8.3.1. Alignment of Gravity with Coordinate Axes

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.

In this chunk, we focus on determining the pressure field in a static fluid. When a fluid is at rest, it doesn't experience shear stress, meaning there's no tangential force acting at the fluid's surface. Instead, the only force acting on the control volume—a defined space we're analyzing—is the normal stress, which corresponds to the pressure. Imagine a cube of fluid suspended in space: the only thing pushing into or pulling out of this cube is pressure acting perpendicular to its surface.

Think of a balloon filled with water. When you hold the balloon still, the water inside exerts pressure outward. There's no movement or shear (like when you squish it), just the pressure from the water on the sides of the balloon. This reflects how, in a control volume at rest, only pressure needs to be considered.

So we can have a two force components. One is the surface force component and other is body force. The body force components if you look it that it will be unit weight multiply the volume of the control volume, which is very simple things. The control volumes, the volume is this much and the unit is ρg and ρ stands for here the density and g stands for accelerations due to gravity.

Here, we identify two types of forces acting on our control volume: surface forces and body forces. Surface forces come from the pressure exerted on the sides of the control volume, while body forces are due to the weight of the fluid itself, which is calculated using the fluid's density (ρ) multiplied by gravity (g). This gives us the overall weight acting on the control volume based on its volume.

Return to the balloon; if you weigh the water inside, that weight represents the body force. However, the pressure pushing outward on the balloon's sides is akin to surface forces. Together, these two help us understand how fluids remain at rest.

Now if you look it over this control volume, all these pressure is going to act it as Pascal law says that pressure acts normal to the surface. So that what we can consider pressures act there normal to the surface. 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.

This chunk explains how pressure behaves in a control volume according to Pascal's law, which states that pressure applied to a confined fluid is transmitted undiminished in every direction. Therefore, any pressure we measure is normal (perpendicular) to the surface of the control volume and varies with position, hence being a function of x, y, and z coordinates. This means that to analyze fluid behavior, we must treat pressure as a multidimensional field.

Think about how a sponge soaks up water. As water seeps into the sponge, the pressure from that water is exerted equally in all directions. If you squeeze the sponge, you can see how the pressure builds up and acts in every direction on the sponge walls, demonstrating how pressure operates in a fluid-filled control volume.

I can use Taylor series to approximate distance of this surface which is a far away from. The pressure at point (x,y,z) on the surface can be expressed as...

In this chunk, we discuss how to utilize a mathematical tool known as Taylor series to estimate pressure values at different points within our control volume. By taking the pressure at a known point and approximating it based on its distance from other surface points in the control volume, we can derive a more complex understanding of pressure distribution within the fluid.

Imagine you have a map of a mountain range. If you're at the peak (where you know the elevation), you can use the map to estimate the elevations (pressures) at points further down the mountain, based on how steep the slopes are. A Taylor series does something similar: it helps us estimate properties of a function based on what we already know about it at a specific point.

If I know P(x, y, z) what will be the pressure at this point (x,y) will have this function. You can look at this negative part is there because this is what distance far away or the opposite of the y direction.

This section emphasizes the importance of maintaining equilibrium in our control volume. The pressure derived using the Taylor series helps us understand how pressure changes as we move between different points in space. This understanding allows us to establish conditions of static equilibrium, which state that the sum of forces acting in any direction must equal zero when the fluid is at rest.

Envision balancing a seesaw. If one side is heavier, it will tilt unless additional weight is added to balance it out. Similarly, for a fluid to remain still (equilibrium), the equal and opposite pressures must act across the control volume surface to maintain balance.

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