The document discusses key concepts in fluid mechanics including:
1. Fluid kinematics deals with fluid motion without reference to forces, using Lagrangian and Eulerian techniques. Different types of flow include steady/unsteady, uniform/non-uniform, and laminar/turbulent.
2. Streamlines, pathlines, and streaklines are defined. Streamlines give the instantaneous flow pattern. Continuity equations are derived for compressible, incompressible, 1D, and 2D steady flows.
3. Discharge is defined as the volume flow rate through a cross-section. Acceleration terms including convective and temporal acceleration are explained for different flow types.
Reynolds number and geometry concept, Momentum integral equations, Boundary layer equations, Flow over a flat plate, Flow over cylinder, Pipe flow, fully developed laminar pipe flow, turbulent pipe flow, Losses in pipe flow
B.TECH. DEGREE COURSE
SCHEME AND SYLLABUS
(2002-03 admission onwards)
MAHATMA GANDHI UNIVERSITY,mg university, KTU
KOTTAYAM
KERALA
Module 1
Introduction - Proprties of fluids - pressure, force, density, specific weight, compressibility, capillarity, surface tension, dynamic and kinematic viscosity-Pascal’s law-Newtonian and non-Newtonian fluids-fluid statics-measurement of pressure-variation of pressure-manometry-hydrostatic pressure on plane and curved surfaces-centre of pressure-buoyancy-floation-stability of submerged and floating bodies-metacentric height-period of oscillation.
Module 2
Kinematics of fluid motion-Eulerian and Lagrangian approach-classification and representation of fluid flow- path line, stream line and streak line. Basic hydrodynamics-equation for acceleration-continuity equation-rotational and irrotational flow-velocity potential and stream function-circulation and vorticity-vortex flow-energy variation across stream lines-basic field flow such as uniform flow, spiral flow, source, sink, doublet, vortex pair, flow past a cylinder with a circulation, Magnus effect-Joukowski theorem-coefficient of lift.
Module 3
Euler’s momentum equation-Bernoulli’s equation and its limitations-momentum and energy correction factors-pressure variation across uniform conduit and uniform bend-pressure distribution in irrotational flow and in curved boundaries-flow through orifices and mouthpieces, notches and weirs-time of emptying a tank-application of Bernoulli’s theorem-orifice meter, ventury meter, pitot tube, rotameter.
Module 4
Navier-Stoke’s equation-body force-Hagen-Poiseullie equation-boundary layer flow theory-velocity variation- methods of controlling-applications-diffuser-boundary layer separation –wakes, drag force, coefficient of drag, skin friction, pressure, profile and total drag-stream lined body, bluff body-drag force on a rectangular plate-drag coefficient for flow around a cylinder-lift and drag force on an aerofoil-applications of aerofoil- characteristics-work done-aerofoil flow recorder-polar diagram-simple problems.
Module 5
Flow of a real fluid-effect of viscosity on fluid flow-laminar and turbulent flow-boundary layer thickness-displacement, momentum and energy thickness-flow through pipes-laminar and turbulent flow in pipes-critical Reynolds number-Darcy-Weisback equation-hydraulic radius-Moody;s chart-pipes in series and parallel-siphon losses in pipes-power transmission through pipes-water hammer-equivalent pipe-open channel flow-Chezy’s equation-most economical cross section-hydraulic jump.
1. Introduction to Kinematics
2. Methods of Describing Fluid Motion
a). Lagrangian Method
b). Eulerian Method
3. Flow Patterns
- Stream Line
- Path Line
- Streak Line
- Streak Tube
4. Classification of Fluid Flow
a). Steady and Unsteady Flow
b). Uniform and Non-Uniform Flow
c). Laminar and Turbulent Flow
d). Rotational and Irrotational Flow
e). Compressible and Incompressible Flow
f). Ideal and Real Flow
g). One, Two and Three Dimensional Flow
5. Rate of Flow (Discharge) and Continuity Equation
6. Continuity Equation in Three Dimensions
7. Velocity and Acceleration
8. Stream and Velocity Potential Functions
Dimension less numbers in applied fluid mechanicstirath prajapati
In dimensional analysis, a dimensionless quantity is a quantity to which no physical dimension is assigned. It is also known as a bare number or pure number or a quantity of dimension one[1] and the corresponding unit of measurement in the SI is one (or 1) unit[2][3] and it is not explicitly shown. Dimensionless quantities are widely used in many fields, such as mathematics, physics, chemistry, engineering, and economics. Examples of quantities, to which dimensions are regularly assigned, are length, time, and speed, which are measured in dimensional units, such as meter , second and meter per second. This is considered to aid intuitive understanding. However, especially in mathematical physics, it is often more convenient to drop the assignment of explicit dimensions and express the quantities without dimensions, e.g., addressing the speed of light simply by the dimensionless number 1.
Reynolds number and geometry concept, Momentum integral equations, Boundary layer equations, Flow over a flat plate, Flow over cylinder, Pipe flow, fully developed laminar pipe flow, turbulent pipe flow, Losses in pipe flow
B.TECH. DEGREE COURSE
SCHEME AND SYLLABUS
(2002-03 admission onwards)
MAHATMA GANDHI UNIVERSITY,mg university, KTU
KOTTAYAM
KERALA
Module 1
Introduction - Proprties of fluids - pressure, force, density, specific weight, compressibility, capillarity, surface tension, dynamic and kinematic viscosity-Pascal’s law-Newtonian and non-Newtonian fluids-fluid statics-measurement of pressure-variation of pressure-manometry-hydrostatic pressure on plane and curved surfaces-centre of pressure-buoyancy-floation-stability of submerged and floating bodies-metacentric height-period of oscillation.
Module 2
Kinematics of fluid motion-Eulerian and Lagrangian approach-classification and representation of fluid flow- path line, stream line and streak line. Basic hydrodynamics-equation for acceleration-continuity equation-rotational and irrotational flow-velocity potential and stream function-circulation and vorticity-vortex flow-energy variation across stream lines-basic field flow such as uniform flow, spiral flow, source, sink, doublet, vortex pair, flow past a cylinder with a circulation, Magnus effect-Joukowski theorem-coefficient of lift.
Module 3
Euler’s momentum equation-Bernoulli’s equation and its limitations-momentum and energy correction factors-pressure variation across uniform conduit and uniform bend-pressure distribution in irrotational flow and in curved boundaries-flow through orifices and mouthpieces, notches and weirs-time of emptying a tank-application of Bernoulli’s theorem-orifice meter, ventury meter, pitot tube, rotameter.
Module 4
Navier-Stoke’s equation-body force-Hagen-Poiseullie equation-boundary layer flow theory-velocity variation- methods of controlling-applications-diffuser-boundary layer separation –wakes, drag force, coefficient of drag, skin friction, pressure, profile and total drag-stream lined body, bluff body-drag force on a rectangular plate-drag coefficient for flow around a cylinder-lift and drag force on an aerofoil-applications of aerofoil- characteristics-work done-aerofoil flow recorder-polar diagram-simple problems.
Module 5
Flow of a real fluid-effect of viscosity on fluid flow-laminar and turbulent flow-boundary layer thickness-displacement, momentum and energy thickness-flow through pipes-laminar and turbulent flow in pipes-critical Reynolds number-Darcy-Weisback equation-hydraulic radius-Moody;s chart-pipes in series and parallel-siphon losses in pipes-power transmission through pipes-water hammer-equivalent pipe-open channel flow-Chezy’s equation-most economical cross section-hydraulic jump.
1. Introduction to Kinematics
2. Methods of Describing Fluid Motion
a). Lagrangian Method
b). Eulerian Method
3. Flow Patterns
- Stream Line
- Path Line
- Streak Line
- Streak Tube
4. Classification of Fluid Flow
a). Steady and Unsteady Flow
b). Uniform and Non-Uniform Flow
c). Laminar and Turbulent Flow
d). Rotational and Irrotational Flow
e). Compressible and Incompressible Flow
f). Ideal and Real Flow
g). One, Two and Three Dimensional Flow
5. Rate of Flow (Discharge) and Continuity Equation
6. Continuity Equation in Three Dimensions
7. Velocity and Acceleration
8. Stream and Velocity Potential Functions
Dimension less numbers in applied fluid mechanicstirath prajapati
In dimensional analysis, a dimensionless quantity is a quantity to which no physical dimension is assigned. It is also known as a bare number or pure number or a quantity of dimension one[1] and the corresponding unit of measurement in the SI is one (or 1) unit[2][3] and it is not explicitly shown. Dimensionless quantities are widely used in many fields, such as mathematics, physics, chemistry, engineering, and economics. Examples of quantities, to which dimensions are regularly assigned, are length, time, and speed, which are measured in dimensional units, such as meter , second and meter per second. This is considered to aid intuitive understanding. However, especially in mathematical physics, it is often more convenient to drop the assignment of explicit dimensions and express the quantities without dimensions, e.g., addressing the speed of light simply by the dimensionless number 1.
Fluid Mechanics Chapter 4. Differential relations for a fluid flowAddisu Dagne Zegeye
Introduction, Acceleration field, Conservation of mass equation, Linear momentum equation, Energy equation, Boundary condition, Stream function, Vorticity and Irrotationality
Fluid Mechanics Chapter 4. Differential relations for a fluid flowAddisu Dagne Zegeye
Introduction, Acceleration field, Conservation of mass equation, Linear momentum equation, Energy equation, Boundary condition, Stream function, Vorticity and Irrotationality
DERIVATION OF THE MODIFIED BERNOULLI EQUATION WITH VISCOUS EFFECTS AND TERMIN...Wasswaderrick3
In this book, we use conservation of energy techniques on a fluid element to derive the Modified Bernoulli equation of flow with viscous or friction effects. We derive the general equation of flow/ velocity and then from this we derive the Pouiselle flow equation, the transition flow equation and the turbulent flow equation. In the situations where there are no viscous effects , the equation reduces to the Bernoulli equation. From experimental results, we are able to include other terms in the Bernoulli equation. We also look at cases where pressure gradients exist. We use the Modified Bernoulli equation to derive equations of flow rate for pipes of different cross sectional areas connected together. We also extend our techniques of energy conservation to a sphere falling in a viscous medium under the effect of gravity. We demonstrate Stokes equation of terminal velocity and turbulent flow equation. We look at a way of calculating the time taken for a body to fall in a viscous medium. We also look at the general equation of terminal velocity.
u can learn what is research, how to do research, research types, research methods, methodology, how to do literature survey, how to give an oral presentation and how to write thesis, research paper
it talks about the introduction of the book of Little book on Stoicism,
it talks mainly about the importance of Stoicism and main components of Stoicism
Actual cycles for internal combustion engines differ from air-standard cycles in many respects.
Time loss factor.
Heat loss factor.
Exhaust blow down factor.
Theoretical cycle based on the actual properties of the cylinder contents is called the fuel air cycle.
The fuel air cycle takes into consideration the following.
The ACTUAL COMPOSITION of the cylinder contents.
The VARIATION OF SPECIFIC HEAT of the gases in the cylinder.
The DISSOCIATION EFFECT.
The VARIATION IN THE NUMBER OF MOLES present in the cylinder as the pressure and temperature change
Automobile Management System Project Report.pdfKamal Acharya
The proposed project is developed to manage the automobile in the automobile dealer company. The main module in this project is login, automobile management, customer management, sales, complaints and reports. The first module is the login. The automobile showroom owner should login to the project for usage. The username and password are verified and if it is correct, next form opens. If the username and password are not correct, it shows the error message.
When a customer search for a automobile, if the automobile is available, they will be taken to a page that shows the details of the automobile including automobile name, automobile ID, quantity, price etc. “Automobile Management System” is useful for maintaining automobiles, customers effectively and hence helps for establishing good relation between customer and automobile organization. It contains various customized modules for effectively maintaining automobiles and stock information accurately and safely.
When the automobile is sold to the customer, stock will be reduced automatically. When a new purchase is made, stock will be increased automatically. While selecting automobiles for sale, the proposed software will automatically check for total number of available stock of that particular item, if the total stock of that particular item is less than 5, software will notify the user to purchase the particular item.
Also when the user tries to sale items which are not in stock, the system will prompt the user that the stock is not enough. Customers of this system can search for a automobile; can purchase a automobile easily by selecting fast. On the other hand the stock of automobiles can be maintained perfectly by the automobile shop manager overcoming the drawbacks of existing system.
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
introduce a novel volumization algorithm, which transforms 2D images into 3D volumetric representations.
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
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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.
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.
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.
About
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
• Remote control: Parallel or serial interface.
• Compatible with MAFI CCR system.
• Compatible with IDM8000 CCR.
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
• Easy in configuration using DIP switches.
Technical Specifications
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
Key Features
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
• Remote control: Parallel or serial interface
• Compatible with MAFI CCR system
• Copatiable with IDM8000 CCR
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
Application
• Remote control: Parallel or serial interface.
• Compatible with MAFI CCR system.
• Compatible with IDM8000 CCR.
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
• Easy in configuration using DIP switches.
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Vaccine management system project report documentation..pdfKamal Acharya
The Division of Vaccine and Immunization is facing increasing difficulty monitoring vaccines and other commodities distribution once they have been distributed from the national stores. With the introduction of new vaccines, more challenges have been anticipated with this additions posing serious threat to the already over strained vaccine supply chain system in Kenya.
3. FLUID KINEMATICS
Kinematics deal with motion of fluid without any reference to cause of motion i.e., force.
The fluid flow is analysed by using two techniques.
1. Langrangian technique
2. Eulerian technique
In Langrangian technique, single fluid particle is taken and the behaviour of this particle is analysed at different
instances of time.
In Eulerian technique, certain section is taken and fluid flow is analysed at that section.
Different types of fluid flow
Steady & Unsteady flow
A flow is said to be steady flow when fluid properties do not change at any cross section at any given time,
otherwise flow is unsteady.
𝐹𝑜𝑟 𝑆𝑡𝑒𝑎𝑑𝑦 𝑓𝑙𝑜𝑤 → [
𝑑𝑣
𝑑𝑡
]
𝑔𝑖𝑣𝑒𝑛 𝑠𝑒𝑐𝑡𝑖𝑜𝑛
= 0 & [
𝑑𝜌
𝑑𝑡
]
𝑔𝑖𝑣𝑒𝑛 𝑠𝑒𝑐𝑡𝑖𝑜𝑛
= 0
Uniform & non-uniform flow
A flow is said to be uniform when fluid properties especially velocity don’t change with space at any given instant
of time, otherwise the flow is non-uniform.
𝐹𝑜𝑟 𝑈𝑛𝑖𝑓𝑜𝑟𝑚 𝑓𝑙𝑜𝑤 → [
𝑑𝑣
𝑑𝑠(𝑥, 𝑦, 𝑧)
]
𝑔𝑖𝑣𝑒𝑛 𝑡𝑖𝑚𝑒
= 0 {𝑠 = 𝑠𝑝𝑎𝑐𝑒(𝑥, 𝑦, 𝑧)}
Laminar & Turbulent flow
In laminar flow fluid particles move in the form of layers, with one layer sliding over the other layer. Laminar flow
generally occurs at low velocities.
In turbulent flow, fluid particles move in highly disorganized manner, leading to rapid mixing of particles.
Turbulent flow generally occurs at high velocities.
Rotational & Irrotational flow
A flow is said to be rotational flow when fluid particles rotate about their own mass centres, otherwise the flow is
irrotational.
Rotation is possible when there is a tangential force, these tangential forces are associated with viscous fluids.
Therefore, real fluids are generally rotational fluids and ideal fluids are irrotational fluids.
Internal & External flows
When the fluid flows through confined passage (Ex- flow of fluid through pipes, ducts) then it is internal flow.
When the fluid flow through unconfined passage (Ex- Flow of fluid (air) over aircraft wing) then flow is external
flow.
Categorization of flow
1. One-dimensional flow
2. Two-dimensional flow
3. Three-dimensional flow
Flow can never be 1-D, because of viscosity.
Stream line
It is an imaginary line or curve drawn in space such that a tangent drawn to it at any point gives velocity vector.
Stream line gives direction of flow as there is no component of velocity in perpendicular direction there is no flow
across the stream line, there is flow only along the stream line. Stream line gives instantaneous snapshot of a flow
pattern. It has no time history. No two stream lines can intersect because velocity is unique at any given instant of
time at a particular time.
4. Equation of Stream line
In 3-D
𝑣⃗ = 𝑢𝑖̂ + 𝑣𝑗̂ + 𝑤𝑘̂
𝑣 = √ 𝑢2 + 𝑣2 + 𝑤2
In 2-D
𝑣⃗ = 𝑢𝑖̂ + 𝑣𝑗̂
𝒖 =
𝑑𝑥
𝑑𝑡
⇒ 𝑑𝑡 =
𝑑𝑥
𝑢
& 𝒗 =
𝑑𝑦
𝑑𝑡
⇒ 𝑑𝑡 =
𝑑𝑦
𝑣
𝒅𝒙
𝒖
=
𝒅𝒚
𝒗
→ 𝐸𝑞𝑢𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑠𝑡𝑟𝑎𝑚 𝑙𝑖𝑛𝑒 𝑖𝑛 2 − 𝐷
𝒅𝒙
𝒖
=
𝒅𝒚
𝒗
=
𝒅𝒛
𝒘
→ 𝐸𝑞𝑢𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑠𝑡𝑟𝑎𝑚 𝑙𝑖𝑛𝑒 𝑖𝑛 3 − 𝐷
Path line
It is the locus of single fluid particle at different instances of time. It follows
Langrangian approach. A path line can intersect with itself.
Streak line
It is the locus of different fluid particles through a fixed point.
5. Unsteady Flow
11:00―11:30 North to South
11:30―12:00 East to West
Steady Flow
11:00 ―12:00North to South
In a steady flow stream lines, streak lines & path lines are identical, whereas in unsteady flow they are different.
Stream lines intersect at stagnation point.
Conservation of mass (Continuity equation)
Generalized Continuity equation
𝑚 = 𝜌 ⋅ 𝑉 → 𝑙𝑛 𝑚 = 𝑙𝑛 𝜌 + 𝑙𝑛 𝑉
Differentiating above equation and simplifying gives
Every fluid flow must satisfy mass conservation or continuity equation. If the fluid flow doesn’t
satisfy continuity equation, then that flow is not possible.
This equation is applicable for any type of fluid flow.
Case-A (Steady flow)
𝐹𝑜𝑟 𝑠𝑡𝑒𝑎𝑑𝑦 𝑓𝑙𝑜𝑤 →
𝜕𝜌
𝜕𝑡
= 0
𝑆𝑜,
𝝏
𝝏𝒙
(𝝆𝒖) +
𝝏
𝝏𝒚
(𝝆𝒗) +
𝝏
𝝏𝒛
(𝝆𝒘) = 𝟎
Case-B (Incompressible flow)
𝐹𝑜𝑟 𝑖𝑛𝑐𝑜𝑚𝑝𝑟𝑒𝑠𝑠𝑖𝑏𝑙𝑒 𝑓𝑙𝑜𝑤 → 𝜌 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡,
𝜕𝜌
𝜕𝑡
= 0
0 +
𝜕
𝜕𝑥
(𝜌𝑢) +
𝜕
𝜕𝑦
(𝜌𝑣) +
𝜕
𝜕𝑧
(𝜌𝑤) = 0 → 𝜌 (
𝜕𝑢
𝜕𝑥
+
𝜕𝑣
𝜕𝑦
+
𝜕𝑤
𝜕𝑧
) = 0
𝝏𝒖
𝝏𝒙
+
𝝏𝒗
𝝏𝒚
+
𝝏𝒘
𝝏𝒛
= 𝟎
This equation is applicable for any type of incompressible flow. (Steady or unsteady)
𝑇ℎ𝑒 𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑖𝑡𝑦 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 𝑓𝑜𝑟 2 − 𝐷 𝑖𝑛𝑐𝑜𝑚𝑝𝑟𝑒𝑠𝑠𝑖𝑏𝑙𝑒 𝑓𝑙𝑜𝑤 𝑖𝑠
𝝏𝒖
𝝏𝒙
+
𝝏𝒗
𝝏𝒚
= 𝟎
Continuity equation for steady 1-Dimensional flow
Flow through pipes, nozzles & diffusers etc…
𝜌 =
𝑚𝑎𝑠𝑠
𝑣𝑜𝑙𝑢𝑚𝑒
𝑚𝑎𝑠𝑠(𝒎) = 𝜌 × 𝑣𝑜𝑙𝑢𝑚𝑒(𝒗)
𝑚̇ =
𝑚
𝑡
=
𝜌 × 𝑣
𝑡
=
𝜌 × (𝐴 × 𝑙)
𝑡
𝒎̇ = 𝝆 × 𝑨 × 𝒗
𝐹𝑜𝑟 𝑠𝑡𝑒𝑎𝑑𝑦 𝑓𝑙𝑜𝑤 → 𝑚1 = 𝑚2
𝝆 𝟏 𝑨 𝟏 𝒗 𝟏 = 𝝆 𝟐 𝑨 𝟐 𝒗 𝟐
𝐼𝑓 𝑡ℎ𝑒 𝑓𝑙𝑜𝑤 𝑖𝑠 𝑖𝑛𝑐𝑜𝑚𝑝𝑟𝑒𝑠𝑠𝑖𝑏𝑙𝑒 → 𝜌1 = 𝜌2
𝐼𝑓 𝑡ℎ𝑒 𝑓𝑙𝑜𝑤 𝑖𝑠 𝑆𝑡𝑒𝑎𝑑𝑦 & 𝐼𝑛𝑐𝑜𝑚𝑝𝑟𝑒𝑠𝑠𝑖𝑏𝑙𝑒,
𝑨 𝟏 𝒗 𝟏 = 𝑨 𝟐 𝒗 𝟐 (𝜌1 = 𝜌2)
𝜕𝜌
𝜕𝑡
+
𝜕
𝜕𝑥
(𝜌𝑢) +
𝜕
𝜕𝑦
(𝜌𝑣) +
𝜕
𝜕𝑧
(𝜌𝑤) = 0
6. Discharge (Q)
Volume flow rate is known as discharge.
𝑄 =
𝑉𝑜𝑙𝑢𝑚𝑒 𝑓𝑙𝑜𝑤𝑖𝑛𝑔 𝑡ℎ𝑟𝑜𝑢𝑔ℎ 𝑎 𝑐𝑟𝑜𝑠𝑠𝑒𝑐𝑡𝑖𝑜𝑛
𝑇𝑖𝑚𝑒
=
𝐴 × 𝑙
𝑡
→ 𝑄 = 𝐴 × 𝑣⃗
In a steady 1-D incompressible flow, discharge remains constant.
Acceleration of a fluid particle
𝑢 = 𝑓(𝑥, 𝑦, 𝑧, 𝑡), 𝑣 = 𝑓(𝑥, 𝑦, 𝑧, 𝑡), 𝑤 = 𝑓(𝑥, 𝑦, 𝑧, 𝑡)
𝑎 =
𝑑𝑣⃗
𝑑𝑡
=
𝑑𝑢
𝑑𝑡
𝑖̂ +
𝑑𝑣
𝑑𝑡
𝑗̂ +
𝑑𝑤
𝑑𝑡
𝑘̂
𝑎 𝑥 =
𝑑𝑢
𝑑𝑡
, 𝑎 𝑦 =
𝑑𝑣
𝑑𝑡
, 𝑎 𝑧 =
𝑑𝑤
𝑑𝑡
𝑎 𝑥 =
𝑑𝑢
𝑑𝑡
=
𝜕𝑢
𝜕𝑥
×
𝜕𝑥
𝜕𝑡
+
𝜕𝑢
𝜕𝑦
×
𝜕𝑦
𝜕𝑡
+
𝜕𝑢
𝜕𝑧
×
𝜕𝑧
𝜕𝑡
+
𝜕𝑢
𝜕𝑡
𝑎 𝑦 =
𝑑𝑣
𝑑𝑡
=
𝜕𝑣
𝜕𝑥
×
𝜕𝑥
𝜕𝑡
+
𝜕𝑣
𝜕𝑦
×
𝜕𝑦
𝜕𝑡
+
𝜕𝑣
𝜕𝑧
×
𝜕𝑧
𝜕𝑡
+
𝜕𝑣
𝜕𝑡
𝑎 𝑧 =
𝑑𝑤
𝑑𝑡
=
𝜕𝑤
𝜕𝑥
×
𝜕𝑥
𝜕𝑡
+
𝜕𝑤
𝜕𝑦
×
𝜕𝑦
𝜕𝑡
+
𝜕𝑤
𝜕𝑧
×
𝜕𝑧
𝜕𝑡
+
𝜕𝑤
𝜕𝑡
Convective Acceleration
The acceleration due to change of velocity with space is known as convective acceleration. For uniform flow
convective acceleration is zero.
Temporal or Local Acceleration
The acceleration due to change of velocity with respective to time is known as temporal acceleration. For steady
flow temporal acceleration is zero.
Type of flow Convective Acceleration Temporal Acceleration
Steady & uniform 0 0
Steady & Non-uniform exists 0
Unsteady & uniform 0 exists
Unsteady & Non-uniform exists exists
Steady flow, 1-Dimensional & incompressible
𝑇𝑒𝑚𝑝𝑜𝑟𝑎𝑙 = 0
𝐴1 𝑣1 = 𝐴2 𝑣2 ⇒ 𝒗 𝟏 = 𝒗 𝟐
Velocity is not changing w.r.t time.
𝐴1 𝑣1 = 𝐴2 𝑣2
𝐴2 < 𝐴1
𝑣2 > 𝑣1
`Stream lines are converging Convective acceleration
𝐴1 𝑣1 = 𝐴2 𝑣2
𝐴2 > 𝐴1
𝑣2 < 𝑣1
Stream lines are diverging deceleration
8. Circulation (Γ)
It is the line integral of tangential component of velocity taken
around a closed curve.
𝛤 = 𝑢 ⋅ 𝑑𝑥 + (𝑣 +
𝜕𝑣
𝜕𝑥
𝑑𝑥) ⋅ 𝑑𝑦 − (𝑢 +
𝜕𝑢
𝜕𝑦
𝑑𝑦) ⋅ 𝑑𝑥 − 𝑣 ⋅ 𝑑𝑦
𝛤 = (
𝜕𝑣
𝜕𝑥
−
𝜕𝑢
𝜕𝑦
) 𝑑𝑥 ⋅ 𝑑𝑦
[𝐴𝑟𝑒𝑎 = 𝑑𝑥 ⋅ 𝑑𝑦]
𝐶𝑖𝑟𝑐𝑢𝑙𝑎𝑡𝑖𝑜𝑛 (𝛤) = 𝑉𝑜𝑟𝑡𝑖𝑐𝑖𝑡𝑦 (2𝜔 𝑧) × 𝐴𝑟𝑒𝑎
In case of irrotational flow, vorticity is zero & circulation is zero.
Velocity Potential function (ϕ)
It is a function of space & time defined in such a manner, that its negative derivative w.r.t space gives velocity in
that direction. The negative sign is taken as the flow is in the direction of decreasing potential.
−
𝜕𝜙
𝜕𝑥
= 𝑢 −
𝜕𝜙
𝜕𝑦
= 𝑣 −
𝜕𝜙
𝜕𝑧
= 𝑤
Velocity potential function can be defined for 2-Dimensional flow
𝜕𝑢
𝜕𝑥
+
𝜕𝑣
𝜕𝑥
=
𝜕
𝜕𝑥
(−
𝜕𝜙
𝜕𝑥
) +
𝜕
𝜕𝑦
(−
𝜕𝜙
𝜕𝑦
) = − (
𝜕2
𝜙
𝜕𝑥2
+
𝜕2
𝜙
𝜕𝑦2
)
Case 1
𝐼𝑓
𝜕2
𝜙
𝜕𝑥2
+
𝜕2
𝜙
𝜕𝑦2
= 0, 𝝓 𝒔𝒂𝒕𝒊𝒔𝒇𝒊𝒆𝒔 𝑳𝒂𝒑𝒍𝒂𝒄𝒆 𝒆𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝑎𝑠 (
𝜕𝑢
𝜕𝑥
+
𝜕𝑣
𝜕𝑥
= 0)
→ Continuity equation is satisfied and flow is possible.
Case 2
𝐼𝑓
𝜕2
𝜙
𝜕𝑥2
+
𝜕2
𝜙
𝜕𝑦2
≠ 0, 𝝓 𝒅𝒐𝒆𝒔𝒏′
𝒕 𝒔𝒂𝒕𝒊𝒔𝒇𝒊𝒆𝒔 𝑳𝒂𝒑𝒍𝒂𝒄𝒆 𝒆𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝑎𝑠 (
𝜕𝑢
𝜕𝑥
+
𝜕𝑣
𝜕𝑥
≠ 0)
→ Continuity equation is not satisfied and flow is not possible.
Case 3
𝜔 𝑧 =
1
2
(
𝜕𝑣
𝜕𝑥
−
𝜕𝑢
𝜕𝑦
) =
1
2
(−
𝜕2
𝜙
𝜕𝑥 ⋅ 𝜕𝑦
+
𝜕2
𝜙
𝜕𝑦 ⋅ 𝜕𝑥
)
𝜔 𝑧 = 0 → 𝐼𝑟𝑟𝑜𝑡𝑎𝑡𝑖𝑜𝑛𝑎𝑙 𝑓𝑙𝑜𝑤
Velocity Potential function exits only for Irrotational flow i.e., the existence of velocity potential function
implies the flow is irrotational. Sometimes irrotational flow are also known as Potential flow.
9. Stream function (Ψ)
It is a function of space & time defined in such a manner that it satisfies continuity equation.
𝑢 = −
𝜕𝜓
𝜕𝑦
𝑣 =
𝜕𝜓
𝜕𝑥
Note Though velocity potential function can be defined for 3-Dimensional flows, it is difficult to define stream
function in 3-Dimensional flows. Therefore, stream functions are generally defined for 2-D flows.
𝜔 𝑧 =
1
2
(
𝜕𝑣
𝜕𝑥
−
𝜕𝑢
𝜕𝑦
) =
1
2
(
𝜕2
𝜓
𝜕𝑥2
+
𝜕2
𝜓
𝜕𝑦2
)
Case 1
𝐼𝑓 𝜓 𝑠𝑎𝑡𝑖𝑠𝑓𝑖𝑒𝑠 𝐿𝑎𝑝𝑙𝑎𝑐𝑒 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 (
𝜕𝑢
𝜕𝑥
+
𝜕𝑣
𝜕𝑦
= 0) &
𝜕2
𝜓
𝜕𝑥2
+
𝜕2
𝜓
𝜕𝑦2
= 0 ⇒ 𝜔𝑧 = 0 → 𝑡ℎ𝑒 𝑓𝑙𝑜𝑤 𝑖𝑠 𝑖𝑟𝑟𝑜𝑡𝑎𝑡𝑖𝑜𝑛𝑎𝑙
Case 2
𝐼𝑓 𝜓 𝑑𝑜𝑒𝑠𝑛′
𝑡 𝑠𝑎𝑡𝑖𝑠𝑓𝑖𝑒𝑠 𝐿𝑎𝑝𝑙𝑎𝑐𝑒 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 (
𝜕𝑢
𝜕𝑥
+
𝜕𝑣
𝜕𝑦
≠ 0) &
𝜕2
𝜓
𝜕𝑥2
+
𝜕2
𝜓
𝜕𝑦2
≠ 0 ⇒ 𝜔𝑧 ≠ 0 → 𝑡ℎ𝑒 𝑓𝑙𝑜𝑤 𝑖𝑠 𝑟𝑜𝑡𝑎𝑡𝑖𝑜𝑛𝑎𝑙
Velocity potential function exists only for rotational flow whereas stream function exists for both rotational &
irrotational flow.
If stream function satisfies Laplace equation, then flow is irrotational.
Significance of Stream Function
𝐸𝑞𝑢𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑎 𝑆𝑡𝑟𝑒𝑎𝑚 𝑙𝑖𝑛𝑒 →
𝑑𝑥
𝑢
=
𝑑𝑦
𝑣
→ 𝑣 ⋅ 𝑑𝑥 = 𝑢 ⋅ 𝑑𝑦 ⇒ 𝑣 ⋅ 𝑑𝑥 − 𝑢 ⋅ 𝑑𝑦 = 0
𝑢 = −
𝜕𝜓
𝜕𝑦
𝑣 =
𝜕𝜓
𝜕𝑥
Substituting we get,
𝜕𝜓
𝜕𝑥
⋅ 𝑑𝑥 − (−
𝜕𝜓
𝜕𝑦
) ⋅ 𝑑𝑦 = 0 ⇒
𝜕𝜓
𝜕𝑥
⋅ 𝑑𝑥 +
𝜕𝜓
𝜕𝑦
⋅ 𝑑𝑦 = 0 → ① → 𝐸𝑞𝑢𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑎 𝑝𝑎𝑟𝑡𝑖𝑐𝑢𝑙𝑎𝑟 𝑠𝑡𝑟𝑒𝑎𝑚 𝑙𝑖𝑛𝑒
𝜕𝜓
𝜕𝑥
⋅ 𝑑𝑥 +
𝜕𝜓
𝜕𝑦
⋅ 𝑑𝑦 = 𝜵𝝍 = 𝑑𝜓 = 0
𝐴𝑠 𝒅𝝍 = 0 → 𝝍 = 𝑪𝒐𝒏𝒔𝒕𝒂𝒏𝒕
For a particular stream line, Stream function remains constant.
𝑄 = 𝐴 ⋅ 𝑣 = (𝑑𝑥 ⋅ 1) ⋅ 𝑣 = 𝑣 ⋅ 𝑑𝑥 =
𝜕𝜓
𝜕𝑥
⋅ 𝑑𝑥
𝑄 =
𝜕𝜓
𝜕𝑥
⋅ 𝑑𝑥 ①
𝑑𝜓 =
𝜕𝜓
𝜕𝑥
⋅ 𝑑𝑥 +
𝜕𝜓
𝜕𝑦
⋅ 𝑑𝑦 (𝑑𝑦 = 0)
𝑑𝜓 =
𝜕𝜓
𝜕𝑥
⋅ 𝑑𝑥 ②
𝐴𝑠 ① = ②,
𝑸 = 𝒅𝝍 (𝐷𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝑖𝑛 𝑆𝑡𝑟𝑒𝑎𝑚 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛)
The difference in stream function gives discharge per unit width.
Relationship between Equipotential lines & Constant Stream Function lines
𝜙(𝑥, 𝑦) = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 → 𝐸𝑞𝑢𝑖𝑝𝑜𝑡𝑒𝑛𝑡𝑖𝑎𝑙 𝑙𝑖𝑛𝑒𝑠 (𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡 𝑙𝑖𝑛𝑒𝑠 ℎ𝑎𝑣𝑒 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡 𝑝𝑜𝑡𝑒𝑛𝑡𝑖𝑎𝑙)
𝑑𝜙 =
𝜕𝜙
𝜕𝑥
⋅ 𝑑𝑥 +
𝜕𝜙
𝜕𝑦
⋅ 𝑑𝑦 = 0 ⇒
𝜕𝜙
𝜕𝑥
⋅ 𝑑𝑥 = −
𝜕𝜙
𝜕𝑦
⋅ 𝑑𝑦 ⇒ −𝑢 ⋅ 𝑑𝑥 = 𝑣 ⋅ 𝑑𝑦 (𝑢 = −
𝜕𝜙
𝜕𝑥
𝑣 = −
𝜕𝜙
𝜕𝑦
)
(𝑆𝑙𝑜𝑝𝑒 𝑜𝑓 𝐸𝑞𝑢𝑖𝑝𝑜𝑡𝑒𝑛𝑡𝑖𝑎𝑙 𝑙𝑖𝑛𝑒)
𝑑𝑦
𝑑𝑥
= −
𝑢
𝑣
=
𝜕𝜙
𝜕𝑥
−
𝜕𝜙
𝜕𝑦
⁄
10. 𝑑𝜓 =
𝜕𝜓
𝜕𝑥
⋅ 𝑑𝑥 +
𝜕𝜓
𝜕𝑦
⋅ 𝑑𝑦 = 0 ⇒
𝜕𝜓
𝜕𝑥
⋅ 𝑑𝑥 = −
𝜕𝜓
𝜕𝑦
⋅ 𝑑𝑦 ⇒ 𝑣 ⋅ 𝑑𝑥 = 𝑢 ⋅ 𝑑𝑦
(𝑆𝑙𝑜𝑝𝑒 𝑜𝑓 𝑆𝑡𝑟𝑒𝑎𝑚 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑙𝑖𝑛𝑒𝑠)
𝑑𝑦
𝑑𝑥
=
𝑣
𝑢
𝑆𝑙𝑜𝑝𝑒 𝑜𝑓 𝐸𝑞𝑢𝑖𝑝𝑜𝑡𝑒𝑛𝑡𝑖𝑎𝑙 𝑙𝑖𝑛𝑒 × 𝑆𝑙𝑜𝑝𝑒 𝑜𝑓 𝑆𝑡𝑟𝑒𝑎𝑚 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑙𝑖𝑛𝑒 = −
𝑢
𝑣
×
𝑣
𝑢
= −1
Equipotential lines & Constant Stream function lines are perpendicular to
each other.
Cauchy―Reimann Equations
𝑢 = −
𝜕𝜙
𝜕𝑥
= −
𝜕𝜓
𝜕𝑦
& 𝑣 = −
𝜕𝜙
𝜕𝑦
=
𝜕𝜓
𝜕𝑥
𝝏𝝓
𝝏𝒙
=
𝝏𝝍
𝝏𝒚
& −
𝝏𝝓
𝝏𝒚
=
𝝏𝝍
𝝏𝒙
BUOYANCY & FLOATATION
Archimedes principle
When a body is submerged either partially or completely, the net vertical upward force exerted by the fluid on the
body is known as buoyancy force (Fb), this buoyancy force is equal to weight of the fluid displaced and this is
known as Archimedes principle.
𝑉𝑓𝑙𝑢𝑖𝑑 𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑑 = 𝑉𝑓𝑑 = 𝑉𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑏𝑜𝑑𝑦 𝑠𝑢𝑏𝑚𝑒𝑟𝑔𝑒𝑑 = 𝐴(𝑥2 − 𝑥1)
(𝑁𝑒𝑡 𝑣𝑒𝑟𝑡𝑖𝑐𝑎𝑙 𝑢𝑝𝑤𝑎𝑟𝑑 𝑓𝑜𝑟𝑐𝑒 𝑒𝑥𝑒𝑟𝑡𝑒𝑑 𝑏𝑦 𝑓𝑙𝑢𝑖𝑑 𝑜𝑛 𝑡ℎ𝑒 𝑏𝑜𝑑𝑦) 𝑭 𝒗⋅𝑵𝒆𝒕
= 𝑤𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑡ℎ𝑒 𝑓𝑙𝑢𝑖𝑑 𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑑
𝑊𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑡ℎ𝑒 𝑓𝑙𝑢𝑖𝑑 𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑑 = 𝜌 𝑓 ⋅ 𝑔 ⋅ (𝑥2 − 𝑥1) ⋅ 𝐴 = 𝜌 𝑓 ⋅ 𝑔 ⋅ 𝑉𝑓𝑑
𝑭 𝒗⋅𝑵𝒆𝒕 = 𝑭 𝒃𝒖𝒐𝒖𝒂𝒏𝒄𝒚 = 𝑊𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑡ℎ𝑒 𝑓𝑙𝑢𝑖𝑑 𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑑 = 𝜌 𝑓 ⋅ 𝑔 ⋅ 𝑉𝑓𝑑
Centre of Buoyancy (B)
It is the point which the Buoyancy force is supposed to be acting, and this buoyancy
force will act at the centroid of the displacement volume. Therefore, centre of buoyancy will lie at the centroid of
displaced volume.
Note
When a homogenous body is completely submerged, then the centre of gravity of body & centre of
buoyancy coincide.
For a floating homogenous body, centre of buoyancy is below the centre of gravity.
For a non-homogenous body (heterogenous), centre of buoyancy and centre of gravity may not coincide
even if it’s completely submerged.
Principle of Flotation
For a floating body to be in equilibrium, Weight of the body must be EQUAL to Weight of fluid displaced and the
line of action of these 2 forces must be same.
𝑊𝑏𝑜𝑑𝑦 = 𝐹𝑏 → 𝐹𝑏 = 𝑊𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑡ℎ𝑒 𝑓𝑙𝑢𝑖𝑑 𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑑 = 𝑤𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑡ℎ𝑒 𝑏𝑜𝑑𝑦
𝑊𝑏𝑜𝑑𝑦 = 𝑊𝑓𝑙𝑢𝑖𝑑 𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑑 → 𝑏𝑜𝑑𝑦 𝑖𝑠 𝑖𝑛 𝑒𝑞𝑢𝑙𝑖𝑏𝑟𝑖𝑢𝑚
Types of Equilibrium
11. Stability conditions for completely submerged bodies
A completely submerged body will be in stable
equilibrium, when the centre of buoyancy is above centre
of gravity.
A completely submerged body will be in unstable
equilibrium when the centre of buoyancy is below centre
of gravity.
A completely submerged body will be in neutral
equilibrium, when centre of buoyancy coincides with
centre of gravity.
Metacentre (M)
It’s the point of intersection normal axis of the body to the new line of action of buoyancy force when the body is
tilted.
Metacentric height
The distance between centre of gravity and Metacentre (M) measured along the normal axis is called as
Metacentric height.
For stable equilibrium Metacentric height is positive, unstable negative.
Stability conditions for partially submerged/floating bodies
A floating body will be in stable
equilibrium, when metacentre is above
centre of gravity.
A floating body is said to be in unstable
equilibrium when the meta centre is
below centre of gravity.
A floating body is said to be in neutral
equilibrium when the meta centre
coincides with centre of gravity.
Mathematical condition for Stable equilibrium
For more stable equilibrium conditions, BM or GM must be as large as possible.
12. 𝐼𝑙𝑙 =
𝑙𝑏3
12
𝐼𝑡𝑡 =
𝑏𝑙3
12
𝑙 > 𝑏 → 𝐼𝑡𝑡 > 𝐼𝑙𝑙
𝐵𝑀 =
𝐼
𝑉𝑓⋅𝑑
𝐵𝑀𝑙−𝑙 =
𝐼𝑙𝑙
𝑉𝑓⋅𝑑
𝐵𝑀𝑡−𝑡 =
𝐼𝑡𝑡
𝑉𝑓⋅𝑑
𝑩𝑴𝒍−𝒍 < 𝑩𝑴𝒕−𝒕 (𝐼𝑡𝑡 > 𝐼𝑙𝑙)
From design point of view the least BM is calculated, i.e., BM about longitudinal axis is calculated. As BMt-t>BMl-l
the body will be more stable when ot oscillates about transverse axis (t-t) than longitudinal axis (l-l).
Oscillation about longitudinal axis are known as Rolling and
transverse axis is known as Pitching.
𝐵𝑀𝑟𝑜𝑙𝑙𝑖𝑛𝑔 < 𝐵𝑀 𝑝𝑖𝑡𝑐ℎ𝑖𝑛𝑔
If rolling is taken care of, then pitching is already taken care of.
Time period of Oscillation
𝑇 = 2𝜋√
𝑘 𝑔
2
𝑔(𝐺𝑀)
(𝑘 𝑔 = 𝑅𝑎𝑑𝑖𝑢𝑠 𝑜𝑓 𝑔𝑦𝑟𝑎𝑡𝑖𝑜𝑛 = √
𝐼
𝐴
)
For more stable equilibrium conditions, metacentric height must be larger, but larger GM results in smaller time
period of oscillation i.e., more number of oscillations in a given time. Therfore passengers are not comfortable
under such conditions. Therefore, for passenger ships, metacentric height is not very high. In case of war ships
stability of ship is more important than comfort, so metacentric height is larger than passenger ships.
Weight lost due to Buoyancy
𝑊𝑒𝑖𝑔ℎ𝑡 𝑙𝑜𝑠𝑠 = 𝑇 − 𝑇1 = 𝑊 − (𝑊 − 𝐹𝑏) = 𝐹𝑏
𝑊𝑒𝑖𝑔ℎ𝑡 𝑙𝑜𝑠𝑠 = 𝐵𝑢𝑜𝑦𝑎𝑛𝑐𝑦 𝑓𝑜𝑟𝑐𝑒
As density of air is very small, the buoyancy effects are negligible in air.
Therefore, correct weight of body is obtained when it is submerged in air.
13. PRESSURE MEASUREMENT
Pressure
It is defined as external normal force exerted in unit area. The area can be real or imaginary.
The unit of pressure is Newton (N)/mm2.
Pressure is a representative of no. of collisions per second.
Mohr’s circle for a Static fluid
For a static fluid there is no shear stress and there are only normal forces
(pressure). Therefore, Mohr’s circle is a point as shown in figure.
Pascals Law
According to Pascal’s Law, pressure at any point in a static fluid is equal in all directions. Conversely if pressure is
applied in a static fluid it is transmitted equally in all directions.
Applications― Hydraulic Lift, Hydraulic brakes etc…
𝐹
𝑎
=
𝑊
𝐴
⇒
𝐴
𝑎
=
𝑊
𝐹
> 1
𝑾 > 𝑭 𝑎𝑠 𝐴 > 𝑎
As W>F, by applying small force large weights can be raised. This doesn’t
mean energy conservation is violated because smaller force moves
through larger distance and larger force moves through smaller distance.
Atmospheric Pressure
Pressure exerted by environmental mass is known as atmospheric pressure. It is around 1.013 bar.
Gauge Atmospheric pressure (Pguage)
The pressure measured w.r.t atmospheric pressure is known as Gauge Pressure.
Absolute Pressure
The pressure measured w.r.t zero pressure is known as absolute pressure.
Vacuum Pressure
The pressure less than atmospheric pressure is known as vacuum pressure. There can be positive gauge or
negative gauge pressure, but there can’t be negative absolute pressure.
𝑉𝑎𝑐𝑢𝑢𝑚 𝑃𝑟𝑒𝑠𝑠𝑢𝑟𝑒 = 𝑃𝑎𝑡𝑚 − 𝑃𝑎𝑏𝑠𝑜𝑙𝑢𝑡𝑒
14. Hydrostatic Law
𝑃 ⋅ 𝑑𝐴 + 𝜌𝑔 ⋅ 𝑑𝐴 ⋅ 𝑑ℎ = (𝑃 + 𝑑𝑃) ⋅ 𝑑𝐴
𝑃 + 𝜌𝑔 ⋅ 𝑑ℎ = 𝑃 + 𝑑𝑃
𝑑𝑃
𝑑ℎ
= 𝜌𝑔 → 𝐻𝑦𝑑𝑟𝑜𝑠𝑡𝑎𝑡𝑖𝑐 𝐿𝑎𝑤
Hydrostatic Law gives variation of pressure in the vertical direction. For a static fluid, the forces acting on liquid
element are pressure & gravity forces.
𝐼𝑓 𝑃𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑖𝑠 𝑡𝑎𝑘𝑒𝑛 𝑖𝑛 𝑢𝑝𝑤𝑎𝑟𝑑 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛, 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑑𝑒𝑐𝑟𝑒𝑎𝑠𝑒𝑠 𝑤𝑖𝑡ℎ ℎ𝑒𝑖𝑔ℎ𝑡,
𝑑𝑃
𝑑ℎ
= −𝑤 = −𝜌𝑔
Pressure at any depth h
𝐴𝑡 𝑓𝑟𝑒𝑒 𝑠𝑢𝑟𝑓𝑎𝑐𝑒 (ℎ = 0)𝑃 = 𝑃𝑎𝑡𝑚
𝑑𝑃
𝑑ℎ
= 𝑤 → 𝑑𝑃 = 𝑤 ⋅ 𝑑ℎ → 𝑃 = 𝑤 ⋅ ℎ + 𝑐
𝑃 = 𝑤 ⋅ ℎ + 𝑃𝑎𝑡𝑚 (ℎ = 0 → 𝑃 = 𝑃𝑎𝑡𝑚)
𝑷 𝒈𝒂𝒖𝒈𝒆 = 𝒘 ⋅ 𝒉 = 𝝆𝒈𝒉 (𝑃𝑎𝑡𝑚 = 0 𝑓𝑜𝑟 𝑔𝑎𝑢𝑔𝑒 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒)
𝑃 = 𝜌𝑔ℎ 𝑖𝑠 𝑏𝑎𝑠𝑒𝑑 𝑜𝑛 𝑡ℎ𝑒 𝑎𝑠𝑠𝑢𝑚𝑝𝑡𝑖𝑜𝑛 𝑡ℎ𝑎𝑡 𝑡ℎ𝑒 𝑑𝑒𝑛𝑠𝑖𝑡𝑦(𝜌)𝑖𝑠 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
Sometimes the pressure is expressed in height column (h) because ρ & g are almost constants and pressure vary
directly with height column.
Barometer
Barometer is used for measuring Atmospheric pressure.
𝑃𝑎𝑡𝑚 = 0 + 𝜌𝑔ℎ ⇒ 𝑃𝑎𝑡𝑚 = 𝜌𝑔ℎ
h calculated is found to be 0.76m.
𝑃𝑎𝑡𝑚 = 𝜌𝑔ℎ = 13.6 × 9.81 × 0.76 = 1.01325 × 105
𝑁
𝑚𝑚2
= 1.01325 𝑏𝑎𝑟
If water is used instead of mercury, the corresponding height will be 10.3 metres. Mercury is used because of its
high density.
Conversion of one fluid column into other fluid column
𝑃1 = 𝑃2 ⇒ 𝜌1 ⋅ 𝑔 ⋅ ℎ1 = 𝜌2 ⋅ 𝑔 ⋅ ℎ2 ⇒ 𝝆 𝟏 ⋅ 𝒉 𝟏 = 𝝆 𝟐 ⋅ 𝒉 𝟐
Assume both are liquids
𝜌1ℎ1
𝜌 𝐻2 𝑂
=
𝜌2ℎ2
𝜌 𝐻2 𝑂
⇒ 𝒔 𝟏 ⋅ 𝒉 𝟏 = 𝒔 𝟐 ⋅ 𝒉 𝟐
If both are gases
𝜌1ℎ1
𝜌 𝑎𝑖𝑟
=
𝜌2ℎ2
𝜌 𝑎𝑖𝑟
⇒ 𝒔 𝟏 ⋅ 𝒉 𝟏 = 𝒔 𝟐 ⋅ 𝒉 𝟐
ℎ2 =
𝑠1
𝑠2
⋅ ℎ1
15. Piezometers
It is a device which is open at both the ends with one end connected at a point where pressure
is to be calculated and another end is open to atmosphere.
𝑃𝑔 𝑎𝑢𝑔𝑒 = 𝜌𝑔ℎ
Piezometers are not suitable for measuring high pressures like gas at high pressures. They are
suitable for moderate liquid pressures
Manometer
They are used for measuring pressure, they are based on balancing of liquid column.
They are divided into 2 types
1. Simple U-Tube (Pressure at a point)
2. Differential (Measure Pressure differences)
Simple U-tube Manometer
Jumping of fluid technique
𝑃 + 𝜌𝑔𝑦 − 𝜌 𝐻𝑔 𝑔𝑥 − 𝑃𝑎𝑡𝑚 = 0
𝑃𝑔 𝑎𝑢𝑔𝑒 = 𝜌 𝐻𝑔 𝑔𝑥 − 𝜌𝑔𝑦
Datum line technique
𝑃𝐴 = 𝑃𝐵
𝑃𝐴 = 𝑃 + 𝜌𝑔𝑦 𝑃𝐵 = 𝜌 𝐻𝑔 𝑔𝑥
𝑃 + 𝜌𝑔𝑦 = 𝜌 𝐻𝑔 𝑔𝑥
𝑃 = 𝜌 𝐻𝑔 𝑔𝑥 − 𝜌𝑔𝑦
Multi U-tube manometers are used for measuring High Pressures
16. FLUID MECHANICS
Fluid
Fluid is a substance which is capable of moving or deforming under the action of shear force.
As long as there is shear force, the fluid flows or deforms continuously.
Examples- Liquids, Gases etc…
Difference between Solids & Fluids
In case of solids under the action of shear force, there is deformation and this deformation doesn’t change with
time. Therefore, deformation dθ is important when this shear force is removed, solids will try to come back to its
original position.
In case of fluids, the deformation is continuous as long as there is shear force, this deformation changes with time.
In fluids the rate if deformation (dθ/dt) is important than dθ. After the removal of shear force the fluid will never
try to come back to its original position. For a static fluid shear force is zero.
Fluid Properties
Density
It is defined as ratio of mass of fluid to its volume. It actually represents the quantity of matter in a given volume.
Its unit is Kg/m3.
Density of water for all calculation purposes is taken as 1000 Kg/m3.
Density depends on temperature and Pressure. As temperature increases density decreases and as pressure
increases density increases.
Specific Weight/ Weight density (w)
It is defined as the ratio of weight of the fluid to its volume. Its unit is N/mm3.
𝑤 =
𝑤𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑡ℎ𝑒 𝑓𝑙𝑢𝑖𝑑
𝑣𝑜𝑙𝑢𝑚𝑒
=
𝑚𝑔
𝑉
= 𝜌𝑔
𝑤 = 𝜌𝑔
Density is an absolute quantity, whereas specific weight is not an absolute quantity, because it varies from
location to location.
Specific gravity
It is defined as the ratio of density of fluid to density of standard fluid.
In case of liquids the standard fluid is water.
In case if gases the standard fluid is either Hydrogen or air at a given temperature and pressure.
Specific gravity of water is one. If the specific gravity of a liquid is less than one it is lighter than water, if greater
than one liquid is heavier than water.
Relative density
It is the ratio of density of one fluid to other fluid.
All Specific gravities are relative densities, but not all relative densities are not specific gravities.
Compressibility (β)
It is the measure of the change of volume (or) change of density w.r.t pressure on a given mass of fluid.
Mathematically it is defined as reciprocal of Bulk Modulus (K)
𝛽 =
1
𝐾
17. 𝐾 =
𝑑𝑃
−𝑑𝑉
𝑉
= −𝑉 ⋅
𝑑𝑃
𝑑𝑉
= 𝜌 ⋅
𝑑𝑃
𝑑𝜌
(
−𝑑𝑉
𝑉
=
𝜌
𝑑𝜌
) → 𝐾 =
𝜌 ⋅ 𝑑𝑃
𝑑𝜌
𝛽 =
𝑑𝜌
𝜌 ⋅ 𝑑𝑃
Liquids are generally treated as incompressible and gases as compressible.
Isothermal compressibility of Ideal gas
𝑃 = 𝜌 ⋅ 𝑅 ⋅ 𝑇 (𝑇 = 𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡)
𝑑𝑃
𝑑𝜌
= 𝑅𝑇
𝐾 𝑇(𝑇 − 𝑐𝑜𝑛𝑠𝑡. ) =
𝜌 ⋅ 𝑑𝑃
𝑑𝜌
= 𝜌 ⋅
𝑑𝑃
𝑑𝜌
= 𝜌 ⋅ 𝑅 ⋅ 𝑇 = 𝑃
𝑲 𝑻 = 𝑷
Adiabatic Bulk Modulus/ Isentropic Bulk Modulus of an Ideal gas
𝑃𝑉 𝛾
= 𝐶1
𝑃𝑉 𝛾
= 𝑃 (
𝑚
𝜌
)
𝛾
= 𝐶1 ⇒
𝑃
𝜌 𝛾
=
𝐶1
𝑚 𝛾
= 𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡 = 𝐶 ⇒ 𝑃 = 𝐶𝜌 𝛾
𝑑𝑃
𝑑𝜌
= 𝐶 ⋅ 𝛾 ⋅ 𝜌 𝛾−1
𝐾𝑎(𝐴𝑑𝑖𝑎𝑏𝑎𝑡𝑖𝑐) = 𝜌 ⋅
𝑑𝑃
𝑑𝜌
= 𝜌 ⋅ (𝐶 ⋅ 𝛾 ⋅ 𝜌 𝛾−1) = 𝛾 ⋅ 𝐶 ⋅ 𝜌 𝛾
= 𝛾𝑃
𝑲 𝒂 = 𝜸 ⋅ 𝑷
As γ>1, Adiabatic bulk modulus is greater than isothermal bulk modulus.
Bulk Modulus is not constant and it increases with increase in Pressure,
because at higher pressure the fluid offers more resistance to further
compression.
Viscosity
The Internal resistance offered by one layer of fluid to the adjacent layer is called Viscosity.
Need to define viscosity
Though the density of oil and water are almost same, their flow behaviour is not same and hence a property is
required to define flow behaviour. This property for defining flow behaviour is Viscosity.
𝑉𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝐺𝑟𝑎𝑑𝑖𝑒𝑛𝑡 =
𝑑𝑢
𝑑𝑦
𝑑𝜃
𝑑𝑡
=
𝑑𝑢
𝑑𝑦
𝜏 = 𝜇
𝑑𝜃
𝑑𝑡
= 𝜇
𝑑𝑢
𝑑𝑦
18. → 𝜇 =
𝜏
𝑑𝜃
𝑑𝑡
=
𝜏
𝑑𝑢
𝑑𝑦
𝐼𝑓
𝑑𝜃
𝑑𝑡
𝑖𝑠 𝑙𝑎𝑟𝑔𝑒, 𝑓𝑙𝑜𝑤 𝑖𝑠 𝑒𝑎𝑠𝑦 𝑎𝑠 𝑣𝑖𝑠𝑐𝑜𝑠𝑖𝑡𝑦 𝑖𝑠 𝑙𝑒𝑠𝑠 𝑖. 𝑒. , 𝑟𝑒𝑠𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑡𝑜 𝑓𝑙𝑜𝑤 𝑖𝑠 𝑙𝑒𝑠𝑠
𝐼𝑓
𝑑𝜃
𝑑𝑡
𝑖𝑠 𝑠𝑚𝑎𝑙𝑙, 𝑓𝑙𝑜𝑤 𝑖𝑠 𝑑𝑖𝑓𝑓𝑖𝑐𝑢𝑙𝑡 𝑎𝑠 𝑣𝑖𝑠𝑐𝑜𝑠𝑖𝑡𝑦 𝑖𝑠 ℎ𝑖𝑔ℎ 𝑖. 𝑒. , 𝑟𝑒𝑠𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑡𝑜 𝑓𝑙𝑜𝑤 𝑖𝑠 ℎ𝑖𝑔ℎ
⇒
𝑑𝜃
𝑑𝑡
= 𝑅𝑎𝑡𝑒 𝑜𝑓 𝐴𝑛𝑔𝑢𝑙𝑎𝑟 𝑑𝑒𝑓𝑜𝑟𝑚𝑎𝑡𝑖𝑜𝑛 (𝑜𝑟)𝑅𝑎𝑡𝑒 𝑜𝑓 𝑆ℎ𝑒𝑎𝑟 𝑠𝑡𝑟𝑎𝑖𝑛
Unit of Viscosity is Newton Second/ square metre (N⋅ s/m2)
Unit of Viscosity in CGS system is poise.
1 poise = 0.1 N⋅ s/m2
Variation of Viscosity with temperature
In case of liquids, the intermolecular distance is small, hence cohesive forces are large. In case of gases,
intermolecular distance is small and cohesive forces are negligible.
With increase in temperature, cohesive forces decrease and resistance to flow also decreases. Therefore, viscosity
of liquid decreases with increase in temperature.
With increase in temperature molecular disturbance increases and resistance to flow increases. Viscosity of gases
increase with temperature.
Newtonian Fluid
Fluids which obey Newton’s law of viscosity are known as Newtonian fluids. According to Newton’s Law of
viscosity, shear stress is directly proportional to rate of shear strain.
𝜏 ∝
𝑑𝜃
𝑑𝑡
⇒ 𝜏 ∝
𝑑𝑢
𝑑𝑦
𝜏 = 𝜇
𝑑𝑢
𝑑𝑦
→ 𝑁𝑒𝑤𝑡𝑜𝑛𝑖𝑎𝑛 𝐹𝑙𝑢𝑖𝑑 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛
μ is the slope in the graph.
Examples- Air, Water, Diesel, Kerosene, Oils, Mercury etc...
Note
For a Newtonian fluid, Viscosity doesn’t change with rate of deformation.
19. Non-Newtonian Fluids
Fluids which don’t obey Newton’s law of viscosity are known as Non-Newtonian fluids.
𝑇ℎ𝑒 𝑔𝑒𝑛𝑒𝑟𝑎𝑙 𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑠ℎ𝑒𝑎𝑟 𝑠𝑡𝑟𝑒𝑠𝑠 (𝜏) & 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑔𝑟𝑎𝑑𝑖𝑒𝑛𝑡 (
𝑑𝑢
𝑑𝑦
) 𝑖𝑠 𝜏 = 𝐴 ⋅ (
𝑑𝑢
𝑑𝑦
)
𝑛
+ 𝐵
Case 1
𝐵 = 0; 𝑛 > 1 𝐷𝑖𝑙𝑎𝑡𝑎𝑛𝑡 𝑓𝑙𝑢𝑖𝑑
A fluid is said to be dilatant fluid for which the apparent viscosity increases with rate of deformation.
Examples- Rice Starch, Sugar in water
As the apparent viscosity is increasing with deformation, these fluids are known
as Shear Thickening Fluids.
𝜏 = 𝐴 ⋅ (
𝑑𝑢
𝑑𝑦
)
𝑛
+ 0 = 𝐴 ⋅ (
𝑑𝑢
𝑑𝑦
)
𝑛
𝜏 = 𝐴 ⋅ (
𝑑𝑢
𝑑𝑦
)
𝑛−1
⋅
𝑑𝑢
𝑑𝑦
= 𝜇 𝑎𝑝𝑝𝑎𝑟𝑒𝑛𝑡 ⋅
𝑑𝑢
𝑑𝑦
Case 2
𝐵 = 0; 𝑛 < 1 𝑃𝑠𝑒𝑢𝑑𝑜 𝑝𝑙𝑎𝑡𝑖𝑐 𝑓𝑙𝑢𝑖𝑑𝑠
For a pseudo plastic fluid, apparent viscosity decreases with rate of deformation.
Examples- Blood, milk, Colloidal Solutions etc…
As the apparent viscosity is decreasing with deformation, these fluids are known
as Shear Thinning Fluids.
Case 3
𝐵 ≠ 0; 𝑛 = 1 𝐵𝑖𝑛𝑔ℎ𝑎𝑚 𝑃𝑙𝑎𝑠𝑡𝑖𝑐
Example- Toothpaste
In case of Bingham plastic fluids, certain minimum shear stress is required for
causing the flow of fluid. Below this shear stress there is no flow and therefore it
acts like a solid. After that it behaves like a fluid. Such substances that behaves
as both solids and fluids are known as Rheological substances and the study of
these substances is known as Rheology.
Ideal Fluid
A fluid which is non-viscous and incompressible is known as ideal fluid though
there’s no ideal fluid, it’s introduced to bring simplicity to analysis.
𝜇 𝐻2 𝑂, 20℃ = 1 𝐶𝑒𝑛𝑡𝑖 𝑃𝑜𝑖𝑠𝑒 = 10−3
𝐾𝑔 𝑚 − 𝑠⁄ 𝜇 𝐻𝑔 = 1.55 𝐶𝑒𝑛𝑡𝑖𝑝𝑜𝑖𝑠𝑒
𝜇 𝐻2 𝑂 = (50 − 55)𝜇 𝑎𝑖𝑟
20. Equation for Linear Velocity Profile
The velocity profile can be approximated as a linear velocity profile, if the gap between plates is very small
(narrow passages).
𝑡𝑎𝑛 𝜃 =
𝑑𝑢 ⋅ 𝑑𝑡
𝑑𝑦
=
𝑣 ⋅ 𝑑𝑡
𝑦
𝑑𝑢
𝑑𝑦
=
𝑣
𝑦
𝜏 = 𝜇
𝑑𝑢
𝑑𝑦
= 𝜇
𝑣
𝑦
𝐹 = 𝜏 ⋅ 𝐴 = 𝜇 ⋅ 𝐴 ⋅
𝑣
𝑦
Kinematic Viscosity (υ)
In fluid mechanics the term (μ/ρ) appears frequently and for convenience this term is taken as Kinematic
viscosity.
𝜐 =
𝜇
𝜌
𝐼𝑡𝑠 𝑢𝑛𝑖𝑡𝑠 𝑎𝑟𝑒
𝑚2
𝑠
𝑖𝑛 𝑆. 𝐼 &
𝑐𝑚2
𝑠
𝑜𝑟 𝑆𝑡𝑜𝑘𝑒.
𝜏 = 𝜇
𝑑𝑢
𝑑𝑦
=
𝜇
𝜌
𝑑(𝜌𝑢)
𝑑𝑦
= 𝜐
𝑑(𝜌𝑢)
𝑑𝑦
Significance of Kinematic Viscosity
Kinematic viscosity represents the ability of fluid to resist momentum. Therefore, it’s a measure of momentum
diffusivity.
Surface Tension
Consider the molecule A which is below the surface of liquid, this molecule is surrounded by various corresponding
molecules and hence under the influence of various cohesive forces, it will be in equilibrium.
Now consider molecule B which is on the surface of liquid, this molecule is under the influence of net downward
force, because of this there seems to be a layer formed which can resist small tensile loads.
This phenomenon is known as Surface Tension. It’s a line force i.e., it acts normal to the line drawn on the surface
and it lies in the plane of surface.
It is denoted by σ.
As Surface tension is basically due to unbalanced cohesive forces and with increase in temperature, cohesive
forces decrease decreasing Surface tension. At critical point surface tension is zero. Surface tension is very small,
so it is neglected in further fluid mechanics analysis.
Surface tension for water air interface at 20ᵒC is 0.0736 N/m.
Liquid droplets assume Spherical shape due to surface tension.∆P
21. Pressure in liquid drop in excess of Atmospheric pressure
𝐹𝑝 = 𝛥𝑃 ⋅ 𝐴 = 𝛥𝑃 ⋅
𝜋
4
𝑑2
𝜎 =
𝐹𝑠
𝐿
⇒ 𝐹𝑠 = 𝜎 ⋅ 𝐿 = 𝜎 ⋅ 𝜋𝑑
𝐹𝑜𝑟 𝐸𝑞𝑢𝑙𝑖𝑏𝑟𝑖𝑢𝑚 ⇒ 𝐹𝑝 = 𝐹𝑠 ⇒ 𝛥𝑃 ⋅
𝜋
4
𝑑2
= 𝜎 ⋅ 𝜋𝑑
Pressure forces tries to separate the droplet whereas surface tension tries to contract the droplet i.e., surface
tension force tries to minimize surface area.
Droplets take spherical shape because sphere has minimum surface area for a given volume.
Capillarity
Capillarity is the effect of surface tension. It’s not a property.
The rise or fall of a liquid when a small diameter tube is introduced in it is known as capillarity.
The capillary rise is due to adhesion and capillary rise is due to cohesion. Water is an example for adhesion and
mercury for cohesion.
Expression for capillary rise/fall in a glass tube
𝑊𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑡ℎ𝑒 𝑙𝑖𝑞𝑢𝑖𝑑 𝑏𝑟𝑜𝑢𝑔ℎ𝑡 𝑢𝑝 = 𝑣𝑒𝑟𝑡𝑖𝑐𝑎𝑙 𝑐𝑜𝑚𝑝𝑜𝑛𝑒𝑛𝑡 𝑜𝑓 𝐹𝑠
𝑊𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑙𝑖𝑞𝑢𝑖𝑑 = 𝑆𝑝𝑒𝑐𝑖𝑓𝑖𝑐 𝑤𝑒𝑖𝑔ℎ𝑡 × 𝑉𝑜𝑙𝑢𝑚𝑒
𝑊𝑒𝑖𝑔ℎ𝑡 = 𝓌 × 𝑉 = 𝓌 ×
𝜋𝑑2
ℎ
4
𝑉𝑒𝑟𝑡𝑖𝑐𝑎𝑙 𝑐𝑜𝑚𝑝𝑜𝑛𝑒𝑛𝑡 𝑜𝑓 𝐹𝑠 = 𝐹𝑠 ⋅ 𝑐𝑜𝑠 𝜃 = 𝜎𝜋𝑑 ⋅ 𝑐𝑜𝑠 𝜃
𝓌 ×
𝜋𝑑2
ℎ
4
= 𝐹𝑠 ⋅ 𝑐𝑜𝑠 𝜃
ℎ =
4𝜎 𝑐𝑜𝑠 𝜃
𝓌𝑑
𝜟𝑷 =
𝟒𝝈
𝒅
→ 𝑳𝒊𝒒𝒖𝒊𝒅 𝒅𝒓𝒐𝒑
𝜟𝑷 =
𝟖𝝈
𝒅
→ 𝑺𝒐𝒂𝒑 𝑩𝒖𝒃𝒃𝒍𝒆
𝜟𝑷 =
𝟐𝝈
𝒅
→ 𝑳𝒊𝒒𝒖𝒊𝒅 𝒋𝒆𝒕
22. Expression for capillary rise in the annulus of 2 concentric tubes
𝑊𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑡ℎ𝑒 𝑙𝑖𝑞𝑢𝑖𝑑 𝑏𝑟𝑜𝑢𝑔ℎ𝑡 𝑢𝑝 = 𝑣𝑒𝑟𝑡𝑖𝑐𝑎𝑙 𝑐𝑜𝑚𝑝𝑜𝑛𝑒𝑛𝑡 𝑜𝑓 𝐹𝑠
𝓌 ×
𝜋
4
(𝑑 𝑜
2
− 𝑑𝑖
2
)ℎ = 𝜎𝜋(𝑑 𝑜 + 𝑑𝑖) ⋅ 𝑐𝑜𝑠 𝜃
ℎ =
4𝜎 𝑐𝑜𝑠 𝜃
𝓌(𝑑 𝑜 − 𝑑𝑖)
Expression for capillary rise between two parallel plates
𝑊𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑡ℎ𝑒 𝑙𝑖𝑞𝑢𝑖𝑑 𝑏𝑟𝑜𝑢𝑔ℎ𝑡 𝑢𝑝 = 𝑣𝑒𝑟𝑡𝑖𝑐𝑎𝑙 𝑐𝑜𝑚𝑝𝑜𝑛𝑒𝑛𝑡 𝑜𝑓 𝐹𝑠
𝑤𝑒𝑖𝑔ℎ𝑡 = 𝓌 × 𝑏ℎ𝑡
𝐹𝑠 = 𝜎(𝑏 + 𝑏) = 2𝜎𝑏
𝓌 × 𝑏ℎ𝑡 = 2𝜎𝑏 ⋅ 𝑐𝑜𝑠 𝜃
ℎ =
2𝜎 ⋅ 𝑐𝑜𝑠 𝜃
𝓌𝑡
Work done in stretching a surface
𝑊𝑜𝑟𝑘 = 𝐹𝑜𝑟𝑐𝑒 × 𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 = (𝜎 ⋅ 𝐿) × 𝑥 = 𝜎 × 𝐼𝑛𝑐𝑟𝑒𝑎𝑠𝑒 𝑖𝑛 𝑆𝑢𝑟𝑓𝑎𝑐𝑒 𝐴𝑟𝑒𝑎 (𝐿 ⋅ 𝑥)
𝑊𝑜𝑟𝑘 = 𝜎 × 𝛥𝐴
Note
The angle of contact between water & glass is 22ᵒ.
The angle of contact between pure water & clean glass tube is 0ᵒ.
The angle of contact between mercury & glass is 130ᵒ.
If the height of capillary tube is not sufficient for possible rise, the liquid will rise up to
top and stops because for further rise there is no glass molecules so, it stops.
If the top of the capillary tube is close, then capillary rise will decrease because the air
trapped at top exerts pressure in the downward direction.
Vapour Pressure
Let us consider a closed container with liquid partially filled in it. The surface molecules due to additional energy
overcomes cohesive forces of liquid below surface. This process occurs until the space above the liquid is
saturated. Under equilibrium the no. of molecules leaving the surface is equal to no. of molecules joining surface.
Under these conditions the pressure exerted by vapour on surface of
liquid is called Vapour pressure.
Vapour pressure increases with increases in temperature because at
higher temperature molecular activity is high.
High Volatile liquids (petrol) have high vapour pressure. Mercury has
least vapour pressure and because of this property it is used in
Manometers.
23. FLUID DYNAMICS
Generally, the forces acting on fluid element are pressure force Fp, gravity force Fg & viscous force Fv.
In Navier stokes equation all these forces are taken into consideration. In Euler’s analysis viscous forces are
neglected, only pressure & gravity forces are taken into consideration.
Navier stokes equation is momentum conservation equation.
Euler’s Equation
Assumption- Flow is Non-viscous
𝑤𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑓𝑙𝑢𝑖𝑑 = 𝓌 × 𝑉 = 𝜌𝑔 × 𝑑𝐴 ⋅ 𝑑𝑆
𝑤 = 𝑚𝑔 → 𝑚 = 𝜌𝑉 = 𝜌 ∙ 𝑑𝐴 ⋅ 𝑑𝑆
→ 𝑤 = 𝜌𝑔 × 𝑑𝐴 ⋅ 𝑑𝑆
𝐴𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑠𝑡𝑟𝑒𝑎𝑚 = 𝑎 𝑠 = 𝑣
𝜕𝑣
𝜕𝑠
+
𝜕𝑣
𝜕𝑡
𝑆𝑡𝑟𝑒𝑎𝑚 𝐹𝑜𝑟𝑐𝑒 (𝐹𝑠) = 𝑚 ∙ 𝑎 𝑠 = 𝜌 ∙ 𝑑𝐴 ⋅ 𝑑𝑆 × 𝑎 𝑠
𝐹𝑠 = 𝜌 ∙ 𝑑𝐴 ⋅ 𝑑𝑆 × (𝑣
𝜕𝑣
𝜕𝑠
+
𝜕𝑣
𝜕𝑡
)
𝑐𝑜𝑠 𝜃 =
𝑑𝑧
𝑑𝑆
⇒ 𝑑𝑧 = 𝑑𝑆 ⋅ 𝑐𝑜𝑠 𝜃
𝑤 ⋅ 𝑐𝑜𝑠 𝜃 = 𝜌𝑔 ∙ 𝑑𝐴 ⋅ 𝑑𝑆 ⋅ 𝑐𝑜𝑠 𝜃 = 𝜌𝑔 ∙ 𝑑𝐴 ⋅ 𝑑𝑧
𝑆𝑡𝑟𝑒𝑎𝑚 𝐹𝑜𝑟𝑐𝑒 (𝐹𝑠) = 𝑃 ⋅ 𝑑𝐴 − (𝑃 + 𝑑𝑃) ⋅ 𝑑𝐴 − 𝑤 ⋅ 𝑐𝑜𝑠 𝜃
𝜌 ∙ 𝑑𝐴 ⋅ 𝑑𝑆 × (𝑣
𝜕𝑣
𝜕𝑠
+
𝜕𝑣
𝜕𝑡
) = 𝑃 ⋅ 𝑑𝐴 − (𝑃 + 𝑑𝑃) ⋅ 𝑑𝐴 − 𝜌𝑔 ∙ 𝑑𝐴 ⋅ 𝑑𝑧
The above equation is Euler’s Equation.
Bernoulli’s Equation (Conservation of Energy equation)
Assumptions
1. Flow is non-viscous
2. Flow is along a stream line
3. No energy is supplied and no energy is taken out from the fluid during the flow
4. Steady flow & Incompressible
𝑑𝑃 + 𝜌𝑔 ∙ 𝑑𝑧 + 𝜌 ⋅ 𝑑𝑆 × (𝑣
𝜕𝑣
𝜕𝑠
+
𝜕𝑣
𝜕𝑡
) = 0 → 𝑑𝑃 + 𝜌𝑔 ∙ 𝑑𝑧 + 𝜌 ⋅ 𝑑𝑆 × 𝑣
𝑑𝑣
𝑑𝑠
= 0 → 𝑑𝑃 + 𝜌𝑔 ∙ 𝑑𝑧 + 𝜌𝑣 ∙ 𝑑𝑣 = 0
→
𝑑𝑃
𝜌
+ 𝑔 ⋅ 𝑑𝑧 + 𝑣 ⋅ 𝑑𝑣 = 0
After Integration,
→
1
𝜌
∫ 𝑑𝑃 + 𝑔 ∫ 𝑑𝑧 + 𝑣 ∫ 𝑑𝑣 = ∫ 0
𝑃
𝜌
+ 𝑔𝑧 +
𝑣2
2
= 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 → 𝐶𝑙𝑎𝑠𝑠𝑖𝑐𝑎𝑙 𝐵𝑒𝑟𝑛𝑜𝑢𝑙𝑙𝑖’𝑠 𝐸𝑞𝑢𝑎𝑡𝑖𝑜𝑛
In above equation, each term represents energy of the fluid per unit mass.
𝑑𝑃 + 𝜌𝑔 ∙ 𝑑𝑧 + 𝜌 ⋅ 𝑑𝑆 × (𝑣
𝜕𝑣
𝜕𝑠
+
𝜕𝑣
𝜕𝑡
) = 0
Steady flow
24. Bernoulli’s Theorem
In a steady incompressible non-viscous flow along a stream line, the sum of pressure, kinetic & potential energy is
constant.
𝐸𝑛𝑒𝑟𝑔𝑦
𝑚𝑎𝑠𝑠
→
𝑃
𝜌
+
𝑣2
2
+ 𝑔𝑧 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
𝐸𝑛𝑒𝑟𝑔𝑦
𝑚𝑎𝑠𝑠 × 𝑔
→
𝐸𝑛𝑒𝑟𝑔𝑦
𝑤𝑒𝑖𝑔ℎ𝑡
→
𝑃
𝜌𝑔
+
𝑣2
2𝑔
+ 𝑧 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
𝑃
𝓌
+
𝑣2
2𝑔
+ 𝑧 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
In this equation, each term represents energy per unit weight.
Various heads in Fluid mechanics
Pressure Head
The height by which fluid rises due to pressure when a piezometer is
connected is known as pressure head.
𝑃 = 0 + 𝜌𝑔ℎ = 𝓌ℎ
ℎ =
𝑃
𝓌
= 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 ℎ𝑒𝑎𝑑
Velocity Head/ Kinematic energy head
It’s the height by which fluid falls in a frictionless environment to reach to a particular height.
𝑣 = √2𝑔ℎ
ℎ =
𝑣2
2𝑔
= 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 ℎ𝑒𝑎𝑑
Potential Energy head (z)
It’s the vertical distance with respect to a reference line.
Piezometric Head
The sum of pressure and potential energy is known as piezometric head.
𝑃𝑖𝑒𝑧𝑜𝑚𝑒𝑡𝑟𝑖𝑐 ℎ𝑒𝑎𝑑 =
𝑃
𝓌
+ 𝑧
Relationship between first law of thermodynamics & Bernoulli’s equation
ℎ1 +
𝑣1
2
2
+ 𝑧1 𝑔 + 𝑞 = ℎ2 +
𝑣2
2
2
+ 𝑧2 𝑔 + 𝑤
ℎ = 𝑠𝑝𝑒𝑐𝑖𝑓𝑖𝑐 𝑒𝑛𝑡ℎ𝑎𝑙ℎ𝑦, 𝑞 =
ℎ𝑒𝑎𝑡 𝑡𝑟𝑎𝑛𝑠𝑓𝑒𝑟
𝑚𝑎𝑠𝑠
, 𝑤 =
𝑊𝑜𝑟𝑘
𝑚𝑎𝑠𝑠
Assumptions
1. Steady flow & incompressible
2. No heat transfer & work transfer
3. No change in Internal energy
ℎ = 𝑢 + 𝑃𝓋 = 𝑢 +
𝑃
𝜌
𝑢1 +
𝑃1
𝜌
+
𝑣1
2
2
+ 𝑧1 𝑔 + 𝑞 = 𝑢2 +
𝑃2
𝜌
+
𝑣2
2
2
+ 𝑧2 𝑔 + 𝑤
(𝑢1 = 𝑢2, 𝑞 = 𝑤 = 0)
𝑃
𝓌
+
𝑣2
2𝑔
+ 𝑧 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 → 𝐵𝑒𝑟𝑛𝑜𝑢𝑙𝑙𝑖′𝑠𝐸𝑞𝑢𝑎𝑡𝑖𝑜𝑛
25. Bernoulli’s equation for a Horizontal Stream line
𝑧1 = 𝑧2
𝑃1
𝜌𝑔
+
𝑣1
2
2𝑔
+ 𝑧1 =
𝑃2
𝜌𝑔
+
𝑣2
2
2𝑔
+ 𝑧2
𝑃1
𝜌𝑔
+
𝑣1
2
2𝑔
=
𝑃2
𝜌𝑔
+
𝑣2
2
2𝑔
Bernoulli’s equation for a real fluid flow problem
𝑃1
𝜌𝑔
+
𝑣1
2
2𝑔
+ 𝑧1 =
𝑃2
𝜌𝑔
+
𝑣2
2
2𝑔
+ 𝑧2 + ℎ 𝐿
(ℎ 𝐿 = 𝐻𝑒𝑎𝑑 𝐿𝑜𝑠𝑠)
In case of irrotational flow, Bernoulli’s equation can be applied between any 2 points (throughout the flow field),
because the stream line constants are same for different streamlines in irrotational flow.
In case of rotational flow, Bernoulli’s must be applied only for a particular stream line, because the stream line
constants are different for different stream lines.
Bernoulli’s equation is not the total energy conservation equation because heat transfer and work transfer are
not taken into consideration. Therefore, Bernoulli’s equation is known as Mechanical Energy Conservation.
Applications of Bernoulli’s Equation
Venturimeter
It’s used for calculating Discharge.
𝑃1
𝓌
+
𝑣1
2
2𝑔
+ 𝑧1 =
𝑃2
𝓌
+
𝑣2
2
2𝑔
+ 𝑧2
(
𝑃1
𝓌
−
𝑃2
𝓌
) + (𝑧1 − 𝑧2) =
𝑣2
2
2𝑔
−
𝑣1
2
2𝑔
= ℎ (𝑃𝑖𝑒𝑧𝑜𝑚𝑒𝑡𝑟𝑖𝑐 ℎ𝑒𝑖𝑔ℎ𝑡)
(
𝑃1
𝓌
+ 𝑧1) − (
𝑃2
𝓌
+ 𝑧2) =
𝑣2
2
2𝑔
−
𝑣1
2
2𝑔
= ℎ
𝐹𝑜𝑟 𝐻𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙 𝐶𝑎𝑠𝑒(𝑧1 = 𝑧2),
(
𝑃1
𝓌
−
𝑃2
𝓌
) =
𝑣2
2
2𝑔
−
𝑣1
2
2𝑔
= ℎ → 𝑣1
2
− 𝑣2
2
= 2𝑔ℎ
𝑄 = 𝐴1 ⋅ 𝑣1 = 𝐴2 ⋅ 𝑣2 → 𝑣1 =
𝑄
𝐴1
→ 𝑣2 =
𝑄
𝐴2
26. 𝑣1
2
− 𝑣2
2
= (
𝑄
𝐴1
)
2
− (
𝑄
𝐴2
)
2
= 2𝑔ℎ → 𝑄2
(
1
𝐴1
2 −
1
𝐴2
2) = 2𝑔ℎ
𝑄 =
𝐴1 ⋅ 𝐴2 √2𝑔ℎ
√𝐴1
2
− 𝐴2
2
As no losses were assumed while deriving this equation, this discharge is known as ideal
discharge or theoretical discharge.
𝑄𝑡ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙 =
𝐴1 ⋅ 𝐴2 √2𝑔ℎ
√𝐴1
2
− 𝐴2
2
→
𝑃1
𝓌
+ 𝐻 + 𝑥 −
𝑥 ⋅ 𝑠 𝐻𝑔
𝑠
− 𝐻 =
𝑃2
𝓌
→
𝑃1
𝓌
−
𝑃2
𝓌
= 𝑥 ⋅ (
𝑠 𝐻𝑔
𝑠
− 1) = ℎ (𝑃𝑖𝑒𝑧𝑜 ℎ𝑡. )
→ 𝐴1 𝑣1 = 𝐴2 𝑣2 → 𝐴2 < 𝐴1 ⇒ 𝑣2 > 𝑣1
→
𝑃1
𝓌
+
𝑣1
2
2𝑔
=
𝑃2
𝓌
+
𝑣2
2
2𝑔
→ 𝑣2 > 𝑣1 ⇒ 𝑃2 < 𝑃1
Principle of Venturimeter
By reducing the area in a steady incompressible flow, velocity increases. This results in decrease of pressure. Due
to this pressure difference, there will be manometric deflection when differential manometer is connected. By
measuring the 𝑥, the discharge will be calculated.
Coefficient of discharge (Cd)
It is defined as the ratio of actual discharge to theoretical discharge.
Cd depends on type of flow (Reynolds no.) and area ratio.
As Venturimeter is gradually converging and diverging device, losses are less and hence Cd is 0.94―0.98.
𝑄𝑡ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙 =
𝐴1 ⋅ 𝐴2 √2𝑔ℎ
√𝐴1
2
− 𝐴2
2
𝑄 𝑎𝑐𝑡𝑢𝑎𝑙 = 𝐶 𝑑 ×
𝐴1 ⋅ 𝐴2 √2𝑔ℎ
√𝐴1
2
− 𝐴2
2
𝑃1
𝓌
+
𝑣1
2
2𝑔
=
𝑃2
𝓌
+
𝑣2
2
2𝑔
+ ℎ𝑙𝑜𝑠𝑠𝑒𝑠 → (
𝑃1
𝓌
−
𝑃2
𝓌
) − ℎ𝑙 =
𝑣2
2
2𝑔
−
𝑣1
2
2𝑔
ℎ − ℎ𝑙 =
𝑣2
2
− 𝑣1
2
2𝑔
⇒ 𝑣2
2
− 𝑣1
2
= 2𝑔(ℎ − ℎ𝑙)
(
𝑄
𝐴1
)
2
− (
𝑄
𝐴2
)
2
= 2𝑔(ℎ − ℎ𝑙) (𝑣1 =
𝑄
𝑎1
, 𝑣2 =
𝑄
𝑎2
)
𝑄 𝑎𝑐𝑡𝑢𝑎𝑙 =
𝐴1 ⋅ 𝐴2 √2𝑔(ℎ − ℎ𝑙)
√𝐴1
2
− 𝐴2
2
27. 𝐶 𝑑 ×
𝐴1 ⋅ 𝐴2 √2𝑔ℎ
√𝐴1
2
− 𝐴2
2
=
𝐴1 ⋅ 𝐴2 √2𝑔(ℎ − ℎ𝑙)
√𝐴1
2
− 𝐴2
2
𝐶 𝑑 = √
ℎ − ℎ𝑙
ℎ
General Properties of a Venturimeter
𝑑2 = (
1
3
𝑡𝑜
1
2
) · 𝑑1
𝐴𝑛𝑔𝑙𝑒 𝑜𝑓 𝑐𝑜𝑛𝑣𝑒𝑟𝑔𝑒𝑛𝑐𝑒 = 20ᵒ − 22ᵒ
𝐴𝑛𝑔𝑙𝑒 𝑜𝑓 𝑑𝑖𝑣𝑒𝑟𝑔𝑒𝑛𝑐𝑒 = 7ᵒ
The angle of divergence is generally kept less than 7ᵒ, in order to avoid flow separation.
Orifice meter
The device is used for finding out discharge and it is the cheapest measurement for calculating discharge.
It is based on same principle as Venturimeter.
It’s a circular disc with a circular hole.
𝐶𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 𝑜𝑓 𝐶𝑜𝑛𝑡𝑟𝑎𝑐𝑡𝑖𝑜𝑛 ( 𝐶𝑐) =
𝐴2
𝐴 𝑜
=
𝑉𝑒𝑛𝑎 𝑐𝑜𝑛𝑡𝑟𝑎𝑐𝑡𝑎 𝐴𝑟𝑒𝑎
𝑂𝑟𝑖𝑓𝑖𝑐𝑒 𝐴𝑟𝑒𝑎
𝐶𝑐 =
𝐴2
𝐴 𝑜
→ 𝐴2 = 𝐶𝑐 ⋅ 𝐴 𝑜
𝑄 = 𝐴1 ⋅ 𝑣1 = 𝐴2 ⋅ 𝑣2 → 𝑣1 =
𝐴2 ⋅ 𝑣2
𝐴1
𝑣1 =
𝐶 𝑐 ⋅ 𝐴 𝑜 ⋅ 𝑣2
𝐴1
𝑃1
𝓌
+
𝑣1
2
2𝑔
=
𝑃2
𝓌
+
𝑣2
2
2𝑔
→ (
𝑃1
𝓌
−
𝑃2
𝓌
) =
𝑣2
2
2𝑔
−
𝑣1
2
2𝑔
= ℎ → 𝑣2
2
− 𝑣1
2
= 2𝑔ℎ
𝑣2
2
− (
𝐶 𝑐 ⋅ 𝐴 𝑜 ⋅ 𝑣2
𝐴1
)
2
= 2𝑔ℎ ⇒ 𝑣2
2
(1 − (
𝐶 𝑐 ⋅ 𝐴 𝑜
𝐴1
)
2
) = 2𝑔ℎ
𝑣2 =
√2𝑔ℎ
√1 −
𝐶𝑐
2 ⋅ 𝐴 𝑜
2
𝐴1
2
𝑄 = 𝐴2 ⋅ 𝑣2 = 𝐶𝑐 ⋅ 𝐴 𝑜 ⋅
√2𝑔ℎ
√1 −
𝐶𝑐
2 ⋅ 𝐴 𝑜
2
𝐴1
2
= 𝐶𝑐 ⋅ 𝐴 𝑜 ⋅
√2𝑔ℎ
√1 −
𝐶𝑐
2 ⋅ 𝐴 𝑜
2
𝐴1
2
×
√1 −
𝐴 𝑜
2
𝐴1
2
√1 −
𝐴 𝑜
2
𝐴1
2
(𝑀𝑢𝑙𝑡𝑝𝑙𝑖𝑛𝑔 𝑎𝑛𝑑 𝑑𝑖𝑣𝑖𝑑𝑖𝑛𝑔 𝑏𝑦 √1 −
𝐴 𝑜
2
𝐴1
2 𝑎𝑛𝑑 𝑟𝑒𝑎𝑟𝑟𝑎𝑛𝑔𝑖𝑛𝑔)
29. If the specific gravity of manometric fluid is less than specific gravity of following fluid, inverted differential
U―tube manometer is used.
𝑣 𝑎𝑐𝑡𝑢𝑎𝑙 = 𝐶𝑣 ⋅ √2𝑔𝑥 ⋅ (
𝑆 𝐻𝑔
𝑆
− 1) = 𝐶𝑣 ⋅ 𝑣𝑡ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙
Device Shape Losses Cd Cost
Venturimeter Low High High
Flow Nozzle Medium Medium Medium
Orifice meter High Low Cheap
Force on Pipe Bends
Momentum Equation
𝛴𝐹 = 𝑚 ⋅ 𝑎 = 𝑚 (
𝑣 − 𝑢
𝑡
) = 𝑚̇ ⋅ (𝑣 − 𝑢)
(𝑚̇ = 𝜌 ⋅ 𝐴 ⋅ 𝑣⃗)
𝛴𝐹 = 𝜌𝑄(𝑣 − 𝑢) Momentum Equation
Applying Momentum equation in x-direction,
𝑃1 ⋅ 𝐴1 + 𝐹𝑥 − 𝑃2 ⋅ 𝐴2 ⋅ 𝑐𝑜𝑠 𝜃 = 𝜌𝑄(𝑣2 ⋅ 𝑐𝑜𝑠 𝜃 − 𝑣1)
Momentum equation in y-direction,
𝐹𝑦 − 𝑃2 ⋅ 𝐴2 ⋅ 𝑠𝑖𝑛 𝜃 = 𝜌𝑄(𝑣2 ⋅ 𝑠𝑖𝑛 𝜃 − 0)
30. VORTEX MOTION
The motion of the fluid along the curved path is known as vortex motion.
1. Vortex motion is of 2 types, Forced Motion & Free Vortex
Forced Vortex Motion
The motion if a fluid in a curved path under the influence of external agency is known as forced vortex motion. As
there is a continuous expenditure of energy in forced vortex motion, Bernoulli’s equation is not applicable. The
equation 𝑣=r⋅ω, is applicable for forced vortex motion.
Example- Liquid in a container when rotated, motion of fluid in impeller of a centrifugal pump.
Forced vortex motion is Rotational Flow.
Free Vortex motion
In free vortex motion, the fluid moves in curved path due to internal fluid action. But not due to external torque.
As there is no expenditure of energy. Therefore, Bernoulli’s equation is applicable for free vortex motion.
𝑑
𝑑𝑡
(𝑚𝑣𝑟) = 𝑇 = 0 → 𝑚𝑣𝑟 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 → 𝑣𝑟 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
𝑣 ⋅ 𝑟 = 𝐾 → 𝑓𝑟𝑒𝑒 𝑣𝑜𝑟𝑡𝑒𝑥 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛
Example – Motion of fluid in diffuser of centrifugal pump. Flow of fluid in pipe bends, whirl pool, flow of liquid in wash
basin.
Free Vortex is an Irrotational flow
Generalized equation for Vortex Motion
𝑉𝑜𝑙𝑢𝑚𝑒 = 𝑑𝐴 ⋅ 𝑑𝑟 𝑀𝑎𝑠𝑠 = 𝑣𝑜𝑙𝑢𝑚𝑒 ⋅ 𝜌
𝑚 = 𝜌 ⋅ 𝑑𝐴 ⋅ 𝑑𝑟
𝑃 ⋅ 𝑑𝐴 +
𝜌 ⋅ 𝑑𝐴 ⋅ 𝑑𝑟 ⋅ 𝑣2
𝑟
= (𝑃 +
𝜕𝑃
𝜕𝑟
𝑑𝑟) 𝑑𝐴
𝑃 +
𝜌 ⋅ 𝑑𝑟 ⋅ 𝑣2
𝑟
= 𝑃 +
𝜕𝑃
𝜕𝑟
𝑑𝑟
𝜌 ⋅ 𝑣2
𝑟
𝑑𝑟 =
𝜕𝑃
𝜕𝑟
𝑑𝑟
𝜌 ⋅ 𝑣2
𝑟
=
𝜕𝑃
𝜕𝑟
This equation gives the variation of pressure in radial direction.
𝜕𝑃
𝜕𝑧
= −𝓌 = −𝜌𝑔
𝑑𝑃 =
𝜕𝑃
𝜕𝑟
⋅ 𝑑𝑟 +
𝜕𝑃
𝜕𝑧
⋅ 𝑑𝑧
𝑑𝑃 =
𝜌 ⋅ 𝑣2
𝑟
⋅ 𝑑𝑟 − 𝜌𝑔 ⋅ 𝑑𝑧
32. LAMINAR FLOW
(Viscous flow of incompressible fluids)
Reynolds Number
It is the ratio of inertia force to viscous force.
𝑅 𝑒 =
𝜌𝑣𝐿
𝜇
𝐿 → 𝑐ℎ𝑎𝑟𝑒𝑐𝑡𝑒𝑟𝑖𝑠𝑡𝑖𝑐 𝑑𝑖𝑚𝑒𝑛𝑠𝑖𝑜𝑛
Significance of L
It is such a dimension over which significant changes in properties occur.
For flow through pipes, characteristic dimension is pipe diameter. For flow over a flat plate, characteristic
dimension is distance from leading edge (𝑥).
Reynold found from his experiment for flow through pipes,
Re < 2000 Laminar
2000 < Re <4000 Transition
Re > 4000 Turbulent
𝑃1
𝜌𝑔
+
𝑣1
2
2𝑔
+ 𝑧1 =
𝑃2
𝜌𝑔
+
𝑣2
2
2𝑔
+ 𝑧2 + ℎ 𝐿
𝑃1
𝜌𝑔
=
𝑃2
𝜌𝑔
+ ℎ 𝐿 (𝑣1 = 𝑣2, 𝑧1 = 𝑧2)
𝑃1 − 𝑃2
𝜌𝑔
= ℎ 𝐿
The pressure decreases in the direction of flow in order to overcome loses i.e.,
pressure gradient is negative in the direction of flow.
Darcy-Weisbach equation
This equation is used for calculating head loss due to friction.
→ ℎ 𝐿 =
𝑓𝐿𝑣2
2𝑔𝐷
𝑓 → 𝐷𝑎𝑟𝑐𝑦 𝑓𝑟𝑖𝑐𝑡𝑖𝑜𝑛 𝑓𝑎𝑐𝑡𝑜𝑟 (𝑜𝑟)𝑀𝑜𝑜𝑑𝑦 𝑓𝑟𝑖𝑐𝑡𝑖𝑜𝑛 𝑓𝑎𝑐𝑡𝑜𝑟
→ ℎ 𝐿 =
4𝑓′𝐿𝑣2
2𝑔𝐷
𝑓′
→ 𝐹𝑎𝑛𝑛𝑖𝑛𝑔𝑠 𝑓𝑟𝑖𝑐𝑡𝑖𝑜𝑛 𝑓𝑎𝑐𝑡𝑜𝑟
𝑓 = 4𝑓′
This equation is applicable for Laminar or turbulent flow, horizontal, inclined or vertical pipes, but the flow must
be steady.
Fully developed flow
A flow is said to be a fully developed flow if the velocity profile doesn’t change in longitudinal direction and
pressure gradient (dP/dx) remains constant.
33. Laminar flow through Circular pipes (Hagen- Poiseuille flow)
Assumptions-
1. Steady flow
2. Fully developed flow
𝛴𝐹 = 𝑚𝑎 = 0
(𝑓𝑟𝑜𝑚 𝑡ℎ𝑒 𝑎𝑠𝑠𝑢𝑚𝑝𝑡𝑖𝑜𝑛 𝑆𝑝𝑎𝑡𝑖𝑎𝑙 𝑎𝑛𝑑 𝑡𝑒𝑚𝑝𝑜𝑟𝑎𝑙 𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛 𝑎𝑟𝑒 𝑧𝑒𝑟𝑜)
𝑃 ⋅ 𝜋𝑟2
− (𝑃 +
𝜕𝑃
𝜕𝑥
𝑑𝑥) 𝜋𝑟2
− 𝜏 ⋅ 2𝜋𝑟 ⋅ 𝑑𝑥 = 0
𝑃 ⋅ 𝑟 − (𝑃 +
𝜕𝑃
𝜕𝑥
𝑑𝑥) 𝑟 − 2𝜏 ⋅ 𝑑𝑥 = 0 ⟶ −
𝜕𝑃
𝜕𝑥
𝑑𝑥 = 2𝜏 ⋅ 𝑑𝑥
𝜏 = −
𝜕𝑃
𝜕𝑥
⋅
𝑟
2
𝐹𝑜𝑟 𝑓𝑢𝑙𝑙𝑦 𝑑𝑒𝑣𝑒𝑙𝑜𝑝𝑒𝑑 𝑓𝑙𝑜𝑤,
𝜕𝑃
𝜕𝑥
= 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
𝑆𝑜, 𝜏 ∝ 𝑟
As the shear stress is zero at the centre of pipe, therefore, viscous forces are zero at the
centre and hence Bernoulli’s equation can be applied along the axis of the pipe.
In a Laminar flow through pipes, shear stress varies linearly from zero at the centre to
the maximum at the pipe wall.
Velocity Distribution
𝜏 = 𝜇
𝑑𝑢
𝑑𝑦
→ 𝜏 = 𝜇
𝑑𝑢
−𝑑𝑟
= −𝜇
𝑑𝑢
𝑑𝑟
(𝑟 + 𝑦 = 𝑅 → 𝑑𝑦 = −𝑑𝑟)
𝜏 = −
𝜕𝑃
𝜕𝑥
⋅
𝑟
2
Equating we get,
−
𝜕𝑃
𝜕𝑥
⋅
𝑟
2
= −𝜇
𝑑𝑢
𝑑𝑟
𝑑𝑢
𝑑𝑟
=
𝑟
2𝜇
(
𝜕𝑃
𝜕𝑥
) ⟶ 𝑑𝑢 =
1
2𝜇
(
𝜕𝑃
𝜕𝑥
) ⋅ 𝑟 ⋅ 𝑑𝑟
Integrating we get,
𝑢 =
1
4𝜇
(
𝜕𝑃
𝜕𝑥
) ⋅ 𝑟2
+ 𝐶
𝐴𝑡 𝑡ℎ𝑒 𝑃𝑖𝑝𝑒 𝑤𝑎𝑙𝑙, 𝑟 = 𝑅 & 𝑢 = 0
0 =
1
4𝜇
(
𝜕𝑃
𝜕𝑥
) ⋅ 𝑅2
+ 𝐶 ⟶ 𝐶 = −
1
4𝜇
(
𝜕𝑃
𝜕𝑥
) ⋅ 𝑅2
36. Shear Factor (V*)
𝜏 = −
𝜕𝑃
𝜕𝑥
⋅
𝑟
2
𝜏 𝑜 = −
𝜕𝑃
𝜕𝑥
⋅
𝑅
2
= −
(𝑃2 − 𝑃1)
(𝑥2 − 𝑥1)
⋅
𝐷
4
=
𝑃2 − 𝑃1
𝐿
⋅
𝐷
4
𝑃1 − 𝑃2 = 𝓌 ⋅ ℎ 𝐿 =
𝜌 ⋅ 𝑓 ⋅ 𝐿 ⋅ 𝑣2
2𝐷
𝑆𝑢𝑏𝑠𝑡𝑖𝑡𝑢𝑡𝑖𝑛𝑔 (𝑃2 − 𝑃1) 𝑖𝑛 𝜏 𝑜,
𝜏 𝑜 =
𝜌 ⋅ 𝑓 ⋅ 𝐿 ⋅ 𝑣2
2𝐷
⋅
𝐷
4𝐿
𝜏 𝑜 =
𝜌 ⋅ 𝑓 ⋅ 𝑣2
8
𝜏 𝑜
𝜌
=
𝑓 ⋅ 𝑣2
8
√
𝜏 𝑜
𝜌
= √
𝑓
8
⋅ 𝑣
𝑆ℎ𝑒𝑎𝑟 𝑉𝑒𝑙𝑜𝑐𝑖𝑡𝑦 (√
𝜏 𝑜
𝜌
) = 𝑉∗
𝑉∗
= √
𝑓
8
⋅ 𝑣
BOUNDARY LAYER THEORY
Boundary layer theory was proposed by Prandtl in 1904.
When a real fluid flows past a solid object, the velocity of the fluid will be same that of object when it comes in
contact with the object. If the object is stationary, the fluid will also have zero velocity. Away from the object the
fluid velocity increases and at some distance from the object, the fluid velocity will be free stream velocity. This
distance from the object where there are velocity gradients is known as Boundary layer thickness and this region
is known as boundary layer region.
In the boundary layer region, the flow is viscous & rotational, as the flow is viscous in boundary layer region. As
the flow is non-viscous outside the boundary layer region, the Bernoulli’s equation can be applied.
37. Development of Boundary Layer over a flat Plate
When a real fluid flows past a flat plate, the velocity of fluid on plate will be same as that of the plate velocity. If
the plate is at rest, the fluid will also have zero velocity. The boundary layer thickness grows as the distance from
the leading-edge increases. Up to a certain distance from the leading edge the flow in Boundary layer is laminar.
As the laminar boundary layer grows instability occurs and the flow changes from laminar to turbulent through
transition. It’s found that even in turbulent boundary layer region close to the plate, the flow is laminar, this
region is known as laminar sub-layer region. Laminar sublayer region exists in turbulent boundary region.
Boundary Conditions
𝐴𝑡 𝑥 = 0 → 𝛿 = 0
𝐴𝑡 𝑦 = 0 → 𝑢 = 0
𝐴𝑡 𝑦 = 𝛿 → 𝑢 = 𝑢∞
𝐴𝑡 𝑦 = 𝛿 →
𝑑𝑢
𝑑𝑦
= 0
Boundary Layer thickness (δ)
It is the distance from the boundary to the point in y-direction, where the velocity is 99% of free stream velocity.
For all calculations, 𝐴𝑡 𝑦 = 𝛿 → 𝑢 = 𝑢∞
Displacement thickness (δ*)
𝑚̇ 𝑖𝑑𝑒𝑎𝑙 = 𝜌𝐴𝑣⃗ = 𝜌 ⋅ (𝑑𝑦 ⋅ 1) ⋅ 𝑢∞
𝑚̇ 𝑟𝑒𝑎𝑙 = 𝜌 ⋅ (𝑑𝑦 ⋅ 1) ⋅ 𝑢
𝑅𝑒𝑑𝑢𝑐𝑡𝑖𝑜𝑛 𝑖𝑛 𝑚𝑎𝑠𝑠 𝑓𝑙𝑜𝑤 𝑟𝑎𝑡𝑒 𝑑𝑢𝑒 𝑡𝑜 𝐵𝑜𝑢𝑛𝑑𝑎𝑟𝑦 𝑙𝑎𝑦𝑒𝑟 𝑔𝑟𝑜𝑤𝑡ℎ = 𝑚̇ 𝑖𝑑𝑒𝑎𝑙 − 𝑚̇ 𝑟𝑒𝑎𝑙
𝑚̇ 𝑖𝑑𝑒𝑎𝑙 − 𝑚̇ 𝑟𝑒𝑎𝑙 = 𝜌 ⋅ 𝑑𝑦 ⋅ 𝑢∞ − 𝜌 ⋅ 𝑑𝑦 ⋅ 𝑢 = 𝜌 ⋅ (𝑢∞ − 𝑢) ⋅ 𝑑𝑦
𝑇𝑜𝑡𝑎𝑙 𝑚𝑎𝑠𝑠 𝑟𝑒𝑑𝑢𝑐𝑡𝑖𝑜𝑛 = ∫ 𝜌 ⋅ (𝑢∞ − 𝑢) ⋅ 𝑑𝑦
𝛿
0
Displacement thickness is the thickness by which boundary should be displaced in order to compensate for mass
flow rate due to boundary layer growth.
𝑇𝑜𝑡𝑎𝑙 𝑚𝑎𝑠𝑠 𝑟𝑒𝑑𝑢𝑐𝑡𝑖𝑜𝑛 = ∫ 𝜌 ⋅ (𝑢∞ − 𝑢) ⋅ 𝑑𝑦
𝛿
0
= 20
38. 𝜌 ⋅ (𝛿∗
⋅ 1) ⋅ 𝑢∞ = 20
𝜌 ⋅ (𝛿∗
⋅ 1) ∙ 𝑢∞ = ∫ 𝜌 ⋅ (𝑢∞ − 𝑢) ⋅ 𝑑𝑦
𝛿
0
𝛿∗
= ∫ (1 −
𝑢
𝑢∞
) ⋅ 𝑑𝑦
𝛿
0
Momentum thickness (θ)
It is the distance by which boundary should be displaced in order to compensate the momentum due to boundary
layer growth.
𝜃 = ∫
𝑢
𝑢∞
⋅ (1 −
𝑢
𝑢∞
) ⋅ 𝑑𝑦
𝛿
0
Energy thickness (δE)
It is the distance by which boundary should be displaced in order to compensate for the reduction in Kinetic
energy due to boundary layer growth.
𝛿 𝐸 = ∫
𝑢
𝑢∞
⋅ (1 −
𝑢2
𝑢∞
2
) ⋅ 𝑑𝑦
𝛿
0
𝑄) 𝑉𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑑𝑖𝑠𝑡𝑟𝑖𝑏𝑢𝑡𝑖𝑜𝑛 𝑖𝑛 𝑏𝑜𝑢𝑛𝑑𝑎𝑟𝑦 𝑙𝑎𝑦𝑒𝑟 𝑖𝑠 𝑔𝑖𝑣𝑒𝑛 𝑏𝑦
𝑢
𝑢∞
=
𝑦
𝛿
.
Displacement thickness(δ*)
𝛿∗
= ∫ (1 −
𝑢
𝑢∞
) ⋅ 𝑑𝑦
𝛿
0
= ∫ (1 −
𝑦
𝛿
) ⋅ 𝑑𝑦
𝛿
0
= ∫ 1 ⋅ 𝑑𝑦
𝛿
0
− ∫
𝑦
𝛿
⋅ 𝑑𝑦
𝛿
0
= (𝛿 − 0) −
1
𝛿
⋅
(𝛿2
− 0)
2
= 𝛿 −
𝛿
2
=
𝛿
2
𝛿∗
=
𝛿
2
Momentum thickness
𝜃 = ∫
𝑢
𝑢∞
(1 −
𝑢
𝑢∞
) 𝑑𝑦
𝛿
0
= ∫
𝑦
𝛿
(1 −
𝑦
𝛿
) 𝑑𝑦
𝛿
0
= ∫
𝑦
𝛿
𝑑𝑦
𝛿
0
− ∫
𝑦2
𝛿2
𝑑𝑦
𝛿
0
=
1
𝛿
⋅
(𝛿2
− 0)
2
−
1
𝛿2
⋅
(𝛿3
− 0)
3
=
𝛿
2
−
𝛿
3
=
𝛿
6
𝜃 =
𝛿
6
𝛿 > 𝛿∗
> 𝜃
Note
𝑇ℎ𝑒 𝑠ℎ𝑎𝑝𝑒 𝑓𝑎𝑐𝑡𝑜𝑟 𝑜𝑓 𝑎 𝑏𝑜𝑢𝑛𝑑𝑎𝑟𝑦 𝑙𝑎𝑦𝑒𝑟 𝑖𝑠 𝑒𝑞𝑢𝑎𝑙 𝑡𝑜 𝐻 =
𝛿⋆
𝜃
, 𝑡ℎ𝑖𝑠 𝑡𝑒𝑟𝑚 𝑖𝑠 𝑢𝑠𝑒𝑑 𝑖𝑛 𝑎𝑛𝑎𝑙𝑦𝑠𝑖𝑠 𝑜𝑓 𝑓𝑙𝑜𝑤 𝑠𝑒𝑝𝑒𝑟𝑎𝑡𝑖𝑜𝑛.
For linear velocity profiles, the shape factor is 3.
Drag force (FD)
It is the force exerted by fluid on plate in direction parallel to relative motion. When angle of incidence of plate is
zero, then drag is due to shear only.
39. Von-Karman Integral equation
Assumptions
1. Steady flow
2. Incompressible flow
3. 2―D flow
4.
𝑑𝑃
𝑑𝑥
= 0 (𝑇ℎ𝑖𝑠 𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛 𝑖𝑠 𝑣𝑎𝑙𝑖𝑑 𝑜𝑛𝑙𝑦 𝑓𝑜𝑟 𝑒𝑥𝑡𝑒𝑟𝑛𝑎𝑙 𝑓𝑙𝑜𝑤𝑠)
From Newton second law of motion, Von-Karman equation can be derived.
𝜏 𝑜
𝜌𝑢∞
2
=
𝑑𝜃
𝑑𝑥
→ 𝑉𝑜𝑛 − 𝐾𝑎𝑟𝑚𝑎𝑛 𝑖𝑛𝑡𝑒𝑔𝑟𝑎𝑙 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛
( 𝜏 𝑜 → 𝑠ℎ𝑒𝑎𝑟 𝑠𝑡𝑟𝑒𝑠𝑠 𝑜𝑛 𝑡ℎ𝑒 𝑠𝑢𝑟𝑓𝑎𝑐𝑒 𝑜𝑓 𝑝𝑙𝑎𝑡𝑒, 𝜃 → 𝑚𝑜𝑚𝑒𝑛𝑡𝑢𝑚 𝑡ℎ𝑖𝑐𝑘𝑛𝑒𝑠𝑠, 𝑥 → 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑓𝑟𝑜𝑚 𝑙𝑒𝑎𝑑𝑖𝑛𝑔 𝑒𝑑𝑔𝑒)
Significance of Van-Karman equation
1. With the help of Van-Karman equation, Boundary layer thickness δ can be calculated.
2. The shear stress on the surface of plate can be calculated.
3. The drag force on the plate can be calculated.
𝑅𝑒𝑦𝑛𝑜𝑙𝑑𝑠 𝑛𝑢𝑚𝑏𝑒𝑟 𝑖𝑠
𝜌𝑢∞ 𝑥
𝜇
(𝑥 𝑖𝑠 𝑡ℎ𝑒 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑓𝑟𝑜𝑚 𝑡ℎ𝑒 𝑙𝑒𝑎𝑑𝑖𝑛𝑔 𝑒𝑑𝑔𝑒)
For flow over the flat plate, if the Reynolds number is less than
5⨯105, then the flow is taken as Laminar.
If the flow is greater than 5⨯105, then the flow is taken as
turbulent.
Average Drag force coefficient (Cd)
𝐶 𝑑 =
𝐹𝐷
1
2
𝜌 ⋅ 𝐴 ⋅ 𝑢∞
2
With the help of average drag force coefficient, drag force can be calculated.
Local drag coefficient (or) Skin friction coefficient (Cfx)
𝐶𝑓𝑥 =
𝜏 𝑜
1
2
𝜌 ⋅ 𝑢∞
2
𝑄) 𝑭𝒐𝒓 𝒂 𝒗𝒆𝒍𝒐𝒄𝒊𝒕𝒚 𝒑𝒓𝒐𝒇𝒊𝒍𝒆 𝒇𝒐𝒓 𝒂 𝒍𝒂𝒎𝒊𝒏𝒂𝒓 𝒃𝒐𝒖𝒏𝒅𝒂𝒓𝒚 𝒍𝒂𝒚𝒆𝒓
𝒖
𝒖∞
=
𝟑𝒚
𝟐𝜹
−
𝒚 𝟑
𝟐𝜹 𝟑
Find boundary Layer thickness (δ), shear stress on surface of plate, Drag force, Average drag coefficient
in terms of Reynolds number?
→ 𝜃 = ∫
𝑢
𝑢∞
⋅ (1 −
𝑢
𝑢∞
) ⋅ 𝑑𝑦
𝛿
0
→ 𝜃 = ∫ (
3𝑦
2𝛿
−
𝑦3
2𝛿3
) ⋅ (1 − (
3𝑦
2𝛿
−
𝑦3
2𝛿3
)) ⋅ 𝑑𝑦
𝛿
0
⇒ 𝜃 =
39𝛿
280
→
𝜏 𝑜
𝜌𝑢∞
2
=
𝑑𝜃
𝑑𝑥
=
𝑑
𝑑𝑥
(
39𝛿
280
) =
39
280
𝑑𝛿
𝑑𝑥
→ 𝜏 𝑜 = 𝜌𝑢∞
2
⋅
39
280
𝑑𝛿
𝑑𝑥
①
𝜏 𝑜 → 𝑠ℎ𝑒𝑎𝑟 𝑠𝑡𝑟𝑒𝑠𝑠 𝑜𝑛 𝑡ℎ𝑒 𝑠𝑢𝑟𝑓𝑎𝑐𝑒 𝑜𝑓 𝑝𝑙𝑎𝑡𝑒 (𝑎𝑡 𝑦 = 0)
𝜏(𝑠ℎ𝑒𝑎𝑟 𝑠𝑡𝑟𝑒𝑠𝑠) = 𝜇
𝑑𝑢
𝑑𝑦
→ 𝜏 𝑜(𝑠ℎ𝑒𝑎𝑟 𝑠𝑡𝑟𝑒𝑠𝑠 𝑜𝑛 𝑡ℎ𝑒 𝑠𝑢𝑟𝑓𝑎𝑐𝑒 𝑜𝑓 𝑝𝑙𝑎𝑡𝑒) = 𝜇
𝑑𝑢
𝑑𝑦
|
𝑦=0
45. Total Gradient Line
The line joining total heads at various points in a flow is known as total energy line.
𝑇𝑜𝑡𝑎𝑙 ℎ𝑒𝑎𝑑 =
𝑃
𝓌
+ 𝑧 +
𝑣2
2𝑔
The distance between TEL & HGL gives velocity head.
In a flow hydraulic gradient line can rise or fall, but total energy line can never rise as long as there I no external
energy input i.e., total energy line will rise in case of pumps & compressors.
Pipes in Series
𝑄1 = 𝑄2 = 𝑄3 = 𝑄4 = 𝑄
ℎ 𝐿 =
𝑓 ⋅ 𝑙1 ⋅ 𝑄2
12 ⋅ 𝑑1
5 +
𝑓 ⋅ 𝑙2 ⋅ 𝑄2
12 ⋅ 𝑑2
5 +
𝑓 ⋅ 𝑙3 ⋅ 𝑄2
12 ⋅ 𝑑3
5 +
𝑓 ⋅ 𝑙4 ⋅ 𝑄2
12 ⋅ 𝑑4
5 + ⋯
ℎ 𝐿 = ℎ𝑙1
+ ℎ𝑙2
+ ℎ𝑙3
+ ℎ𝑙4
+ ⋯
48. Centre of Pressure
It’s the point through which total hydro static force is supposed to be acting.
Case-1 (Inclined Surface)
From the principle of Moments, the centre of pressure can be calculated,
𝑥 𝑐⋅𝑝 = 𝑥̅ +
𝐼 𝐺𝐺
𝐴 ⋅ 𝑥̅
⋅ 𝑠𝑖𝑛2
𝜃
𝐼 𝐺𝐺 =
𝑏 ⋅ 𝑑3
12
, 𝐴 = 𝑏 ⋅ 𝑑
IGG is the moment of Inertia about centroidal axis, which is parallel to OO’.
θ is the angle made by surface with respect to free surface.
The centre of pressure is below C.G because the pressure increases with depth.
Case-2 (Plane Vertical surface)
Put θ=90ᵒ in case 1
𝐹 = 𝓌 ⋅ 𝐴 ⋅ 𝑥̅
𝑥 𝑐.𝑝 = 𝑥̅ +
𝐼 𝐺𝐺
𝐴 ⋅ 𝑥̅
⋅ 𝑠𝑖𝑛2
𝜃 → 𝑥 𝑐.𝑝 = 𝑥̅ +
𝐼 𝐺𝐺
𝐴 ⋅ 𝑥̅
⋅ 𝑠𝑖𝑛2
90ᵒ
→ 𝑥 𝑐.𝑝 = 𝑥̅ +
𝐼 𝐺𝐺
𝐴 ⋅ 𝑥̅
Case-3 (Plane Horizontal surface)
Put θ=0ᵒ in case 1
𝑥 𝑐.𝑝 = 𝑥̅ +
𝐼 𝐺𝐺
𝐴 ⋅ 𝑥̅
⋅ 𝑠𝑖𝑛2
𝜃 → 𝑥 𝑐.𝑝 = 𝑥̅ +
𝐼 𝐺𝐺
𝐴 ⋅ 𝑥̅
⋅ 𝑠𝑖𝑛2
0ᵒ → 𝑥 𝑐.𝑝 = 𝑥̅
Case Force Centre point
Inclined 𝓌Ax̅
𝑥̅ +
𝐼 𝐺𝐺
𝐴 ⋅ 𝑥̅
⋅ 𝑠𝑖𝑛2
𝜃
Vertical 𝓌Ax̅
𝑥 𝑐.𝑝 = 𝑥̅ +
𝐼 𝐺𝐺
𝐴 ⋅ 𝑥̅
Horizontal 𝓌Ax̅ 𝑥 𝑐.𝑝 = 𝑥̅
49. Hydrostatic force on Curved surfaces
𝑑𝐹 = 𝑃 ⋅ 𝑑𝐴 → 𝑑𝐹 = 𝜌 ⋅ 𝑔 ⋅ 𝑥 ⋅ 𝑑𝐴
𝑑𝐹 𝐻 = 𝑑𝐹 ⋅ 𝑠𝑖𝑛 𝜃 = 𝜌 ⋅ 𝑔 ⋅ 𝑥 ⋅ 𝑑𝐴 ⋅ 𝑠𝑖𝑛 𝜃
The horizontal component of force on curved surface is equal to hydrostatic force on vertical projection area, and
this force acts at the centre of pressure of corresponding area.
𝑑𝐹𝑣 = 𝑑𝐹 ⋅ 𝑐𝑜𝑠 𝜃
𝑑𝐹𝑣 = 𝜌 ⋅ 𝑔 ⋅ 𝑥 ⋅ 𝑑𝐴 ⋅ 𝑐𝑜𝑠 𝜃 = 𝜌 ⋅ 𝑔 ⋅ (𝑥 ⋅ 𝑑𝐴 𝑐𝑜𝑠 𝜃) = 𝜌 ⋅ 𝑔 ⋅ 𝑉 = 𝑚𝑔
𝑑𝐹𝑣 = 𝑤𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑓𝑙𝑢𝑖𝑑
The vertical component of force on the surface is equal to weight of the liquid contained by curved surface taken
up to free surface and this weight will act from the centre of gravity of corresponding weight.
Special Cases
50. Turbulent flow
In turbulent flow, as there is a continuous mixing of fluid particles, velocity fluctuates continuously. Hence no
turbulent flow is purely steady.
In turbulent flow the shear stress is due to fluctuation of velocity in flow as well as in the transverse direction.
Head loss in turbulent is proportional to (v1.75 to v2), where as in laminar flow head loss is proportional to v.
Boussinesq equation
𝜏 = 𝜇
𝑑𝑢
𝑑𝑦
+ 𝜂
𝑑𝑢
𝑑𝑦
(𝜂 = 𝑒𝑑𝑑𝑦 𝑣𝑖𝑠𝑐𝑜𝑠𝑖𝑡𝑦)
It’s very difficult to find eddy viscosity. Hence this equation is not used.
Reynold developed an equation for turbulent shear stress, τ = ρ⋅ u’⋅ v’, where u’ & v’ are fluctuating component of
velocity in x & y directions respectively.
Prandtl’s mixing theory
Mixing length is that length in transverse direction, where in fluid particles after colliding lose excess momentum
and reach momentum of new region. According to Prandtl mixing length ‘l’ is equal to 0.4y, where y is distance
measured from pipe wall.
At the pipe wall mixing length is zero.
𝑢′
= 𝑣′
= 𝑙 ⋅
𝑑𝑢
𝑑𝑦
𝜏 = 𝜌 ⋅ 𝑢′
⋅ 𝑣′
= 𝜌 ⋅ (𝑙 ⋅
𝑑𝑢
𝑑𝑦
) ⋅ (𝑙 ⋅
𝑑𝑢
𝑑𝑦
) = 𝜌 ⋅ 𝑙2
⋅ (
𝑑𝑢
𝑑𝑦
)
2
𝜏
𝜌
= 𝑙2
⋅ (
𝑑𝑢
𝑑𝑦
)
2
→ √
𝜏
𝜌
= 𝑙 ⋅ (
𝑑𝑢
𝑑𝑦
)
𝑊𝑒 𝑘𝑛𝑜𝑤 𝑡ℎ𝑎𝑡 √
𝜏
𝜌
= 𝑉∗ (𝑠ℎ𝑒𝑎𝑟 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦)
𝑉∗
= 𝑙 ⋅ (
𝑑𝑢
𝑑𝑦
) = 0.4𝑦 ⋅ (
𝑑𝑢
𝑑𝑦
) (𝑙 = 4𝑦)
𝑉∗
0.4
×
𝑑𝑦
𝑦
= 𝑑𝑢
Integrating,
∫
𝑉∗
0.4
×
𝑑𝑦
𝑦
= ∫ 𝑑𝑢
𝑢 = (
5
2
𝑉∗
⋅ 𝑙𝑛 𝑦) + 𝐶
The velocity distribution in turbulent flow is logarithmic nature.