7.3.1. Pressure Variations in Fluid at Rest

Now let us come to the very basic concept what we are talking about the fluid at rest. So the basically we are talking about now, the fluid at rest, okay? If it is a fluid is at rest, it is a very simplified problem now. Like as I said it any fluid flow problems we look at either the pressure field, the velocity field, or the density and temperature field. When the fluid is rest now, very simply way the velocity vectors becomes zero and if I consider incomprehensible the density is a constant and if the temperature is not very much I need not need a thermodynamics first laws to define the problems. Then only the left is that the pressure field.

In fluid mechanics, when we say 'fluid at rest', we mean that the fluid is not moving. This simplifies the analysis greatly because if the fluid isn't moving, the velocity vectors are zero. In this scenario, we don't need to consider how velocity changes over time or space, simplifying our calculations. Generally, we analyze three main fields: pressure, velocity, and density. However, when a fluid is not moving (i.e., at rest), we focus only on the pressure field. Further, if we consider the fluid to be incompressible, the density remains constant which also simplifies our analysis. The temperature can usually be treated as a constant in basic fluid static problems.

Consider a bucket of water sitting still on the ground. The water is not moving, so the speed of the water is zero. Because of this, the only thing we need to think about is the pressure exerted by the water on the bottom of the bucket, which comes from the weight of the water above it.

Since there is no velocity, there is no velocity gradient, definitely as Newton's second law says that there is no velocity gradients that means no shear stress. So when a fluid is at rest, there is no shear stress acting on that.

In a fluid at rest, because there is no movement, it follows that there can be no changes in velocity within the fluid, which means there is no velocity gradient. According to Newton's second law, the absence of a velocity gradient means there is no shear stress being applied to the fluid. Shear stress occurs when layers of fluid move relative to each other, and since this isn't happening, we can say shear stress is zero. This simplification allows us to focus on pressure forces without worrying about shear forces, making calculations much easier.

Think of a jar filled with honey that you leave untouched. The honey does not move, and consequently, no internal layers slide over each other. This absence of flow means there are no shear stresses causing the honey to change position, illustrating that in a stationary fluid, shear stress is effectively zero.

So you can take a surface or take a control volume. Over that control surface you can define it the shear stress components become zero. That is very simplified now that any control volume you consider it over that control volume surface as the fluid is at rest conditions, there is no shear stress acting on the control surface. So that is the very easy problems what we have now. And what the two forces we have? The gravity force and the force due to the pressure distribution.

When we analyze a control volume (a specific region in space where we examine the behavior of fluid), we can simplify our analysis greatly since the shear stress is zero. In a fluid at rest, the only two forces acting on the control volume are the gravitational force pulling downward (due to the weight of the fluid) and the pressure force acting upwards (due to the pressure at the bottom of the control volume). This means we can balance these forces to understand how pressures behave throughout the fluid.

Imagine a tall glass of water. The water exerts pressure on the bottom of the glass due to the weight of the water above it. At the same time, gravity is pulling down on the water. The balance between these two forces determines the pressure at any point within that glass.

So we have just considered this fluid element or the control volumes at the equilibrium conditions. Similar way, rearranging the above equation gives P = P + ρgz.

When we analyze a fluid at rest, we assume that it is in equilibrium, meaning all forces are balanced and there are no net movements. In hydraulics, we often express this equilibrium condition in terms of pressure change with depth, derived from the balance between gravitational force and pressure force. The equation P = P + ρgz indicates that the pressure at a certain depth (P) increases as you go deeper in a fluid, taking into account the weight of the fluid above it. This relationship is crucial in applications like calculating pressures in dams and underwater structures.

This concept can be visualized if you think of diving underwater. As you swim deeper in a pool, you can feel the pressure increasing around you because more water is above you pressing down due to gravity. The formula demonstrates how that pressure increase relates to water depth.

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