8.1.1. Introduction to Control Volume

Now if I go for the next ones that how to get the pressure field when fluid is at rest. That means I am just looking the what could be the functions of the P, P = P (x, y, z). If it that it is now again I am considering a very simple case, let us consider a control volume like this okay. This is what my control volume. As I said it when the fluid is at rest the shear stress become zero.

In fluid mechanics, the control volume is a defined space where we can analyze fluid behavior. When the fluid is at rest, it does not experience shear stress, meaning there are no forces trying to slide one layer of fluid over another. Instead, we deal exclusively with normal stresses, which are related to pressure. The pressure can be expressed as a function of coordinates (x, y, z), indicating how it varies in different spatial locations.

Think of a swimming pool after everyone has left and the water is still. The water doesn't 'flow' or 'slide' in any direction, so there are no forces pushing the water layers over each other. If you measure the pressure at different points in the pool, you'll find it varies with depth, but not horizontally, demonstrating that pressure acts uniformly in still water.

At that point, the gravity force is acting it which is the body force component. The gravity force what will be there? It will be, unit weight multiply the volume of the control volume, which is V = Lx * Ly * Lz which is very simple things. The control volumes, the volume is this much and the unit is ρ * g, where ρ stands for here the density and g stands for accelerations due to gravity.

In this context, we consider the effects of gravity as a body force acting on the control volume. The total force due to gravity can be calculated by multiplying the volume of the control volume (determined by its dimensions: length, width, height) by the product of the fluid's density and acceleration due to gravity. This is crucial for understanding how pressure contributes to forces in a fluid at rest.

Imagine you have a cube of water (your control volume) floating in the ocean. The weight of the water exerts pressure on the cube's surfaces from the water above it. The deeper you go, the heavier the water column above you, simulating the effect of gravity as a body force pushing down on the fluid.

Now if you look it over this control volume, all these pressures are going to act as Pascal’ law says that pressure acts normal to the surface.

According to Pascal's law, the pressure in a fluid at rest acts equally in all directions and is perpendicular to the surface of any object within the fluid. This means that if we measure the pressure at different points on our control volume, we can establish a pressure gradient. A pressure gradient is a rate of pressure change per unit distance and can influence fluid movement if conditions change.

Think of taking a balloon filled with water. When you squeeze it from the outside, the pressure applied is exerted equally in all directions, pushing against the inner walls of the balloon. Similarly, in a fluid at rest, changes to the pressure in one place affect the pressure equally across the entire volume.

Then we need to define now as a pressure as a x, y, z is a pressure field what we are looking it which is a function of x, y, z I can use Taylor series to approximate distance of this surface which is far away.

In mathematical analysis, Taylor series can approximate complex functions. Here, we can use the Taylor series to represent pressure as a function of spatial coordinates. This allows us to estimate pressure changes with respect to variations in position, aiding in modeling how pressure changes inside our control volume.

Consider the way sound ripples through water. If you drop a stone into a calm pond, the ripples expand outward. By looking at the time and distance from the stone, we can predict how high the water rises at various points using a mathematical model similar to a Taylor series.

So similar way, we can derive the pressure component, the force component due to the pressures in x directions and the z directions respectively.

In the analysis of the control volume, we can determine forces acting along the x, y, and z axes by integrating pressure components across those axes. These integrations help ensure we account for every force acting on our control volume, particularly when it comes to predicting fluid behavior and equilibrium.

Think of balancing a seesaw. If you want to keep the seesaw level, you have to ensure that the weight is evenly distributed on both sides. By calculating the force due to weight on each side, similar to how we analyze pressure in our control volume, you can maintain balance.

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