6.4.1. Streamlines

So for a very simple case, we can get analytical solutions like u and v and w and the pressure and that analytical solutions can help us to know the velocity, the pressure distribution of these problems and it satisfies conservation equations, mass conservation equations, linear momentum equations. Also, it satisfies the boundary conditions at the floor also flow inject what is coming it.

In fluid dynamics, we can derive analytical solutions for various velocity components (u, v, w) and pressure. These solutions allow us to understand how fluid flows in different situations. Analytical solutions also ensure that certain laws, like the conservation of mass and momentum, are upheld. For example, in a flowing river, we can predict how fast the water will move and how the pressure will distribute along the riverbed based on these equations.

Think of a water slide; the faster you go down (velocity), the more pressure you feel against your back. Analyses of how the water flows on the slide can predict speed and pressure distribution just like in fluid dynamics.

The velocity field, when we are talking about we are talking about this velocity as a vector quantity, which varies in space in case of the Cartesian coordinate system of x, y, z and the time. But most often for easy point of view, we resolve this velocity vector component into its scalar component in Cartesian coordinate systems like the i and j and k.

A velocity field describes how the speed and direction of fluid flow change from one point to another in space. It is represented as a vector, which means it has both magnitude (speed) and direction. In a three-dimensional space (x, y, and z axes), the velocity components are typically expressed in terms of unit vectors in these directions (i, j, k) to ease calculations and visualizations.

Imagine a snow globe. As you move the globe, the snowflakes (representing particles of fluid) drift through different layers of water (representing the velocity field). Each snowflake moves at different speeds and angles, which we can represent using coordinate systems.

A streamline is a line everywhere tangent to the velocity vector at a given instant. That means time equal to zero or we take a snapshot. At each point, if I draw this the tangent, that tangent should give directions of the velocity vector.

Streamlines visually represent fluid flow. If you were to take a still image of the fluid at a moment in time, the lines you draw that are tangent to the flow's velocity vectors are called streamlines. They show how fluid moves without mixing and help predict the future positions of particles in the flow.

Think of a school of fish swimming together in a pond. If you paused the water and took a picture, the invisible lines showing the paths the fish took would look similar to streamlines. They represent how and where the fish move in relation to each other without clashing.

Pathline is the actual path traversed by a given fluid particle. A streakline is the locus of the particles that have earlier passed through a prescribed point.

Pathlines illustrate the journey of individual fluid particles over time, reflecting how they move within a flow. Streaklines, conversely, show the path that particles have taken through a specific point in space over time. They help visualize how fluid moves from one location to another, which can be particularly useful in experimental situations, such as dye injection in a flowing river.

Consider a ribbon being waved in the wind. If you let go of the ribbon at one point, the path it took while flying through the air represents a pathline. Now, if you were to apply colored spray paint at the same point, the trail left by the spray represents streaklines, showing where particles exited from that location.

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