11.3. Fluid at Rest

Let us come back to the topics what we have been discussing that fluid at the rest okay. As I told it when the fluid is at rest, we have two force components. One is the force due to pressure and the second is the gravity force. There are no velocity components, and there is no shear stress component.

In fluid mechanics, a fluid at rest experiences two primary forces: the pressure force and the gravitational force. When we say that a fluid is at rest, it means that it is not moving, so we do not consider any effects of velocity or shear stress. This simplifies the analysis of forces acting on the fluid. The pressure force acts uniformly in all directions and is dependent on the depth of the fluid, while gravity is constant. The balance of these forces determines the behavior of the fluid in static conditions.

Think of a still pond of water. In this case, the water does not move. The water exerts pressure downwards due to its weight, and gravity pulls it downwards. There are no currents, waves, or disturbances, much like how a book rests flat on a table without sliding off.

Now if you look it if you have to determine what could be the forces on a curved surface? If you take any point, as you know it the pressure is a scalar quantity, and will have the pressure component normal to the point where you were considering the surface.

When analyzing forces on curved surfaces submerged in fluid, it is essential to understand how pressure varies at different points due to the curvature. Pressure acts perpendicular to any surface, and because the surface is curved, the pressure direction changes at different points. To compute the total force acting on the curved surface, we can resolve it into components along three axes (x, y, z) and integrate over the entire surface. Simplifying the analysis often requires breaking the curved surface down into vertical and horizontal components for easier calculations.

Imagine a balloon that is partly submerged in water. The water exerts pressure on the balloon's surface. Since the balloon is curved, the pressure at the bottom of the balloon is higher than at the top. To find out how much force the water is pushing on the balloon, we consider those differences in pressure as we calculate the total force acting on the balloon's surface.

Now let us come to very interesting topics what is we use in buoyancy as you know it the famous scientist Archimedes what invented this buoyancy concept when he was in a bathtub experienced the lighter weight because of the buoyancy forces.

Buoyancy refers to the upward force that a fluid exerts on an object submerged in it, which was famously described by Archimedes. According to his principle, the buoyant force is equal to the weight of the fluid displaced by the object. Thus, when an object is placed in a fluid, it experiences a force upward that makes it feel lighter. If the weight of the object is less than the buoyant force, it will float. If it is more, it will sink. The key to buoyancy is understanding that this force acts through the center of buoyancy, which is located at the centroid of the displaced fluid volume.

Consider a rubber duck floating in a bathtub. The rubber duck is displacing water equal to its weight. If the weight of the duck is less than the weight of the water it displaces, it floats. If you were to add weights to the duck until it becomes heavier than the water it displaces, it would begin to sink. This balance of forces is fundamental to understanding buoyancy.

Now if you look at another point, we also look at the center of buoyancy. That means the line of actions of the buoyant force like we try to compute for the other force component.

The center of buoyancy is the theoretical point where the buoyant force acts, and it is located at the centroid of the displaced volume of fluid. This point is crucial in determining the stability of floating or submerged bodies. In practical scenarios, the center of buoyancy shifts depending on the orientation and shape of the underwater object, affecting whether it maintains balance or tips over. If the center of buoyancy is not aligned with the center of gravity, the object may rotate or capsize.

Think of a swimmer trying to float. When a swimmer is horizontal in the water, their center of gravity is aligned well with their center of buoyancy, allowing them to stay balanced. However, if they tilt or move in a way that shifts either one, they may start to tip over unless they adjust their body to restore balance.

Let us understand the principles of the stability of floating objects. There will be gravity force which will act at the CG of the floating object, and there will be the force which is the buoyant force acting as the CG of the displaced liquid...

Floating objects can be in three states of equilibrium: natural, stable, or unstable. Natural equilibrium occurs when the center of gravity and center of buoyancy are both in the same vertical line, allowing the object to remain in a balanced position regardless of slight disturbances. Stable equilibrium is when any small disturbance will cause the object to return to its original positioning, due to the buoyant force producing a restoring torque. Conversely, unstable equilibrium is when any small disturbance leads to a tipping point, causing the object to capsize. These concepts are critical when designing ships or other floating vessels to ensure their safety and stability.

Picture a small boat on a lake. If it is perfectly upright and a gust of wind hits it but the boat stays upright and corrects itself, it is in stable equilibrium. However, if the boat tips slightly and continues to tip further until it capsizes, it is in unstable equilibrium. Understanding these states can be akin to balancing a pencil on your finger; if it's perfectly vertical, it will stay upright. But even a slight tilt will send it crashing down.

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