1.4. Lecture – 48: Viscous Fluid Flow

Welcome students. So, this is the week 10, lecture number 1, here we are going to study about the topic that is mentioned in this slide, this is about viscous fluid flow. Actually, you have, we have gone through this topic before but in a much more crude manner.

The introduction sets the stage for the lecture on viscous fluid flow, indicating that this is a continuation of a topic students have encountered before. However, this time, it will be explored in depth.

Think of it like revisiting a familiar recipe but diving deeper into the cooking techniques. Just like you might have made a cake before, but now you're learning how to perfectly whisk egg whites to achieve that fluffy finish.

The main objective of this module is going to be able to derive Navier Stokes equation from scratch, so how to start from the beginning and derive the Navier Stokes equation.

This chunk outlines the goal of the module, which is to derive the Navier-Stokes equation. This equation is fundamental in fluid mechanics as it describes the motion of viscous fluid substances. Understanding its derivation will provide insights into how fluids behave under various conditions.

Imagine you're trying to understand how a car moves through the air. You need to break down the forces acting on the car, much like you would derive the Navier-Stokes equation to understand fluid motion thoroughly.

We will go very slow in this and we will complete this module in lecture 3 to 4, so now I will minimize this and use this board now.

The lecturer emphasizes a hands-on approach to learning, suggesting that derivation of the Navier-Stokes equations will be done manually rather than relying heavily on slides. This method is believed to enhance understanding for a complex topic like fluid dynamics.

It's like learning to ride a bicycle. Initially, you won't just watch others; you'd get on that bike, feel its balance, and understand how to steer by practicing instead of just watching a video.

In fluid mechanics, a matter is classified into fluids and non-fluids. So, in thermodynamics, the normal definition is classification in solids, liquids and gases.

This chunk clarifies how matter is classified differently in fluid mechanics compared to thermodynamics. In fluid mechanics, the focus is on fluids (which include liquids and gases) versus non-fluids, primarily solids. This distinction is crucial for understanding the behaviors of materials under different forces.

Think of a crowd at a concert: the crowd fluidly moves as a whole (fluids), while the solid structures of the concert stage or the ground (non-fluids) don’t move with the crowd.

Another basic revision is properties of fluid, so first is kinematic property, that is, velocity, acceleration, vorticity, rate of strain, angular velocity etc.

This segment introduces various properties of fluids, starting with kinematic properties. These are characteristics that describe the motion of fluids without considering the forces acting upon them, including parameters like velocity and acceleration.

Consider how you track the speed of a car (velocity) as it goes around a bend. The car's speed and path are like the fluid’s kinematic properties.

There are many other properties like this kinematic. Transport properties, you know, are viscosity, thermal conductivity, mass diffusivity.

Transport properties help understand how fluids respond to external stimuli. For instance, viscosity measures a fluid's resistance to flow, which affects how it behaves in different scenarios, such as flowing through pipes or over surfaces.

Think of syrup versus water. Syrup is more viscous, meaning it flows slowly compared to water. This unique property illustrates how transport properties affect fluid behavior in real-world applications.

There are some other miscellaneous properties. Miscellaneous properties includes surface tension, it is important to know these properties, at least know what those are vapour, pressure.

Miscellaneous properties such as surface tension and vapor pressure play critical roles in fluid behaviors. Surface tension, for example, explains why some insects can walk on water while vapor pressure is crucial in understanding boiling and evaporation.

Imagine how a water droplet forms on a leaf due to surface tension. That phenomenon showcases how unique properties of fluids affect everyday experiences.

Now we go into little bit more detail in kinematic properties.

This transition indicates a deeper examination of kinematic properties, where terms such as velocity, strain rates, and angular velocity will be discussed. These concepts are essential for analyzing fluid motion mathematically.

Consider how a soccer ball travels through the air. By analyzing its velocity and how it changes direction (angular velocity), we can better understand the fluid dynamics surrounding it.

So, the idea of this particular lecture today is that we are going to write what actually material derivatives are and we see the rotation you know how the fluid particle gets rotated and try to obtain the strain rates.

This section introduces material or substantial derivatives, defining how they quantify changes in fluid properties for individual particles moving through a flow field. This concept is vital for understanding how fluids change over time and space.

Imagine a leaf floating on a stream. The material derivative helps analyze how the leaf's position and state change as it moves with the water.

A fluid element can undergo the following 4 types of motion or deformation: translation, rotation, extensional strain or dilation, and shear strain.

The basic types of motion a fluid can experience include translation (moving from one location to another), rotation (spinning), extensional strain (stretching), and shear strain (sliding). Each of these motions influences how fluid flows and behaves under different conditions.

Think of a dance performance; the dancers may move across the stage (translation), spin (rotation), stretch into formations (extensional strain), or slide past each other (shear strain), analogous to the movements of fluid particles.

So, we are going to derive the strain rates, so one thing I would like to take your attention to, is this figure.

Here, the focus is on deriving strain rates from fluid deformation. The process involves analyzing how the shape of a fluid element changes during motion, which is crucial for predicting fluid behavior under various stresses.

Picture dough being kneaded; as you stretch and fold it, the strain rates inform you how much the dough's shape is changing. Likewise, fluid elements undergo similar transformations under motion.

With reference to the above figure, we can observe the following transformations, I mean, of transformation or deformation in the fluid element within the time interval of t + dt.

This segment emphasizes the observation of a fluid element over a time interval, noting how different points within the fluid move relative to each other. Understanding these transformations helps in analyzing fluid flow and the resulting stresses.

Imagine capturing a video of a balloon being inflated. Watching the points on the balloon's surface move as it expands is akin to tracking transformations of a fluid element over time.

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