Lecture - 53
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Introduction to Viscous Fluid Flow
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Welcome back, students! Today we will derive the Navier-Stokes equations for viscous fluid flow. Can anyone remind me what viscous flow is?
Isn't it the flow of fluids that have a measurable viscosity, showing resistance to flow?
And I think it involves shear stress and deformation, right?
Exactly! Viscous flow is characterized by the internal friction due to viscosity. This internal friction can be described using equations we will derive today.
Difference Between Thermodynamic and Mechanical Pressure
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Now, let's discuss the difference between thermodynamic pressure and mechanical pressure. Who can explain how they differ?
I believe thermodynamic pressure relates to the state of the fluid, while mechanical pressure depends on stresses in the fluid?
Right! The mechanical pressure is the average of normal stresses in the fluid. Can someone tell me the formula for mechanical pressure?
It's p bar = -(1/3)(τxx + τyy + τzz) where τxx, τyy, and τzz are the normal stresses!
Great job! As we've discussed, these pressures only align under specific conditions, including the Stokes Hypothesis.
Deriving Navier-Stokes Equations
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Next, we will derive the Navier-Stokes equations. Who can remind us why we assume incompressible flow?
Because in many hydraulic applications, like water flow, the density remains constant!
And it makes our calculations simpler, especially with the divergence of velocity being zero.
Exactly! Incompressible flow allows us to simplify the equations. Let’s see how that works in our equations.
Inviscid Flow and Euler's Equation
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Finally, we explore inviscid flow. What happens to our equations if we neglect the viscous terms?
We get the Euler equation, which applies to frictionless flows!
Correct! It's a first-order equation that simplifies our computations even further. It leads us to Bernoulli's equation.
Can you remind us what Bernoulli’s equation represents?
Of course! Bernoulli’s equation relates pressure, velocity, and height in a flowing fluid. Excellent participation today, everyone!
Introduction & Overview
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Quick Overview
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In this lecture, the Navier-Stokes equations for viscous flow are derived, highlighting the distinction between thermodynamic pressure and mechanical pressure. The Stokes Hypothesis is introduced to explain conditions under which these pressures can be considered the same, especially for incompressible flow.
Detailed
Summary of Lecture - 53
In this lecture, Prof. Mohammad Saud Afzal elaborates on the derivation of the Navier-Stokes equations that govern the behavior of viscous fluids. The instructor begins by revisiting the deformation law for Newtonian viscous fluids. It emphasizes the distinction between thermodynamic pressure and mechanical pressure, defining the mechanical pressure derived from the average of normal stresses. The concepts of the Stokes Hypothesis are introduced, highlighting its rare satisfaction in real fluids.
The importance of incompressible flow is then discussed, revealing that the Navier-Stokes equations can be simplified when certain assumptions (such as constant viscosity and density) are made. Finally, the lecture covers the concept of inviscid flow and derives the Euler equation leading to Bernoulli's equation, establishing the foundation for computational fluid dynamics to be discussed in subsequent modules.
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Understanding Pressure in Fluid Mechanics
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So 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
In fluid mechanics, there are two types of pressures referred to: thermodynamic pressure and mechanical pressure. Thermodynamic pressure is the pressure related to the state of the fluid (temperature, volume, etc.), whereas mechanical pressure arises from the stresses within the fluid. In the context of Navier–Stokes equations, the mechanical pressure calculated is often different from the usual thermodynamic pressure that we might expect.
Examples & Analogies
Think of a tire. The air pressure inside a tire (mechanical pressure) might be different from what you consider when you look at the temperature effects on that air's ability to hold pressure (thermodynamic pressure). They are interrelated but not the same.
Mechanical Pressure Calculation
Chapter 2 of 8
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The mechanical pressure p bar is negative one-third of sum of three normal stresses. 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
The mechanical pressure (represented as p bar) in a fluid can be calculated using the average of three normal stress components acting in different directions: τ xx, τ yy, and τ zz. The formula indicates that mechanical pressure is considered as a negative one-third of the sum of these stress components, reflecting the internal forces distributed in the fluid.
Examples & Analogies
Imagine a sponge submerged in water. The sponge absorbs stress from all sides—top, bottom, and sides. The average stress it experiences, similar to the average pressure exerted on the sponge from the water, can be thought of in this manner.
Conditions for Pressure Equivalence
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...if you look at the above equation p bar will be = p either lambda + 2/3 mu = 0 or divergence of V = 0.
Detailed Explanation
For mechanical pressure to be equal to thermodynamic pressure, certain conditions must be met. This can occur if either the term involving lambda and mu equals zero or if the divergence of velocity (divergence of V) is zero. This condition is often valid in cases of incompressible flow where certain simplifications can be made.
Examples & Analogies
Think of water flowing through a hose. If there's no change in the water's flow (velocity remains constant), the pressures can be thought of as balanced. In this state, the mechanical pressure can equal the thermodynamic pressure, similar to how fluids behave in pipelines.
Stokes Hypothesis Explanation
Chapter 4 of 8
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Now coming back to this Stokes Hypothesis it says even in case of compressible fluid is lambda + 2/3 mu = 0 then we can have.
Detailed Explanation
The Stokes Hypothesis suggests that the relationship between the viscous coefficient (lambda) and the shear viscosity (mu) can determine whether both mechanical and thermodynamic pressures can be equal in compressible fluids. However, experiments often show that this condition is rarely satisfied, as lambda values are generally positive in practical scenarios.
Examples & Analogies
Think of a filled balloon that might change its shape depending on the air pressure inside (lambda) and how flexible the balloon's material is (mu). If the conditions aren’t just right, the pressures won’t balance out as we expect.
Introduction to Navier–Stokes Equations
Chapter 5 of 8
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So now going back to the objective of this module the Navier–Stokes equations. The 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 Navier–Stokes equations describe how fluids move by relating the velocity, pressure, density, and viscosity of the fluid. These equations are derived from the fundamental laws of motion (Newton’s laws) and take into account how the fluid deforms under various forces.
Examples & Analogies
Consider how water flows in rivers versus how it behaves when poured from a bottle. The Navier-Stokes equations help scientists understand the different flow behaviors, like turbulence in rivers, as opposed to the smooth flow of water from a bottle.
Incompressible Flow in Navier–Stokes Equations
Chapter 6 of 8
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When it comes to Navier–Stokes equations for practical purposes this is the equation that we are going to use.
Detailed Explanation
In the case of incompressible flow, where the fluid density remains constant and the divergence of velocity is zero, the Navier–Stokes equations simplify. This simplified form is often used in hydraulic applications, making it more practical for engineers.
Examples & Analogies
Think of how water, which we assume is incompressible, flows through various systems like pipes and channels. The simplicity of the Navier-Stokes equations in these situations helps engineers design systems efficiently without complicating factors like compressibility.
Understanding Inviscid Flow
Chapter 7 of 8
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So actually we are going to study wave mechanics as part of Inviscid flow because there we will have potential theory and other things.
Detailed Explanation
Inviscid flow refers to flow where viscous forces are negligible. This allows us to simplify the fluid equations significantly. The study of inviscid flow leads to understanding wave dynamics and phenomena that occur in ideal fluids, which lack internal friction.
Examples & Analogies
Imagine a smooth, frictionless slide in a playground. If a child were to slide down, they would move with relatively little resistance, analogous to how fluids behave in inviscid flow situations.
Euler Equation and Bernoulli's Equation
Chapter 8 of 8
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Chapter Content
This equation is first order right in velocity and pressure. Thus, it is simpler than another change is in viscous fluid flow.
Detailed Explanation
When discussing inviscid flow, we can derive the Euler equation, which governs the flow behavior of fluids without considering viscosity. This leads to the Bernoulli equation, an important principle in fluid dynamics that relates pressure, velocity, and height in a flowing fluid, especially in applications involving energy conservation.
Examples & Analogies
Consider the Aerodynamics of a plane. As air flows over the wings, the pressure differs above and below the wings, allowing the plane to lift off the ground. Bernoulli's equation helps explain this phenomenon clearly, linking speed with pressure in the flow.
Key Concepts
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Navier-Stokes Equations: Foundational equations of fluid dynamics that incorporate viscosity.
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Thermodynamic Pressure: Pressure that stems from the state of the fluid in thermodynamic equilibrium.
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Mechanical Pressure: Pressure derived from the fluid's internal stresses.
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Incompressible Flow: Flow where density remains constant, allowing simplification of the Navier-Stokes equations.
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Inviscid Flow: Flow described by Euler's equations, where viscous effects are negligible.
Examples & Applications
A water flow in a pipe can be analyzed using Navier-Stokes equations to predict how it will behave under various conditions.
Airfoil design in aerodynamics often uses Euler's equations to determine the pressure over wings, crucial for flight mechanics.
Memory Aids
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Rhymes
Flow so smooth, no stress in view, Euler equations tell us what fluids do.
Stories
Imagine a river flowing down a mountain. The water travels swiftly (like inviscid flow) and adjusts with minimal friction, gathering speed before flowing into calm waters below (Bernoulli's principle in effect).
Memory Tools
P.V.V - Pressure, Velocity, Viscosity - Remember the key components when dealing with fluid behavior!
Acronyms
NAVIER - Newtonian, Average pressure, Viscous terms, Incompressible relation, Equations of motion, Relevant to fluid flow.
Flash Cards
Glossary
- Viscous Fluid
A fluid that exhibits viscosity, leading to resistance to flow.
- NavierStokes Equations
Equations that describe the motion of viscous fluid substances.
- Thermodynamic Pressure
Pressure associated with the thermal state of a fluid.
- Mechanical Pressure
Pressure determined by the stresses within the fluid.
- Stokes Hypothesis
A hypothesis stating that mechanical pressure equals thermodynamic pressure under specific conditions.
- Euler Equation
An equation governing the motion of inviscid (non-viscous) fluids.
- Incompressible Flow
Flow in which the density of the fluid remains constant.
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