12.2.3. Rigid Body Rotations and Centrifugal Forces

If I have a half filled liquid container. It has a free surface and this is the container. If I accelerate it with acceleration a. Okay, I have a tank and I am just accelerating with acceleration a. [...] the liquid will have a different free surface and after that there will be no change of the velocity gradients and no shear strain formation.

When a liquid container is accelerated, the liquid inside undergoes a behavior change. Initially, if the container is moving, the liquid sloshes around. However, once the acceleration is steady, the liquid settles into a new free surface, making it behave as if it were a rigid body. This means that the liquid is no longer changing shape internally due to shear forces, as no velocity gradients are present. Thus, it behaves uniformly.

Imagine riding a car on a straight road. When you accelerate suddenly, you feel your body push back against the seat, just like how the liquid sloshes. But if the car moves at a constant speed, your body feels stable, similar to how the liquid settles at a new surface.

So the problem is now it is quite simplified and it becomes a just as if a rigid body motions. That means, the liquid is there but we can consider because there is no shear stress, only this have. That means what are the force components are there? The force components are one is force due to the pressure, gravity force and force due to this acceleration component.

In the context of the fluid in an accelerating container, there are three main forces acting: gravitational force, pressure force, and the force due to acceleration. Since there is no shear stress involved in uniform acceleration, the fluid can be treated as a static body with respect to these forces. The pressure gradient is directly influenced by both gravitational pull and the acceleration of the fluid.

Think of a game where you fill a balloon with water and then run. When you stop running, the water in the balloon settles and takes the shape of the balloon. The forces acting on the water (gravity pulling down and your running causing pressure) are similar to the forces on the fluid.

Now if you look at this interesting equations, which is the simplified equations what we got it for this the liquid containers that you can see that the change of the pressure gradient and (g – a), are the pressure gradient acts in the direction of (g – a).

The pressure within an accelerating fluid behaves such that the pressure gradient (change in pressure with height) is influenced by the difference between the gravitational force and the accelerating force. This understanding allows for predicting how the fluid pressure changes as the container accelerates, with the direction of the pressure gradient being aligned against the resultant force (gravity minus acceleration).

Consider mixing drinks in a tilted cup; the liquid will rise on one side and lower on the other due to gravity and the cup's tilt. Just like that, in an accelerating container, the pressure distribution changes due to forces acting in different directions.

Now we are talking about the rotations. If you look it any chemical industry, the dairy industry many times we do the mixing of the two liquids. What we do it we actually do the uniform rotations of the liquid contents okay. [...] it is again as if like a rigid body motions.

When liquid containers rotate uniformly, the contents experience a centrifugal force that changes the distribution of the liquid's surface. Over time, this motion leads to a steady shape that mimics rigid body behavior, meaning there are no internal velocity gradients causing movement within the fluid, similar to how a solid body behaves.

Think about the Earth spinning. Water in a large rotating drum experiences centrifugal force moving it outward, leading to a stable surface similar to how the ocean's water levels are influenced by gravitational pull and the rotation of the Earth.

If you look it that, if you have a constant translations accelerations or uniform rotations, the gradient of P is equal to rho times of the vectorical difference between acceleration due to gravity and the centrifugal acceleration component.

In a rotating frame, the pressure gradient relates to the density of the fluid multiplied by the net effect of gravitational acceleration and the centrifugal acceleration. This relationship helps in understanding how pressure differences create potential flows within the fluid body.

Consider giving a spin to a merry-go-round. The children on the edge feel pushed outward due to centrifugal forces, just as fluids experience pressure variations in rotating systems, influenced by gravitational forces at play.

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