19.1.5. Total Force Acting on Control Volume

Now, if I have the stress component there and I have the normal vectors, if I resolve the force components, I will have the scalar product between the stress tensor and the n vectors, that is how we do it. And for the total surface area we do surface integrals to compute it.

This introduction sets the stage for understanding how to calculate the total forces acting on a defined control volume. It highlights that we must consider both the stress components, which represent how forces are distributed across the surface, and the normal vectors, which indicate the direction perpendicular to the surface area. By applying the scalar product between the stress tensor (which contains information about the forces) and the normal vectors, we can effectively determine the resultant forces on that surface. To compute the total force, surface integrals across the entire control area are performed.

Think of it like measuring how much pressure a balloon feels when you press on its surface. The stress tensor acts like a map that tells you how much force is being applied at each point on the balloon's surface, while the normal vectors indicate the direction in which you are pressing. By combining these two concepts, you can calculate the total pressure on the balloon.

The body force component will have volume integrals of rho g dV, g is the vector quantity as we consider the g, acceleration due to gravity can have a vector commodity with three scalar components of gx, gy, gz.

Here, we learn about the body force components acting on the control volume, which are forces that originate from the volume of the fluid itself, such as gravitational forces. This force is calculated using the formula involving density (rho), gravitational acceleration (g), and an infinitesimal volume element (dV). Gravitational force can be represented in three dimensions as different components (gx, gy, gz) to account for varying directions of this force in space. Thus, volume integrals help accumulate these infinitesimal forces throughout the entire fluid volume in the control system.

Imagine holding a box in a room. The weight of the box is a direct representation of gravity acting downwards. Similar to how you calculate the total weight of all objects in a box by adding them up, we calculate the total body force inside a control volume by integrating the effects of gravity acting on every tiny portion of the fluid.

The total force acting on the control volume will have the body force component and surface force component.

This chunk emphasizes the two main components contributing to the total force acting on the control volume: the body force, which is derived from volume integrals, and the surface forces, which arise from pressure and viscous stresses acting on the surfaces bounding the fluid volume. Understanding that both sets of forces must be included for a complete picture of the total forces is crucial for solving fluid mechanics problems.

Think of a swimming pool. The water inside the pool experiences two types of forces: the gravitational force pulling it down (body force) and the pressure exerted from water pushing against the sides of the pool (surface forces). To fully understand how the water will behave under different conditions, both sets of forces need to be taken into account.

Now, if I divide the force components, that means, the total force will be one component from the body force, it is gravity force component. The surface force component we can resolve it into the force due to the pressure, force due to viscosity, force due to the other reactions.

Here, we can further break down the total forces into more specific components. The body's gravity force acts downward, while the surface force can be further separated into pressure forces and viscous forces. By simplifying these components, engineers can focus on relevant forces depending on the nature of their fluid dynamics problem, such as neglecting certain forces if they are dominantly small.

Consider a car moving through water. The weight of the car (gravity) is one force acting down. The water also pushes against the car's surface (pressure), and friction from the water sliding past the car (viscous force) is another. When trying to analyze how the car moves, one might prioritize understanding the pressure forces affecting it rather than the gravitational force, depending on the situation.

When you do surface integrals of this constant, pressure distribution in a closed control volume, that is supposed to be 0 as it happens here. All these direction angles cancel out each other.

This chunk introduces the idea that in many fluid mechanics problems involving control volumes, atmospheric pressure is often canceled out when calculating the net forces. Since atmospheric pressure acts equally in all directions on a closed surface, when integrating over that surface, the contributions from opposing directions balance out to zero. Thus, engineers often focus on gauge pressure, which is the pressure above atmospheric pressure, for their calculations.

Imagine blowing up a balloon. When the balloon is in a room, it feels both the external air pressure and the internal pressure caused by the air inside it. When you're calculating the forces on the balloon, you can ignore the uniform pressure acting from the outside because it impacts all sides equally, similar to how we treat atmospheric pressure in fluid mechanics.

It is engineering skill or art to be developed by the students how to use the control volume concept. How to define the control volume for a flow system so that you can easily solve it.

The ability to select and define an appropriate control volume is key in fluid mechanics. This chunk emphasizes that selecting a suitable control volume requires practice and skill. Different configurations can make the calculations simpler and help one accurately analyze the forces at play. The positioning of control surfaces must align appropriately with fluid velocity to make calculations straightforward and effective.

Think of it like framing a photograph. The way you choose to frame the picture can emphasize its beauty or make it look cluttered. Similarly, in fluid dynamics, choosing your control volume correctly allows you to highlight the relevant forces and makes the problem-solving process much clearer and more effective.

--