6.1.2. Velocity Field

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 satisfy conservation equations, mass conservation equations, linear momentum equations.

In fluid mechanics, we can derive analytical solutions for velocity (u, v, w) and pressure in a fluid flow. These solutions are valuable because they help us understand how velocity and pressure are distributed in a flow system. The solutions must also satisfy the conservation laws, which include mass conservation and linear momentum equations. Essentially, these laws ensure that mass and momentum are conserved in a flowing fluid.

Think of a flowing river. Just as water moves from a higher elevation to a lower one, fluid dynamics must obey certain rules—like maintaining a consistent volume (mass conservation) and moving steadily without sudden jerks (momentum conservation).

We can simplify it and we can like it a two-dimensional incompressible steady flow, this is what the total simplification of problems or these assumptions are hold good for these type of problems.

To analyze fluid flow more easily, we often simplify the problem by assuming two-dimensional flow (meaning the fluid moves in only two directions, like left/right and up/down) that is incompressible (the fluid's density does not change). These assumptions make it easier to apply mathematical equations to find solutions for velocity and pressure.

Imagine a flat, calm pond where you throw a stone. The ripples created can be seen as two-dimensional waves, and if we consider the water's density to remain constant, it simplifies understanding how the waves travel outward.

How to get this u and v equations and the pressure that what we will discuss later, but at present you know that we can get it the functional relationship of u, v with respect to a Cartesian coordinate of x, y, z and these problems becomes a steady problem.

We can express the velocity components (u, v, w) as functions of spatial coordinates (x, y, z) and time (t). In a steady flow, the fluid properties at any fixed point do not change over time, allowing for clearer determination of how velocity varies with respect to position in space.

If you imagine pushing a toy car along a table, the car's speed and direction change based on where you push it. Measuring where the car is (x, y coordinates) at any moment gives you a clear relationship between its position and speed—just like we relate velocity components to coordinates in fluid dynamics.

When we are talking about we are talking this velocity as a vector quantity, which vary in a 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.

The velocity field in fluid mechanics is described as a vector quantity, meaning it has both magnitude (speed) and direction, which varies in space (x, y, z) and over time (t). Typically, we break this vector down into its scalar components. In Cartesian coordinates, these components are represented as u (x-direction), v (y-direction), and w (z-direction).

Think of driving a car. Your speed (magnitude) and steering direction (angle) represent your velocity. When determining your position on a grid (like a map), you can break this down into components: moving forward (u), turning left (v), and changing altitude if you're flying (w).

V(x, y, z, t) = u(x, y, z, t) + v(x, y, z, t) + w(x, y, z, t)

The velocity field can be mathematically expressed by combining the individual velocity components at any point in the three-dimensional space and at a given moment in time. This equation helps us visualize how the total velocity at any point is affected by its components u, v, and w.

Consider a flowing river. The total speed of the water at a specific location can be seen as the combination of different currents pushing the water in various directions. Each current adds up to give you the water's overall behavior at that point.

Now as already I discussed that we talk about the pressures which is very dynamic variables, the pressure distribution play the major roles because as you know it the flow is come from high energy to the low energy.

Pressure in fluid flow is a crucial factor because it indicates where and how the fluid moves. Typically, fluids flow from areas of higher pressure (or high energy) to areas of lower pressure. Understanding pressure distribution helps predict the fluid's behavior in various conditions.

Consider a balloon. If you press on one side, the air inside moves to an area where there's less pressure (the open space) because the push from the higher pressure causes it to flow. Similarly, understanding pressure in fluids helps engineers design systems that manage fluid flow effectively.

If I know the velocity field, we can compute it what will be the acceleration, the rate of change of the velocity gradient, velocity.

Once we have established the velocity field, we can determine the acceleration by calculating how the velocity changes over time and across different points in the fluid. This change in velocity is essential for understanding the motion of the fluid.

Think of a car speeding up at a stoplight. The faster it accelerates (the change in speed), the more quickly it covers distance. Similarly, in fluids, examining changes in speed allows us to understand how quickly and effectively the fluid is moving.

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