19.6.1. Summary of Key Points

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.

This chunk introduces the concept of surface forces in relation to stress tensors. Surface forces are forces acting on a surface, and we can define these forces mathematically using a 'stress tensor.' A stress tensor is a mathematical construct that represents internal forces distributed over an area. In the 3-dimensional space (Cartesian coordinates), a stress tensor can be described with nine components, corresponding to the forces acting in three dimensions: x, y, and z. Understanding stress tensors is fundamental in both solid and fluid mechanics, as both fields rely on similar principles to describe these forces.

Think of a sponge filled with water. When you push on the sponge from different sides, you're applying forces at the surface. These forces can be analyzed using the concept of stress tensors. Just as the sponge responds in different ways depending on where it's pressed, the stress tensor helps us predict how materials (both solid and fluid) will respond to different forces.

So, basically these are the viscous terms. So, over the surface we can define it, which is the shear stress components or the viscous stress component. So, these nine components of stress in Cartesian coordinate is defined in this surface. So, we can solve the problems considering the surface force defined as stress tensors and defining as normal stress and the shear stress component.

In this chunk, the text differentiates between normal and shear stresses. Normal stresses act perpendicular to the surface (like compression due to pressure), while shear stresses act parallel (like friction). The normal stress component is a combination of pressure forces and viscosity, implying how the fluid behaves under pressure. By breaking down the stress into these two categories, we make it easier to analyze fluid behavior and fluid-structure interactions in engineering problems.

Imagine trying to rub your hands together in water. The force you feel against your hands is due to shear stress as you move them. However, the pressure of the water that pushes against your hands is similar to normal stress. By understanding these forces separately, engineers can design better swimming suits, boats, and underwater devices.

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, okay?

This chunk explains how to calculate total surface force by using the stress tensor alongside normal vectors. The normalization makes it simpler to compute the interaction between stress and the direction of force acting on the surface (through a scalar product). To get the total forces acting on a control surface, we use surface integrals. This means summing up the contributions from all points on a surface to find out how much force is applied across that whole surface.

Consider holding a large flat sponge against a strong wind. The total force that pushes on the sponge involves looking at many tiny points across the surface and how the wind pushes at each point. Here, you’d sum up all those tiny pushes to get the total force pushing on the sponge, similar to how integrals sum contributions to find total forces in mechanics.

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 . That means what I am defining is g = g I + g j + g k.

This chunk distinguishes between surface forces (acting on a surface due to external conditions) and body forces (forces that act throughout the volume of the material). The body force is calculated through volume integrals that consider the density of the fluid (C1) and the gravitational force (B3). It shows how gravity affects every part of the fluid regardless of position, by integrating these effects over the entire volume.

Think of the weight of water in a bathtub. The gravitational force affects every drop of water, which illustrates the concept of body force. Just like how every droplet is affected by gravity, a fluid in a tank feels gravitational body forces, affecting how that fluid flows in different situations.

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

In this chunk, the emphasis is on understanding that the total force acting on a fluid system is the combined result of both body forces (like gravity) and surface forces (like pressure and shear). When engineers assess a fluid system, they must account for both types of forces to fully understand how the fluid behavior will change under different conditions.

Imagine a swimming pool with a floating object. The object feels the downward pull of gravity (body force) and the upward push of buoyancy (surface force). Both these forces together determine if the object sinks, floats, or moves. Similarly, when analyzing fluids, both forces must be considered for accurate predictions.

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