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3.6. Inviscid flow and Euler equation
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Let's explore the difference between mechanical and thermodynamic pressure. Mechanical pressure is derived from normal stress components in a fluid, while thermodynamic pressure relates to the kinetic energy of the fluid's molecules.
How do we calculate mechanical pressure?
Great question! Mechanical pressure, denoted as p bar, is calculated as negative one-third of the sum of three normal stresses. The relationship can be expressed as p bar = - (1/3)(τ_xx + τ_yy + τ_zz).
What does it mean if both pressures are equal?
If both pressures are equal, it typically means we assume certain conditions, such as divergence of velocity being zero, representing an incompressible flow.
What about the Stokes Hypothesis?
The Stokes Hypothesis asserts that lambda + 2/3 mu must equal zero for both pressures to be the same. This is only true under very specific conditions.
So, is it commonly satisfied in fluid mechanics?
Not really, as lambda is typically positive. Hence, the Stokes Hypothesis is rarely satisfied.
To summarize, mechanical pressure is derived from stresses, and while it can equal thermodynamic pressure, specific conditions, including those of the Stokes Hypothesis, are necessary for that equality.
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Now, let’s talk about how the Navier-Stokes equations are simplified in inviscid flow conditions.
What does inviscid flow mean?
Inviscid flow refers to fluid flow where viscous effects are negligible. In this scenario, we can simplify the Navier-Stokes equations.
What will the key outcomes be?
By dropping the viscosity effects, we derive the Euler equation, which is less complex since it describes flow without friction.
Can you explain what the Euler equation looks like?
Certainly! The Euler equation expresses the motion of fluid particles without viscosity, offering a first-order relationship in pressure and velocity.
And how does this relate to Bernoulli's equation?
When integrated along a streamline, the Euler equation leads us to Bernoulli's equation, which describes the conservation of mechanical energy in fluids.
To recap, invoking inviscid flow allows us to use the simpler Euler equation and is essential for deriving Bernoulli's principles.
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Finally, let’s look at how the Euler equation is used in practical applications.
What are some real-life examples?
The Euler equation is fundamental in calculating trajectories of projectiles and analyzing fluid flows in pipes where friction is minimal.
Are there conditions where it cannot be used?
Absolutely. The Euler equation is not applicable when viscous effects significantly influence flow, such as in boundary layer regions.
So, it's better for high-speed scenarios?
Exactly! It's most accurate in scenarios involving incompressible, high-speed flows where viscosity is negligible.
What about turbulent flows?
In turbulent flows, viscosity plays a substantial role, hence the Navier-Stokes equations would be more appropriate.
In summary, the Euler equation is vital in predicting fluid dynamics under inviscid conditions and has notable applications in high-speed fluid scenarios.
Overview
Short Summary
This section discusses the derivation of the Euler equation for inviscid flow and the relationship between mechanical and thermodynamic pressure within fluid dynamics.
Medium Summary
The content elaborates on the derivation of the Euler equation from the Navier-Stokes equations under the assumption of inviscid flow. It also explores the differences between mechanical and thermodynamic pressure, introducing concepts like the divergence of velocity and the implications of the Stokes Hypothesis.
Detailed Summary
Inviscid Flow and Euler Equation
Inviscid flow refers to fluid motion where viscosity is negligible, allowing us to simplify the Navier-Stokes equations to derive the Euler equations. This section begins by clarifying the distinction between mechanical pressure, derived from stress components, and thermodynamic pressure, emphasizing that under certain conditions, both can be regarded as equal.
The mechanical pressure is defined as negative one-third of the sum of normal stresses and is represented mathematically. The section explains the Stokes Hypothesis, which suggests the condition under which both types of pressure match, particularly the divergence of velocity being zero in incompressible flow scenarios. The derivation continues to show how the Navier-Stokes equations can be simplified for incompressible and inviscid fluids.
The Euler equation, derived from these simplifications, is highlighted next, showcasing its importance in analyzing fluid behavior without viscous effects. The significance of this equation lies in its simpleness compared to the Navier-Stokes equations, notably allowing for a no-slip condition removal, which results in flow behavior that differs from viscous conditions. Finally, the derivation culminates in a detailed explanation of Bernoulli's equation, tying together the insights gained from the discussion on inviscid flows.
Audio Book
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Create a free accountSo now moving to the last part again that is called Inviscid flow. 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.
Detailed Explanation
In inviscid flow, we focus on fluid dynamics in which the viscosity of the fluid is negligible. This can simplify the equations that govern fluid motion. The study typically includes topics like wave mechanics and potential theory, which help us understand how fluids move without the influence of viscous forces. The Euler equations and Bernoulli’s theorem originate in this context, framing our understanding of ideal fluid behavior.
Examples & Analogies
Think of inviscid flow like the way an idealized, smooth slide allows children to slide down without resistance. In this scenario, the slide is perfectly smooth (no friction), similar to how inviscid fluid has no viscosity affecting the flow.
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Create a free accountIf we assume that viscous terms are negligible as well in equation 21 then Navier–Stokes NS can be reduced to...Very famously called Euler equation for Inviscid flow.
Detailed Explanation
When we ignore the effects of viscosity (because it's negligible), we can simplify the Navier-Stokes equations. This leads us to the Euler equation, which describes the motion of inviscid fluids. The reduction highlights that the Euler equation is simpler and primarily deals with pressure and velocity changes within the fluid but does not consider resistance caused by viscosity.
Examples & Analogies
Imagine riding a bike down a hill. If the ground is perfectly smooth (no friction), you can think of yourself moving in a way similar to an inviscid flow, where your speed changes only due to gravity and air resistance, just like fluid influences pressure and velocity without viscosity.
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Create a free accountSo this equation is first order right in velocity and pressure all right. Thus, it is simpler than another change is in viscous fluid flow we have already been assuming no slip condition that you have seen also in the boundary layer theory and turbulent and laminar flows.
Detailed Explanation
Euler's equation is a first-order equation, which means it is linear in terms of velocity and pressure, simplifying the equations governing fluid motion. Unlike viscous flows, where we must consider the no-slip condition (where fluid at the boundary moves with the boundary), the Euler equation allows for slipping across boundaries due to the absence of viscosity. This characteristic differentiates the behavior of inviscid flows from those with viscosity.
Examples & Analogies
Think of sliding across a smooth ice surface—your movement isn’t hindered by any friction (similar to viscous forces). This allows you to coast along freely, as in inviscid flow without restrictions at the boundaries.
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Create a free accountEuler equation for steady incompressible frictionless flow integrates along a streamline between points 1 and 2, to give Bernoulli's equation.
Detailed Explanation
When we apply Euler's equation to steady, incompressible, and frictionless flow, we can integrate it along a streamline. This integration results in Bernoulli's equation, which relates the pressure, velocity, and height of fluid at different points along a flow path. Bernoulli’s equation is crucial in predicting how fluids behave in various engineering applications.
Examples & Analogies
Imagine a flowing river where water's speed and height change as it flows over rocks and into pools. By applying Bernoulli's principle, we can predict how fast the water will flow in a narrow section versus a wider section, much like how a garden hose speeds up when narrowed.
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Key concepts
Core takeaways and short definitions to help you quickly recall the key ideas from this section.
- Inviscid Flow:
A type of fluid flow where viscosity is negligible.
- Euler Equation:
The governing equation for inviscid flow, derived from the Navier-Stokes equations.
- Mechanical Pressure:
Pressure calculated from normal stresses.
- Thermodynamic Pressure:
Pressure associated with the fluid's thermal properties.
- Stokes Hypothesis:
Conditions that lead to equality between mechanical and thermodynamic pressure.
Examples
Memory aids
Inviscid flow is smooth as silk, with no viscosity to spill; Euler's equations guide the way, in fluid dynamics, they hold sway.
Imagine a river flowing gracefully, its surface untouched by stickiness, embodying inviscid flow. The Euler equation watches over, ensuring no friction disrupts its dance.
I.E.T.S: Invincible Euler Theorem States - this aids in recalling inviscid flow principles.
Flash Cards
Glossary
Inviscid Flow
Fluid motion where viscosity is negligible.
Euler Equation
A simplified form of the Navier-Stokes equations that describes inviscid flow.
Mechanical Pressure
Derived pressure from fluid normal stresses.
Thermodynamic Pressure
Pressure related to the thermal state of a fluid's molecules.
Stokes Hypothesis
A condition under which mechanical and thermodynamic pressures can be considered equivalent.
Bernoulli's Equation
An expression of the conservation of energy for fluid flow.