1.7. Pressure Forces and Shear Forces

We have kept this figure for the forces in the right hand side. So, if you see, there will be pressure forces acting P x from the left, P x + P pressure force are at x + dx from the right and then there will be shear forces acting here, in this direction and there will be shear forces acting at Tr + dr, you know, and if we apply the force balance.

In a fluid flow situation, different forces are acting on a small element of fluid. On one side, there is a pressure force exerted from the left side (denoted as Px), while on the right side, there is a pressure force acting at the point a short distance away (denoted as Px + dP). Shear forces, which arise from the friction between the fluid layers, also act on the fluid element. Understanding these forces is crucial for analyzing how the fluid will move or behave under different conditions.

Imagine you are pushing a box across a frictionless surface. The force you apply to push the box is like the pressure force. However, if the surface were sticky, there would also be a resistance force that opposes your push. This resistance is akin to the shear forces acting on the fluid in our scenario.

Now, this one, force acting in this direction which is positive in sign and there is another force acting in the negative direction, so, that is what we have done. So, again this is a shear force, so, we have multiplied this tau here, the one here, with 2 pi r dx, because this is the thickness over which it is, I mean, so, this is the length over which it is acting.

When calculating the net effect on the fluid element, we consider both the positive and negative forces at play. The positive force represents the pressure force pushing the fluid element forward, while the negative force represents the opposing shear force. The shear force is calculated by multiplying the shear stress (tau) by the surface area over which it acts, which in this case is represented as 2πr dx (where r is the radius of the fluid element and dx is a small length along the flow). This helps in determining how these forces affect the overall motion of the fluid.

Consider a heavy book resting on a table. The weight of the book exerts a downward force due to gravity (like the pressure force), while the friction between the book and the table provides an opposing force (like shear forces). If you try to slide the book, you're experiencing both of these forces, and your net effort to move the book depends on how these forces interact.

Now, equation number 2 can be written as simply you see, so this is pressure at x + dx - P x by dx, so it can be written as, dP dx and this can be written as, d r into tau dr.

After doing the calculations, we can express the net force acting on the fluid element in terms of changes in pressure over distance (dP/dx) and shear stresses (du/dr). This is a mathematical way to highlight how the pressure and shear forces affect the behavior of the fluid, relying on calculus to describe the changes continuously across small intervals.

Think of water flowing through a long garden hose. As water moves through, the pressure changes at different points along the hose. If you could measure the difference in pressure every few centimeters, you'd find a gradient that tells you how tightly it's flowing (like dP/dx). This idea helps visualize how forces affect fluid motion over different lengths of the hose.

Now what we get? We can get, mu by r d dr of r du dr is equal to dP dx because this minus will make it come on this side and this dP dx can be on the left side, right side.

In laminar flow, shear stress is typically proportional to the velocity gradient (i.e., how quickly velocity changes in different layers of the fluid). This relationship lets us substitute the shear stress into our equations so we can relate the shear effects to the overall pressure changes in the flow system. The factor mu signifies the fluid's viscosity, which indicates how much the fluid resists flow.

Imagine spreading peanut butter on bread. If the peanut butter is thick (high viscosity), it will take more effort to spread it smoothly across the surface compared to a thinner substance like butter (low viscosity). The change in how easily these spreads can flow represents the dynamics of shear stresses and how they relate to the overall flow in a fluid.

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