2.9. Relationship Between Potential and Stream Functions

In a 2D flow, this type of flow considers two stream lines S1 and S2. The flow rate per unit depth of an incompressible fluid across two stream lines is constant and independent of the path.

The concept of stream function is crucial in fluid mechanics. In a two-dimensional flow, we can visualize the flow as consisting of several streamlines. A stream line is a line that is tangent to the velocity vector of the flow at every point. This means that the fluid particles follow these lines without crossing them. When we say that the flow rate per unit depth across two stream lines is constant, it means that no matter which path (S1 or S2) from point A to point B you take, the amount of fluid flowing through these lines remains the same, which is a fundamental feature of incompressible fluids like water.

Imagine a busy highway with two lanes (the streamlines). Cars (the fluid particles) move from one city (point A) to another (point B). It doesn't matter if a car takes the left lane (S1) or the right lane (S2); as long as both lanes have the same traffic flow rate, the number of cars arriving at each point is the same.

Based on that, a stream function psi is defined that it is constant along the stream line and the difference of these stream line for the 2 streamline is equal to the flow rate between them.

The stream function, denoted by psi (ψ), is a mathematical construct that simplifies the analysis of fluid flow. It is defined such that its value remains constant along a streamline. This means if you pick any streamline and measure the value of the stream function along that streamline, it won't change. Additionally, the difference in the values of the stream function between two different streamlines is directly proportional to the flow rate between these streamlines. This gives us a relationship between the geometry of the flow and its dynamics.

Think of the stream function like the elevation on a topographic map. Just like each point can have a constant elevation along a contour line (the elevation remains the same), the stream function remains constant along a streamline. The difference in elevation between two contour lines (different streamlines) tells you how steep the terrain is, similar to how the difference in stream function tells us about the flow rate.

So, if there is a stream function psi, u is given by del psi / del y and v is given as del psi / del x.

In fluid dynamics, the velocity components of a flow can be expressed in terms of the stream function. Specifically, in a two-dimensional flow, the velocity in the x-direction (u) is equal to the partial derivative of the stream function with respect to y, and the velocity in the y-direction (v) is equal to the negative partial derivative of the stream function with respect to x. This relationship interlinks the flow velocity with the geometry defined by the stream function and provides a way to derive flow velocities from known stream functions.

Imagine you are on a water ride at an amusement park where the water flow creates waves. The waves represent the streamlines, and the speed of the water flow can change based on how steep the waves are. Here, the stream function is like the pattern of the waves, and the speed of the water moving left or right is related to how high or low the waves are, just as the velocity components relate to changes in the stream function.

For an irrotational flow del v / del x - del u / del y is equal to 0, and if we substitute v as - del psi / del x and u as del psi / del y, we are going to get the Laplace equation.

In fluid dynamics, irrotational flow refers to fluid motion where the flow field does not contain any rotation about an axis. Mathematically, this condition is expressed by stating that the difference between the partial derivatives of the velocity components is zero: del v / del x - del u / del y = 0. When we incorporate the relationships of the velocities in terms of the stream function, we derive the Laplace equation, which helps us understand the potential flow characteristics of an incompressible fluid. This forms the foundation of potential flow theory.

Consider a perfectly smooth water surface on a calm lake – it forms an irrotational flow condition. If you were to observe the movement of the water, the surface remains flat, and there are no whirlpools or eddies. The calm surface represents a scenario where the velocity components satisfy the Laplace equation, illustrating a flow pattern without rotational disturbance.

Lines of constant phi velocity potential are called equipotential lines.

Equipotential lines represent locations within the flow field where the potential function has the same value. Just like streamlines represent paths that fluid particles follow, equipotential lines represent the spatial distribution of the potential function. In the context of fluid flow, equipotential lines are orthogonal to streamlines, which means they intersect at right angles, indicating where the energy in the flow field remains constant. Understanding equipotential lines is crucial in analyzing fluid flow because they help in visualizing how energy is distributed across the flow.

Think of equipotential lines like the contour lines on a map that indicate height or elevation above sea level. If you walk along a contour line, your elevation remains the same (like potential remains constant). If you visualize a mountain landscape (the flow field) with various peaks and valleys, knowing the elevation contours (equipotential lines) helps you understand the lay of the land without necessarily walking through every point.

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