5.6.1. Numerical Examples

Now let me commit that is what I am talking about that we have a volumetric flux what is going in through this stream to conjugative streamlines that is what will be coming to these ones. flow can cross, no flow can cross a streamline. It is combined with these two streamlines. That is what I explain it.

In fluid mechanics, when we talk about streamline patterns, it's crucial to understand that fluid cannot cross these streamlines. This means when analyzing fluid flow through a channel formed by two streamlines, we must consider that the volume of fluid entering one section of the channel will equal the volume exiting another section, ensuring conservation of mass. This principle is crucial in analyzing flow rates and understanding how fluids behave in various scenarios.

Imagine a water slide at an amusement park. The water cannot jump over the slide's edges (analogous to streamlines), so all the water entering the slide at the top must flow out at the bottom. If at one point the slide narrows, the water speeds up as it has to fit through the tighter space, much like how velocity changes at varying distances between streamlines.

Let I solve three simple problems how we can really solve these. These are the problems comes in the gate or engineering service levels. Let you have this a two-dimensional the steady problems in compressible flow in xy planes which define a stream functions as these functions. So, and a, b, c are constants and that values are given to us to verify this flow field does it satisfy the continuity equations.

In this section, we will tackle numerical examples where we will verify if the stream functions given represent a valid flow field according to continuity equations. When working with fluid flow in two dimensions, it's essential to validate that the calculated stream function aligns with mass conservation principles by checking if the velocity divergence is zero.

Consider a game of tug-of-war where two teams pull against each other on a rope (streamlines). The rope remains taut and doesn’t break, similar to how a stream function must maintain flow integrity without 'leaking' into other regions. Mathematically, checking the continuity equation ensures that no matter how the teams (fluid) flow, the total force (or mass) remains balanced.

Let me commit to the examples of gate 2010 questions which is give a stream functions find out the velocity factors. okay it is a quite easy. So it is given the stream functions you have to find out the velocity factors okay which will be velocity factors. That means you need to remember it okay that is what you say get. So U scalar component is a gradient of the y directions, v is a minus of gradient of the x directions.

To determine the velocity components from the given stream functions, we compute the gradients of the stream functions. In essence, the U velocity component can be derived from the partial derivative of the stream function with respect to the y direction, while the V component is obtained as the negative of the partial derivative with respect to the x direction. This method will help ascertain how fast and in which direction the fluid moves at any given point in our flow field.

Think of a conveyor belt in a factory. The speed of objects on different sections can vary based on how the belt is adjusted (like streamlines and their gradients). To understand how fast an item moves (the fluid velocity components), we would measure how fast it travels at different points on the belt, mirroring how we analyze stream functions to derive velocity.

Another very interesting problems it is not that difficult it is a 90 degree corners okay. Let me come back to that we have the streamline of a steady ideal fluid flows okay 90 degree corners defined by the functions 12 x y okay 12 x y you can get the unit of stream functions is meter square per second which is a indicating of accelerations close to the accelerations.

This problem examines the behavior of fluid flow around sharp corners. Stream functions help visualize how fluid behaves when it encounters obstacles, like 90-degree corners. By analyzing the provided stream functions, we can derive velocity components and infer how the flow accelerates or decelerates as it navigates these corners.

Consider a car making a sharp turn at a street corner. As it rounds the corner, the force exerted by the tires on the road must change (acceleration) to maintain its path without sliding off the road. Similarly, the fluid adjusts its velocity as it navigates around the corner, which we analyze through our stream functions.

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