I am Walter G. I am an Electromechanics Assignment Expert at eduassignmenthelp.com. I hold a Master’s Degree in Electromechanics, from Oxford University, UK. I have been helping students with their assignments for the past 8 years. I solve assignments related to Electromechanics.
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1 ECE 6340 Fall 2013 Homework 8 Assignment.docxjoyjonna282
1
ECE 6340
Fall 2013
Homework 8
Assignment: Please do Probs. 1-9 and 13 from the set below.
1) In dynamics, we have the equation
E j Aω= − −∇Φ .
(a) Show that in statics, the scalar potential function Φ can be interpreted as a voltage
function. That is, show that in statics
( ) ( )
B
AB
A
V E dr A B≡ ⋅ = Φ −Φ∫ .
(b) Next, explain why this equation is not true (in general) in dynamics.
(c) Explain why the voltage drop (defined as the line integral of the electric field, as
defined above) depends on the path from A to B in dynamics, using Faraday’s law.
(d) Does the right-hand side of the above equation (the difference in the potential
function) depend on the path, in dynamics?
Hint: Note that, according to calculus, for any function ψ we have
dr dx dy dz d
x y z
ψ ψ ψ
ψ ψ
∂ ∂ ∂
∇ ⋅ = + + =
∂ ∂ ∂
.
2) Starting with Maxwell’s equations, show that the electric field radiated by an impressed
current density source J i in an infinite homogeneous region satisfies the equation
( )2 2 iE k E E j Jωµ∇ + = ∇ ∇⋅ + .
Then use Ampere’s law (or, if you prefer, the continuity equation and the electric Gauss
law) to show that this equation may be written as
( )2 2 1 i iE k E J j J
j
ωµ
σ ωε
∇ + = − ∇ ∇⋅ +
+
.
2
Note that the total current density is the sum of the impressed current density and the
conduction current density, the latter obeying Ohm’s law (J c = σE).
Explain why this equation for the electric field would be harder to solve than the equation
that was derived in class for the magnetic vector potential.
3) Show that magnetic field radiated by an impressed current density source satisfies the
equation
2 2 iH k H J∇ + = −∇× .
Explain why this equation for the magnetic field would be harder to solve than the
equation that was derived in class for the magnetic vector potential.
4) Show that in a homogenous region of space the scalar electric potential satisfies the
equation
2 2
i
v
c
k
ρ
ε
∇ Φ + Φ = − ,
where ivρ is the impressed (source) charge density, which is the charge density that goes
along with the impressed current density, being related by
i ivJ jωρ∇⋅ = −
Hint: Start with E j Aω= − −∇Φ and take the divergence of both sides. Also, take the
divergence of both sides of Ampere’s law and use the continuity equation for the
impressed current (given above) to show that
1 ii v
c c
E J
j
ρ
ωε ε
∇⋅ = − ∇⋅ = .
Note: It is also true from the electric Gauss law that
vE
ρ
ε
∇⋅ = ,
but we prefer to have only an impressed (source) charge density on the right-hand side of
the equation for the potential Φ. In the time-harmonic steady state, assuming a
homogeneous and isotropic region, it follows that ρv = ρvi. That is, there is no charge
3
density arising from the conduction current. (If there were no impressed current sources,
the total charge density would therefore be ze ...
1 ECE 6340 Fall 2013 Homework 8 Assignment.docxjoyjonna282
1
ECE 6340
Fall 2013
Homework 8
Assignment: Please do Probs. 1-9 and 13 from the set below.
1) In dynamics, we have the equation
E j Aω= − −∇Φ .
(a) Show that in statics, the scalar potential function Φ can be interpreted as a voltage
function. That is, show that in statics
( ) ( )
B
AB
A
V E dr A B≡ ⋅ = Φ −Φ∫ .
(b) Next, explain why this equation is not true (in general) in dynamics.
(c) Explain why the voltage drop (defined as the line integral of the electric field, as
defined above) depends on the path from A to B in dynamics, using Faraday’s law.
(d) Does the right-hand side of the above equation (the difference in the potential
function) depend on the path, in dynamics?
Hint: Note that, according to calculus, for any function ψ we have
dr dx dy dz d
x y z
ψ ψ ψ
ψ ψ
∂ ∂ ∂
∇ ⋅ = + + =
∂ ∂ ∂
.
2) Starting with Maxwell’s equations, show that the electric field radiated by an impressed
current density source J i in an infinite homogeneous region satisfies the equation
( )2 2 iE k E E j Jωµ∇ + = ∇ ∇⋅ + .
Then use Ampere’s law (or, if you prefer, the continuity equation and the electric Gauss
law) to show that this equation may be written as
( )2 2 1 i iE k E J j J
j
ωµ
σ ωε
∇ + = − ∇ ∇⋅ +
+
.
2
Note that the total current density is the sum of the impressed current density and the
conduction current density, the latter obeying Ohm’s law (J c = σE).
Explain why this equation for the electric field would be harder to solve than the equation
that was derived in class for the magnetic vector potential.
3) Show that magnetic field radiated by an impressed current density source satisfies the
equation
2 2 iH k H J∇ + = −∇× .
Explain why this equation for the magnetic field would be harder to solve than the
equation that was derived in class for the magnetic vector potential.
4) Show that in a homogenous region of space the scalar electric potential satisfies the
equation
2 2
i
v
c
k
ρ
ε
∇ Φ + Φ = − ,
where ivρ is the impressed (source) charge density, which is the charge density that goes
along with the impressed current density, being related by
i ivJ jωρ∇⋅ = −
Hint: Start with E j Aω= − −∇Φ and take the divergence of both sides. Also, take the
divergence of both sides of Ampere’s law and use the continuity equation for the
impressed current (given above) to show that
1 ii v
c c
E J
j
ρ
ωε ε
∇⋅ = − ∇⋅ = .
Note: It is also true from the electric Gauss law that
vE
ρ
ε
∇⋅ = ,
but we prefer to have only an impressed (source) charge density on the right-hand side of
the equation for the potential Φ. In the time-harmonic steady state, assuming a
homogeneous and isotropic region, it follows that ρv = ρvi. That is, there is no charge
3
density arising from the conduction current. (If there were no impressed current sources,
the total charge density would therefore be ze ...
23Network FlowsAuthor Arthur M. Hobbs, Department of .docxeugeniadean34240
23
Network Flows
Author: Arthur M. Hobbs, Department of Mathematics, Texas A&M Uni-
versity.
Prerequisites: The prerequisites for this chapter are graphs and trees. See
Sections 9.1 and 10.1 of Discrete Mathematics and Its Applications.
Introduction
In this chapter we solve three very different problems.
Example 1 Joe the plumber has made an interesting offer. He says he has
lots of short pieces of varying gauges of copper pipe; they are nearly worthless
to him, but for only 1/5 of the usual cost of installing a plumbing connection
under your house, he will use a bunch of T- and Y-joints he picked up at a
distress sale and these small pipes to build the network shown in Figure 1. He
claims that it will deliver three gallons per minute at maximum flow. He has
a good reputation, so you are sure the network he builds will not leak and will
cost what he promises, but he is no mathematician. Will the network really
deliver as much water per minute as he claims?
408
Chapter 23 Network Flows 409
Figure 1. A plumber’s nightmare.
Example 2 We want to block access to the sea from inland town s on
river R. We can do this by dropping mines in the river, but because the river
spreads out in a wide delta with several outlets, the number of mines required
depends on where we drop them. The number of mines required in a channel
ranges from a high of 20 mines in R to a low of 1 in some channels, as shown
in Figure 2. In that figure, each channel is shown with a number indicating
how many mines will block it. What is the smallest number of mines needed to
block off s’s access to the sea, and where should the mines be placed?
Figure 2. Delta system of river R and numbers of mines
needed to close channels.
Example 3 At Major University, Professor Johnson is asked to hire graders
for 100 sections spread among 30 different courses. Each grader may work for
one, two, or three sections, with the upper bound being the grader’s choice,
but the number actually assigned being Professor Johnson’s choice. Professor
Johnson contacts the potential graders, learns from each both his choice of
number of sections and which courses he is competent to grade, and makes a
table showing this information, together with the number of sections of each
course being offered. Because the real problem is too large to use as an example
here, Table 1 gives a smaller example. How should the assignment of graders
be made?
In this chapter, we begin with Example 1, solve Example 2 on the way to
solving Example 1, and then solve Example 3 by using the theory developed.
Many more kinds of problems are solved using the methods of this chapter in
the book Flows in Networks by Ford and Fulkerson [4].
410 Applications of Discrete Mathematics
Course Course Course Course Max # Sec.
1 2 3 4 Wanted
Student 1 yes yes no no 3
Student 2 yes no yes yes 2
Student 3 no yes yes yes 3
# Sec. Needed 3 1 1 2
Table 1. Graders and courses.
Flow Graphs
In Example 1, we have a network of pipes th.
Aiims previous year sample paper set 2Prabha Gupta
Find here previous year AIIMS question papers and solved sample question bank with answer keys at Aglasem. Visit @ https://admission.aglasem.com/aiims-mbbs-2019/
The Effect of High Zeta Potentials on the Flow Hydrodynamics in Parallel-Plat...CSCJournals
This paper investigates the effect of the EDL at the solid-liquid interface on the liquid flow through a micro-channel formed by two parallel plates. The complete Poisson-Boltzmann equation (without the frequently used linear approximation) was solved analytically in order to determine the EDL field near the solid-liquid interface. The momentum equation was solved analytically taking into consideration the electrical body force resulting from the EDL field. Effects of the channel size and the strength of the zeta-potential on the electrostatic potential, the streaming potential, the velocity profile, the volume flow rate, and the apparent viscosity are presented and discussed. Results of the present analysis, which are based on the complete Poisson-Boltzmann equation, are compared with a simplified analysis that used a linear approximation of the Poisson-Boltzmann equation.
I am Calisto Dante. Currently associated with eduassignmenthelp.com as a Digital Communication Assignment Help Expert. After completing my Master's in Digital Communication from, OCAD University in Toronto, Canada. I was in search of an opportunity that expands my area of knowledge hence I decided to help students with their assignments. I have written several digital communication assignments to date to help students overcome numerous difficulties they face.
23Network FlowsAuthor Arthur M. Hobbs, Department of .docxeugeniadean34240
23
Network Flows
Author: Arthur M. Hobbs, Department of Mathematics, Texas A&M Uni-
versity.
Prerequisites: The prerequisites for this chapter are graphs and trees. See
Sections 9.1 and 10.1 of Discrete Mathematics and Its Applications.
Introduction
In this chapter we solve three very different problems.
Example 1 Joe the plumber has made an interesting offer. He says he has
lots of short pieces of varying gauges of copper pipe; they are nearly worthless
to him, but for only 1/5 of the usual cost of installing a plumbing connection
under your house, he will use a bunch of T- and Y-joints he picked up at a
distress sale and these small pipes to build the network shown in Figure 1. He
claims that it will deliver three gallons per minute at maximum flow. He has
a good reputation, so you are sure the network he builds will not leak and will
cost what he promises, but he is no mathematician. Will the network really
deliver as much water per minute as he claims?
408
Chapter 23 Network Flows 409
Figure 1. A plumber’s nightmare.
Example 2 We want to block access to the sea from inland town s on
river R. We can do this by dropping mines in the river, but because the river
spreads out in a wide delta with several outlets, the number of mines required
depends on where we drop them. The number of mines required in a channel
ranges from a high of 20 mines in R to a low of 1 in some channels, as shown
in Figure 2. In that figure, each channel is shown with a number indicating
how many mines will block it. What is the smallest number of mines needed to
block off s’s access to the sea, and where should the mines be placed?
Figure 2. Delta system of river R and numbers of mines
needed to close channels.
Example 3 At Major University, Professor Johnson is asked to hire graders
for 100 sections spread among 30 different courses. Each grader may work for
one, two, or three sections, with the upper bound being the grader’s choice,
but the number actually assigned being Professor Johnson’s choice. Professor
Johnson contacts the potential graders, learns from each both his choice of
number of sections and which courses he is competent to grade, and makes a
table showing this information, together with the number of sections of each
course being offered. Because the real problem is too large to use as an example
here, Table 1 gives a smaller example. How should the assignment of graders
be made?
In this chapter, we begin with Example 1, solve Example 2 on the way to
solving Example 1, and then solve Example 3 by using the theory developed.
Many more kinds of problems are solved using the methods of this chapter in
the book Flows in Networks by Ford and Fulkerson [4].
410 Applications of Discrete Mathematics
Course Course Course Course Max # Sec.
1 2 3 4 Wanted
Student 1 yes yes no no 3
Student 2 yes no yes yes 2
Student 3 no yes yes yes 3
# Sec. Needed 3 1 1 2
Table 1. Graders and courses.
Flow Graphs
In Example 1, we have a network of pipes th.
Aiims previous year sample paper set 2Prabha Gupta
Find here previous year AIIMS question papers and solved sample question bank with answer keys at Aglasem. Visit @ https://admission.aglasem.com/aiims-mbbs-2019/
The Effect of High Zeta Potentials on the Flow Hydrodynamics in Parallel-Plat...CSCJournals
This paper investigates the effect of the EDL at the solid-liquid interface on the liquid flow through a micro-channel formed by two parallel plates. The complete Poisson-Boltzmann equation (without the frequently used linear approximation) was solved analytically in order to determine the EDL field near the solid-liquid interface. The momentum equation was solved analytically taking into consideration the electrical body force resulting from the EDL field. Effects of the channel size and the strength of the zeta-potential on the electrostatic potential, the streaming potential, the velocity profile, the volume flow rate, and the apparent viscosity are presented and discussed. Results of the present analysis, which are based on the complete Poisson-Boltzmann equation, are compared with a simplified analysis that used a linear approximation of the Poisson-Boltzmann equation.
I am Calisto Dante. Currently associated with eduassignmenthelp.com as a Digital Communication Assignment Help Expert. After completing my Master's in Digital Communication from, OCAD University in Toronto, Canada. I was in search of an opportunity that expands my area of knowledge hence I decided to help students with their assignments. I have written several digital communication assignments to date to help students overcome numerous difficulties they face.
I am Shane Bruce. I love exploring new topics. Academic writing seemed an interesting option for me. After working for many years with eduassignmenthelp.com, I have assisted many students with their assignments. I can proudly say, each student I have served is happy with the quality of the solution that I have provided. I have acquired my Master’s Degree in MATLAB, from Curtin University in Perth, Australia.
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I am John Klok. I am an Electromechanics Assignment Expert at eduassignmenthelp.com. I hold a Master’s Degree in Electromechanics, from Royal Roads University, Canada. I have been helping students with their assignments for the past 6 years. I solve homework related to Electromechanics. Visit eduassignmenthelp.com or email info@eduassignmenthelp.com. You can also call on +1 678 648 4277 for any assistance with Electromechanics Assignments.
I am Moffat D. I am a Magnetic Materials Assignment Expert at eduassignmenthelp.com. I hold a Master’s Degree in Electromagnetic, from The University of Liverpool, UK. I have been helping students with their assignments for the past 8 years. I solve assignments related to Magnetic Materials.
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I am Nathan J. I am a Magnetic Materials Assignment Expert at eduassignmenthelp.com. I hold a Master’s Degree in Electromagnetic, from The University of Queensland, Australia. I have been helping students with their assignments for the past 12 years. I solve assignments related to Magnetic Materials.
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I am Baddie K. I am a Magnetic Materials Assignment Expert at eduassignmenthelp.com. I hold a Masters's Degree in Electro-Magnetics, from The University of Malaya, Malaysia. I have been helping students with their assignments for the past 12 years. I solve assignments related to Magnetic Materials.
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I am Arnold H. I am a Magnetic Materials Assignment Expert at eduassignmenthelp.com. I hold a Ph.D. in Electro-Magnetics, from The University of Hertfordshire, UK. I have been helping students with their assignments for the past 6 years. I solve assignments related to Magnetic Materials.
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I am Carl D. I am a Bimolecular Homework Expert at eduassignmenthelp.com. I hold a Master's in MSc, from DePaul University, Chicago. I have been helping students with their homework for the past 7 years. I solve homework related to Biomolecular.
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I am Craig D. I am a Physical Chemistry Assignment Expert at eduassignmenthelp.com. I hold a Ph.D. in Physical Chemistry, from The University of Queensland. I have been helping students with their homework for the past 9 years. I solve assignments related to Physical Chemistry.
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I am Grey N. I am a Physical Chemistry Assignment Expert at eduassignmenthelp.com. I hold a Ph.D. in Physical Chemistry, from Calgary, Canada. I have been helping students with their homework for the past 6 years. I solve assignments related to Physical Chemistry.
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How to Split Bills in the Odoo 17 POS ModuleCeline George
Bills have a main role in point of sale procedure. It will help to track sales, handling payments and giving receipts to customers. Bill splitting also has an important role in POS. For example, If some friends come together for dinner and if they want to divide the bill then it is possible by POS bill splitting. This slide will show how to split bills in odoo 17 POS.
Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
Instructions for Submissions thorugh G- Classroom.pptxJheel Barad
This presentation provides a briefing on how to upload submissions and documents in Google Classroom. It was prepared as part of an orientation for new Sainik School in-service teacher trainees. As a training officer, my goal is to ensure that you are comfortable and proficient with this essential tool for managing assignments and fostering student engagement.
2024.06.01 Introducing a competency framework for languag learning materials ...Sandy Millin
http://sandymillin.wordpress.com/iateflwebinar2024
Published classroom materials form the basis of syllabuses, drive teacher professional development, and have a potentially huge influence on learners, teachers and education systems. All teachers also create their own materials, whether a few sentences on a blackboard, a highly-structured fully-realised online course, or anything in between. Despite this, the knowledge and skills needed to create effective language learning materials are rarely part of teacher training, and are mostly learnt by trial and error.
Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
This webinar will introduce you to my framework, highlighting the key competencies I identified from my research. It will also show how anybody involved in language teaching (any language, not just English!), teacher training, managing schools or developing language learning materials can benefit from using the framework.
Ethnobotany and Ethnopharmacology:
Ethnobotany in herbal drug evaluation,
Impact of Ethnobotany in traditional medicine,
New development in herbals,
Bio-prospecting tools for drug discovery,
Role of Ethnopharmacology in drug evaluation,
Reverse Pharmacology.
We all have good and bad thoughts from time to time and situation to situation. We are bombarded daily with spiraling thoughts(both negative and positive) creating all-consuming feel , making us difficult to manage with associated suffering. Good thoughts are like our Mob Signal (Positive thought) amidst noise(negative thought) in the atmosphere. Negative thoughts like noise outweigh positive thoughts. These thoughts often create unwanted confusion, trouble, stress and frustration in our mind as well as chaos in our physical world. Negative thoughts are also known as “distorted thinking”.
The Roman Empire A Historical Colossus.pdfkaushalkr1407
The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
Culturally, the Romans were eclectic, absorbing and adapting elements from the civilizations they encountered, particularly the Greeks. Roman art, literature, and philosophy reflected this synthesis, creating a rich cultural tapestry. Latin, the Roman language, became the lingua franca of the Western world, influencing numerous modern languages.
Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
Palestine last event orientationfvgnh .pptxRaedMohamed3
An EFL lesson about the current events in Palestine. It is intended to be for intermediate students who wish to increase their listening skills through a short lesson in power point.
1. For any help regarding Magnetic Materials Assignment Help
Visit :- https://www.eduassignmenthelp.com/
Email :- info@eduassignmenthelp.com
call us at :- +1 678 648 4277
Electromechanics Assignment Help
2. Incompressible In viscid Fluids
Depth W into paper
Fig. 12P.3 = 0
12.3. A magnetic rocket is shown in Fig. 12P.3. A current source 10 (distributed over the
depth W) drives a circuit composed in part of a movable piston. This piston drives an
incompressible, inviscid, nonconducting fluid through two orifices, each of height d and
depth W. Because D >> d, the flow is essentially steady.
(a) Find the exit velocity V as a function of 4.
(b) (b) What is the thrust developed by the rocket? (You may assume that it is under
static test.)
eduassignmenthelp.com
3. For Section 7.2: Prob. 7.2.1 In Sec. 3.7, ci is defined such that in the conservative
subsystem, Eq. 3.7.3 holds. Show that ai satisfies Eq. 7.2.3 with p+ai. Further, show that
if a "specific" property Si is defined such that Bi pai, then by virture of conservation of
mass, the convective derivative of Si is zero.
For Section 7.6:
Prob. 7.6.1 Show that Eq (hb of Tah1P 7_6_2 is correct
Prob. 7.6.2 Show that Eqs. (j) and (2) from
Table 7.6.2 are correct.
Prob. 7.6.3 A pair of bubbles are formed with
the tube-valve system shown in the figure.
Bubble 1 is blown by closing valve V2 and
opening Vl. Then, Vl is closed and V2 opened so that
the second bubble is filled. Each bubble can be
regarded as having a constant surface tension y.
With the bubbles having the same initial radius Eo, when t = 0, both valves are opened
(with the upper inlet closed off). The object of the following steps is to describe the
resulting dynamics.
(a) Flow through the tube that connects the bubbles is modeled as being fully developed
and viscous dominated. Hence, for a length of tube k having inner radius R and with a
viscosity of the gas the volume rate of flow is related to the pressure difference by
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4. The inertia of the gas and bubble is ignored, as is that of the surrounding air. Find an
equation of motion for the bubble radius E1 .
(b) With the bubbles initially of equal radius Eo , there is a slight departure of the radius
of one of the bubbles from eauilibrium. What hapDens?
(c) In physical terms, explain the result of (b).
For Section 7.8:
Prob. 7.8.1 A conduit forming a closed loop consists of a pair of
tubes having cross-sections with areas Ar and Ay . These are
arranged as shown with a fluid having density pb filling the lower
half and a second fluid having density Pa filling the upper half.
The object of the following steps is to determine the dynamics of
the fluid, specifically the time dependence of the interfacial
positions Er and E .
(a) Use mass conservation to relate the displacements (Er', E )
to each other and to the fluid velocities (vr, vk) on the right
and left respectively. Assume that the fluid is inviscid and has
a uniform profile over the cross-section of a tube.
(b) Use Bernoulli's equation, Eq. 7.8.5 to relate quantities
evaluated at the interfaces in the lower fluid, and in the upper
fluid.
(c) Write the boundary conditions that relate quantities across
the interfaces.
(d) Show that these laws combine to give an equation of motion
for the right interface having the form
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5. Prob. 3.10.3 Fig. P3.10.3 shows a circular cylindrical tube of
inner radius a into which a second tube of outer radius b
projects half way. On top of this inner tube is a "blob" of liquid
metal (shown inside th broken-line box) having an arbitrary
shape, but having a base radius equal to that of the inner
tube. The outer and inner tubes, as well a the blob, are all
essentially perfectly conducting on the time scale o interest.
When t=0 , there are no magnetic fields. When t=O+, the ou -
t. tube is used to produce a magnetic flux which has density
Bo z a distan 2 >> a above the end of the inner tube.
What is the magnetic flux den ity over the cross section of the
annulus between tubes a distance 2 (2 >> a) below the end
of the inner tube? Sketch the distribution of surface
current.on the perfect conductors (outer and inner tubes and
blob), indicating the relative densities. Use qualitative
arguments to state whether the vertical magnetic force on the
blob acts upward or downward. Use the stress tensor to find
the magnetic force acting on the blob in the z direction. This
expression should be exact if 2 >> a, and be written in terms
of a, b, Bo and the permeability of free space yo.
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6. Prob. 3.10.4 The mechanical configuration is as in Prob. 3.10.3. But, instead of the
magnetic field, an electric field is produced by making the outer cylinder have the
potential Vo relative to the inner one. Sketch the distribution of the electric field, and
give qualitative arguments as to whether the electrical force on the blob is upward or
downward. What is the electric field in the annulus at points well removed from the tip
of the inner cylinder? Use the electric stress tensor to determine the z-directed electric
force on the blob. + 4 .
Prob. 3.10.5 In an EQS system with polarization, the force density is not F = PpE +
PfE, where Pp is the polarization charge. Nevertheless, this force density can be used
to correctly determine the total force on an object isolated in free space. The proof
follows from the argument given in the paragraph following Eq. 3.10.4. Show that the
stress tensor associated with this force density is
Show that the predicted total force will agree with that found by any of the force
densities in Table 3.10.1. Prob. 3.10.6 Given the force density of Eq. 3.8.13, show that
the stress tensor given for this force density in Table 3.10.1 is correct. It proves helpful
to first show that
Prob. 3.10.7 Given the Kelvin force density, Eq. 3.5.12, derive the consistent stress
tensor of Table 3.10.1. Note the vector identity given in Prob. 3.10.6.
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7. Prob. 3.10.8 Total forces on objects can sometimes be found by the energy method
"ignoring" fringing fields and yet obtaining results that are "exact." This is because the
change in total energy caused by a virtual displacement leaves the fringing field
unaltered. There is a "theorem" than any configuration that can be described in this
way by an energy method can also be-described by integrating the stress tensor over
an appropriately defined surface. Use Eqs. 3.7.22 of Table 3.10.1 to find the force
derived in Prob. 2.13.2.
For Section 3.11:
Prob. 3.11.1 An alternative to the derivation represented by Eq. 3.11.7 comes from
exploiting an integral theorem that is analogous to Stokes's theorem.1
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8. � �
Thus, the combination of Eqs. 1 and 4 give a first order differential equation
describing the evolution of ξ1 or ξ2. In normalized terms, that expression is
� �
Problem 7.6.3
Mass conservation requries that
(1)
With the pressure outside the bubbles defined as p0, the pressure inside the
respective bubbles are
(2)
so that the pressure difference driving fluid between the bubbles
once the valve is opened is
.
(3)
The flow rate between bubbles given by differentiating Eq. 1 is then equal to
Qv and hence to the given expression for the pressure drop through the
connecting tubing.
(4)
Solution
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9. Thus, the velocity is a function of ξ1, and can be pictured as shown in the figure. It
is therefore evident that if ξ1 increases slightly, it will tend to further increase. The
static equilibrium at ξ1 = ξ0 is unstable. Physically this results from the fact that γ
is constant. As the radius of curvature of a bubble decreases, the pressure
increases and forces the air into the other bubble. Note that this is not what would be
found if the bubbles were replaced by most elastic membranes. The example is
useful for giving a reminder of what is implied by the concept of a surface tension.
Of course, if the bubble can not be modelled as a layer of liquid with interior and
exterior interfaces comprised of the same material, then the basic law may not apply.
In the figure, note that all variables are normalized. The asymptote comes at the
radius where the second bubble has completely collapsed.
where
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10. Figure 1: Bubble velocity versus bubble displacement from Eq. (5) (Image by MIT
OpenCourseWare.)
Courtesy of James R. Melcher. Used with permission.
Solution to Problem 7.6.3 in Solutions Manual for Continuum
Electromechanics, 1982, pp. 7.3-7.4
Problem 3.10.3
(a)
The magnetic field is “trapped” in the region between tubes. For an infinitely long pair
of coaxial conductors, the field in the annulus is uniform. Hence, because the total
flux πa2B0 must be constant over the length of the system, in the lower region
a2B0
B z =
a2 −b2
(6)
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11. (b)
The distribution of surface current is as sketched below. It is determined by the
condition that the magnetic flux at the extremities be as found in (a) and by the
condition that the normal flux density on any of the perfectly conducting surfaces
vanish.
(c)
Using the surface force density K× < B >, it is reasonable to expect the net
magnetic force in the z direction to be downward.
(d)
One way to find the net force is to enclose the “blob” by the control volume shown in
the figure and integrate the stress tensor over the enclosing surface.
�
fz = Tzj nj da
S
Contributions to this integration over surfaces (4) and (2) (the walls of the inner
and outer tubes which are perfectly conducting) vanish because there is no shear
stress on a perfectly conducting surface. Surface
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12. Figure 2: Magnetic field lines and stress tensor enclosing surface (dashed) (Image by
MIT OpenCourseWare.)
(5) cuts under the “blob” and hence sustains no magnetic stress. Hence, only
surfaces (1) and (3) make contributions, and on them the magnetic flux density is
given and uniform. Hence, the net force is
Note that, as expected, this force is negative.
Problem 3.10.4
The electric field is sketched in the figure. The force on the cap should be upward. To
find this force use the surface S shown to enclose the cap. On S1 the field is zero. On
S2 and S3 the electric shear stress is zero because it is an equi-potential and hence
can support no tangential E. On S4 the field is zero. Finally, on S5 the field is that of
infinite coaxial conductors.
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13. Thus, the normal electric stress is
and the integral for the total force reduces to
(8)
(9)
(10)
Figure 3: Electric field lines and stress tensor enclosing surface (dashed) (Image by
MIT OpenCourseWare.)
Courtesy of James R. Melcher. Used with permission.
Solution to Problem 3.10.4 in Solutions Manual for Continuum
Electromechanics, 1982, p. 3.8.
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14. (11)
The exit velocity at each orifice is obtained by using Bernoulli’s law just to the left
of the piston and at either orifice, from which we obtain
(12)
(13)
(14)
Problem 4
(a)
From the results of problem 12.2, we have that the pressure p, acting just to the left
of the piston, is
at each orifice.
(b)
The thrust is
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15. (c)In the steady state, we choose to integrate the momentum theorem, Eq. (12.1.29),
around a rectangular surface, enclosing the system for −L ≤ x1 ≤ +L.
−ρV 2 a + ρ[V (L)]2b = p0a −p(L)b + F,
(15)
0
where F is the x1 component force per unit length which the walls exert on the fluid.
We see that there is no x1 component of force from the upper wall, therefore F is
the force purely from the lower wall.
In the steady state, conservation of mass, (Eq. 12.1.8), yields
(17)
(18)
(19)
(16)
Bernoulli’s equation gives us
Solving (17) for P(L), and then substituting this result and that of (16) into (15), we
finally obtain
The problem asked for the force on the lower wall, which is just the negative of F.
Thus
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