This document discusses the equations of motion for rigid fluids subjected to different types of acceleration.
1) The general equation of motion for a rigid fluid is derived from applying Newton's second law to a small fluid element. This relates the fluid acceleration to the pressure gradient and gravitational force.
2) For specific cases of a fluid at rest, in free fall, and linearly accelerating, the pressure gradient terms in the equation are simplified.
3) For a rotating fluid contained in a cylindrical tank, the equation is derived in cylindrical coordinates and relates the radial pressure gradient to the centripetal acceleration from rotation.
Surface Tension
Surface Tension in Detail
Surface Tension on Liquid Droplet
Surface Tension on Hollow Droplet
Surface Tension on Liquid Jet
Capillary
Expression for Capillary Rise
Expression for Capillary Fall
Sources
Surface Tension
Surface Tension in Detail
Surface Tension on Liquid Droplet
Surface Tension on Hollow Droplet
Surface Tension on Liquid Jet
Capillary
Expression for Capillary Rise
Expression for Capillary Fall
Sources
Hybrid optimization of pumped hydro system and solar- Engr. Abdul-Azeez.pdffxintegritypublishin
Advancements in technology unveil a myriad of electrical and electronic breakthroughs geared towards efficiently harnessing limited resources to meet human energy demands. The optimization of hybrid solar PV panels and pumped hydro energy supply systems plays a pivotal role in utilizing natural resources effectively. This initiative not only benefits humanity but also fosters environmental sustainability. The study investigated the design optimization of these hybrid systems, focusing on understanding solar radiation patterns, identifying geographical influences on solar radiation, formulating a mathematical model for system optimization, and determining the optimal configuration of PV panels and pumped hydro storage. Through a comparative analysis approach and eight weeks of data collection, the study addressed key research questions related to solar radiation patterns and optimal system design. The findings highlighted regions with heightened solar radiation levels, showcasing substantial potential for power generation and emphasizing the system's efficiency. Optimizing system design significantly boosted power generation, promoted renewable energy utilization, and enhanced energy storage capacity. The study underscored the benefits of optimizing hybrid solar PV panels and pumped hydro energy supply systems for sustainable energy usage. Optimizing the design of solar PV panels and pumped hydro energy supply systems as examined across diverse climatic conditions in a developing country, not only enhances power generation but also improves the integration of renewable energy sources and boosts energy storage capacities, particularly beneficial for less economically prosperous regions. Additionally, the study provides valuable insights for advancing energy research in economically viable areas. Recommendations included conducting site-specific assessments, utilizing advanced modeling tools, implementing regular maintenance protocols, and enhancing communication among system components.
Sachpazis:Terzaghi Bearing Capacity Estimation in simple terms with Calculati...Dr.Costas Sachpazis
Terzaghi's soil bearing capacity theory, developed by Karl Terzaghi, is a fundamental principle in geotechnical engineering used to determine the bearing capacity of shallow foundations. This theory provides a method to calculate the ultimate bearing capacity of soil, which is the maximum load per unit area that the soil can support without undergoing shear failure. The Calculation HTML Code included.
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Final project report on grocery store management system..pdfKamal Acharya
In today’s fast-changing business environment, it’s extremely important to be able to respond to client needs in the most effective and timely manner. If your customers wish to see your business online and have instant access to your products or services.
Online Grocery Store is an e-commerce website, which retails various grocery products. This project allows viewing various products available enables registered users to purchase desired products instantly using Paytm, UPI payment processor (Instant Pay) and also can place order by using Cash on Delivery (Pay Later) option. This project provides an easy access to Administrators and Managers to view orders placed using Pay Later and Instant Pay options.
In order to develop an e-commerce website, a number of Technologies must be studied and understood. These include multi-tiered architecture, server and client-side scripting techniques, implementation technologies, programming language (such as PHP, HTML, CSS, JavaScript) and MySQL relational databases. This is a project with the objective to develop a basic website where a consumer is provided with a shopping cart website and also to know about the technologies used to develop such a website.
This document will discuss each of the underlying technologies to create and implement an e- commerce website.
Forklift Classes Overview by Intella PartsIntella Parts
Discover the different forklift classes and their specific applications. Learn how to choose the right forklift for your needs to ensure safety, efficiency, and compliance in your operations.
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Cosmetic shop management system project report.pdfKamal Acharya
Buying new cosmetic products is difficult. It can even be scary for those who have sensitive skin and are prone to skin trouble. The information needed to alleviate this problem is on the back of each product, but it's thought to interpret those ingredient lists unless you have a background in chemistry.
Instead of buying and hoping for the best, we can use data science to help us predict which products may be good fits for us. It includes various function programs to do the above mentioned tasks.
Data file handling has been effectively used in the program.
The automated cosmetic shop management system should deal with the automation of general workflow and administration process of the shop. The main processes of the system focus on customer's request where the system is able to search the most appropriate products and deliver it to the customers. It should help the employees to quickly identify the list of cosmetic product that have reached the minimum quantity and also keep a track of expired date for each cosmetic product. It should help the employees to find the rack number in which the product is placed.It is also Faster and more efficient way.
Saudi Arabia stands as a titan in the global energy landscape, renowned for its abundant oil and gas resources. It's the largest exporter of petroleum and holds some of the world's most significant reserves. Let's delve into the top 10 oil and gas projects shaping Saudi Arabia's energy future in 2024.
Welcome to WIPAC Monthly the magazine brought to you by the LinkedIn Group Water Industry Process Automation & Control.
In this month's edition, along with this month's industry news to celebrate the 13 years since the group was created we have articles including
A case study of the used of Advanced Process Control at the Wastewater Treatment works at Lleida in Spain
A look back on an article on smart wastewater networks in order to see how the industry has measured up in the interim around the adoption of Digital Transformation in the Water Industry.
Immunizing Image Classifiers Against Localized Adversary Attacksgerogepatton
This paper addresses the vulnerability of deep learning models, particularly convolutional neural networks
(CNN)s, to adversarial attacks and presents a proactive training technique designed to counter them. We
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When combined with 3D convolution and deep curriculum learning optimization (CLO), itsignificantly improves
the immunity of models against localized universal attacks by up to 40%. We evaluate our proposed approach
using contemporary CNN architectures and the modified Canadian Institute for Advanced Research (CIFAR-10
and CIFAR-100) and ImageNet Large Scale Visual Recognition Challenge (ILSVRC12) datasets, showcasing
accuracy improvements over previous techniques. The results indicate that the combination of the volumetric
input and curriculum learning holds significant promise for mitigating adversarial attacks without necessitating
adversary training.
Immunizing Image Classifiers Against Localized Adversary Attacks
Linear Acceleration
1. Rigid-body Subjected to Accelerations 75
Basic Equation for Rigid-Body Subjected to Acceleration
Although we have proved that the pressure at a given point has the
same magnitude in all directions, and thus it is a scalar functions
question, we are now faced with an equally important question- how
does the pressure in a fluid in which there are no shearing stresses vary
from point to point?. To answer this question, we consider a small
element of fluid removed from some arbitrary position within the mass of
fluid of interest as illustrated in Fig. (1). There are two types of forces
acting on this element:
Surface forces (due to the pressure), and
Body force (equal to the weight of the element)
x
y
z
z
y
x
g
z
x
2
y
y
p
p
j
i
k
y
x
2
z
z
p
p
y
x
2
z
z
p
p
z
x
2
y
y
p
p
x
y
z
Fig. (1) – The Surface and Body Forces acting on a Differential Fluid
Element in Vertical and Horizontal Directions
If we let the pressure at the center of the element be designated as p,
then the average pressure on the various faces can be expressed in
2. Rigid-body Subjected to Accelerations 76
terms of P and its derivation, as shown in Fig. 1 (the surface forces in
the x-direction are not shown). The resultant surface force in the y-
direction is
z
x
2
y
y
p
p
z
x
2
y
y
p
p
Fy
or
z
y
x
y
p
Fy
Similarly, for the x- and z-directions the resultant surface forces are:
z
y
x
x
p
FX
and z
y
x
z
p
FZ
The resultant surface force acting on the element can be expressed as:
k
F
j
F
i
F
F z
y
x
s
or
z
y
x
k
z
p
j
y
p
i
x
p
Fs
The pressure gradient in vector form
where i
, j
, and k
are the unit vectors along the coordinate axes.
The group of terms in parentheses can be written as:
p
k
z
p
j
y
p
i
x
p
The symbol is the gradient or “del” vector operation. Thus, the
resultant surface force per unit volume can be expressed as
p
z
y
x
Fs
3. Rigid-body Subjected to Accelerations 77
Since the z axis is vertical, the weight of the element is
z
y
x
g
k
W
where the negative sign indicates that the force due to the weight is
downward (in the negative z direction).
Newton’s second law, applied to the fluid element, can be expressed as
a
m
F
where:
F : represents the resultant force acting on the element,
a : is the acceleration of the element, and
m
: is the element mass, which can be written as z
y
x
g
a
m
k
W
F
F s
or
a
)
z
y
x
(
k
)
z
y
x
(
g
)
z
y
x
(
p
and cancelling z
y
x
, the general equation of motion for a fluid that
acts as a rigid body (no shearing stresses) is determined to be
a
k
g
p
……………….. (1)
this can be expressed in scalar form in the three orthogonal directions,
as
x
a
x
p
, y
a
y
p
and )
a
g
(
z
p
z
…………….. (2)
4. Rigid-body Subjected to Accelerations 78
Application (i) – Rigid-Body at Rest
For fluid at rest or moving on a straight path at constant velocity, all
components of acceleration are zero, and the relation in Eq. (2) reduces
to
0
x
p
, 0
y
p
, and g
z
p
(fluid at rest) ……. (3)
which confirm that, in fluid at rest, the pressure remains constant in
horizontal direction (p is independent of x and y) and varies only in the
vertical direction as a result of gravity.
Application (ii) – Rigid-Body in Free Fall
A freely falling body accelerates under the influence of gravity, Fig. (2).
When the air resistance is negligible, the acceleration of the body equals
the gravitational acceleration and accelerate in any horizontal direction is
zero. Therefore,
0
a
a y
x
, and g
az
(fluid accelerates downward) ……. (4)
Then, Eq. (2) reduces to
0
z
p
y
p
x
p
, (p = constant)……. (5)
When the direction of motion is reversed and the fluid is forced to
accelerate vertically with g
az
by placing the fluid container in an
elevator, the pressure gradient in the z-direction is g
2
z
p
.
5. Rigid-body Subjected to Accelerations 79
g
az
h
Z
g
az
h
Z
1
p 1
p
1
2 p
p h
g
2
p
p 1
2
Density Density
Free Fall
of a Liquid az = - g
Upward Acceleration
of a Liquid az = + g
Free Fall of a Fluid Body
Fig. ( ) - The Effect of Acceleration on the Pressure of a Liquid
During Free Fall and Upward Acceleration
Fig. (2) – The Effect of Acceleration on the Pressure of a
Fluid during Free Fall and Upward Acceleration.
Therefore, the pressure difference across a fluid layer now doubles
relative to the stationary fluid case.
az 0 az 0
g h g h
h
Fig. ( ) - Variation of the Magnitude of Pressure with the Variation of
az Variation.
Fig. (3) – Variation of the Magnitude of Pressure with the Variation of az
.
Application (iii) – Rigid-Body Acceleration
Many fluids such as water, gasoline and milk are transported in tankers.
In an accelerated tanker, the fluid rushes to the tank, and some initial
splashing occurs but then a new free surface (usually horizontal) is
6. Rigid-body Subjected to Accelerations 80
formed, each fluid particle assumes the same acceleration, and the
entire fluid moves like a rigid body. Rigid body-motion of a fluid also
occurs when the fluid is contained in a tank rotates about an axis.
We consider an open container of a liquid that is translating along a
straight path with a constant acceleration “a” as illustrated in Fig (4).
Since 0
ax it follows from Eqs .(2) that the pressure gradient in the x-
direction is zero ( 0
x
p
) . In the y- and z-directions
y
a
y
p
, and )
g
a
(
z
p
z
……. (6)
The change in pressure between two closely spaced points located at y,
z and dy
y , dz
z can be expressed as
dz
z
p
dy
y
p
dp
……..…. (7)
or in terms of the results from Eqs.(6) and (7)
dz
)
g
a
(
dy
a
dp z
y
……..…. (8)
Along a line of constant pressure, dp = 0, and therefore from Eq. (8) it
follows that the slope of this line is given by the relation
tan
g
a
a
y
d
z
d
z
x
……..…. (9)
7. Rigid-body Subjected to Accelerations 81
g
aZ
y
a
.
const
p1
.
const
p2
.
const
p3
Free surface slope = dz/dy
y
x
z
y
a
z
a a
Fig. (4) – Lines of Constant Pressure (which are the Projections of
the Surfaces of Constant Pressure on the y - z plane) in a
Linearly Accelerating Liquid, and the Vertical Rise.
Application (iv) –Rigid-Body Rotation
As a second case, consider rotation of the fluid about the z axis without
any translation as sketched in Fig. (5). We know from experience that
when a glass filled with water is rotated about its axis, the fluid is forced
outward as a result of the so-called centrifugal force, and the free
surface of the liquid becomes concave. This is known as the forced
vortex motion.
Consider a vertical cylindrical container partially filled with a liquid. The
container is now rotated about its axis at a constant angular velocity of
, as shown in Fig. (5). After initial transients, the liquid will move as a
rigid-body together with the container.
8. Rigid-body Subjected to Accelerations 82
2
r r
a
Still Water Level
r
Axis of rotation
R
Fig. ( ) – Rigid-body Rotation of a Liquid in a Tank.
z
Fig. (5) – Rigid-body Rotation of a Liquid in a Tank.
This problem is best analyzed in cylindrical coordinates (r, , z), with z
taken along the centerline of the container directed from the bottom
towards the free surface, since the shape of the container is a cylinder,
and the fluid particles undergo a circular motion.
The centripetal acceleration of a fluid particle rotating with a
constant angular velocity at a distance r from the axis of
rotation is 2
r and is directed radially towards the axis of
rotation (negative r- direction). That is, 2
r r
a
There is a symmetry about the z-axis, which is the axis of
rotation, and thus there is no dependence.
Then p = p ( r, z ) and 0
a
. Also, since there is no motion in
the z-direction, 0
az
9. Rigid-body Subjected to Accelerations 83
Then the equation of motion for rotating fluids (Eq. ) reduce to
2
r
r
p
, 0
p
, g
z
p
……..…. (10)
Then the total difference of )
,
r
(
p
p
, which is
dz
z
p
dr
r
p
dp
……..…. (11)
becomes,
dz
g
dr
r
dp 2
……..…. (12)
Along a surface of constant pressure, such as the free surface, dp = 0,
so that from Eq. (12)
g
2
r
r
d
z
d 2
r
.
const
p1
.
const
p2
.
const
p3
z
.
const
p4
C
o
n
s
t
a
n
t
p
r
e
s
s
u
r
e
l
i
n
e
s
g
2
/
r2
2
Axis of rotation
2
2
max r
g
2
z
Fig. (6) – Surfaces of Constant Pressure in a Rotating Liquid
10. Rigid-body Subjected to Accelerations 84
Integration of this result gives the equation for the surface of constant
pressure as
t
tan
cons
g
2
r
z
2
2
……..…. (13)
This equation reveals that these surfaces of constant pressure are
parabolic, as illustrated in Fig. (6).
Integration of Eq. (12) yields
z
d
g
dr
r
p
d 2
or
t
tan
cons
z
g
2
r
p
2
2
……..…. (14)
where the constant of integration can be expressed in terms of a
specified pressure at some arbitrary point ro, zo. This result shows that
the pressure varies with the distance from the axis of rotation, but at a
fixed radius, the pressure varies hydrostatically in the vertical direction
as shown in Fig. (6).
Since the volume of a parabolic is one-half the base area times its
height, as shown in Fig. (7), the still-water level is exactly halfway
between the height and low points of the free surface. The center of the
fluid drops an amount g
4
R
2
h 2
2
and the edges raise an equal
amount.
11. Rigid-body Subjected to Accelerations 85
h
R
2
Volume 2
g
2
R
h
2
2
2
h
2
h
Still Water Level
R
Axis of rotation
Fig. (7) – Determining the free-surface Position for Rotation
of a Cylinder of Fluid about its Central Axis.
12. Rigid-body Subjected to Accelerations 86
Fluid Masses subjected to Accelerations
ax
az
z
x
dz
dx
g
z
P
Fz
dz
dx
x
P
Fx
Fig. ( )-
Applying Newton's second law to the fluid element of Fig. (1) which is
being accelerated in such a way that its components of acceleration are
x
a and z
a . The summation of force components on such an element is
dz
dx
x
P
Fx
…………..(1)
dz
dx
g
y
P
Fy
With the mass of the element equal to ( dz
dx
) the component forms of
Newton's second law may be written:
13. Rigid-body Subjected to Accelerations 87
dz
dx
a
dz
dx
x
P
x
………………………………..…..(2)
dz
dx
a
dz
dx
g
y
P
y
Eq. (2) in component form, based on rectangular coordinates with the
positive z axis being vertically upward, can be expressed as
x
a
x
P
……………………………………………………………..(3)
)
g
a
(
z
P
z
These equations characterize the pressure variation through an
accelerated rigid-body of fluid and with them specific application may be
studied.
The change in pressure between two closely spaced points located at x,
z and x+dx, z+dz can be expressed as
dz
y
P
dx
x
P
dp
………….. (4)
and substituting the above expression for x
P
and z
P
, we
obtain
dz
)
g
a
(
dx
a
dp z
x
………….. (5)
However, along a line of constant pressure dP = 0 and hence, for such a
line
14. Rigid-body Subjected to Accelerations 88
)
g
a
(
a
x
d
z
d
z
x
………….. (6)
a- For constant Linear Acceleration with ax = 0
Here a container of liquid is accelerated vertically upward,
0
x
P
, and with no change of pressure with x, eq. (5) becomes
)
g
a
(
z
P
z
………….. (7)
Eq. (7) shows that the pressure will vary linearly with depth, but the
variation is due to the combined effects of gravity and the
externally induced acceleration, )
a
g
( z
rather than simply
specific weight g
.
B- For constant Linear Acceleration with ay = 0
Here the slope free surface (which is a line of constant pressure dP = 0)
is given by Eq.( )
ax
Az+g
z
x
dx
dz
15. Rigid-body Subjected to Accelerations 89
Along a free surface the pressure is constant, so that for the accelerating
mass shown in Fig. ( ) the free surface will be inclined as 0
ax . In
addition, all lines of constant pressure will be parallel to the free surface
as illustrated.
Free surface of slope = dz/dx
P1
P2
P3
Constant
pressure
lines
ax
az
Fig. ( ) – Linear Acceleration with a Free Surface.