2.7. Stream Function

In a 2D flow or a 2 dimensional 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.

A stream function is a mathematical tool used to describe the flow of fluids in a two-dimensional space. In this context, '2D flow' means that the motion of the fluid can be described using just two dimensions—typically the horizontal (x) and vertical (y) directions. When considering two streamlines, S1 and S2, we note that the flow rate (the volume of fluid passing through a unit area per unit time) remains constant as the fluid moves across these streamlines. Importantly, this flow rate is independent of the specific path taken between the two points on the streamlines.

Think of a river as a two-dimensional flow field. Just like a swimmer can choose various paths to move between two points on the riverbank, the flow rate across different paths between two streamlines in our example remains the same. If one swimmer takes a shortcut while another takes a longer, winding route, both will still experience the same flow rate of water over their respective paths.

Based on that a stream function psi is so 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, often denoted by the symbol ψ (psi), is defined such that its value does not change along a particular streamline. This means if you pick any point on a streamline, the value of the stream function remains constant while moving along that streamline. Moreover, the difference in the values of the stream function between two different streamlines is directly related to the flow rate. This relationship helps in analyzing fluid flows efficiently because if you know the stream function values at two points, you can deduce the flow rate between those points.

Imagine two rivers (streamlines) flowing next to each other but having different flow rates depending on their width and speed. If you think of the height of the water at each point as the value of the stream function, then the difference in water height between the two rivers (streamlines) can indicate how fast water would flow from one river to another. Thus, the stream function helps visualize and quantify the behavior of the fluid.

Thus, phi A - phi B is equal to flow rate between S 1 and S 2. The flow from left to right is taken as positive, in the sign convention. The velocity is u and v in x and y directions are given by, this is important. So, if there is a stream function psi. So, u is given by del psi / del y and v is given as del psi / del x.

In fluid mechanics, understanding the velocity of the fluid at any point can be achieved using the stream function. The changes in the stream function (ψ) allow us to compute the horizontal component of velocity (u) and the vertical component of velocity (v). Specifically, u is calculated as the partial derivative of the stream function with respect to y, and v is the partial derivative with respect to x. This formulation is incredibly useful because it not only simplifies computations but also adheres to the characteristics of incompressible flows.

To relate this to a real-world scenario, think about a water slide. As you slide down, your horizontal and vertical speeds can change depending on how steep the slide is (affecting the y direction) and where you are on the slide (affecting the x direction). The stream function helps us predict how fast you're going in both directions based on the shape and incline of the slide.

For an irrotational flow del v / del x - del u / del y is equal to 0. Therefore, if we substitute v as, what we saw v was you go back and see v was - del psi / del x. And u was del psi / del y. Now, put v as this and u as this you are going to get and this is the Laplace equation.

In the context of fluid dynamics, an irrotational flow means that the flow has no rotation, or 'vorticity'. For such flows, the change in velocity components must satisfy a specific mathematical condition called the Laplace equation. When we express the velocities in terms of the stream function (where v is derived from the change in the stream function with respect to x, and u is derived with respect to y), we find that these relationships lead us to the conclusion that the velocities must adhere to the Laplace equation, which describes how these functions behave.

Consider a flat, calm pond where a gentle rain creates ripples—there's no spinning of water, just straightforward flow patterns. In such a scenario of irrotational flow, the way the ripples propagate can be mathematically analyzed using the Laplace equation, similar to how we relate velocity changes through the stream function in fluid mechanics, underscoring the predictable nature of calm, irrotational fluid behavior.

In an irrotational flow the velocity can be written as, gradient of scalar function phi called the velocity potential that is for an irrotational flow.

In fluid dynamics, particularly in the study of irrotational flows, we introduce a concept called the potential function (denoted by φ, phi). This function serves as a scalar representation of the flow's characteristics. It allows us to express the fluid's velocity components as gradients of this scalar function. Essentially, the velocity of the fluid in the x direction is the change in the potential function with respect to x, and similarly for the y direction. This theoretical framework helps in simplifying calculations and understanding flow patterns in irrotational fluids.

Visualize standing atop a hill—if you imagine the height as the potential energy (the potential function) of a sliding marble, when you release it, it rolls down the hill not only following the path of steepest descent (the gradient) but also conserving energy. Similarly, in irrotational flows, we can use the potential function to describe the movement of fluids flowing along the 'slopes' defined by this scalar function.

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