Incompressible flow and Navier–Stokes equations
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Introduction to Navier–Stokes Equations
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Today, we will be deriving the Navier–Stokes equations, which are fundamental in understanding viscous fluid flows. Can anyone tell me what you understand about these equations?
Are they used to model how fluids flow, especially when they are viscous?
Exactly! The Navier–Stokes equations provide a mathematical formulation for these fluid movements. They take into account viscosity, which is crucial for modeling flows like water in pipes.
How do they relate to pressure in fluids?
Great question! We differentiate between thermodynamic pressure and mechanical pressure. Mechanical pressure can be represented by the negative one-third sum of the normal stresses in the fluid.
So mechanical pressure isn't the same as thermodynamic pressure?
Correct! They are not identical but can be equal under certain conditions, which we will explore shortly. Let's remember this as the 'pressure paradox' — both types can appear similar but aren't always the same.
What conditions make them equal?
If lambda + 2/3 mu equals zero or if the divergence of velocity is zero, that is when mechanical and thermodynamic pressure can be considered equivalent. This condition, known as the Stokes Hypothesis, applies often in hydraulic studies.
To recap, the Navier–Stokes equations take into account various forces acting on fluid elements, including pressure differences, and represent the motion of viscous fluids, laying a foundation for understanding fluid dynamics.
Incompressible Flow
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Now let's talk about incompressible flow. Can anyone explain what happens in incompressible flow conditions?
The density of the fluid is constant, right?
Correct! For incompressible flow, the divergence of velocity is also zero. This helps simplify our Navier–Stokes equations.
Is this why water is often assumed to have incompressible flow in hydraulic engineering?
Exactly! When we assume that viscosity is constant, we can further simplify our equations to better model real-world applications like flow in pipes or rivers.
What’s the significance of simplifying to constancy in viscosity?
It allows us to uncouple fluid behavior from temperature variations—keeping our analyses simpler under steady conditions. This leads us to applying the same equations across various hydraulic systems.
Can you summarize the Navier–Stokes equations for incompressible flow?
Certainly! When applying these equations to incompressible flow, we assume constant viscosity and can express the dynamics without considering temperature effects, yielding critical insights for practical fluid mechanics.
From Navier–Stokes to Bernoulli
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Last, let's explore how the Navier–Stokes equations can be related to Bernoulli's equation. Does anyone know what conditions are necessary for this?
It has to be frictionless flow, right?
Absolutely! When we look at inviscid flow, we can reduce our framework dramatically. For steady, incompressible, and frictionless flow, the Navier–Stokes equations yield the well-known Bernoulli equation.
So this is how the two concepts connect?
Precisely! The conditions allow us to integrate across a streamline, leading us to Bernoulli’s principles, which describe conservation of energy in fluids.
Could we see a practical example of Bernoulli’s equation?
Of course! A classic example is the lift generated on an airplane wing, where airflow speeds up above the wing, leading to lower pressure and hence lift.
Could you summarize our discussion?
In summary, the Navier–Stokes equations form the foundation for fluid mechanics, allowing for the derivation of other important equations, including Bernoulli's, under specific assumptions of flow conditions like incompressibility and lack of viscosity.
Introduction & Overview
Read summaries of the section's main ideas at different levels of detail.
Quick Overview
Standard
This section delves into the derivation of the Navier–Stokes equations, emphasizing the conditions under which they apply to incompressible flow. It highlights the relationship between mechanical and thermodynamic pressures and introduces the Stokes Hypothesis, which simplifies analyses in hydraulic engineering.
Detailed
In this section, we derive the Navier–Stokes equations specifically for Newtonian viscous fluid flow, contrasting mechanical and thermodynamic pressures. Mechanical pressure is expressed as the negative one-third sum of three normal stresses, while the conditions for incompressibility are introduced, particularly through the Stokes Hypothesis, suggesting that divergence of velocity approaches zero. The section extends into the implications of assuming constant viscosity in hydraulic applications and discusses how the Navier–Stokes equations reduce to the Euler equations for inviscid flow. By illustrating the relationship among various equations, including Bernoulli's equation, the section emphasizes the conceptual depth and practical significance of these fundamental laws in hydraulic engineering.
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Understanding Mechanical vs. Thermodynamic Pressure
Chapter 1 of 4
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Chapter Content
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. So the 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. or it is written lambda + 2/3 mu divergence of V and this is equation number 19. If you substitute the value of τ xx, τ yy and τ zz during the previous equation this is what you are going to get okay. So what does this mean? This means that mean pressure in deforming viscous fluid is not = thermodynamic property called pressure.
Detailed Explanation
In fluid mechanics, there are two types of pressure discussed: thermodynamic pressure and mechanical pressure. While thermodynamic pressure is a property associated with the fluid's thermodynamic state, mechanical pressure (denoted as p bar) is derived from the behavior of stresses within the fluid. Specifically, mechanical pressure relates to the average of three normal stresses, which are pressure components acting on the fluid at various orientations. The equation shows this relationship, where p bar is equal to the negative third of the sum of three normal stresses, and further interprets the influence of the fluid's properties like the viscosity (mu) and the divergence of velocity (V). This distinction illustrates that pressure within a deforming viscous fluid isn't purely a thermodynamic property.
Examples & Analogies
Imagine a balloon filled with air. The air pressure inside is thermodynamic pressure, which changes with temperature and the amount of air. When you press the balloon, the internal forces generate mechanical pressure that reacts to the deformation and stress applied. So, while you can measure the air pressure (thermodynamic) in the balloon, the stresses created by your hand pressing on it can be considered mechanical pressure.
Stokes Hypothesis and Conditions for Same Pressure
Chapter 2 of 4
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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. So let us say =0 actually this is called Stokes Hypothesis all right. The second way is divergence of V can be if it is 0 then also this is possible. This is possible this is more commonly possible because divergence of V 0 for incompressible flow. So that is when we deal with water and hydraulics you see I mean all the time we assume incompressible flow.
Detailed Explanation
For both thermodynamic pressure and mechanical pressure to equate in a fluid, certain conditions must be satisfied. The first condition is captured in the Stokes Hypothesis, which states that the coefficient pertaining to shear stresses (lambda) and the divergence of velocity must equal zero. If the divergence of velocity (often denoted as V) is zero, this implies that the fluid is incompressible, which is a standard assumption in hydraulic applications such as water flow. This understanding is crucial because it helps simplify the Navier–Stokes equations for incompressible flow.
Examples & Analogies
Think of a swimming pool filled with water. When you dive in and move around, the water doesn’t compress; it flows and moves around you without a significant change in density. Here, we consider the water as incompressible, and thus, we can simplify the calculations of flow using conditions like divergence of velocity being zero, which aligns with the idea that there's a consistent flow of water around your movements.
Navier–Stokes Equations for Incompressible Flow
Chapter 3 of 4
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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. So if you look at the Navier–Stokes equations above this will be 0 this the last term divergence of V all right. 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
In the context of the Navier–Stokes equations, analyzing the scenario of incompressible flow is essential. Incompressible flow implies that the fluid's density (rho) remains constant. This results in the divergence of velocity (V) equating to zero. By substituting this into the Navier–Stokes equations and also assuming viscosity (mu) is constant—typical in hydraulics—we can simplify the equations significantly, yielding a more manageable form specifically tailored for incompressible flow conditions.
Examples & Analogies
Consider how water flows from a garden hose. The flow is smooth, and the viscosity of water doesn’t change significantly as it moves through the hose. We assume it remains incompressible, allowing us to use simplified equations (like the Navier–Stokes equations for incompressible flow) to predict how the water will behave under different conditions.
Deriving the Euler Equation
Chapter 4 of 4
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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, but here in Inviscid the Euler it (()) (13:14) to discuss about mention Euler and the Bernoulli theorem how it has its origin okay. So if we assume that viscous terms are negligible as well in equation 21 then Navier–Stokes NS can be reduced to. This is equation number 22.
Detailed Explanation
When we consider inviscid flow (flow without viscosity), the Navier–Stokes equations can be further simplified. By neglecting the viscous terms, we derive the Euler equation, which is significant in the study of potentials in fluid mechanics. The Euler equation facilitates discussions about concepts like wave mechanics and the origins of the Bernoulli theorem, providing insights into fluid behavior under ideal conditions where viscosity is not a factor.
Examples & Analogies
Imagine a perfectly smooth waterslide with no water resistance (like what happens in inviscid flow). The motion of a person sliding down can be described using simplified rules (Euler equations), ignoring the frictional forces present in real-life scenarios. This allows for a clearer understanding of how fluid moves under ideal conditions.
Key Concepts
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Navier–Stokes equations: Mathematical foundation for fluid flow.
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Incompressible flow: A condition where fluid density remains constant.
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Mechanical vs. thermodynamic pressure: Different types of pressure that can be equal under certain conditions.
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Stokes Hypothesis: An assumption that simplifies the Navier–Stokes equations for certain fluids.
Examples & Applications
Water flowing through a pipe under constant pressure indicates incompressible flow.
Airplane wings generating lift through decrease in pressure, illustrated by Bernoulli's equation.
Memory Aids
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Rhymes
In viscous flow so grand, Navier–Stokes takes a stand. Stokes Hypothesis will help to see, pressure's nature, just let it be.
Stories
Imagine a river flowing steadily. The water doesn’t compress, each droplet sings its hymns of flow. As it glides past rocks, the pressure changes—not random, but governed by the great equations of fluid dynamics.
Memory Tools
To remember pressure types: 'Mechanical Stays Pressure Nice To Each,' indicating mechanical and thermodynamic distinctions.
Acronyms
P-M-T for Pressures
for mechanical
for thermodynamic
for their differences
highlighting that they aren't the same unless certain conditions are met.
Flash Cards
Glossary
- Navier–Stokes equations
A set of equations that describe the motion of viscous fluid substances.
- Incompressible flow
Flow where the fluid density remains constant throughout the motion.
- Mechanical pressure
Pressure derived from stress in a fluid, represented mathematically for incompressible flows.
- Thermodynamic pressure
The pressure of a fluid as defined by thermodynamics, related to temperature and volume.
- Stokes Hypothesis
An assumption in fluid dynamics stating that the second coefficient of viscosity is not significant, simplifying the application of Navier–Stokes equations.
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