11.1.6. Second Example

Example 1 says that what is the irrotational velocity field associated with the potential functions as given here. We have to find out the irrotational velocity field, does this flow field satisfy incompressible continuity equations?

In this example, the goal is to determine the irrotational velocity field that corresponds to certain potential functions. An irrotational flow is one in which the flow does not have any rotation, meaning there are no vortices present in the fluid. To find the irrotational velocity field, we can derive it from the velocity potential function given. This is done by taking the partial derivatives of the potential function with respect to the spatial coordinates. If the flow is irrotational, it should also fulfill the incompressible continuity equation, which is a core principle of fluid mechanics stating that the mass flow rate must remain constant from one cross-section of a pipe to another.

Imagine a calm lake where the water flows smoothly without any ripples or spins—this represents an irrotational flow. In contrast, if you were to throw a stone into the lake, you would create ripples and rotations in the water—this represents rotational flow. Understanding flow types is crucial when designing hydraulic systems to ensure smooth and efficient operations.

The first steps were to find out what is the irrotational velocity field? ... So those terms become 0.

To find the velocity components of the flow field, you take the negative partial derivative of the velocity potential function with respect to the x-coordinate to find the u-component of velocity and with respect to the y-coordinate for the v-component. The calculations yield specific values for these components. If certain terms in this process do not contain the variable of interest, they effectively become zero, simplifying the calculations.

Think of a pristine river flowing downstream—it has a defined speed (velocity) but no eddies or other spinning motions. You can visualize measuring the flow speed directly at different points to understand how the whole system operates, much like deducing the velocity components from the potential function.

Now if we look at it these are the velocity fields that is what is the part number a what we are looking for irrotational field. ... So substituting this value what will get it -6 -6 is -12. That means it is not satisfied.

In the second part of the example, we determine whether the flow field satisfies the incompressible continuity equation by checking if the sum of the partial derivatives of the velocity components equals zero. The calculations show that the sum does not equate to zero (-12 instead of 0), indicating that the flow does not satisfy the continuity equation, thus confirming some properties of the flow could have rotative components or it might not be incompressible.

Imagine if water is constantly being added to a hose but not being released at the other end; this represents a failure to meet the continuity equation since water volume would accumulate. Analogously, in our flow, if all components do not balance out, it indicates changes such as potential blockage or turbulence.

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