Question

The minimum pressure required to force the blood of density $1.04\text{ g}\text{.c}{{\text{m}}^{-3}}$ from the heart to the top of the head (vertical distance = 60 cm) is

Answer 1

Daniel Bernoulli's theory states that as the speed of a flowing fluid (liquid or gas) increases, the pressure inside the fluid decreases. The total mechanical energy of the moving fluid, which includes the gravitational potential energy of elevation, fluid pressure energy, and fluid motion kinetic energy, remains constant. Also, we can say that the mass of fluid flowing through different cross sections is equal if the fluid is in a streamline flow and is incompressible and this is known as principle of continuity.

Complete step-by-step solution:
In this question, we need to find out the minimum pressure required to force the blood from heart to the top of the head and we have been given few quantities such as
Density of Blood $(\rho )$ = $1.04\text{ g}\text{.c}{{\text{m}}^{-3}}$
Also, ${{h}_{2}}-{{h}_{1}}=60\text{ cm}$
Now, by using Bernoulli Equation
${{P}_{1}}+\dfrac{1}{2}\rho V_{1}^{2}+\rho g{{h}_{1}}=\text{ Constant}$
Or we can write
${{P}_{1}}+\dfrac{1}{2}\rho V_{1}^{2}+\rho g{{h}_{1}}=\text{ }{{P}_{2}}+\dfrac{1}{2}\rho V_{2}^{2}+\rho g{{h}_{2}}$
${{P}_{1}}-{{P}_{2}}=\rho g({{h}_{2}}-{{h}_{1}})+\dfrac{1}{2}\rho (V_{2}^{2}-V_{1}^{2})$
Also, ${{V}_{1}}={{V}_{2}}$
Putting the values, we get
${{P}_{1}}-{{P}_{2}}=\rho g({{h}_{2}}-{{h}_{1}})$
${{P}_{1}}-{{P}_{2}}=(1.04)(981)(60)$
On solving this, we get
${{P}_{1}}-{{P}_{2}}=6.1\text{ dyne c}{{\text{m}}^{-2}}$
Hence, this is the minimum pressure required to force the blood of density $1.04\text{ g}\text{.c}{{\text{m}}^{-3}}$ from the heart to the top of the head.

Note: Bernoulli's concept is used in the theory of ocean surface waves and acoustics to analyze the unsteady potential flow. It is often used to approximate fluid parameters such as pressure and speed. For example, when we are standing at a train station and a train arrives, we appear to collapse in front of it. This is explained by Bernoulli's theorem, which states that as the train passes us, the velocity of the air between us and the train increases. As a result of the equation, the friction decreases, and the pressure from behind drives us towards the train. The Bernoulli effect is the basis for this.