Water flows through a long horizontal tube. Suppose P_A and P_B...

For an ideal incompressible fluid flowing through a horizontal tube, Bernoulli's equation states that:

$P + \dfrac{1}{2}\rho v^2 = \text{constant}$

Applying this principle at points $A$ and $B$ within the tube:

$P_A + \dfrac{1}{2}\rho v_A^2 = P_B + \dfrac{1}{2}\rho v_B^2$

For the pressures at both points to be equal ($P_A = P_B$), the velocities of the fluid at these points must also be equal ($v_A = v_B$).

According to the equation of continuity for an incompressible fluid:

$A_A v_A = A_B v_B$

Substituting $v_A = v_B$ into the continuity equation gives:

$A_A = A_B$

Therefore, the pressures at points $A$ and $B$ will be equal only if the cross-sectional areas at these two points are identical.