4.1.1 - Microscopic Point of View
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Introduction to Fluid Flow and Velocity Gradients
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Welcome, class! Today, we'll explore how fluids behave under the influence of shear stress, particularly when flowing between parallel plates. Let's start with the no-slip condition. Can anyone remind me what that means?
It means that the fluid at the surface of the stationary plate has zero velocity.
Exactly! And what about the fluid adjacent to the moving plate?
That fluid moves at the velocity V.
Correct! So, as we move from point B to point A, there is a linear increase in fluid velocity. Now, why do you think this gradient is important?
I guess it relates to how shear stress develops in the fluid.
Right! The velocity gradient indicates how quickly the fluid is deforming and leads us to the concept of shear stress. Let's remember the term 'velocity gradient' as VG—understand its role in shear stress formation.
Got it! VG for velocity gradient.
Great! As we continue, let's also think about how these concepts relate to viscosity.
Shear Stress and Viscosity
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Now that we've established the velocity gradient, let's discuss shear stress. Can someone define shear stress for us?
Shear stress is the force per unit area acting parallel to the surface.
Exactly! And how does this relate to viscosity in fluids?
I think viscosity is a measure of a fluid's resistance to flow, right?
Yes! According to Newton's law of viscosity, shear stress is proportional to the shear strain rate. If we represent the shear strain rate with the symbol 'du/dy', how would we express this relationship mathematically?
It would be τ = μ * (du/dy), where τ is shear stress and μ is viscosity.
Well done! Understanding this equation—τ = μ * du/dy—helps us grasp fluid dynamics better. Just remember the acronym 'TSV' for shear stress, viscosity, and velocity gradient.
TSV, nice! It helps to remember those key relationships.
Effects of Temperature and Pressure on Viscosity
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Alright! Let’s move on to temperature effects on viscosity. What happens to a fluid’s viscosity when you increase the temperature?
I believe it decreases for liquids because the intermolecular forces become weaker.
Exactly! Lower binding forces lead to lower viscosity. And what about gases?
For gases, viscosity increases with temperature due to greater molecular movement.
Right again! Let's summarize our findings: the key term to remember here is 'temperature effect.' For liquids, increasing temperature leads to decreased viscosity, while for gases, it leads to increased viscosity. We can remember this by the mnemonic 'LiTe-Visc'—Lower viscosity for Liquids with Temperature increases, and Viscosity increases for gases.
Got it, LiTe-Visc!
Understanding Non-Newtonian Fluids
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So far, we’ve focused on Newtonian fluids. What do we call fluids that don't follow the simple rules of Newtonian flow?
Those are called non-Newtonian fluids!
That's right! Can anyone give me an example of a non-Newtonian fluid?
Toothpaste is a good one; it behaves differently under different forces.
Perfect example! Non-Newtonian fluids can have varying viscosity depending on the shear rate, making them more complex to study. Let’s use the acronym 'NNP' for Non-Newtonian Properties to remind ourselves of this complexity.
NNP stands for Non-Newtonian Properties!
Practical Applications of Viscosity
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Finally, how are these concepts related to practical applications? For instance, why is understanding viscosity important in engineering?
It affects how fluids are transported in pipelines and manufacturing processes.
Exactly! The efficiency of pumping, mixing, and applying fluids depends heavily on their viscosity. We can summarize this by remembering 'Visc-PUMP'—Understanding Viscosity is crucial for PUMPing efficiency!
Visc-PUMP is a neat way to remember!
I’m glad you find it helpful! By linking theory to applications, you'll better understand the significance of these principles in real-world scenarios.
Introduction & Overview
Read summaries of the section's main ideas at different levels of detail.
Quick Overview
Standard
Focusing on fluid behavior at a microscopic level, particularly in parallel plate systems, this section examines how shear stress is developed due to velocity gradients and how viscosity relates to temperature and pressure conditions. It also touches on how different fluids respond under various shear forces.
Detailed
Microscopic Point of View
In this section, we delve into the microscopic perspective of fluid dynamics, particularly focusing on fluid flow between two parallel plates. We begin with the no-slip condition, where one plate remains stationary while the other moves at velocity V. As we analyze a fluid element defined by points A, B, C, and D in the region between the plates, we observe a linear velocity distribution that increases from B (velocity = 0) to A (velocity = V).
As fluid flows over time, deformation occurs, leading to angular deformations in the fluid element. By considering the relationships between angular deformations, shear strain rate, and velocity gradients, we derive that shear stress is proportional to the shear strain rate based on Newton’s law of viscosity. This crucially differentiates fluid mechanics from solid mechanics, where stress correlates to strain rather than strain rate.
Next, we explore how temperature and pressure affect viscosity, highlighting that temperature increases lead to reduced intermolecular forces for liquids, thus lowering their viscosity. In contrast, gases exhibit an increase in viscosity with temperature due to enhanced molecular movement.
Finally, we discuss the relation of viscosity's temperature sensitivity, providing insights into Sutherland's correlation and its empirical constants for different fluids. This section effectively sets the groundwork for understanding how fluid behavior under various conditions affects applications in engineering and physics.
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![The Macroscopic Study of Fluids [Physics of Fluid Mechanics #4]](https://img.youtube.com/vi/7q3-xQs4Hj8/mqdefault.jpg)
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Fluid Flow and Velocity Variation
Chapter 1 of 6
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Chapter Content
If I consider the same fluid flow through a parallel plate, okay. One of the plate the velocity is zero which is at the rest conditions and other the top plate is moving with a velocity V. So as a microscopic point of view...
Detailed Explanation
In this chunk, we are examining fluid flow between two parallel plates. One plate is stationary, and the other is moving at a certain speed, denoted as V. In such a scenario, the fluid experiences varying velocities depending on its position between the plates. This situation is characterized by no-slip conditions where the fluid directly in contact with the stationary plate has zero velocity (meaning it doesn’t move at all). As you move away from this plate towards the moving plate, the fluid gains velocity in a linear manner from 0 at the stationary plate to V at the moving plate. The velocity changes uniformly, which can be represented by a straight line when graphed.
Examples & Analogies
Imagine spreading butter on a piece of bread. The section of butter closest to the bread (like the fluid at the stationary plate) does not move, while the butter further away from the knife (like the fluid closer to the moving plate) moves quicker as you spread it. Similarly, this illustrates how fluid velocity varies in this parallel plate scenario.
Angular Deformations in Fluid Elements
Chapter 2 of 6
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Chapter Content
If as a linear velocity distribution as you assume it as we got it some of the parallel flow conditions after ∆t time these fluid element will be deformed as angular deformations will happen...
Detailed Explanation
This chunk talks about the behavior of fluid elements over time within the given fluid flow. As the fluid flows and experiences different velocities, it undergoes deformation. Specifically, after a small time increment (denoted as ∆t), the fluid element becomes distorted and experiences angular deformations. This means the shape of the fluid element—a segment of our fluid in the flow—starts to change from its original shape as the flow progresses, demonstrating the dynamic nature of fluid motion. The notion of angle θ arising from this deformation shows how fluid mechanics involves not just straight motion but also changes in shape.
Examples & Analogies
Think of a soft jelly cube on a plate. If you start to tilt and push it gently, the gel will not just slide around but also start to lose its original square shape and form other angles as it deforms. This deformation resembles what the fluid elements experience as they flow and change shape.
Velocity Gradient and Shear Stress Relationship
Chapter 3 of 6
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Chapter Content
So now we are coming to the exactly the solid mechanics concept the stress having the proportionality with the strain...
Detailed Explanation
Here, we define the relationship between shear stress (the stress resulting from forces applied parallel to the surface of a material) and velocity gradient (how quickly velocity changes in space). In fluids, this relationship is expressed as shear stress being proportional to the rate of change of velocity—this is known as the shear rate. Essentially, the faster the velocity changes across different layers of the fluid, the greater the shear stress acting on the fluid. This connection highlights a key difference between fluid and solid mechanics; in fluids, shear stress depends not on the total strain but rather on how quickly the strain is changing over time.
Examples & Analogies
Consider stirring a thick cake batter. The harder you stir (which can be thought of as applying shear), the smoother and more consistent the batter becomes. Here, the more vigorous the stirring (greater gradient of speed) leads to a higher stress on the batter—similar to how shear stress is related to speed variation across layers in fluids.
Newton's Laws of Viscosity
Chapter 4 of 6
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Chapter Content
That is the basic difference between the fluid mechanics and the solid mechanics. So same shear strain relationship we are doing it but in this case, we are telling it that the shear stress is a proportionality to the shear strain rate and that is the Newton's laws of viscosity that shear strain is having a...
Detailed Explanation
This chunk explains Newton's laws of viscosity, which describe how fluids behave when shear stress is applied. In simple terms, for Newtonian fluids, the shear stress is directly proportional to the shear strain rate (the rate at which deformation occurs). This means that as more force is applied, the fluid flows more easily, akin to how oil flows faster than syrup when pushed. If there is no applied shear, the fluid doesn’t flow — which is the basis of understanding viscosity.
Examples & Analogies
Picture two types of syrup: one is thin like water, and the other is thick and sticky. When you pour both onto a surface, you notice that the thin syrup spreads quickly while the thick syrup requires more force to spread. This showcases how viscosity affects fluid flow, aligning with Newton's law of viscosity.
The Effect of Temperature on Viscosity
Chapter 5 of 6
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Chapter Content
Now let us commit that how does the temperature effect on the coefficient of the viscosity. Now we have to look at the molecular levels okay...
Detailed Explanation
This chunk discusses how temperature influences the coefficient of viscosity in both liquids and gases. In general, as temperature increases, the molecular motion also increases which results in a decrease in viscosity for liquids. This means that hot liquids flow more easily than cold ones. For gases, the situation is a bit different: usually, increasing the temperature causes viscosity to increase because molecules move more and collide more, leading to an increase in friction between gas particles.
Examples & Analogies
Think about honey; when it's cold, it's thick and doesn’t pour well. However, when you warm it up, it becomes runnier and flows much more easily. This illustrates how temperature can drastically alter the viscosity of liquids.
Understanding Non-Newtonian Fluids
Chapter 6 of 6
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Chapter Content
Most of the common fluid flow problems also follows Newtonian fluids but some of the case it does not follow the Newtonian fluids...
Detailed Explanation
In this section, we explore non-Newtonian fluids, which do not have a constant viscosity regardless of the shear rate. Instead, their viscosity can change depending on the applied shear rate. For example, some fluids become thinner (like ketchup) when stirred quickly but thickens under slow movements or pressure. This behavior is distinct from Newtonian fluids where viscosity remains consistent regardless of the forces applied. Understanding non-Newtonian behavior is important in many applications where the fluid's behavior under different stress conditions is critical.
Examples & Analogies
Consider a tube of ketchup; when you shake it, it flows smoothly, but when you’re trying to pour it out, it hardly budges. This shows that the viscosity is dynamically changing in response to how much force is applied, a classic case of a non-Newtonian fluid!
Key Concepts
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Shear Stress: The force that causes layers of fluid to slide past each other, essential for understanding fluid flow.
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Viscosity: A property that quantifies a fluid's internal resistance to flow and affects applications in engineering.
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No-Slip Condition: Important in fluid mechanics, denoting that there is no motion of fluid along stationary surfaces.
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Velocity Gradient: Represents how rapidly velocity changes within a fluid and is crucial for the calculation of shear stress.
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Non-Newtonian Fluids: Fluids that do not have a constant viscosity; their properties change under different shear rates.
Examples & Applications
Water flowing between two parallel plates exhibits a linear velocity profile, indicating shear stress and viscosity.
Toothpaste is a non-Newtonian fluid; it has a higher viscosity when at rest and flows easily when squeezed.
Memory Aids
Interactive tools to help you remember key concepts
Rhymes
If your fluid’s slow and thick, viscosity is the trick!
Stories
Imagine a river flowing gently through rocks. At one point it’s calm, but as you reach a narrow passage, it speeds up and flows easily. This is much like how viscosity changes with shear stress.
Memory Tools
Remember 'TSV' for Shear stress, Viscosity, and Velocity gradient—these are the relationships we must connect!
Acronyms
LiTe for liquids—Increasing Temperature, Lower viscosity!
Flash Cards
Glossary
- Shear Stress
The force per unit area acting parallel to the surface of a material.
- Viscosity
A measure of a fluid's resistance to flow, influenced by intermolecular interactions.
- Velocity Gradient
The rate of change of velocity with respect to distance; crucial for determining shear stress.
- NonNewtonian Fluid
A fluid whose viscosity changes with the shear rate, unlike Newtonian fluids.
- NoSlip Condition
A condition where fluid velocity at a solid boundary is zero.
- Sutherland Correlation
An empirical relationship that connects dynamic viscosity with temperature for gases.
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