An ideal liquid of density rho is pushed with velocity V through the c...
An ideal liquid of density rho is pushed with velocity V through the c...
Understanding Fluid Dynamics in the Tube
The scenario involves an ideal liquid of density \(\rho\) flowing through a horizontal tube with a central limb and two side limbs, each having equal cross-sectional area \(A\). To determine the force exerted by the liquid on the tube, we can apply Bernoulli's principle and the continuity equation.
Continuity Equation
- Since the flow is streamline and the cross-sectional areas are equal, the velocity of the liquid in each section must remain constant.
- Therefore, the velocity \(V\) in the central limb is the same in the side limbs.
Application of Bernoulli’s Principle
- According to Bernoulli’s equation, the pressure difference in the tube can be derived from the kinetic energy per unit volume of the fluid:
\[
P + \frac{1}{2} \rho V^2 = \text{constant}
\]
- In the side limbs (where the velocity is lower), the pressure will be higher due to the conservation of energy.
Calculating the Force
1. **Pressure in the Central Limb**: Let \(P_c\) be the pressure in the central limb.
2. **Pressure in the Side Limbs**: Let \(P_s\) be the pressure in the side limbs. Using Bernoulli's principle, we can express \(P_s\) as:
\[
P_s = P_c + \frac{1}{2} \rho V^2
\]
3. **Force Calculation**: The force exerted by the liquid on the tube wall can be calculated using the pressure difference across the side limbs.
\[
F = (P_s - P_c) \cdot A = \left(\frac{1}{2} \rho V^2\right) \cdot A
\]
Conclusion
- The liquid exerts a force \(F\) on the tube, which can be calculated using the difference in pressures derived from Bernoulli's principle. This force is a result of the velocity of the liquid and its density, influencing the pressure exerted on the walls of the tube.
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