Question

Water is flowing with a speed of 2 m/s in a horizontal pipe with cross- sectional area decreasing from $2\times {{10}^{-2}}{{m}^{2}}$ to 0.01 ${{m}^{2}}$ at pressure $4\times {{10}^{4}}Pa$. What will be the pressure at a small cross-section?

Answer 1

Here, the area is decreasing and the water we know under ordinary conditions is an incompressible liquid. So, we can make use of equation of continuity to find out the velocity at the other end where area has decreased. After finding out the velocity our task is to find the pressure at the other hand and for that we can make use of the Bernoulli’s theorem.

Formula used:
Equation of continuity, ${{A}_{1}}{{v}_{1}}={{A}_{2}}{{v}_{2}}$
Bernoulli’s theorem, $P+\rho gh+\dfrac{\rho {{v}^{2}}}{2}=k$, where k is a constant

Complete step by step answer:
Using equation of continuity,
{{A}_{1}}{{v}_{1}}={{A}_{2}}{{v}_{2}} \\\ \Rightarrow 2\times {{10}^{-2}}\times 2=0.01\times {{v}_{2}} \\\ \Rightarrow {{v}_{2}}=4m/s \\\
Now, we know the velocity at the other hand and we can make use of Bernoulli’s theorem to find out the pressure.
{{P}_{1}}+\dfrac{\rho {{v}_{1}}^{2}}{2}={{P}_{2}}+\dfrac{\rho {{v}_{2}}^{2}}{2} \\\ \Rightarrow {{P}_{2}}=-\dfrac{\rho {{v}_{2}}^{2}}{2}+{{P}_{1}}+\dfrac{\rho {{v}_{1}}^{2}}{2} \\\ \Rightarrow {{P}_{2}}={{P}_{1}}-\dfrac{\rho ({{v}_{1}}^{2}-v_{2}^{2})}{2} \\\ \because \rho =1000kg/{{m}^{3}} \\\ \Rightarrow {{P}_{2}}=4\times {{10}^{4}}-\dfrac{1000({{2}^{2}}-{{4}^{2}})}{2} \\\ \therefore {{P}_{2}}=3.4\times {{10}^{4}}Pa \\\
So, the pressure comes out to be $3.4\times {{10}^{4}}Pa$.

Additional Information:
The ability of fluids to transmit pressure is the key point which holds Pascal's law. This law has many applications in our everyday lives. The working of hydraulic lift, hydraulic jack, hydraulic press and forced amplification is used in the braking system of most cars is based on Pascal’s law. Hydraulic pumps also work on the same principle. This law has helped significantly and immensely in weighing heavy loads with ease.

Note: The Bernoulli’s principle holds when the liquid is incompressible. This is a very important law of fluid dynamics. This is a result of law of conservation of energy. The theorem holds the law of conservation of mechanical energy. Also, we assume that there is no loss of energy as there are no viscous forces in the fluid. The flow must be streamlined.