11.3. Applications and Examples

This is the field what we used to solve it, we have already solved this part. Today we are looking at what will be the wall stress, shear stress at this point, also this point, stream function, vorticity, velocity potential and the average velocity if known the velocity field. So in the last class we estimate this velocity field from Navier-Stokes equations that we assume it vw is a 0 okay, neglecting the gravity force components.

In this chunk, the focus is on understanding how velocity fields are analyzed within the context of fluid mechanics. The discussion references the use of Navier-Stokes equations to determine various fluid parameters such as wall stress, shear stress, and average velocity. The assumption that gravitational forces are negligible simplifies our calculations. We also learned in the previous class how these equations can estimate the velocity field based on given conditions and parameters.

Imagine trying to determine how fast a river flows at various points; you could use equations that consider the slope of the riverbed (analogous to pressure gradients) and ignore the small influence of wind (similar to neglecting gravitational forces in this context). Just like those equations would help you understand water flow, the Navier-Stokes equations help us understand fluid behaviors in complex scenarios.

Now if you look it I am just substituting wall shear stress using the Newton's laws of viscosity we can get tau xy wall fluid kinematics we have done this part mu is equal to del u by del y del v by del x at wall y equal to plus minus h.

This section dives into the calculation of wall shear stress, a vital part of fluid dynamics. It employs Newton's laws of viscosity, which relates shear stress (c) to the velocity gradient. The equation describes how the viscosity () and the changes in velocity in the direction perpendicular to the wall contribute to the shear stress. In simpler terms, when fluid moves past a solid surface, the friction (shear stress) depends on how quickly the fluid's speed changes near the surface.

Think about spreading butter on a piece of bread. As you spread the butter (the fluid), how thick or smoothly it spreads depends on how soft it is (viscosity) and how quickly you are moving your knife (velocity gradient). Similarly, the wall shear stress gauges how well a fluid can slide over a surface.

If you look it as the plane okay it is a 2 dimensional plane flow that is what so it can have the stream flow okay. Stream functions we can get it because it is a 2 dimensional flow okay as the x and y flow is steady and incompressible.

In this segment, the focus is on stream functions which are crucial in analyzing two-dimensional steady and incompressible flows. A stream function allows us to visualize how fluid particles move within a flow field, and it's particularly useful because it simplifies the equations governing the flow. Vorticity, a measure of the local rotation of the fluid, is also introduced in this context as it relates to the flow characteristics.

Think of stream functions as the paths that leaves follow on a river. They illustrate how liquid flows along specific paths in a given direction without changing flow significantly. If we imagine swirling leaves in a creek, the leaves represent particles of fluid, and vorticity describes how the water is spinning around them.

So we are getting the stream functions we are getting the stream function as we are considering the centralized stream functions is equal to 0 we can setting that value then we can be then we can get it what will be the stream functions value at both the fixed walls.

This chunk discusses the calculation of velocity potential functions, which are particularly relevant for irrotational flows. By setting the potential function to zero at a central point, we often find it easier to compute how velocities are distributed at boundaries (like fixed walls). Understanding where these functions equal zero helps us establish a clearer picture of flow behavior around obstacles.

Consider a water slide where kids are seated—their height (analogous to the velocity potential) determines how fast they go. If we mark a reference point at the top of the slide as zero (like setting the potential function to zero), we can easily calculate their speed at the bottom based on how high they started.

Then you have to compute the average velocity it is a very easy that for a if I considering this discharge which is u dA velocity in area integrations from minus h to plus h divide by this area I will get it the average velocity.

This section talks about how to compute average velocity across a fluid flow using integrals. By summing up all the velocities at various points over a defined area (between -h and +h) and dividing by that area, we arrive at an average value which gives us a good indication of the flow characteristics. This technique is common in fluid mechanics to simplify complex fluid fields into a manageable understanding of the average behavior.

Think of measuring the average speed of a car going through a tunnel. If you record the speed at several points along the tunnel, you can sum those speeds, and then divide by the number of measurements you took to determine the average speed of the car during that tunnel. In fluid mechanics, this method helps us gauge the flow uniformly.

But in case of the differential approach, we need to do the integrations and differentiations to get the pressure field, to get wall shear stress, to get stream functions, velocity potential function with a lot of series of approximations.

In this chunk, boundary layer approximations are discussed, emphasizing their importance within the framework of fluid mechanics. When solving complex flow problems using differential equations—like the Navier-Stokes equations—many approximations must be made, particularly when dealing with the boundary layer where the effects of viscosity are pronounced. These approximations simplify calculations, making complex equations more manageable.

Imagine trying to predict how a car will handle at high speeds; knowing the rough road might influence your predictions (like approximations in fluid dynamics). Just as a driver accounts for various factors while estimating performance, fluid engineers use approximations to navigate through the complexities of fluid behavior in boundary layers.

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