19.3. Applying Linear Momentum Equations

Now, if you come back to the surface forces, like for example, for tetrahedral structures like this where you have dx, dy, and dz components are there, and your axis is like this, x direction, y direction, and z direction. So, you can define the surface forces as a stress tensor. Stress means what, force per unit area. So, you can define as a stress tensor. So, this stress will have nine components. You could have this knowledge in solid mechanics. I am just repeating it. There is not much difference between solid mechanics and fluid mechanics when you consider at stress level.

Stress tensors are mathematical objects that help us describe how forces are distributed over a surface area. In three-dimensional space, we have three axes: x, y, and z. Each axis can have components acting on it, leading to a total of nine components of stress in the tensor. The diagonal components of the tensor represent normal stresses (forces acting perpendicular to the surface), while the off-diagonal components represent shear stresses (forces acting parallel to the surface). Understanding this concept is essential because it provides a foundation for analyzing surface forces on objects in both solid and fluid mechanics.

Think of pushing down on a soft sponge that has been cut into a cube. The pressure from your hand exerts a stress on the sponge's surface. If you apply a force on one face, the sponge compresses and the stress can be described with a tensor, showing how the force is distributed. If the same sponge was in water, similar principles apply, but now we consider fluid pressures acting on all faces of the sponge. This understanding helps engineers design structures that can withstand different types of forces.

So, we have stress tensors in order to describe all surface force components. That is what will have nine components which will have, as you know the subscript describes that. The stress in the z direction acting on the face whose normal is eight directions. This is similar notation to what we use in solid mechanics. So, you have the stress tensor coordinate systems defining nine stress components...

The nine components of the stress tensor are crucial for understanding how different forces act on a given surface. By recognizing which forces are normal or shear, engineers can calculate the resultant forces using integrals over the surface. This involves resolving these stress components depending on their direction and magnitude, which is necessary for analyzing forces on structures or fluid interfaces.

Imagine a tomato sitting on a cutting board. The pressure you apply from above creates normal stress directed downward through the tomato, and if you were to slice it, the knife would create shear stresses as the blade moves through the flesh. By understanding how these stresses work on a surface, chefs and engineers can better design cutting tools or food processing equipment.

Total surface force acting on a control surface. 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 g, g, g...

In fluid mechanics, forces acting on a control volume are categorized into body forces (forces that act throughout the volume, like gravity) and surface forces (forces that act on the boundaries of the volume, like pressure or viscous forces). The body force due to gravity is represented by the integral of the weight across the control volume, while the surface forces can often be quantified using stress tensors. Understanding these forces allows for better predictions of how fluids will behave under different conditions.

Consider a swimming pool filled with water. The weight of the water provides a body force acting downwards due to gravity. Each side of the pool also bears pressure forces acting due to the water pressing against the walls. Engineers consider both these types of forces when designing structures to ensure they can withstand the weight of the water and the forces exerted by it.

Now, if I take a control volume, everywhere I will have atmospheric pressure, then the absolute pressure from some locations. So, if I consider the atmospheric pressure is acting throughout this control surface and do surface integral, if the pressure is having the direction, and if I do a surface integrals over this, it will be cancelled out and becomes 0...

Atmospheric pressure acts uniformly in all directions over a control volume. When performing calculations for fluid dynamics, it is common to neglect this pressure during surface integrals because its effects effectively cancel out when integrated over a closed surface. Instead, engineers focus on gauge pressure, which is the pressure relative to atmospheric pressure. This simplification allows for more manageable calculations without losing accuracy in the results.

When you're inflating a balloon, the air pressure inside is higher than the atmospheric pressure outside. However, if you're measuring the change in pressure while inflating, all you care about is how much pressure is over and above the atmospheric level (gauge pressure). Just like that, we often ignore the atmospheric pressure when calculating forces in fluids since it exerts the same force in all directions.

Now, another point is how to choose control volume, because that is what the art is, like you do free body diagrams in solving solid mechanics problems. Similar way, drawing the appropriate control volume is an art...

Choosing an appropriate control volume is key to simplifying fluid dynamics problems. A well-defined control volume will allow you to clearly identify inflow and outflow conditions, making it easier to apply conservation laws. The location and shape of the control volume selected will significantly determine how straightforward the problem is to solve. Engineers often practice this skill to become proficient in applying fluid dynamics principles effectively.

Imagine trying to find out how much water is flowing out of a garden hose. If you select a control volume that only includes the end of the hose, it's easy to measure and understand the flow rate. However, if your control volume extends back into the tap, the water pressure and movement changes complicate your readings and calculations. Selecting a proper control volume simplifies such problems.

Now, coming back to applying the Reynolds transport theorem we have to write the linear momentum equations. At this systems level force is equal to mass into acceleration...

The Reynolds transport theorem relates the change in momentum of a system to the flow of mass and momentum through its boundaries. It forms the basis for deriving momentum equations used in fluid dynamics, allowing us to analyze complex systems. By incorporating both body and surface forces into a mathematical framework, engineers can model and predict fluid behavior under various conditions accurately.

Think of a moving car that changes speed or direction. The forces acting on it, like engine power and wind resistance, can be analyzed using similar principles as Reynolds transport theorem in fluid mechanics. This analogy helps illustrate how changes in flowing fluids relate to forces acting on objects. Just as we can derive equations of motion for cars, we can derive equations for fluids using this theorem.

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