Department of Civil and Environmental Engineering
CVNG 1001: Mechanics of Fluids
FUNDAMENTALS
Difference between units and dimensions
Units express the dimensions of a given quantity, whereas dimensions express a physical
property. For example, length is a dimension that may be expressed in metres (m), feet
(ft), kilometres (km) etc.
Additionally, the units used to express the magnitude of a physical property must indicate
its dimensions.
Q1: What are some other dimensions?
Fundamental Dimensions
A fundamental dimension is one which can not be separated into other dimensions. The
choice of which dimensions are fundamental is very subjective; however, seven
dimensions have been accepted internationally as the fundamental dimensions. These are:
ƒ Length
ƒ Mass
ƒ Time Interval
ƒ Electric Current
ƒ Temperature
ƒ Luminous Intensity
ƒ Amount of substance
In fluid mechanics, many applications use the fundamental dimensions of mass, length
and time only.
These fundamental dimensions may also be referred to as reference or primary
dimensions. The units that express these fundamental dimensions are called base units or
reference units.
The table below shows the SI units associated with these fundamental dimensions, and
the internationally accepted symbol for those units.
Quantity
Length
Mass
Time Interval
Electric Current
Temperature
Luminous Intensity
Amount of substance
Unit
metre
kilogram
second
ampere
kelvin
candela
mole
Symbol
m
kg
s
A
K
cd
mol
Dimensions that are determined from these fundamental dimensions are called derived or
secondary dimensions.
Q2: What are some derived dimensions?
Academic year: 2006-2007
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Department of Civil and Environmental Engineering
CVNG 1001: Mechanics of Fluids
Dimensional Formulae
If any quantity is described in terms of its fundamental dimensions, then this expression
is termed the dimensional formula of the quantity. For example the dimensional formula
of velocity is given as the ratio of length to time. (NB: The expression of a quantity in its
base dimensions may be facilitated by using the units of the quantity).
The fundamental dimensions in the dimensional formula are usually written as a capital
letter enclosed in square brackets, for example [L], [T] and [M] are used for length, time
and mass respectively.
The power to which the fundamental dimension is raised in the dimensional formula is
called the dimension with respect to that given fundamental dimension.
Any magnitude (or quantity) that may be expressed by only a numerical value (a
numeric) has no dimension and is said to be dimensionless or non-dimensional.
Q3: Write the dimensional formula for the following quantities:
ƒ Area
ƒ Force
ƒ Mass density
ƒ Pressure
ƒ Dynamic viscosity
ƒ Strain
2
A3: [L ], [MLT-2], [ML-3], [ML-1T-2], [ML-1T-1], and dimensionless
Dimensional Homogeneity
If all terms in a given physical equation (or expression) have identical dimensional
formulae, then the equation (or expression) is said to be dimensionally homogenous.
Alternatively, an equation is dimensionally homogenous if the form of the equation does
not depend on the units of the variables within the equation.
2
Q4: The equation h = Ax 3 (where h is the depth of water, and x is the distance offshore)
is dimensionally homogenous. What is the dimensional formula of A?
Q5: The equation for discharge over a sharp-crested weir is Q = 2 3 bC d 2 g H
3
2
and is
dimensionally homogenous. What is the dimensional formula of Cd?
Q6: The equation F = G
M 1M 2
r2
is dimensionally homogenous. What is the dimensional
formula of G?
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Department of Civil and Environmental Engineering
CVNG 1001: Mechanics of Fluids
DIMENSIONAL ANALYSIS
What is Dimensional Analysis?
Dimensional Analysis is a rational procedure for combining physical variables into
dimensionless products, thereby reducing the number of variables that need to be
considered for an unknown physical process.
Dimensional Analysis involves the following main steps:
1. Identification of the important independent variables of the process. This step is
essential and is achieved only by clearly defining the problem.
2. Decide which variable is to be the dependent variable.
3. Determine how many independent dimensionless products can be found from the
variables.
4. Reduce the system of variables to the proper number of independent
dimensionless variables. Always attempt to obtain standard dimensionless terms.
These standard terms will be identified in the section on Dynamic Similarity.
Elaboration of the four Main Steps
Step 1
ƒ Clearly define the problem and determine the main variables of interest,
avoiding unnecessary variables.
ƒ Consider what basic laws govern the physical process, even if only crudely.
ƒ List the variables in terms of broad categories, such as geometric material
properties, spatial effects or temporal effects.
ƒ Include physical parameters that may be considered constant but may play an
important role.
ƒ Ensure all variables are independent, for example, specific weight, γ, is equal to
the product of mass density, ρ, and the acceleration due to gravity, g. Consider ρ
and g instead of γ, as ρ is independent of g.
Step 2
ƒ Determine the parameter that is to be under investigation, for example force on
an object or discharge over a weir.
ƒ It is usual that this parameter appears only once in the dimensionless terms
which will be generated in step 4
Step 3
ƒ Determine the number of dimensionless terms expected.
ƒ This stage is facilitated by a theorem known as the Buckingham PI Theorem.
The Buckingham PI Theorem states that:
In a dimensionally homogenous equation involving “n” variables, the
number of dimensionless products (or PIs) that can be formed from “n”
variables is “n-r” where “r” is the number of fundamental dimensions
encompassed by all the variables.
Step 4
ƒ Reduce variables to the dimensionless ratios using either the Algebraic (also
called Equation or Rayleigh) Method or the Tabular Method.
ƒ Standard dimensionless ratios should be used where possible
ƒ Check PIs (Π1, Π2, ….) to ensure that these ratios are truly dimensionless
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Department of Civil and Environmental Engineering
CVNG 1001: Mechanics of Fluids
Q7a: A process is governed by the variables Q (discharge), N (revolutions per second), D
(diameter), P (pressure) and ρ (mass density). Write the dimensional formula of each
variable.
Q7b: How many fundamental dimensions are there?
Q7c: How many dimensionless PIs are there?
Q7d: Obtain the dimensionless PIs.
A7a: Q [L3T-1], N [T-1], D [L], P [ML-1T-2] and ρ [ML-3]
(Hint: use the units as a guide)
A7b: 3 fundamental dimensions of mass length and time
A7c: According to the Buckingham PI theorem 2 dimensionless PIs are expected
A7d: Using the Tabular Method to reduce variables to dimensionless PIs
First tabulate the variables and insert the dimension of the fundamental dimensions as in
the table below.
Q
N
D
P
ρ
[M]
0
0
0
1
1
[L]
3
0
1
-1
-3
[T]
-1
-1
0
-2
0
The aim is to eliminate dimensional variables and obtain the dimensionless terms. This is
achieved by multiplying or dividing variables by a selected variable to remove the
dependence of one fundamental dimension. The dependence of the variables on the other
fundamental dimensions is removed in a similar manner, one fundamental dimension at a
time. The process is complete when the table contains only zeroes.
The first fundamental dimension chosen to remove variable dependence is a matter of
personal choice, but for efficiency the dimension corresponding to column with the most
zeroes is selected. Therefore select [M] column.
To make variables independent of the mass dimension, divide or multiply by a variable.
The only variables with a mass dimension are P and ρ. Choose ρ.
Q
N
D
P/ρ
ρ/ρ
[M]
0
0
0
0
0
[L]
3
0
1
2
0
[T]
-1
-1
0
-2
0
The hatched row need not be considered again as the parameter ρ/ρ is dimensionless and
cannot be used again as a multiplier or divider. Consider the length dimension next, and
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Department of Civil and Environmental Engineering
CVNG 1001: Mechanics of Fluids
use D as a multiplier or divider. The variable being investigated should not be chosen as a
divider or multiplier. In this case Q is the variable being investigated.
3
Q/D
N
D/D
P/ρD2
[M]
0
0
0
0
[L]
0
0
0
0
[T]
-1
-1
0
-2
Again, the hatched row need not be considered again as the parameter D/D is
dimensionless and cannot be used again as a multiplier or divider. Finally consider the
time dimension, and use N as the multiplier or divider.
Q/ND3
N/N
P/ρN2D2
[M]
0
0
0
[L]
0
0
0
[T]
0
0
0
Again, the hatched row with the parameter N/N need not be considered again.
Finally the two dimensionless PIs are Π1 and Π2 where Π 1 =
Q
ND
3
and Π 2 =
P
.
ρN 2 D 2
Note: If Q is the variable being investigated (the dependent variable) then one can
write Q = f (N , D, P, ρ ) , where f is some function of the variables. Once the
dimensionless PIs are obtained, then one can write Π 1 = g (Π 2 ) , where g is some function
of the dimensionless PIs. (Π1 usually contains the variable being investigated).
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Department of Civil and Environmental Engineering
CVNG 1001: Mechanics of Fluids
SIMILARITY
Introduction to Physical Similarity
Physical modelling is used quite often to investigate phenomena that cannot be described
completely by an analytical method. Generally the model is scaled down, but sometimes
the model may be larger depending on the phenomena being investigated. Physical
modelling can only be valid if the prototype and the model are said to be physically
similar.
Two systems are said to be physically similar in respect to certain specified physical
quantities when the ratio of corresponding magnitudes of these quantities between the
two systems is everywhere the same.
Types of Physical Similarity
Geometric Similarity
Geometric similarity implies similarity of shape. Therefore the ratio of any length in one
system to the corresponding length in the other system is everywhere the same. This ratio
for a given length (e.g. length of the longest side of a rectangle) is called the scale ratio
or scale factor. Models that maintain geometric similitude, for all length parameters, are
referred to as undistorted models. Distorted models are physical models in which the
scale ratios for different length parameters (between the model and the prototype) are
different. For example, in some distorted models the horizontal length scale ratios and the
vertical length scale ratios are different. In cases where the surface finish (expressed
perhaps as the roughness height) may be relevant, it is necessary to also scale the
roughness height. This is not always possible, resulting in a distorted model.
Note that in scaling the dimensions of the model, forces that may have been negligible in
the prototype may assume significance at the model scale. Example is in modelling the
dimensions of rivers where the chosen scale is applied to both the horizontal and vertical
dimensions. For consistency of the scale the depth assumes such small values that the
flow regime may be laminar, and surface tension forces become significant. In such a
case, the use of a distorted model might be required to circumvent these problems.
Kinematic Similarity
Kinematic similarity implies similarity of motion. This means that both the similarity of
lengths and of time intervals must be satisfied. Firstly, kinematic similarity is similarity
of velocities as the corresponding lengths in the two systems are in a fixed ratio and
corresponding time intervals are also in a fixed ratio. Secondly, kinematic similarity
means similarity of accelerations. Geometric similarity does not imply kinematic
similarity. However, if two systems are to be kinematically similar, then a necessary prerequisite is that the systems be geometrically similar.
Dynamic Similarity
Dynamic similarity implies similarity of forces. For two systems to be dynamically
similarity, the magnitude of forces at similarly located points in each system are in a
fixed ratio. Dynamic similarity also implies that the magnitude ratio of any two forces in
one system is the same as the magnitude ratio of the same two forces in the other system.
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Department of Civil and Environmental Engineering
CVNG 1001: Mechanics of Fluids
In addition, for systems that vary with time, the magnitude ratio of any two forces must
be the same at corresponding points of the two systems and at corresponding times.
If one flow system is to be compared with another flow system then the behaviour of the
fluid must be similar in both systems. Therefore, if the motion of the fluid behaviour is
similar, then that implies that kinematic similarity exists. As stated earlier, one
requirement for kinematic similarity is that geometric similarity exists, and for flow
systems this means geometric similarity of boundaries. Additionally, kinematic similarity
implies that the direction of any fluid particle is similar. However, the direction is
controlled by the resultant forces acting on the particles. Since the direction is similar,
then the ratio of forces in one system must be the same as the ratio in the other.
Therefore, for complete similarity, flow systems must be dynamically similar. Therefore
in dynamically similar flows, the polygon of forces for individual fluid particles, are
geometrically similar. Therefore, the component forces have the same ratio of magnitude
between the two flow systems. Dynamic similarity then produces geometric similarity of
the flow patterns.
It is usually impossible to satisfy all the conditions of dynamic similarity, so only those
conditions that are dominant are usually satisfied. As a result, non-compliance with all
the aspects of similarity produces scale effects.
The types of similarity identified above are not exhaustive, and other similarities exist,
such as thermal similarity and chemical similarity.
Ratio of Forces Arising in Dynamic Similarity
Dynamic similarity is of major importance in fluid mechanics. The types of forces that
may control the behaviour of fluids are:
1. Forces due to differences in piezometric pressure between different points in the
fluid.
2. Forces resulting from the action of viscosity.
3. Forces acting from outside the fluid, e.g. gravity.
4. Forces due to surface tension.
5. Elastic forces, i.e. those due to the compressibility of the fluid.
6. Inertia forces, taken as the resultant of all the forces acting on the fluid. Inertial
forces are always present in flow problems and must be balanced by the other
forces.
Dynamic Similarity of Flow with Dominant Viscous Forces
In this system, only two forces are considered to be dominant: viscous forces and inertia
forces. Therefore, the ratio of the inertial and the viscous forces provides the relative
influence of the two forces in the flow system. If this ratio is the same in two systems,
then the systems are said to be dynamically similar with respect to the dominant forces.
An example of this type of system is flow in a full, closed conduit, or the flow of water
past a submarine submerged deep enough as not to produce waves at the surface.
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Department of Civil and Environmental Engineering
CVNG 1001: Mechanics of Fluids
The inertia force on a fluid particle, Fi, may be expressed in its physical units by first
indicating that this force is a product of the fluid particle mass and fluid particle
acceleration. The mass is the product of density, ρ, and volume. Volume is proportional
to the cube of some characteristic length, l. Therefore, mass is proportional to ρl3.
u
t
Acceleration is the rate of change of velocity and is given by , where ‘u’ is some
particular velocity divided by some particular interval, ‘t’. However, ‘t’ can be expressed
l
u
as , which is a chosen characteristic length, ‘l’, divided by a characteristic velocity, ‘u’.
Therefore, acceleration is given by
proportional to ρl2u2.
u
u2
=
l
l
u
( )
, and the magnitude of the inertia force is
Recall that the shear stress is the product of the viscosity and the rate of shear, which
is µ
u
, and the area over which this stress acts is proportional to l2. So the magnitude of
l
the viscous forces is proportional to product of the shear stress and the area, µ
is given by µul.
Therefore,
This ratio,
Inertia Force
is proportional to
Viscous Force
u 2
xl , and
l
ρl 2 u 2 ρlu
.
=
µul
µ
ρul
, is known as the Reynolds number.
µ
Dynamic Similarity of Flow with Dominant Gravity Forces
In this system, only two forces are considered to be dominant: gravity forces and inertia
forces. Therefore, the ratio of the inertial and the gravity forces provides the relative
influence of the two forces in the flow system.
An example of this type of system is where there is a water-air interface (free surface) or
where there is an interface between two immiscible fluids. Flows of this type are open
channel flow or the wave motion caused by the passage of a ship.
The magnitude of the inertia forces, as determined above, is proportional to ρl2u2, and the
magnitude of the gravity force is its weight, and is proportional to ρl3g. Thus,
Inertia Force
Gravity Force
is proportional to
square root of this ratio
Academic year: 2006-2007
u
lg
ρl 2 u 2 u 2
=
. However, it is more convenient to use the
lg
ρl 3 g
. This ratio,
u
, is known as the Froude number.
lg
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Department of Civil and Environmental Engineering
CVNG 1001: Mechanics of Fluids
Dynamic Similarity of Flow with Dominant Surface Tension Forces
In this system, only two forces are considered to be dominant: surface tension forces and
inertia forces. The ratio of the inertial and the surface tension forces provides the relative
influence of the two forces in the flow system. The magnitude of the surface tension force
is proportional to γl, where ‘γ’ is the surface tension on a line element of characteristic
length, ‘l’. Thus,
1
⎛ ρl ⎞ 2
ρl 2 u 2 ρlu 2
=
. The square root of this ratio, u⎜⎜ ⎟⎟ ,
is proportional to
γl
γ
Surafce Tension Force
⎝γ ⎠
Inertia Force
is usually known as the Weber number.
Dynamic Similarity of Flow with Dominant Elastic Forces
In this system, only two forces are considered to be dominant: elastic forces and inertia
forces. The ratio of the inertial and the elastic forces provides the relative influence of the
two forces in the flow system. The magnitude of the elastic force is proportional to Kl2,
where ‘K’ is the bulk modulus of elasticity. Thus,
Inertia Force
Elastic Force
is proportional to
ρl 2 u 2
Kl 2
=
ρu 2
K
. This ratio,
ρu 2
K
, is known as the Cauchy
number.
Dynamic Similarity of Flow with Dominant Pressure Forces
Although pressure forces are always present, in this system the pressure forces and inertia
forces are considered to be dominant. The ratio of the pressure and the inertia forces
provides the relative influence of the two forces in the flow system. The magnitude of the
pressure force is proportional to pl2, where ‘p’ is the pressure at a point in the flow. Thus,
Pr essure Force
Inertia Force
is proportional to
pl 2
ρl u
2
2
=
p
ρu
2
. This ratio,
p
ρu 2
, is known as the Euler
number.
Q8a: A process is governed by the variables F (force), v (velocity), D (diameter), µ
(dynamic viscosity) and ρ (mass density). Write the dimensional formula of each
variable.
Q8b: How many fundamental dimensions are there?
Q8c: How many dimensionless PIs are there?
Q8d: Obtain the dimensionless PIs.
Q9a: A process is governed by the variables F (force), v (velocity), l (length), g
(acceleration due to gravity) and ρ (mass density). Write the dimensional formula of each
variable.
Q9b: How many fundamental dimensions are there?
Q9c: How many dimensionless PIs are there?
Q9d: Obtain the dimensionless PIs.
Academic year: 2006-2007
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Department of Civil and Environmental Engineering
CVNG 1001: Mechanics of Fluids
Application of Dynamic Similarity
A. Aircraft Design
Use of dynamic similarity is made extensively in aircraft design. The reasons are for (i)
cost savings, as it would be cheaper to construct a model of the plane and perform tests
within a wind tunnel than to construct the actual aircraft for testing in a flight; (ii) safety,
as human lives will be placed at risk by having an untested aircraft manned by a pilot.
For tests conducted on the model in the wind-tunnel to be effective, the pattern of air
flow around the model must be dynamically similar to the air flow round the prototype.
The application of dynamic similarity will yield a result of the form:
Π1 = φ(Π2, Π3, Π4, ...) in which Πs are dimensionless groups.
A test is made to discover the value of Π1 for the model at particular values of the
dimensionless groups. If each of the independent group Π2, Π3, Π4, ... have the same
value for the prototype as for the model, then the result is equally applicable to the
prototype.
Complete similarity may hardly ever be achieved owing to conflicting conditions
required say to make Π2 and Π3 the same. While this is so, usually, the effect of several
forces may be negligible and so only few dimensionless groups may be relevant to the
comparison.
As an example, consider the testing of an aircraft model to test the drag force F on the
prototype. The surface tension forces may be ignored as the fluid is completely immersed
in the fluid and so there is no interface between two fluids. The gravity forces may also
be ignored as the density of air is small and there is very little vertical movement over the
body of the model. Lastly, the compressibility effects (or elastic forces) may be ignored
as low-speed aircrafts are being considered.
Therefore the viscous and inertia forces are dominant and the relevant dimensionless
number is the Reynolds Number. For dynamic similarity, the ratio of the magnitude of
the dominant forces at any corresponding point must be the same for both systems, and
therefore the Reynolds number must be the same. Therefore the Reynolds number in the
model must equal the Reynolds number in the prototype.
⎛ ρul ⎞
⎛ ρul ⎞
⎟⎟
⎟⎟
= ⎜⎜
.
µ
⎝
⎠ prototype ⎝ µ ⎠ mod el
And one may write: ⎜⎜
ρ p u p l p ρ mumlm
⎛ ρul ⎞
⎛ ρul ⎞
⎟⎟ = ⎜⎜
⎟⎟ , and
=
.
µp
µm
⎝ µ ⎠ p ⎝ µ ⎠m
Or using shorthand notation ⎜⎜
If the fluid in the model and the prototype is the same, then ρp =ρm, and µp = µm, and the
above equation reduced to u p l p = u m l m , and
defined as N L =
lm
lp
lp
lm
=
um
up
. The length scale ratio, NL may be
, and all scale ratios must then be the ratio of the parameter in the
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Department of Civil and Environmental Engineering
CVNG 1001: Mechanics of Fluids
model to that of the prototype. Therefore, the velocity scale ratio, NV, is defined
as N V =
um
up
. And for
lp
lm
=
um
up
, one can write N V =
1
NL
.
It must be noted that there may be physical limitations to achieving the required model
conditions and so consideration might have to be given to altering other aspects of the
experiments, such as using different density fluids or experimental temperatures, that is
N ρ ≠ 1 and N µ ≠ 1 .
B. Free Surface Flow (Coastal Applications)
For this type of system, gravity forces are dominant and scaling is done according to the
Froude number. Therefore the Froude number in the model must equal the Froude
number in the prototype.
⎛ u ⎞
⎛
⎞
⎟ = ⎜ u ⎟ , which one can obtain N = N
And one may write: ⎜
N L . For most
V
g
⎜ gl ⎟
⎝
⎠p
⎜ gl ⎟
⎝
⎠m
purposes, the gravity scale ratio, Ng =1, and the equation may be reduced to N V = N L .
Q10: For Froude scaling in an undistorted model show that N T = N L .
REFERENCES
Massey, B.S. 1995. Mechanics of Fluids. 6th ed. London: Chapman & Hall
Hamill, L. 1995. Understanding Hydraulics. 5th ed. London: Macmillan
Douglas, J.F., J.M. Gasiorek, and J.A. Swaffield. 1995. Fluid Mechanics. 3rd ed.
Singapore: Longman
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Department of Civil and Environmental Engineering
CVNG 1001: Mechanics of Fluids
A8a: F [MLT ], v [LT ], D [L], µ [ML-1T-1] and ρ [ML-3]
(Hint: use the units as a guide)
A8b: 3 fundamental dimensions of mass length and time
A8c: According to the Buckingham PI theorem 2 dimensionless PIs are expected
A8d: Using the Tabular Method to reduce variables to dimensionless PIs
-2
-1
F
v
D
µ
ρ
[M]
1
0
0
1
1
[L]
1
1
1
-1
-3
[T]
-2
-1
0
-1
0
F/ρ
v
D
µ/ρ
ρ/ρ
[M]
0
0
0
0
0
[L]
4
1
1
2
0
[T]
-2
-1
0
-1
0
[M]
0
0
0
0
[L]
2
0
1
1
[T]
0
0
0
0
[M]
0
0
0
[L]
0
0
0
[T]
0
0
0
2
F/ρv
v/v
D
µ/ρv
2
2
F/ρv D
D/D
µ/ρvD
Finally the two dimensionless PIs are Π1 and Π2 where Π 1 =
F
ρv 2 D 2
and Π 2 =
µ
.
ρvD
⎛ ρvD ⎞
⎛ µ ⎞
F
F
⎟⎟ = h(Re )
⎜⎜
⎟
⎜
=
h
=
g
or
2
2
⎜ ρvD ⎟
ρv D
ρv 2 D 2
⎝ µ ⎠
⎠
⎝
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Page 12 of 13
Department of Civil and Environmental Engineering
CVNG 1001: Mechanics of Fluids
A9a: F [MLT ], v [LT ], l [L], g [LT-2] and ρ [ML-3]
(Hint: use the units as a guide)
A9b: 3 fundamental dimensions of mass length and time
A9c: According to the Buckingham PI theorem 2 dimensionless PIs are expected
A9d: Using the Tabular Method to reduce variables to dimensionless PIs
-2
-1
F
v
l
g
ρ
[M]
1
0
0
0
1
[L]
1
1
1
1
-3
[T]
-2
-1
0
-2
0
F/ρ
v
l
g
ρ/ρ
[M]
0
0
0
0
0
[L]
4
1
1
1
0
[T]
-2
-1
0
-2
0
F/ρg
v/√g
l
g/g
[M]
0
0
0
0
[L]
3
½
1
0
[T]
0
0
0
0
F/ρgl3
v/√(gl)
l/l
[M]
0
0
0
[L]
0
0
0
[T]
0
0
0
Finally the two dimensionless PIs are Π1 and Π2 where Π 1 =
Therefore,
F
ρgl 3
and Π 2 =
u
.
gl
⎛ u ⎞
F
⎟ = h(Fr ) .
= h⎜
3
⎜
⎟
ρgl
⎝ gl ⎠
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Page 13 of 13