Question

When you vomit your stomach forces a fluid flow from your mouth. Treating your stomach, esophagus, and mouth as a continuous vertical tube of radius 1 cm and length 60 cm, what gauge pressure must your stomach generate to cause vomit to move at 1.5 m/s out of your mouth, in Pa

Answer 1

Answer:

The gauge pressure as calculated is 5862.36 Pa

Solution:

As per the question:

Radius, r = 1 cm = 0.01 m

Length, l = 60 cm = 0.6 m

Velocity of discharge of fluid from the mouth, v = 1.5 m/s

Now,

By using the continuity equation:

Av = A'v'                       (1)

where

A = Area of the mouth

A' = Area of the tube

v' = Velocity of the fluid inside the stomach

Since, the question assumes the stomach, mouth and esophagus as continuous vertical tube, then:

A = A'

Thus

From eqn (1):

v = v' = 1.5 m/s

Now,

With the help of Bernoulli's equation:

$P + \frac{1}{2}\rho v^{2} + \rho gh = P' + \frac{1}{2}\rho v'^{2} + \rho gh'$

$P + \frac{1}{2}\rho v^{2} + \rho g(h - h') = P' + \frac{1}{2}\rho v^{2} +$

$\rho g(h - h') = P' - P$

where

h - h' = l = 60 am = 0.6 m

P = Pressure at mouth = 1 atm = $1\times 10^{5}\ Pa$

g = acceleration due to gravity

Density of water, $\rho = 997\ kg/m^{3}$

P' = Pressure inside the stomach

P' - P = Gauge Pressure

$P' - P = 997\times 9.8\times 0.6 = 5862.36\ Pa$