7.2.1. Linear Momentum Equations

In vector form if you remember it which is the linear momentum equations deriving with the components like rho by rho t rho v is velocity plus is equal to del dot product of rho v is equal to rho g is a vectors plus we have the del product or divergence of the stress tensor.

The Cauchy equations represent the fundamental principle of linear momentum in a fluid. Here, we consider the rate of change of momentum within a fluid element over time. This equation shows that the change in momentum (left side) is due to forces acting on it, such as gravity (rho * g), and the stresses within the fluid (represented by the stress tensor). The equation essentially balances the inertial forces against the stress forces acting on the volume.

Imagine a soccer ball (the fluid element) being kicked (force applied). The change in its motion (velocity) over time is governed by the forces (like the kick and gravity) acting on it, similar to how the Cauchy equations balance forces in a fluid.

If you look at that del and V the velocity divergence it is talking about the net volume outflow from a control volume.

Velocity divergence quantifies how much fluid flows out of a given control volume relative to the volume of the fluid. If velocity divergence is positive, it means more fluid is exiting the control volume than entering. Conversely, if it’s negative, there’s more fluid entering than leaving. This concept is critical for understanding fluid dynamics, particularly in analyzing flow rates and behavior in confined spaces or through boundaries.

Think of a balloon being inflated. As air fills the balloon, there’s a net outflow (divergence) at the opening where air enters, leading to an increase in volume. If you pressed on the balloon’s side, creating a net inflow at the opening, this would illustrate the concept of negative divergence.

These two can indicating for us two force component one is body force another is the surface force that is what body force and the surface force component.

In fluid mechanics, forces acting on fluid elements can be categorized into two main types: body forces and surface forces. Body forces (like gravity) act throughout the volume of the fluid, while surface forces (like pressure and shear stress) act on the surfaces of the fluid elements. Understanding the balance between these two types of forces is crucial for analyzing motion and flow within a fluid.

Consider a river (the fluid) flowing downhill (affected by gravity, a body force). The water's surface (where it meets the bank) experiences friction, which is a surface force. Both forces affect the river’s flow, similar to how gravity and friction affect a sled sliding down a snowy hill.

If you look at that even if you do a vector notations we are deriving the very basic Newton's second law is that force is equal to mass into acceleration.

The relationship we derive in fluid dynamics mirrors Newton’s second law, which states force equals mass times acceleration. In the context of fluid mechanics, we apply this principle to control volumes. The equations are set up such that the forces acting on a fluid (both body and surface forces) are balanced against the fluid’s mass and acceleration over time. This foundational principle helps us understand how fluids move and respond to various forces.

Think of driving a car. When you press the accelerator (force), the car (mass) gains speed (acceleration). Similarly, in fluids, applying a force results in acceleration, whether it’s pushing water through a hose or air flowing through a vent.

What I can suggest you please look the dimensions okay. So the basic equations what I have rho dv by dt is equal to rho g del dot sigma j okay.

In deriving Cauchy’s equation, we emphasize the importance of dimensions and units. Ensuring that both sides of an equation have the same dimensions validates our calculations. The left side (density times change in velocity) represents the inertial force, while the right side includes forces due to pressure and viscous stress. By ensuring dimensional consistency, we can confirm the physical accuracy of our derived equations.

Consider cooking and measuring ingredients. For example, when adding flour to a recipe, ensuring the right amount (like using cups or grams) is crucial for the recipe to turn out well. Similarly, in our equations, checking dimensions ensures the 'recipe' for fluid behavior is correct.

The Navier-Stokes equations does that approximations to reduce this unknowns the 10 unknowns to the 4 unknowns.

The journey from Cauchy’s equations to the Navier-Stokes equations involves making assumptions to simplify complex relationships. By assuming the fluid is incompressible and the flow is isothermal, we can reduce the number of unknowns significantly. This process helps create a manageable set of equations that describe fluid motion more efficiently, allowing for practical applications in computational fluid dynamics.

Think of organizing a complex event, like a wedding. Initially, you might have a long list of tasks and responsibilities (unknowns). By deciding on a simpler format, like a buffet instead of plated dinners, you can streamline the planning process, similar to how simplifying fluid equations helps manage their complexity.

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