines, and Streaklines for the case of a smoke being continuously emitted by a chimney at
point P, in the presence of a shifting wind.
In a steady flow, streamlines, pathlines, and streaklines all coincide. Here all can be marked by the smoke line
Flow Visualization
Pathlines Future Flow
history (t-dependent)
Streamlines: Flow history at
each instant (t-independent)
Streaklines: Past Flow
history (t-dependent)
Background: Basic Physics of Fluids
Background: Basic Physics of Fluids
The Equations of Fluid Motion
Lagrangian view of fluid motion
Suitable for fluid mechanics
Coordinate system fixed, and within which fluid
properties are measures as functions of time
as the flow passes fixed spatial locations.
More suitable for practical purposes- all
engineering analyses of fluid flow.
Can’t produce “total” acceleration along the
direction of motion of fluid parcels as needed for
use of Newton’s second law.
Suitable for solid mechanics
Watching the path of each parcel as it moves
from some initial location, = “placing a
coordinate system on each fluid parcel” and
“riding on that parcel as it travels through the
fluid.”
The equations simple as can be obtained
from Newton’s second law.
For complete description a very large
number of fluid parcels is necessary-a
significant burden.
Eulerian view of fluid motion
Analysis of traffic flow along a freeway
Eulerian Viewpoint: A certain length of freeway may
be selected for study and called the field of flow.
Obviously, as time passes, various cars will enter and
leave the field, and the identity of the specific cars
within the field will constantly be changing. The traffic
engineer ignores specific cars and concentrates on
their average velocity as a function of time and
position within the field, plus the flow rate or number
of cars per hour passing a given section of the
freeway. This engineer is using an eulerian
description of the traffic flow.
Langrangian Viewpoint: Other investigators, such
as the police or social scientists, may be interested in
the path or speed or destination of specific cars in the
field. By following a specific car as a function of time,
they are using a lagrangian description of the flow.
Length of Freeway
Practical Example 1
Lagrangian Current Observations Eulerian Curent Observations
Practical Example 2: Measurements of ocean current
We follow a parcel of fluid as it moves.
Examples would be to measure
temperatures from a weather balloon,
or from a free-floating buoy.
To make measurements is to have an observation site
geographically fixed.
For example, we can measure temperature at a fixed
weather station or from an anchored buoy in the ocean.
Lagrangian view
Eulerian View
The short and
straight arrows
near shore
represent current
measurements
taken by fixed
Eulerian stations
either on shore,
on ocean towers,
or on moored
bouys.
Shows the
tracks of many
drifters through
time.
Practical Example 3: Measurements of ocean current
The substantial derivative: Derivation
Eulerian frame does not produce “total” acceleration along the direction of motion of fluid
parcels as needed for use of Newton’s second law.
This difficulty is overcome by expressing acceleration in an Eulerian reference frame in
terms of those in a Lagrangian system for deriving the equation of motion.
= The substantial derivative of a field
property f ( x, y, z, t ) in a flow field velocity
vector U = (u, v, w).
The x-direction acceleration is given by
The substantial derivative: Example 1
The substantial derivative: Example 1
Physical Interpretation: The radiation of a quantity from a
volume equals the flux of the quantity coming through the
surface of the volume.
Uses: Additionally, the theorem provides a way to transform
potentially complicated integration over surfaces to much
less complicated ones over volumes and vice versa.
1-D case: Line integral from points a to b
3-D case: Volume R and Surface integral on S
F any smooth vector field over a volume R ⊂ R3 with a smooth boundary S
Review of Pertinent Vector Calculus: Gauss’s Divergence Theorem
Transforming line integral to a point
Transforming volume integral to
the surface integral
Review of Pertinent Vector Calculus: Application of Gauss’s Theorem to a Scalar Function
By applying the Gauss’s theorem, we have
Let F be a vector expressed as below.
The integral of the gradient of a scalar function f over a
volume R equal the integral of f over the surface S.
Review of Pertinent Vector Calculus: Transport theorems
Leibnitz’s Formula:
Physical meaning: The time-rate of change of a quantity f in a region =
the rate of change of f within the region
+ the net amount of f crossing the boundary of the region due to movement of the boundary
Purpose: to derive the equations of fluid dynamics, and other transport phenomena by exchanging
integration and differentiation
1-D case:
The General Transport Theorem:
3-D case:
Reynolds Transport Theorem
Review of Pertinent Vector Calculus: Transport theorems
Substantial derivative Velocity of a fluid element Volume of a fluid element
Review of Pertinent Vector Calculus: Reynolds theorems
Hence, we can replace the ordinary derivative on the left-hand-side with the substantial derivative.
This is because R(t) is a fluid element traveling with the fluid velocity (u, v, w).
Conservation of Mas: the Continuity equation
We consider system with a fixed mass m of fluid contained in an arbitrary region R(t), where
boundary S(t) can move with time. Furthermore, the system has the fixed mass. It does not matter
whether it is the same mass all time- only that the amount is the same.
Relation of mass of the system with the density of the fluid comprising it.
To make it convenient, imagine a system where a balloon filled with hot air surrounded by
cooler air. As heat is transferred from the balloon to its surroundings, the temperature of the
air inside the balloon will decrease, and the density will increase. At the same time the size of
the balloon will shrink, corresponding to a change in R(t). But the mass of air inside the
balloon remains constant—at least if there are no leaks.
Decrease of temperature inside the balloon, and decrease of volume, and so change in R(t)
Increase of density of air inside the balloon. Mass of air: constant in time
Derivation of the continuity equation
Conservation of Mas: the Continuity equation
Applying General Transport theorem to exchange
Integration and differentiation.
R (t) is a fluid element with U the fluid velocity of
the flowing fluid (instead of W).
Using Gauss’s theorem and changing the surface
integral S (t) to volume integral R (t)
Substituting in the above equation.
The Differential Continuity Equation
R(t) was arbitrary (i.e., it can be made arbitrarily small— within the confines of the continuum
hypothesis),
the integrand must be zero everywhere within R(t).
If this were not so (e.g., the integral is zero because there are positive and negative contributions
that cancel), we could subdivide R(t) into smaller regions over which the integral was either positive
or negative, and hence violating the fact that it is actually zero.
Conservation of Mas: the Continuity equation
Other forms of the Differential Continuity Equation
Conservation of Mas: the Continuity equation
Continuity Equation: and steady Incompressible
For steady flow
For incompressible flow
b = U (vector), f (scalar) =
constant density
Control volume (integral) analysis of the continuity equation
Differential Form Integral Form
Using Gauss’s theorem
Using Transport theorem
U: Flow velocity in the Control volume R (t); W: velocity of the Control surface S (t).
S (t)
Control volume (integral) analysis of the continuity equation
It is clear from this equation that the only parts of the
control surface that are actually important for analysis
of mass conservation are those through which mass
can either enter or leave the control volume.
Applications of Control-Volume Continuity Equation: Steady Case
Entrance/exit
fixed in space
and time, W = 0
For incompressible fluid
Thus, a steady incompressible
flow must speed up when
going through a constriction
and slow down in an
expansion, simply due to
conservation of mass.
Slow down
Speed up
Practical Application: Speeding up flow through a constriction
Practical Problem: Spray
Applications of Control-Volume Continuity Equation: Unsteady Case
Defined Control Volume: Displayed with a dashed line
– simpler control volume
the entrance and exit are
independent of time and
not in motion, implying
that W = 0 on all of Se.
We next denote the time-rate of change of volume of fluid by
Can write the integral continuity equation as
Once ˙V is known, it can be used to find the time-rate of change of any appropriate physical dimension
associated with the volume of fluid.
Practical Applications of Control-Volume Continuity Equation: Unsteady Case
Applications of Control-Volume Continuity Equation: Exit & Entrance in motion
Engine exhaust nozzle Engine intake
Fuel inlet
Steady flight
The exhaust plane of the engine are moving
with respect to the air, so W is non-zero .
Applications of Control-Volume Continuity Equation: Exit & Entrance in Motion
= 0 as the air is considered to be still
As no component of engine motion in the direction normal to the plane through which fuel must flow, W · n = 0
Practical Problem of Control-Volume Continuity Equation: Exit & Entrance in motion
Derivation of the Navier–Stokes Equations
A general force balance
consistent with Newton’s second
law of motion
Developing a multi-dimensional
form of Newton’s law of
viscosity
Formulation of the Momentum
Equations for a control volume of a
fluid element.
Evaluating surface forces
appearing in the equations of
momentum
Obtaining the differential form of
the momentum equations.
Arguing the physical
significance / description
of the stresses
Arriving at the Navier-Stokes
Equations (unsteady, incompressible)
1-D 3-D
Rotating
Compressing (p)
Normal Stresses Tangential Stresses
A force balance: Newton’s second law of motion
Derivation of the Navier–Stokes Equations
Applicable if a material region viewed as consisting of point masses.
Applicable if a material region NOT viewed as consisting of point masses (e.g. fluid element).
General form of Newton’s second law
Force per unit volume = time-rate change of momentum per unit volume
Density = mass/volume
To calculate the time-rate change of momentum, utilizing Eulerian view of fluid flow.
Allow us to use the substantial derivate to represent acceleration as below.
x-component of momentum per unit volume with the region R (t) as a fluid element.
(U is a complete velocity vector)
Derivation of the Navier–Stokes Equations
are Body and Surface forces
By using Reynolds Transport theorem
By applying Gauss’s theorem
to the surface integral
By applying product-rule for
differentiation
f = and b = U
Divergence free condition of incompressible flow , constant density (divergence of density = 0)
By applying the substantial derivative
definition with f = u
Formulation of the Momentum Equations for a control volume of a fluid element.
Derivation of the Navier–Stokes Equations
Analogous results for y and z momentum allows to write the equation in the vector form as below.
Where , the dot product of a tensor and a vector. Each component of Fs is the (vector) dot product of the
corresponding row of the tensor with the column vector n.
By applying the Gauss’s theorem to the surface integral with F =
R(t) was arbitrary (i.e., it can be made arbitrarily small— within the confines of the continuum
hypothesis),
the integrand must be zero everywhere within R(t).
If this were not so (e.g., the integral is zero because there are positive and negative contributions
that cancel), we could subdivide R(t) into smaller regions over which the integral was either positive
or negative, and hence violating the fact that it is actually zero.
The general, momentum balance equation, valid at all points of any fluid flow
Differential form of the momentum equations
Treatment of Surface Forces
Derivation of the Navier–Stokes Equations
The is a matrix and Fs is a 3-D vector. Thus, must be a 3x3 matrix with total 9 elements.
The elements are two types: Normal and tangential forces.
The forces are on an arbitrary fluid element
In x-component , implies that the x is acting on a face perpendicular to the x-axis.
The second script y indicates the stress is in the y-direction.
Each face must have two components of share stress because of 2 corresponding tangential directions.
The forces/stresses have been displaced only ‘visible’ faces only.
The analogous ones are present also on each of the corresponding three faces.
From the Newton’s law’s of viscosity for 1-D flow in the x-direction
changing only in the y-direction, we have.
Imagine, another fluid element, just above and sliding past the y-
plane in the x-direction, and in contact, then the Newton’s law is
associated the 3-D share stress component
Similarly, if we were to imagine another fluid element sliding past the x
face of the cube in the y direction, we would expect this to generate x-
direction changes in the v component of velocity.
Hence it follows that is associated with expressed in the
partial derivatives for the flow is no more 1-D.
Treatment of Surface Forces
Derivation of the Navier–Stokes Equations
The components of the shear stress are related as follows:
If we consider the two shear stress components described
above and imagine shrinking the fluid element to a very
small size we would see that in order to avoid discontinuities
of along the edge of the cube between the x and y faces it
would be necessary to require
On the x-face, we have
On the x-face, we have
We define the share stress acting on the surface of a
3-D fluid element as
Since we cannot have 2 velocity gradients equal as
As this would imply an irrotational flow and most
flows are not irrotational.
Derivation of the Navier–Stokes Equations
Physical significance of the tangential stresses
According to the 1-D flow above, Newton’s law of
viscosity relates shear stress to rate of angular
deformation through the viscosity.
Therefore, in 3-D to understand the physical
significance of the stresses, we need to find the source
of for each face of our fluid element.
We focus only on the x-face (yz-plane), but the argument
we use will apply to any of the faces.
The figure shows the deformation induced by x- and y-
direction motions caused by fluid elements moving past
this x-face, viewed edgewise such that both contribute to
Changes of v in the x direction,
by y-direction motion, will tend to
distort the fluid element by
moving the x face in a generally
counter-clockwise direction.
Changes of u in the y direction,
by x-direction motion, will tend to
distort the fluid element by
moving the x face in a generally
counter-clockwise direction.
angular deformation rate
Imagine v / u being momentum per unit mass, so its time rate of
change, due to an adjacent fluid element, creates a force which
then produces angular deformation of the fluid element.
Time-rate change (Momentum) = Force produce angular
deformation
Derivation of the Navier–Stokes Equations
Physical significance of the normal stresses & pressure
Two contributions to the normal force:
(i) pressure,
(ii) normal force of viscous origin.
Evidence: Pour very viscous fluids like molasses. It will
flow very slowly because normal viscous forces are able
to support tensile stresses (e.g. surface tension) arising
from gravitational force acting on the falling liquid.
Drop a solid object same density as molasses, and at
the same time pour the molasses. Solid object would fall
to the floor much more quickly because of the absence
of any normal viscous force.
Normal stresses physically relate to stretching or
compressing of the fluid element.
In the presence of the this stress, u is, in general
changing with x throughout the fluid element, so
is different on the two faces; so these 2 contributions
must be added.
But, as the element size is shrunk to zero, these
approach equality.
Construction of the Matrix
In the momentum equation, force contributions come
from taking the divergence of , means that the
first column of should provide the force to balance
time-rate of change of momentum associated with the
u component of velocity. Thus, the information in the
first column of should consist of and
the contribution of pressure p acting on the x face.
In the above matrix, the physical significance of
the pressure p is viewed as compressive – on all
the faces. They act in directions opposite the
respective outward unit normal to the fluid
element. So, al such terms will have minus signs
in the formulas to follow.
Derivation of the Navier–Stokes Equations
Arriving at the Navier-Stokes Equations
Navier-Stokes Equations: time-dependent, constant density and constant viscosity Flows
Analysis of the Navier–Stokes Equations
What to know about analysis of the NS equations?
Mathematical structure of the NS equations
Physical interpretation of each of its terms.
Mathematical Structure: Difficulties
The unknown dependent variables are the three velocity
components u, v and w, and the pressure p. Hence, there
are four unknown functions required for a solution, and only
three differential equations.
The remedy for this is to explicitly invoke conservation of
mass (Continuity equation).
The three momentum equations can be viewed as the
respective equations for the three velocity components,
implying that the divergence-free condition (Continuity)
must be solved for pressure.
This is particularly difficult because pressure does not even
appear explicitly in this equation of continuity.
in addition, boundary (space) and initial (time) conditions
are required to produce particular solutions corresponding
to the physics of specific problems of interest.
Momentum equations
Continuity equation
Advantages
To simplify the equations to treat
specific physical flow situations.
Knowing the physics represented
by each term/s, we can readily
determine which terms can be
omitted.
Even an elementary understanding
of the structure of PDEs and their
solutions can help use the
commercial CFD / codes correctly.
Unfortunately the Commercial
CDF codes /programs have been
written almost with a lack of the
problem formulation.
It comprise a 3-D, time-dependent system
of nonlinear partial differential equations.
The terms such as are nonlinear in
the dependent variable u.
Physical interpretation: Inertial Terms
Analysis of the Navier–Stokes Equations
Substantial Derivative: The left-hand side of this
equation is the substantial derivative of the u
component of velocity,
Inertial Terms: These terms are often called
the “inertial terms” in the context of the N.–S.
equations, and we recall that they consist of two
main contributions: local acceleration and
convective acceleration.
Accelerations: It is useful to view them simply as
accelerations resulting from spatial changes in the
velocity field for it is clear that in a uniform flow
this part of the acceleration would be identically
zero.
Momentum Change: These terms also be viewed
as time-rate of change of momentum per unit
mass.
Uniform flow:
acceleration = 0
On-uniform flow:
with an acceleration
Physical interpretation: Pressure Forces
The first term on the right-hand side of the above
equation represents normal surface forces due
to pressure,
The present form this is actually a force per unit
mass as it must be to be consistent with time-
rate of change of momentum per unit mass on
the left-hand side as below.
Analysis of the Navier–Stokes Equations
Physical interpretation: Viscous Forces
Diffusion of Momentum: Viscosity arises at the
molecular level, and the terms given above are
associated with molecular transport (i.e.,
diffusion) of momentum.
Smoothing Process: In general, second
derivative terms in a differential equation are
usually associated with diffusion, and in both
physical and mathematical contexts this
represents a smearing, or smoothing, or mixing
process.
Effects on for high-viscosity fluids: The profile
varies smoothly coming away from zero velocity
at the wall, and reaching a maximum velocity in
the center of the duct. The core region of high-
speed flow is relatively small, and the outer
(near the wall) regions of low speed flow are
fairly large.
Reason: Large viscosity is able to mediate diffusion of
viscous forces arising from high shear stress near the
solid surfaces far into the flow field, thus smoothing
the entire velocity profile.
Effects on for low-viscosity fluids: In this flow we
see a quite narrow region of low-speed flow near the
solid boundaries and a wider region of nearly
constant-velocity flow in the central region of the duct.
Reason: The low momentum fluid near the wall does
not diffuse away into the central flow region in a
sufficiently short time effect passing fluid parcels; the
speed of this region is nearly uniform, so for a given
mass flow rate it must be lower.
Scaling and Dimensional Analysis
It makes it possible to scale results from model tests to predict corresponding results for
the full-scale prototype.
Scaling the Navier-Stokes equations
Objectives: to attempt to cast the Navier–Stokes equations
in a form that would yield exactly the same solution for two
geometrically similar objects, via scaling.
Scaling the Navier-Stokes equations
We see from this that the scaled continuity
equation is identical in form to the unscaled one,
It should first be observed that all quantities on the
left-hand side of this equation are dimensionless, as
are all derivative terms on the right-hand side. This
implies that it should be the case that
In general we express the Reynolds number as
where U and L are, respectively, velocity and length
scales; nyu, as usual, denotes kinematic viscosity. In
the present case, we have
If we now suppress the “ ∗ ” notation we can express the
2-D momentum equations in dimensionless form as
where Fr is the Froude number, defined as
Scaling the Navier-Stokes equations
The above equations are dimensionless, and their solutions
now depend only on the parameters Re and Fr. In particular,
if flow fields associated with two geometrically similar
objects have the same Reynolds and Froude numbers, then
they have the same scaled velocity and pressure fields.
Scaling the Navier-Stokes equations
Practical Example: Airfoil
Full-scale portion of an actual aircraft wing
A small wind tunnel model
The airfoils are considered to be geometrically similar.
We wish to demonstrate that if Reynolds numbers are the same for the two flows, then the forces
acting on these will exhibit a constant ratio between any two corresponding locations on the two
airfoils.
We will examine only the forces due to pressure.
Scaling the Navier-Stokes equations
We first observe that by geometric similarity we must have
Now we can define pressure scales for the two flows as
Actual pressures acting on the respective elements
of surface area shown in the figure must be
Here, the are dimensionless pressures that
arise as solutions to the governing equations.
Now observe that the second equality holds because if Re is
the same in both flows it must be that the scaled pressures
satisfy ; furthermore are all
constants, the far right-hand side must be a constant. Thus,
we have shown that
and this has been done for a completely arbitrary
element of surface on which pressure is acting.
Applications of the Navier-Stokes Equations
Equations of fluid statics: 2
Static Equilibrium in Fluids: Pressure and
Depth (Alternative Approach): 3
Pascal’s Principle: 4
Buoyancy in static fluids: 5
Applications of Archimedes’ Principle:6
Analysis of Barometer: Atmospheric
Pressure Measurement: 15
Bernoulli’s Equation: Derivation: 7-8
Applications of Bernoulli’s Equation: 9-14
Analysis of Manometer:
Non-atmospheric Pressure Measurement:16
Control—Volume Momentum Equations-
Unsteady Flow: 17-18
Application: Momentum Equations Applied
to a Rapidly-Expanding Pipe Flow: 19-21
Classical Exact Solution to Navier-Stokes Equations:
Plain Poiseuille Flow (Channel Flow): 24-26
Classical Exact Solution to Navier-Stokes Equations:
The Hagen Poiseuille Solution (Pipe Flow): 29
Classical Exact Solution to Navier-Stokes Equations-
Couette Flow: 22-23
Equations of fluid statics
3-D incompressible N.–S. equations in a Cartesian coordinate system with the positive z direction opposite that
of gravitational acceleration.
Equation of continuity Momentum Equations
We have u = v = w ≡ 0, implying that all derivatives of velocity components also are
zero no viscous forces in a static fluid.
The continuity equation is satisfied trivially, and the momentum equations yields to
px = 0 implies p is not a function of x.
py = 0 implies p is not a function of y.
Integrating pz and noticing that C is
not a function of x, neither y, yields.
What causes the pressure to increase as a submarine
dives? The increased pressure is due to the added
weight of water pressing on a submarine as it goes
deeper.
How much does it go up for a given increase
in depth? If the depth increases by the amount h,
the pressure increase by the amount
Static Equilibrium in Fluids: Pressure and Depth (Alternative Approach)
Pascal’s Principle: An external pressure applied to an enclosed fluid is transmitted unchanged
to every point within the fluid.
Piston 1 is pushed down with F1 increasing
the pressure in the cylinder by
and so F2 is
Imagine A2 = 100A1, then pushing down on
Piston 1 with a force F1, we push on piston 2
with a force 100F1- our force has been
magnified 100 times.
In cylinder 1, the Piston 1 displaces a fluid volume A1d1,
and the same volume (A2d2) flows into cylinder 2. If we move piston 1 down a distance d1, piston 2
rises a distance d2= d1/100. Thus our force has
been magnified 100 times, but distance it moves
has been reduced 100 times.
Various Uses for Hydraulic Lifts
http://www.hydraulicmania.com/hydraulic_lift.htm
Buoyancy in static fluids
Buoyant Force? A fluid exerts a
net upward force on any object it
surround.
Archimedes’ Principal? An object completely immersed in a
fluid experiences an upward buoyant force equal in magnitude
to the weight of fluid displaced by the object.
Flotation An object floats when it displaces an
amount of fluid equal to its weight
Applications of Archimedes’ Principle
Maximum
safety load
As you travel from saltwater
(higher density) to freshwater
(lower density), your ship’s
hull will sink deeper into the
water- danger if too heavy.
Since the ship floats higher in
salt water (because of higher
density), bottom line is used
to indicate maximum load for
safety.
Heated air in the balloon
expands the bag, increasing the
volume of the air that the
balloon displaces.
Heated air spills out of the
balloon, decreasing its weight.
The average density of the
balloon + the hot air in it
becomes lower than that of
surrounding air, and it starts to
float.
The water of Dead Sea is
unusually dense because of
its great salt content.
Swimmers can float higher in
the water than they
accustomed and engage in
recreational activities that we
don’t ordinary associate with
a dip in the ocean.
Bernoulli’s Equation: Derivation
The 2-D, incompressible N.–S. equations can be expressed as
Assumption 1: the flow being treated as inviscid- viscous forces can be considered to be
small in compression with the other forces such as inertial, pressure and body forces.
Assumption 2: the flow being treated as steady (time-independent) and irrotatational
on the right-hand side of the above equation (right) because ρ is constant by incompressibility, we can write
, left as
and
(a system known as the Euler equations).
which is shorthand for U · U
But because C must be the same in both equations, the left-hand sides are equal, and we have
the well-known Bernoulli’s equation.
Bernoulli’s equation for gaseous flow or in cases where only elevation is constant
(y1=y2=y) with the omission of
In the above equation indicates that that if the fluid speeds up its pressure decreases. In the
equation above
if U1= U2, then p1=p2
If U2 is greater than U1, then p2 must be smaller than p1
P2 acts to increase the speed of the fluid element, and p2 acts to decrease its speed.
The element will speed up, then, only if p2 is less than p1.
Bernoulli’s Equation: Derivation
Applications of Bernoulli’s Equation
Body’s Density Measurement
Using Water
Using air: 1000
times less dense
than water in a
Bod Pod
Scuba diver
Divers usually wear a buoyancy
compensator, which contains a flexible
bag that can hold air. The bag can be
inflated with compressed air from the
cylinder worn on the back, or deflated by
letting bubbles escape into the water.
The diver can therefore adjust buoyancy
and choose to move up, down or stay at
a certain depth, like the Cartesian Diver.
Cartesian Diver
A small glass tube with an air bubble
trapped inside. As the density of tube +air
is less than the water density, the diver
floats.
When the bottle is squeezed, the pressure
in the water rises, and the air bubble is
compressed to a smaller volume.
Now the overall density of the tube and
the air is greater than that of water, and
diver sinks.
Swim bladder is an air sac
whose volume can be controlled
by the fish.
Adjusting the size of the swim
bladder, the fish can float
without effort at a given depth,
just like a Cartesian Diver.
Applications of Bernoulli’s Equation
Submerged volume of
a floating object
Tips of the iceberg
Most people know that the
bulk of an iceberg lies below
the surface of the water.
But, as with ships and
swimmers, the actual
proportions that is submerged
depends whether the water is
fresh (low density) or salt
(high density).
Submerged iceberg
Applications of Bernoulli’s Equation
Applications of Bernoulli’s Equation
Applications of Bernoulli’s Equation
We often say that a hurricane or tornado “blew the roof of a
house”. However, the house at left lost its roof not because of the
great pressure exerted by the wind, but rather the opposite. In
accordance with the Bernouilli effect, the high speed of the wind
passing over roof created a region of reduced pressure. Normal
atmospheric pressure inside the house then blew up the roof.
Applications of Bernoulli’s Equation: Torricelli’s Law
Analysis of Barometer: Atmospheric Pressure Measurement
Boundary condition Momentum Equation
As the fluid used in
Barometer have very
small vapor pressure
under normal condition
A simple measurement of the height directly gives the desired atmospheric pressure.
Analysis of Manometer: Non-atmospheric Pressure Measurement
The term can be neglected as the
density of the gas is low in
comparison the density of water.
Control—Volume Momentum Equations: Unsteady Flow
Control—Volume Momentum Equations: Steady Flow
Application: Momentum Equations Applied to a Rapidly-Expanding Pipe Flow
Steady (time-independent) and Incompressible (constant density)
Control Volume Continuity Equation
Control Volume Momentum Equation
Rapidly-Expanding Pipe flow is simplified
version of the physical situation as shown.
Such flows occur in numerous engineering
applications, especially in air-conditioning &
ventilation systems, and in plumbing systems
Rapidly-Expanding Pipe Flow Goal of the analysis: to predict the change of the flow resulting from
rapid expansion of the pipe, assuming the upstream flow is known
Continuity Equation
Momentum Equation
With u1, D1, and D3, pressure change through the expansion can be predicted.
Important. In the above equation since D1 < D3, p < 0, which in turn implies that p3 > p1.
That is, the downstream pressure is higher than the upstream pressure..
The continuity eq forces the downstream flow to be slower because of it wider area.
Hence, the flow momentum has decreased between locations 1 and 3.
A force must be applied in the direction opposite the direction of motion of the fluid to have it happen,
The only mechanism for generating this force is an increase in the downstream pressure.
Rapidly-Expanding Pipe Flow: Steady and Incompressible Flow
Why does the pressure decrease in a rapidly expanding pipe ?
Since the velocity is constant
Classical Exact Solution to Navier-Stokes Equations: Couette Flow
Consider the flow of a viscous Newtonian fluid
between two parallel plates located at y = 0 and y = h.
The upper plane is moving with velocity U. Calculate
the flow field.
Navier-Stokes Equations in Cartesian co-
ordinates.
Integrating twice yields:
Important: Exactly the same result is obtained from heuristic physical arguments when studying
Newton’s law of viscosity.
Classical Exact Solution to Navier-Stokes Equations: Couette Flow
Classical Exact Solution to Navier-Stokes Equations: Plain Poiseuille Flow (Channel Flow)
Integrating twice
Classical Exact Solution to Navier-Stokes Equations: Plain Poiseuille Flow (Channel Flow)
Integrating twice
Classical Exact Solution to Navier-Stokes Equations: Plain Poiseuille Flow (Channel Flow)
Classical Exact Solution to Navier-Stokes Equations: The Hagen Poiseuille Solution (Pipe Flow)
1. r-momentum
2. theta-momentum
3. z-momentum
and
4. Continuity equation
and write the following forms:
Classical Exact Solution to Navier-Stokes Equations: Plain Poiseuille Flow (Channel Flow)
r-momentum
theta-momentum
z-momentum
Continuity
Classical Exact Solution to Navier-Stokes Equations: Plain Poiseuille Flow (Channel Flow)
(Continuity equation)
Study of Flow in a Syphon:
a device that can be used to transfer liquids between two containers without using a pump
Objectives: to determine speed of the liquid entering the bucket and pressure in the hose at location 2.
Assume steady behavior = The volume flow rate through the
hose is small compared with the total volume of the tank =
the height of liquid in the tank does not change very rapidly. From the continuity equation since fluid is a liquid, hence
incompressible constant density) , and corresponding areas
are equal, we have
From continuity equation, we have
The area
As for heights, we observe that
The information above will be employed to determine the
speed and pressure at location 2 by applying the Bernoulli’s
equation at locations 1 & 2, and 2 & 3 and then equating them.
Study of Flow in a Syphon:
a device that can be used to transfer liquids between two containers without using a pump
Bernoulli’s equations at locations 1 and 2:
Bernoulli’s equations at locations 2 and 3:
Since
Equating the above equations as is given in the Bernoulli’s equation, we have
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