11.3.4. Wake Formation in Fluid Mechanics

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.

This chunk introduces wall stresses and the velocity field. It establishes that the investigation includes several factors such as wall stress and shear stress. The reference to the Navier-Stokes equations suggests that previous lessons focused on calculating these velocities under controlled assumptions, such as neglecting gravity. This sets a foundation for understanding how the fluid dynamics is analyzed with respect to boundary conditions.

Consider a simple garden hose when it’s turned on. The water flowing through the hose has different speeds at different points. Close to the wall of the hose, the water moves slower due to friction (similar to wall stress in fluid mechanics). This is a practical illustration of how velocity varies across a given section of flow.

We get this u component that is what we did it last class minus dp by dx okay that is what dp by dx is equal to h square by 2 mu types of 1 minus y square by h square. This is the velocity field we got it applying this Navier-Stokes equations as you remember it.

In this chunk, the calculation of the velocity field based on the differential pressure dp/dx is presented. The relationship between pressure gradient and velocity is derived from the Navier-Stokes equations. Key parameters such as dynamic viscosity (mu), channel height (h), and position within the channel (y) are used to establish this relationship, crucial in predicting flow characteristics in a given geometry.

Imagine you're driving a car down a sloped road. The faster you go depends on the slope of the road and the friction between your tires and the asphalt. Similarly, the equations indicate how pressure gradients dictate how quickly fluid moves in a pipe, analogous to how a car's acceleration is determined by road conditions.

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 discusses how to calculate shear stress at the wall using Newton's law of viscosity. The shear stress (tau xy) is related to the velocity gradient near the wall. The focus on partial derivatives (del u/del y) highlights how fluid layers move at different speeds, leading to a shear stress at the boundary that can be quantified through this relationship.

Think about applying pressure to a thick sponge with a thick cream on it. The cream moves but slowly near the surface since it’s 'stuck' to the sponge, similar to how fluid near a wall adheres to it, causing shear stress. The sponge represents the wall, and the cream represents the fluid.

If you look it I 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.

This chunk introduces the concept of stream functions in the context of two-dimensional, incompressible flow. In fluid mechanics, stream functions are useful for visualizing flow patterns. The assumption of steady flow indicates that characteristics do not change over time, allowing for straightforward calculations of flow features and vorticity (the measure of rotation, important in understanding how fluids behave).

Imagine tracing the path of a leaf floating down a river. The path it takes represents a streamline. The concept here is similar; streamlines in a fluid illustrate how particles within it move, allowing engineers to predict flow behavior over surfaces.

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 segment explains how to compute average fluid velocity using integration over the area. By considering the velocity at various points from -h to +h, you can determine the average, providing vital information about the overall flow dynamics. This step is fundamental in analyzing fluid movements and is often pivotal in engineering applications.

When trying to understand how quickly a stream is flowing, one might average the flows at different points, like checking with a stick how deep and fast the water is at various spots. Similarly, this method averages the flow velocity across a defined area.

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