Distinguishing between static and dynamic pressure in fluids I read about the difference between static and dynamic pressure in fluids in the mechanics textbook (part 2) by BM Sharma. It explained me by using a Pitot tube and an ordinary pipe attached to the same main pipe. I applied Bernoulli's theorem at point $A$ and at the datum line:
$$P_1+(1/2)(\rho)v^2 +0 = P_0 + 0 + (\rho)g(h_s).$$

The heights $h_s, h_d$, the datum line and point $A$ and $B$ are indicated in the image.
Similarly, I applied the equation for point $B$:
$$P_1 + (1/2)(\rho)v^2 + 0 = P_0 + 0 + (\rho)gh_d.$$
Thus, I obtained the result that the heights $h_s$ and $h_d$ are identical.
Please explain where my analysis is wrong. Please explain by applying Bernoulli's theorem at point $A$ and the datum level.
 A: $$\underline{\textit{Analysis}}$$
Let $P_A=P_0$, $P_B=P_0$ and $P'_A$, $P'_B$ denote the static pressures at the points $A$, $B$, and their vertical projections $A'$, $B'$, on the datum line respectively. Let $v_A=0$, $v_B=0$ and $v'_A$, $v'_B=0$ denote the horizontal velocity components at the points $A$, $B$, and their vertical projections $A'$, $B'$, on the datum line respectively. We assume that the fluid is incompressible.
In order to analyze the static pressure differences between points in the two columns, we apply Pascal's law. In order to analyze the dynamic pressure differences between two points at two points on the streamline coinciding with the datum line, we apply the Bernoulli's equation. The Pascal's law implies that $$P'_A=P_A+\rho g h_s=P_0+\rho g h_s,$$ $$P'_B=P_B+\rho g h_d=P_0+\frac{1}{2}\rho g h_d,$$ so that $$0<\rho g (h_d-h_s)=P'_B-P'_A,$$ which is the quantitative difference between the static and dynamic pressure heads in the flow. The Bernoulli's equation applied to the to the streamline $A'B'$ implies yields $$P_A'+\frac{1}{2}\rho (v'_A)^2=P_B'+\frac{1}{2}\rho (v'_B)^2$$ so that $$0<P'_B-P'_A=\frac{1}{2}\rho((v'_A)^2-(v'_B)^2)=\frac{1}{2}\rho(v'_A)^2,$$ wherein the quantity on the right hand side of the above expression defined as the dynamic pressure of the flow. Therefore the dynamic pressure head is estimated (on neglecting head losses) by measuring the difference $h_d-h_s$ as $$P_\text{dynamic}:=\frac{1}{2}\rho(v'_A)^2=\rho g (h_d-h_s)=P_B-P_A.$$

$$\underline{\textit{Explanation}}$$
The Bernoulli's equation reads $P+\frac{1}{2}\rho v^2 + \rho g h=constant$ at all points along a streamline in a steady, incompressible and inviscid flow. The variable $P$ corresponds to the physical quantity of static pressure at the associated point, while the expressions $\frac{1}{2}\rho v^2, \rho g h$ are referred to as the dynamic pressure and pressure head due to gravity head at the associated point, respectively. The critical error in the analysis presented in the OP is that the Bernoulli's equation is not applied at points along streamlines, but perpendicular to them. Specifically, the Bernoulli's equation cannot be applied to relate the flow properties across the points $A$ and $A'$ as in the OP, since there is no streamline connecting the points $A$ and $A'$ (in others words, there is no flow occurring across those points since the 'flow' or fluid in the column $AA'$ is static).
