Viscous Fluid Flow (Contd.)
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Introduction to the Navier–Stokes equations
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Welcome to today's discussion! We're dissecting the Navier-Stokes equations. Can anyone tell me why these equations are significant in fluid dynamics?
They describe how fluids move and the forces acting on them.
Exactly! They are essential for predicting flow patterns. Now, who remembers what we discussed about normal stresses in a fluid?
Normal stresses are related to pressure. There are three components: τxx, τyy, and τzz.
Perfect! And can anyone explain how these stresses relate to mechanical pressure?
Mechanical pressure is one-third of the sum of the three normal stresses.
Great job! Remember the acronym PNM, standing for Pressure, Normal stresses, Mechanical pressure. This will help us keep track of the relationships.
Thermodynamic vs. Mechanical Pressure
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Now, let's dive into the difference between thermodynamic and mechanical pressures. Who can state how they differ?
Thermodynamic pressure is a state property, while mechanical pressure can vary under flow conditions.
Spot on! Mechanical pressures derived in fluid motion can sometimes differ from thermodynamic ones. Can anyone suggest conditions under which they can be equal?
When the divergence of velocity is zero!
Exactly! We refer to this as the condition for incompressible flow. Remember, in hydraulics, we often assume incompressible flow. Let's summarize with the mnemonic IMA, representing Incompressible, Mechanical, and Actual.
Navier-Stokes Equations for Incompressible Flow
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Next, we will consider Navier-Stokes equations in the context of incompressible flow. What happens to divergence in this scenario?
The divergence of velocity equals zero!
Exactly! This simplification leads to more applicable forms for practical engineering. Can anyone explain the significance of assuming constant viscosity?
It allows us to ignore temperature variations, making analysis simpler!
That's right! HCM – Hydraulics, Constant viscosity, Mechanical flow is our takeaway here.
Inviscid Flow and the Euler Equation
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Finally, let's transition to inviscid flow! When do we consider flow inviscid?
When viscous forces are negligible!
Exactly! This leads us to the Euler equations. How is the Euler equation simplified from Navier-Stokes?
By eliminating viscous terms from the equation!
Correct! Let’s remember the mnemonic E-V – Euler and Visco – to remind us of this relationship. And can you tie this into Bernoulli’s equation?
Oh! They derive from the Euler equations when integrated along a streamline!
Excellent understanding! Always connect the dots with fundamental principles.
Introduction & Overview
Read summaries of the section's main ideas at different levels of detail.
Quick Overview
Standard
In this section, the lecture elaborates on the derivation of the Navier–Stokes equations, explores the differences between thermodynamic and mechanical pressures, and introduces the conditions under which these pressures equate. Additionally, it provides insights into incompressible flow and the significance of the Euler equations.
Detailed
In this section of the lecture on Viscous Fluid Flow, the focus is primarily on the derivation of the Navier–Stokes equations, key equations that describe the motion of viscous fluid substances. The section begins with a recap of the fundamental deformation law for Newtonian viscous fluids, highlighting the mathematical representation of mechanical pressures derived from normal stresses. It establishes that mechanical pressure is typically negative one-third of the sum of three normal stresses but diverges from thermodynamic pressure under most circumstances.
The Stokes Hypothesis is introduced, where conditions under which mechanical pressure can equate to thermodynamic pressure are explained, specifically in the context of incompressible flow where the divergence of velocity is zero. Moving on, the simplified form of the Navier–Stokes equations for incompressible flow is derived, emphasizing the practicality of this formulation in hydraulic applications. Finally, the section touches on the Euler equations for inviscid flow, elaborating on their derivation and significance in fluid mechanics, ultimately leading to Bernoulli's equation. This summary encapsulates the extensive theoretical and practical applications of these equations in hydraulic engineering.
Audio Book
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Deformation Law Overview
Chapter 1 of 9
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Chapter Content
Welcome back students to the final lecture of this module viscous fluid flow where we are deriving the Navier–Stokes equations. In the last lecture we wrote the general deformation law for Newtonian Viscous Fluid okay this equation we call it equation number 18.
Detailed Explanation
In this section, the professor is introducing the topic of viscous fluid flow and its relation to the Navier-Stokes equations. The deformation law for Newtonian viscous fluids is fundamental, as it describes how these fluids respond to applied forces. It's essential to distinguish this law, referred to as equation number 18, as it will lead to the derivation of the Navier-Stokes equations in this lecture.
Examples & Analogies
Imagine kneading dough. When you apply pressure (force) to the dough, you can see it stretch and change shape. This represents how a viscous fluid deforms under stress, much like how the dough conforms to the forces applied to it.
Mechanical vs. Thermodynamic Pressure
Chapter 2 of 9
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Chapter Content
we will talk a little bit before writing the Navier–Stokes equations we will talk about the difference between the thermodynamic and mechanical pressure. So do you think they both are same, no they are not. So mechanical pressure so the pressure that we derive we find out during the Navier–Stokes equations or any other such equation is the thermodynamic pressure.
Detailed Explanation
The professor is highlighting the distinction between thermodynamic and mechanical pressure. While they might seem similar, in the context of fluid flow and equations like the Navier-Stokes, they refer to different concepts. Thermodynamic pressure is associated with the state of the fluid (like temperature and phase), while mechanical pressure results from the fluid's environment and deformation under stress.
Examples & Analogies
Think of a balloon. The air pressure inside is thermodynamic, as it relates to the gas laws and the state of the air. However, when you squeeze the balloon, the pressure you feel on the outside is mechanical pressure, resulting from the physical deformation of the balloon's material.
Mechanical Pressure Calculation
Chapter 3 of 9
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Chapter Content
So mechanical pressure p bar is negative one-third of sum of three normal stresses all right. What are the three normal stress τ xx, τ yy and τ zz. So p bar can be written as – of 1/3 τ xx + τ yy + τ zz.
Detailed Explanation
This portion introduces the calculation of mechanical pressure in a viscous fluid. The professor explains that the mechanical pressure can be derived as a function of the normal stresses acting on the fluid in three orthogonal directions (x, y, and z). Specifically, it states that the mechanical pressure is equivalent to negative one-third of the sum of these three normal stresses, providing a formula for easy calculation.
Examples & Analogies
Imagine sitting on a soft sofa. The pressure you exert on the sofa can be thought of as the normal stresses. If you sit in three different positions (flat on your back, on your side, and leaning forward), the total pressure felt by the sofa can be viewed as the sum of these pressures. The mechanical pressure of the sofa 'feeling' your weight is similar to how fluids experience internal stresses.
Stokes Hypothesis
Chapter 4 of 9
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Chapter Content
However, if we want both of them to be same two different ways right. One is if you look at the above equation p bar will be = p either lambda + 2/3 mu = 0 or divergence of V = 0. This is called Stokes Hypothesis.
Detailed Explanation
The Stokes Hypothesis provides conditions under which mechanical pressure and thermodynamic pressure can be considered equal. If either the parameters lambda and mu fulfill the specific requirement or if the divergence of velocity is zero, these two pressures can equate, typically observed in incompressible flows. Understanding this hypothesis is crucial in fluid mechanics as it simplifies analysis under certain flow conditions.
Examples & Analogies
Think about a water bottle under pressure. If you keep filling it without allowing any air to escape (zero divergence), the pressure inside (mechanical) can align with the thermodynamic pressure if no heat is exchanged with the environment, representing the conditions outlined in the Stokes Hypothesis.
Navier-Stokes Equations Derivation
Chapter 5 of 9
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Chapter Content
So now going back to the objective of this module the Navier–Stokes equations okay. So desired momentum equation for general Newtonian viscous fluid is obtained by equation 18 that is the deformation law in rewritten Newton’s Law.
Detailed Explanation
The professor emphasizes the goal of the lecture: to derive the Navier-Stokes equations for Newtonian viscous fluids by applying the deformation law discussed previously. This is a significant step in fluid dynamics, as these equations govern the motion of fluid substances under various forces and conditions.
Examples & Analogies
Consider cooking spaghetti. Following a recipe (like Newton's Law) is akin to using the deformation law, then combining all the ingredients methodically to reach the finished dish (Navier-Stokes equations). Just as in cooking, precise steps lead to understanding and mastering fluid dynamics.
Incompressible Flow Considerations
Chapter 6 of 9
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Chapter Content
So to do that we go to a fresh page so incompressible flow what is that if rho is that means incompressible flow that means divergence of V = 0.
Detailed Explanation
In this part, the concept of incompressible flow is introduced. The professor mentions that incompressible flow implies that the fluid density does not change (rho is constant), which leads to the divergence of velocity being zero. This concept is crucial because many fluids, like water, are treated as incompressible in many engineering applications.
Examples & Analogies
Think of a full balloon. When you squeeze it, the volume remains relatively constant, and the shapes change, while the internal pressure adjusts. This is like incompressible flow, where the density stays constant even when shapes and conditions around it change.
Simplified Navier-Stokes for Incompressible Flow
Chapter 7 of 9
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Chapter Content
Therefore, in equation 20 if we assume mu is constant okay as well because in most of the hydraulic purposes we assume mu is constant.
Detailed Explanation
Here, the professor indicates that for incompressible flow, if viscosity (mu) is assumed constant, it simplifies the Navier-Stokes equations further. This simplification allows engineers to analyze most hydraulic systems effectively, as many practical fluid scenarios involve constant viscosity under typical conditions.
Examples & Analogies
Think about oil flowing through pipes. If the oil temperature doesn't change much (constant viscosity), the calculations needed to ensure smooth transport are easier. Just like using a consistent method to successfully bake a cake avoids complications, using constant viscosity simplifies fluid calculations.
Introduction to Inviscid Flow
Chapter 8 of 9
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Chapter Content
And now moving to the last part again that is called Inviscid flow.
Detailed Explanation
This transition marks the introduction to the concept of inviscid flow, where the effects of viscosity are negligible. The discussion will later tie into wave mechanics and the Euler equations for inviscid flows, which are vital in defining how fluids behave when viscosity does not significantly affect their motion.
Examples & Analogies
Picture a smooth sheet of ice. When sliding over it, air moves effortlessly with little resistance, akin to inviscid flow where viscosity is minimal. This concept is essential in fluid dynamics as it often simplifies equations governing fluid behavior, especially in high-speed or low-viscosity scenarios.
Euler Equation for Inviscid Flow
Chapter 9 of 9
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Chapter Content
So if we assume that viscous terms are negligible as well in equation 21 then Navier–Stokes NS can be reduced to…
Detailed Explanation
This discussion leads to the Euler equation for inviscid flow. The professor mentions how the Navier-Stokes equations can be simplified drastically when viscous effects are ignored. This simplification is crucial for understanding inviscid flow behavior, such as in high-speed aerodynamics.
Examples & Analogies
Think of a high-speed train slicing through the air. The air behaves almost as if it has no viscosity due to the train's speed, allowing engineers to simplify the fluid dynamics equations. This likens to observing how a stone thrown into still water creates ripples, while a fast-moving object creates fewer disturbances.
Key Concepts
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Navier-Stokes Equations: Fundamental equations for viscous fluid motion.
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Mechanical Pressure: Derived from normal stresses, can differ from thermodynamic pressure.
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Incompressible Flow: Assumption in hydraulics when fluid density remains constant.
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Stokes Hypothesis: Specific conditions where mechanical and thermodynamic pressures equate.
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Euler Equations: Describe inviscid flow, simplified from Navier-Stokes equations.
Examples & Applications
The application of the Navier-Stokes equations in predicting airflow over airplane wings.
Using Bernoulli's principle to analyze fluid flow in pipelines.
Memory Aids
Interactive tools to help you remember key concepts
Rhymes
Pressure and flow, they connect, Navier-Stokes gives us respect.
Stories
Imagine fluids racing through pipes; they twist and turn, but with Navier-Stokes, we can predict their flight.
Memory Tools
For understanding flow: 'MVI - Mechanical, Viscous, Inviscid - can help recall pressure types'.
Acronyms
PNE - Pressure, Normal stresses, Euler equation helps you to remember key fluid dynamics concepts.
Flash Cards
Glossary
- NavierStokes equations
A set of equations that describe the motion of viscous fluid substances.
- Mechanical Pressure
The pressure derived from the normal stresses in a fluid.
- Thermodynamic Pressure
The pressure that is a property of the state of a fluid, depending on temperature and other conditions.
- Incompressible Flow
A flow where the density of the fluid remains constant.
- Stokes Hypothesis
Condition under which mechanical pressure equals thermodynamic pressure, often applied to incompressible flows.
- Euler Equation
Equation describing fluid motion in inviscid flow, simplified from Navier-Stokes.
- Bernoulli's Equation
An expression describing the conservation of energy for flowing fluids.
Reference links
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