16.1.2. Net Pressure Force

Now just to look it, I am not going detail derivations of this part if you take a fluid element along a stimuli like this is the fluid element okay, this is the stream line which is having dx and dn dimensions, you have the shear stress which is changing at this along the n’th directions and you get it what could be the shear stress. Similar way you can find out the pressure values and all.

In fluid mechanics, the behavior of fluids is studied through elements of fluid. Here, we consider a small portion, or 'element', of fluid, which is influenced by forces acting upon it. This element can be represented along a streamline (the path that a fluid particle follows). The dimensions 'dx' and 'dn' represent tiny changes along these paths. Shear stress, which is a measure of how much a fluid is being 'stretched' or 'sheared', will vary as we move along the fluid element. By analyzing these properties, we can calculate various forces, including shear forces and pressure.

Imagine a slice of bread in a toaster. When you toast it, the edges are heated, causing the center to expand while the edges remain rigid. This scenario is similar to how shear stress changes within a fluid element. Just as the tension in the bread varies across its surface, shear stress within a fluid changes based on the forces acting on it.

Force due to the viscosity (friction) is ... The shear force can be expressed as ... Then you can compute the force due to the viscosity that will be the change of shear stress into the volumetric part.

Viscosity is a measure of a fluid's resistance to deformation or flow; it's akin to 'thickness'. When layers of fluid slide over each other, they exert shear forces due to this viscosity. The relationship between the change in shear stress and the volumetric aspect of the fluid can be derived mathematically. By understanding how shear stress influences flow, we can compute the total viscous force acting on a fluid element.

Consider honey pouring over pancakes. The honey flows slowly because it has high viscosity. This resistance to flow is why it clings to the pancakes rather than flowing straight off. By measuring how the honey flows, we can understand the forces at play and calculate the viscosity.

The net pressure force for steady flow, Inertia force computation ... Similar way the net pressure force acting of this you can see it will be this part and inertia force computation which is the, in case of the steady flow, mass into the acceleration or rate of change of the momentum flux.

The net pressure force in a fluid system considers the forces acting due to pressure differences across surfaces within the fluid. When dealing with steady flow, we also need to factor in inertia forces, which are the result of mass multiplied by acceleration or changes in momentum. This balance of forces helps us predict how fluids behave under different conditions, grounding our understanding of fluid mechanics in fundamental physical laws.

Think of a car driving on a highway. If you're driving at a steady speed, the forces of motion and air pressure against the car are balanced, similar to how net pressure force and inertia force balance in a flowing fluid. If you suddenly accelerate, the inertia forces act upon you, pushing you back into your seat, demonstrating how changes in motion affect the forces experienced by both the car and the fluids around it.

If you equate it and substitute these values in case of loss of dynamic similarities the ratio between these parts ... that when you go to the element level we can derive the ratio between inertia force to viscous force which comes out to be Reynolds numbers.

Dynamic similarity is a concept in fluid mechanics where models of fluid behavior can be compared across systems. By establishing ratios such as the Reynolds number (which compares inertial forces to viscous forces), we can predict how fluids will behave in different scenarios. This helps in designing experiments and applications based on smaller-scale models to understand larger-scale behaviors, such as airflow over wings or water flow through pipes.

This concept is similar to trying to predict the wave patterns on a small pond to understand how waves might behave in an ocean. The smaller body of water can serve as a model to forecast behaviors in a much larger and complex system.

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