Difference between thermodynamic and mechanical pressure
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Understanding Mechanical Pressure
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Let's start with mechanical pressure. It is defined as negative one-third of the sum of three normal stresses in a fluid.
What are those three normal stresses?
Great question! The three normal stresses are τxx, τyy, and τzz, which represent the stress acting on different planes of the fluid.
So, how do we write the equation for mechanical pressure then?
We can write it as p bar = - (1/3)(τxx + τyy + τzz). This equation highlights the relationship between stress and mechanical pressure.
And why is this important?
Understanding this relationship helps us analyze fluid behavior under different flow conditions, especially in hydraulic applications.
So, to recap, mechanical pressure relates to the internal stresses in a fluid. It helps us predict how fluids will behave in systems.
Thermodynamic vs Mechanical Pressure
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Now, let's compare thermodynamic pressure with mechanical pressure. Are they the same?
I think they are different based on what we learned.
Correct! Mechanical pressure is derived from the stress state of a fluid, while thermodynamic pressure is related to the state's equations of state. They are not inherently the same.
Under what conditions can they be considered equal?
Good point! They can be equal when two conditions are met: either lambda + (2/3)mu = 0 or divergence of V = 0. The latter often applies to incompressible flows.
Does that mean we can always apply these equations to hydraulic systems?
Yes! In hydraulic applications, we often assume incompressible flow, making it a critical condition.
To summarize, mechanical and thermodynamic pressures differ fundamentally but can coincide under specific conditions in fluid dynamics.
The Stokes Hypothesis
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Next, let's talk about the Stokes Hypothesis. It plays a significant role in our understanding of fluid behaviors.
What does this hypothesis state?
The Stokes Hypothesis suggests that for mechanical and thermodynamic pressures to be equal, specific stress conditions must be satisfied.
Is it always applicable?
Unfortunately, it is not always satisfied in all fluid flows, especially in compressible fluids where lambda is typically a positive value.
What implications does this have for our work?
Understanding when this hypothesis is applicable allows for better predictions in fluid dynamics, particularly within hydraulic systems and incompressible flow.
So to conclude, the Stokes Hypothesis helps bridge our understanding between thermodynamic and mechanical pressures but holds limitations.
Introduction & Overview
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Quick Overview
Standard
The section discusses the definitions and formulas for thermodynamic and mechanical pressure in the context of viscous fluid flow. It also addresses the Stokes Hypothesis, conditions for equality, and the implications in incompressible fluids.
Detailed
In the study of viscous fluid flow, it is important to differentiate between thermodynamic pressure and mechanical pressure. Thermodynamic pressure is a property derived from the state's equations of state, while mechanical pressure (denoted as p bar) relates to the stress state in the fluid. Specifically, mechanical pressure is defined as negative one-third of the sum of three normal stresses (τxx, τyy, τzz). A key equation captures this relationship: p bar = - (1/3)(τxx + τyy + τzz), which can also be expressed using bulk viscosity (lambda) and dynamic viscosity (mu). The section emphasizes that these pressures are equal under specific conditions, largely articulated by the Stokes Hypothesis, where either lambda + (2/3)mu = 0 or the divergence of velocity (div V) = 0, the latter being especially relevant for incompressible flow. This relationship is significant in hydraulic contexts where incompressible flow is assumed. Finally, the section briefly mentions challenges associated with the Stokes Hypothesis in compressible fluids.
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Understanding Mechanical Pressure
Chapter 1 of 3
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Chapter Content
Mechanical pressure (p bar) is negative one-third of the sum of three normal stresses: \( \tau_{xx}, \tau_{yy}, \tau_{zz} \). This can be expressed as:
\[
p_{bar} = -\frac{1}{3}(\tau_{xx} + \tau_{yy} + \tau_{zz})\]
It can also be represented with the relation:
\[
p_{bar} = \lambda + \frac{2}{3}\mu \nabla \cdot V\]
where \( \lambda \) and \( \mu \) are fluid properties.
Detailed Explanation
Mechanical pressure is a specific measure of pressure that relates to the internal stresses in a fluid. It is calculated by taking the average of three types of stresses that act normal to the surface of the fluid element. This means it considers how much force is applied per unit area due to these stress components. The equation shows that mechanical pressure depends on both a constant called \( \lambda \) and a term that involves the divergence of the velocity field, which affects the fluid's motion. Understanding this is crucial because it helps us differentiate between mechanical pressure and thermodynamic pressure in fluid dynamics.
Examples & Analogies
Imagine trying to measure how hard you can push against a balloon. The 'mechanical pressure' tells you how the air inside is pushing back at your fingers based on its shape and how it's deforming. Just like in our equation, the pressure will change based on the balloon's material properties (like elasticity) and how much you're pushing on it!
The Relationship Between Mechanical and Thermodynamic Pressure
Chapter 2 of 3
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Chapter Content
Mechanical pressure and thermodynamic pressure can be equal under specific conditions:
1. \( \lambda + \frac{2}{3}\mu = 0 \)
2. \( \nabla \cdot V = 0 \)
The first condition is known as the Stokes Hypothesis, and the second is common in incompressible flow, like water.
Detailed Explanation
For both mechanical and thermodynamic pressures to be the same, certain conditions must be fulfilled. The Stokes Hypothesis posits that the relationship between the constants \( \lambda \) and \( \mu \) must be very specific, which isn't often the case in real fluids. Additionally, for conditions that denote incompressibility, like most liquids, the divergence of the velocity must equal zero. This simplifying assumption allows engineers and scientists to work with a more straightforward model while analyzing fluid behavior in various situations.
Examples & Analogies
Think of a scenario where you're trying to find the temperature of the sea. Under certain stable conditions, the surface temperature (which is like thermodynamic pressure) and the temperature at deeper levels (akin to mechanical pressure) can be nearly the same. If the water is calm (equivalent to the divergence being zero), the temperatures equalize. However, as you go deeper and face different pressures (like wind creating waves), these temperatures can start to vary.
Limitations of the Stokes Hypothesis
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Chapter Content
The Stokes Hypothesis is rarely satisfied because \( \lambda \) is usually positive. Therefore, it's uncommon to find situations where \( \lambda + \frac{2}{3}\mu = 0 \).
Detailed Explanation
This hypothesis is an important concept in fluid mechanics, but its practical applicability is limited. Since \( \lambda \) represents a physical property related to fluid compressibility, it's predominantly positive, leading to the conclusion that finding instances where the condition holds true is quite rare. This realization emphasizes that while mathematical models are useful, very few real-world situations can perfectly align with simplified assumptions like those of the Stokes Hypothesis.
Examples & Analogies
Consider a car on different terrains. On a flat highway, the signs (like the Stokes Hypothesis) may apply quite easily. However, when you hit a bumpy off-road path, the situation changes drastically. The simple rules you believed applied start to break down because the conditions change, reflecting how rarely the Stokes Hypothesis is met in actual fluid situations.
Key Concepts
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Mechanical Pressure: The pressure derived from the internal stresses in a fluid.
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Thermodynamic Pressure: The pressure defined by the thermodynamic state of a fluid based on equations of state.
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Incompressible Flow: A fluid flow model where the fluid density is constant.
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Stokes Hypothesis: Conditions under which mechanical and thermodynamic pressures can be considered equal.
Examples & Applications
Example of mechanical pressure calculation in a hydraulic system using τxx, τyy, and τzz.
Illustration of incompressible flow assumption in a water pipe.
Memory Aids
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Rhymes
In fluids, pressure's quite a dance, mechanical's stress, in numbers, it prance. Thermodynamic's the state of strain, both can match, but conditions reign.
Stories
Imagine two friends, Thermo and Mech, they often argue about who’s best. Thermo backs the state while Mech counts the stress, they can collide in special instances, but mostly, they jest.
Memory Tools
Use 'STUD' - Stokes for Conditions, Thermodynamic for State, Uncoupled in Stress, Divergence is vital.
Acronyms
Remember 'M-P-T'
for Mechanical pressure
for Pressure conditions
for Thermodynamic pressure.
Flash Cards
Glossary
- Mechanical Pressure
Defined as negative one-third of the sum of three normal stresses in a fluid.
- Thermodynamic Pressure
The pressure associated with the thermodynamic state of a fluid, derived from the equations of state.
- Incompressible Flow
A flow condition where the fluid's density remains constant; divergence of velocity equals zero.
- Stokes Hypothesis
A hypothesis that states conditions for mechanical and thermodynamic pressures to be equal, specifically related to viscosity.
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