Why does fluid pressure decrease when area decreases and relationship between pressure area and velocity?
You start with connecting velocity to area via continuity (mass conservation). Then you use Bernoulli's equation to connect pressure to area.
This is the most intuitive and easiest way. Of course, you can prove it more rigorously.
Here's how it goes.
1) Continuity for incompressible flows is, essentially, non-variability of the volume flow rate: Q = const. The formula for Q is given by Q = v * A. And we have v * A = const. Example: you have a water flow in a pipe which has a contraction. Continuity will give you v_1 * A_1 = v_2 * A_2. Here subscript 2 refers to a smaller diameter, subscript 1 refers to a bigger diameter. Rearrange: v_1 / v_2 = A_2 / A_1. Since A_2 < A_1, A_2 / A_1 < 1 and, therefore, v_1 / v_2 < 1. That means v_2 > v_1. In other words, if area is smaller, velocity is bigger.
2) Now, let's analyze Bernoulli's equation in light of the previous conclusion. Bernoulli's equation is given p + 0.5 * rho * v^2 + rho * g * h = const. Neglect hydrostatic head: p + 0.5 * rho * v^2 = const. For a pipe that we considered previously (i.e., for a pipe with a big and small diameters), we have: p_1 + 0.5 * rho * v_1^2 = p_2 + 0.5 * rho * v_2^2. Rearrange: p_1 - p_2 = 0.5 * rho * v_2^2 - 0.5 * rho * v_1^2. We have already concluded above that v_2 > v_1. Therefore, 0.5 * rho * v_2^2 - 0.5 * rho * v_1^2 > 0. Therefore, p_1 - p_2 > 0. Therefore, p_1 > p_2. In other words, pressure is bigger when velocity is smaller. Velocity is smaller if the area is bigger. Therefore, pressure is bigger if the area is bigger.
as we know fluid pressure (p=force /area).from this formula we can relate the parameters p~1/area where as p~force .starting from this point as pressure decrease area will be increased vice versa.due to inverse relation.
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