8.1.3. Pressure Field as a Function of Coordinates

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. So there is no shear stress component on this fluid plane. Only this normal stress which is as equivalent to the pressure will act over this control surface. Over the control surface only the pressure is going to act it because the shear stress is zero okay.

In this chunk, we are introduced to the concept of how pressure behaves in a fluid at rest. When a fluid is not moving, it experiences no shear stress because the fluid layers are not sliding past each other. In this state, the only stress acting on the fluid is normal stress, which equates to pressure. Therefore, the focus is on understanding that pressure is a function of three-dimensional coordinates (x, y, z) and that we can define pressure at any point within a control volume using these coordinates.

Imagine a calm lake. The water is not moving, and if you place a small object beneath the surface, the only force acting on it from the surrounding water is the pressure from the water above it. This pressure is due to the weight of the water and can be thought of as 'weight pushing down' uniformly across the object, which aligns with the idea of normal stress acting in a fluid at rest.

The pressure at the centre of the fluid control volume is assumed to be P(x, y, z). 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, ρg × V.

This chunk discusses the force components acting within a control volume, focusing on gravity as a body force. Here, 'ρ' represents the fluid density, 'g' is the acceleration due to gravity, and 'V' is the volume of the control volume. Gravity pulls down on the fluid, and we can calculate the body force using the equation ρg multiplied by the volume. This highlights how pressure in a fluid system involves understanding both internal stresses (pressure) and external forces (gravity).

Think of a water bottle filled with water. The weight of the water at the top pushes down on the water at the bottom, creating pressure at the base of the bottle. This pressure can be calculated by considering the weight of the water column, similar to how we quantify the effects of gravity on our control volume.

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

In this chunk, the use of the Taylor series to express pressure as a function of coordinates is introduced. The Taylor series is a mathematical tool that allows us to approximate complex functions. By expanding the pressure at a point (x, y, z) using Taylor series, we can derive how pressure changes with distance in three-dimensional space, facilitating a better understanding of the pressure field within the fluid.

Imagine you are on a smooth hill. If you want to know the slope at your feet (the pressure at a specific point), you can use the surrounding terrain to estimate how steep it gets as you move a little further away. This estimation is much like using the Taylor series, where we leverage the known pressure at one point to predict it at another nearby point.

If I know P(x, y, z) what will be the pressure at this specific point, and if you equate it finally, we will get this one. The gradient and the volume of the control panel. The negative gradient and the volume of control volume is what we will get. This is the force component acting along the y directions.

This segment emphasizes calculating how changes in pressure correspond to changes in gravitational force acting on a control volume. By determining the gradient of pressure (the rate of change of pressure), we can relate it to the force components acting on the fluid. The 'negative gradient' references how pressure decreases the further from the fluid's surface you go, a fundamental concept in hydrostatics.

Picture being in a swimming pool. As you swim deeper, you feel more pressure pressing down on you because of the water above. The change in pressure from the surface to the bottom illustrates how the pressure gradient influences the forces at play in a fluid.

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