What is the physical meaning of divergence, curl and gradient of a vector field?

Consider a ball in your hand. Now take any point on the ball and imagine a vector acting perpendicular to the ball on that point. That is your gradient in 3D. Now imagine vectors acting on all points of the ball. It would look something like this:

The red arrows perpendicular to the surface of the ball are the gradients (in 3D) of various points on the ball. If we apply gradient function to a 2D structure, the gradients will be tangential to the surface.

For a better understanding of gradient representation in 2D, consider that you are climbing a mountain. You are at a certain point on the mountain:

Now, you have to move in a certain direction to gain or lose altitude (shown in red arrows). These vector lines represent the gradients at your point of location in 2D:

Like is said before, if you are going to represent these gradients in 3D form, they will be perpendicular to the surface of the mountain.  Take a look at the images below:

In 3D form, Gradients are surface normal to particular points.

In 2D format, Gradients  tangents representing the direction of steepest descent or ascent.

Divergence:

Consider water flowing through a large pipe. Now, it has smaller pipes joined to it. Hence, as the water flows, more water is added along the way by the smaller pipes. Hence, the mass flow rate increases as the water flows.

In another case, consider that there is a leakage in the pipe. Hence the mass flow rate decreases as it flows. This change in the flow rate through the pipe, whether it increases or decreases, is called as divergence. Divergence denotes only the magnitude of change and so, it is a scalar quantity. It does not have a direction.

When the initial flow rate is less than the final flow rate, divergence is positive (divergence > 0). If the two quantities are same, divergence is zero. If the initial flow rate is greater than the final flow rate divergence is negative (divergence <0).

Curl:

Imagine pouring water in a cup. The water won’t just low linearly but rather, as it reaches the end of the cup, it will flow in a rotational motion before settling in the cup. Or consider water draining down the sink, it will swirl in a rotational motion before going out. If we plot this rotational flow of water as vectors and measure it, it will denote the Curl.

Curl is a measure of how much a vector field circulates or rotates about a given point. when the flow is counter-clockwise, curl is considered to be positive and when it is clock-wise, curl is negative. Sometimes, curl isn’t necessarily flowed around a single time. It can also be any rotational or curled vector.

Learning about gradient, divergence and curl are important, especially in CFD. They help us calculate the flow of liquids and correct the disadvantages. For example, curl can help us predict the voracity, which is one of the causes of increased drag. By using curl, we can calculate how intense it is and reduce it effectively. Calculating divergence helps us understand the flow rate and correct it to suit our needs.

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