19.2.1. 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.

Stress is essentially the amount of force applied on a surface per unit area. When considering three-dimensional objects (like tetrahedra), we have forces acting on surfaces in three dimensions (x, y, z). The stress tensor captures this by describing how these forces (surface forces) are distributed across these dimensions with nine components altogether. Each component of the tensor corresponds to how much force is exerted in a particular direction on a particular face of the object.

Imagine a balloon filled with air. If you poke at it from different sides, you're applying forces on the surface of this balloon. The stress tensor would help quantify how these forces are distributed across the interior of the balloon, impacting how it deforms based on where you poke.

If you look at that, if you take this diagonal component which is the normal component to this surface... These are compositions of the pressure force and the viscous force component. But the diagonal component what we have is ... and all, which is acting tangentially. So, basically these are the viscous terms.

Stress can be categorized into normal stress and shear stress. Normal stress is the stress component acting perpendicular to the surface, resulting from either pressure or viscous forces. Shear stress, on the other hand, arises from forces acting parallel to the surface and is linked to the fluid's viscosity. Understanding this distinction is crucial for analyzing how materials and fluids behave under external forces.

Think of a deck of cards. If you push down on the cards, you're applying normal stress. However, if you slide your hand across the top of the deck, you're applying shear stress. The cards can slide easily over each other, which represents how fluids might behave under shear stress situations.

Now, if I have the stress component there and I have the normal vectors, if I resolve the force components... to compute it, okay? Please do not be more worried about how we are having a scalar product of stress tensors and normal factors which will be coming to be again a second order vector components.

To calculate the total forces acting on an object using stress tensors, we often utilize scalar products of the stress tensor and normal vectors representing the surfaces of the object. The process involves integrating these products over the surface to determine the total impact of all forces acting on the object. This will help in determining how the object's momentum changes under these forces.

Consider a floor with different tiles. Each tile can be viewed as a surface, and the forces pushing down (weight of furniture, people, etc.) will affect how they interact with one another. Using the stress tensor to model these interactions lets you predict how much overall pressure is distributed across the entire floor.

Total force acting on the control volume will have the body force component and surface force component. The body force component will have volume integrals of rho g dV, ... but many of the times we make alignments in such a way that only this k direction or negative direction and k is in upward direction.

Forces acting on an object or fluid can be categorized as body forces and surface forces. Body forces are those that act throughout the volume of the object (like gravitational force, denoted

Think about a tree: the wind blowing against it represents surface forces, while the gravitational force pulling it downwards illustrates body forces. Both types of forces affect how the tree moves and bends in the environment.

When you apply a linear momentum equations to a control volume, like I have the control surface like this, if you look at this control surface, in these three phases we have the pressure is atmospheric pressure.....

A control volume is a defined region in space used to analyze fluid flow and momentum equations. In analyzing forces, we often consider pressures acting at the boundaries or surfaces of this volume. By defining the control volume carefully, we can simplify calculations by neglecting uniform pressures (like atmospheric pressure) that cancel out when integrated around the volume's surface.

Imagine a blender filled with fruit. The control volume is the blender itself, and as you blend, the forces (like pressure from the rotating blades) act on the fruit inside. If you only cared about the fruit moving around in the blender and not outside forces (like air pressure), you can view the fruit’s behavior more simply.

At this systems level force is equal to mass into acceleration... So, this is the basic equation which is linear momentum equation in Reynolds transport theorem point of view.

The fundamental principle of dynamics is that the total force acting on a system equals the rate of change of its momentum (mass times acceleration). In the context of fluid dynamics, this relationship allows us to derive the linear momentum equations using the Reynolds transport theorem, which helps examine how fluid properties change as they move through a control volume.

Think of a car speeding up on a highway. The force from the engine accelerates the car. If more passengers (mass) get into the car, a greater force must be applied to maintain acceleration. This principle of balancing forces and changes in motion mirrors the equations derived from fluid momentum principles.

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