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Exploring Fluid Dynamics and the Hagen-Poiseuille Equation
Emanuela Joseph, Illeanna Torres, Breeanna Robinson
PHY 2141 Sec 005
Instructor: Hayden Hall
Introduction:
The purpose of this experiment was to explore fluid dynamics and the Hagen-Poiseuille
equation. Fluid dynamics focuses on the flow of liquids and gasses and can be used in real-life
situations in healthcare. The Hagen-Poiseuille equation is used within fluid dynamics to create a
relationship between pressure, fluid resistance, and flow rate. An example of using the Hagen-
Poiseuille equation in real life would be to explain constriction or dilation of capillaries in
respect to blood pressure. Previously gathered videos were used to analyze fluid flow in
microfluidic devices. These videos will be analyzed using ImageJ to find the velocities of 5 um
beads contained in the fluid. The flow of Device A is arranged as a series and the flow of Device
B is in parallel direction. There are wide and narrow channels in each device, being 6.0 mm wide
and 3.0 mm wide, respectively. The rate of flow and velocities were calculated using the Hagen-
Poiseuille equation.
In the recorded videos, the experimenters caused the fluid to move through the devices by
causing a repeatable, consistent force on a plunger syringe. This caused the fluid with the 5 um
beads to flow from the left to the right of the devices and the rate of the motion was calculated
using the Hagen-Poiseuille equation.
For the Series System, we hypothesized that the velocity of the fluid in the narrow
channel will be greater than the wide channel since the wider channel has a larger area. However,
the volumetric flow rate would be equal since the continuity equation states the flow into an area
must be equal to the flow out of the area.
For the Parallel System, we hypothesized that the wide channel would have a greater
fluid velocity than the narrow channel since the larger area of the parallel system causes the
velocity to be faster. Likewise, the volumetric flow rate would also be greater in the wide
channel due to the larger area.
Methods:
We first observed the motion of the beads as they were flowing in the (4)
systems:
Parallel Wide, Parallel Narrow, Series Wide, and Series Narrow. We formulated a hypothesis on
which system would have the greatest fluid velocity and the greatest volumetric flow rate (Q).
Analyzing the videos was the next step as we opened ImageJ and uploaded the videos. The
motion of 10 beads from each of the 4 videos were tracked using the manual tracker plugin. The
average velocity, standard deviation, and the flow rate of each system was calculated using Excel
and the Hagen-Poiseuille equation as well as graphs to give a visual of the collected data.
Analysis
:
Due to the uncertainties, the range of the measurements are 3.848919
± 1.466416, 5.274502 ±
4.985595.
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Due to the uncertainties, the range of measurements for the series narrow and wide are
3.1947566
± 1.1594272, 7.0978655 ± 3.1983043.
Conclusion
:
Our results from our analysis supports our initial hypothesis. In the Parallel
System, the average velocity for the narrow channel was 3.85 um/s whereas in the wide
channel it was 5.27 um/s. This shows that in the parallel system the pressure is
maintained due to the sharing of openings in both channels. The calculated volumetric
flow rate in the wide channel is also greater than the narrow channel due to the same
reasoning. In the series system, the average velocity for the narrow channel was 3.19
um/s, while the wide channel had an average velocity of 7.09 um/s. Looking at the data
you can see that the average velocity from the narrow series 7 channel in comparison to
the wide series 7 channel had practically doubled. Our hypothesis stated that in series 7,
the narrow channel would have greater velocity compared to the wide channel. As we can
see, the results show the opposite effect. The velocity in the wide channel had much
greater velocity than the narrow channel which does not support our initial hypothesis for
the series channels. The ratio of Vwide to Vnarrow in the parallel system is 1.37:1
whereas the ratio in the series system is 2.22:1. These values would describe the
relationship between the velocity of the wide channels compared to the narrow channels.
It is possible for the ratios to be slightly off due to human error throughout the
experiment, but from our knowledge of fluid flow, our hypothesis would stand correct.
With all experiments, there is room for error. There may have been margins of
error in this experiment that could have disrupted the collected data. On the ImageJ
software, we were responsible for following the beads through the frame, so the most
common error would be not following the same bead throughout the whole frame. This
would allow all the data to be slightly skewed if different beads were used on the same
trial. This error may have been present in our data and could have affected the hypothesis.
An example of this would be in the series 7 narrow, the standard deviation is offset due to
the average only being made up of 8 beads instead of 10. This is because 2 beads in the
narrow channel had only a single velocity of -1 um/s.
In this experiment we learned that fluid flow in parallel and series systems are
different. In physics 2130, we learned that fluid flowing in a tube that has a wider area
has high pressure and low velocity whereas the narrow area has low pressure and high
velocity. Even though our calculations did not align with this statement due to human
error, it was still learned. In addition to the flow rate, it is understood that the fluid flow
into a system must equal the fluid flow out of the system due to the continuity equal for a
system of series. In the parallel system, it is known that the fluid velocity should be
greater in the wide channel which our calculations support. This happens because of the
larger area of water that is rushing into the system compared to the narrow area.
As stated in the introduction, fluid dynamics and the HP equation can expand
outside of physics and the experiment we conducted. Fluid dynamics is an important part
of physics used in real-life events, specifically blood circulation. In the introduction, we
briefly mentioned how fluid dynamics and the HP equation can be used in relation to the
constriction and dilation of capillaries. This idea is called hemodynamics and is common
in those with high blood pressure because it describes blood flow and how it’s regulated.
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