7.2. Deriving Cauchy Equations

If you look at from Cauchy equations what we get it just in vector forms. 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. This is what we have derived as a Cauchy equations which is a linear momentum equations for infinitely small control volumes.

Cauchy equations emerge from fluid mechanics and describe the momentum balance in fluid systems. The equation can be broken down into components that relate changes in fluid velocity (rho by rho t rho v) to forces acting on the fluid. The term 'del dot product of rho v' represents the change in momentum due to fluid motion, while 'rho g' represents the gravitational force acting on the mass of the fluid. Finally, the divergence of the stress tensor encompasses the internal forces acting within the fluid, illustrating how pressure and viscous components interact.

Imagine holding a heavy bucket of water. As you lift the bucket, the water inside moves and wants to push against the sides of the bucket due to its weight (analogous to the weight force rho g). If you suddenly stop lifting, the water also pushes back against the bottom of the bucket as it decelerates (related to changes in momentum). The complexities of these forces in motion and at rest are encapsulated in Cauchy’s equations.

If you go back very basic things, the the vectors divergence okay. If you look at that del and V the velocity divergence it is talking about the net volume outflow from a control volume okay.

Velocity divergence quantifies the rate at which fluid is expanding or contracting in a given volume. It is calculated through 'del dot V', where 'V' is the velocity vector. When the divergence is positive, it indicates that more fluid is flowing out of the control volume than flowing in, leading to an expansion of fluid (net outflow). Conversely, a negative divergence suggests that fluid is being 'sucked' into the volume, resulting in a contraction.

Think of a deflating balloon. As the air leaves the balloon, the volume decreases, leading to negative divergence as the air leaves. Conversely, if you were to blow up the balloon, it would expand and have a positive divergence, since more air is entering than leaving.

The same way if I just put it the del into rho into v what is this is indicating is the mass outflow for a unit control volume.

When you multiply the density (rho) by the velocity (V) and apply the divergence operator, you effectively assess mass flow in and out of a control volume. This concept captures how changes in fluid density and velocity contribute to the mass balance within the control volume.

Consider water flowing through a pipe. If you know the speed of the water (velocity) and how dense it is (mass), you can determine how much water is flowing out of or into sections of the pipe. If you observe a section of the pipe that is widening, you might find that as water moves through at the same speed, the amount flowing out might change based on the new area it passes through.

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.

Fluid dynamics entails both body forces (like gravity acting on a fluid mass) and surface forces (resulting from pressure and shear stress at the boundaries of a control volume). Each of these force types impacts fluid flow. Body forces act through the volume of the fluid, whereas surface forces act on the fluid's surface resulting in changes in velocity and direction.

Consider a boat on a lake. Gravity is the body force pulling it down, while the push of water against the base of the boat is the surface force that helps keep it afloat. As the wind blows (an external force), it can alter the boat's movement by affecting its surface in combination with the body forces.

If you look at that what we are deriving this that we are not bringing a new physics to that. The same concept we are talking about that we know from Newton s second laws the force is equal to mass into acceleration.

Cauchy’s equations are fundamentally linked to Newton’s second law of motion, which posits that the net force acting on an object (or control volume) equals its mass multiplied by its acceleration. This classical law translates into fluid dynamics as you consider how forces (both internal and external) affect fluid behavior and momentum changes.

Think about a car accelerating from a stop. The acceleration depends on how much fuel (mass) you have in the tank and how much force the engine generates. Similarly, in fluid dynamics, just like the car can speed up or slow down due to these forces, fluids behave based on their net forces.

Now let me go for the next part which is Navier-Stokes equations okay. What the Navier-Stokes equations are did it and what is the problems was there from Cauchy's equations if the problems or the mathematical challenges.

The Navier-Stokes equations extend the principles described by Cauchy equations, incorporating the effects of viscosity into flow dynamics. They aim to provide a more comprehensive mathematical model for fluid motion under varying conditions. Specifically, they address limitations of Cauchy equations by offering a framework that accounts for complex fluid behavior, allowing scientists and engineers to solve real-world problems in fluid mechanics.

Imagine trying to calculate how honey flows compared to water. While both fluids follow similar gravitational principles, honey’s thickness (viscosity) needs to be accounted for in calculations. The Navier-Stokes equations help provide those calculations that Cauchy's equations, which assume simpler behavior, can't cover.

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