#### Presentation Title

Investigating the Absence of Surface Energy in the Bernoulli Equation

#### Faculty Mentor

Prof. John L. Watkins

#### Start Date

17-11-2018 1:30 PM

#### End Date

17-11-2018 1:45 PM

#### Location

C308

#### Session

Oral 3

#### Type of Presentation

Oral Talk

#### Subject Area

physical_mathematical_sciences

#### Abstract

Our work aims to show that under certain circumstances surface tension must be factored into the Bernoulli equation to accurately model the behavior of a fluid in motion. The Bernoulli equation is an expression of the conservation of energy principle applied to a fluid in motion. Therefore, we propose it must include the energy used in creating the surface of the fluid, or surface tension. In the absence of this term, the Bernoulli equation does not account for all energies and hence cannot accurately predict the behavior of a fluid in motion. To test this claim, we consider the scenario of a vertical laminarly flowing fluid with a free surface. As a fluid falls in the manner described, it takes a recognizable shape. It is currently accepted that this shape can be determined using the Bernoulli equation by plotting the radius at any point along the stream as a function of its vertical distance from a fixed point. To test the accuracy of the accepted method, we compare the resultant shape of a laminar vertical stream predicated with the Bernoulli equation to the actual shape of a real fluid flowing under the same conditions. In order to accomplish this, we built a device that puts out a constant laminar stream when a fluid runs through it. As the device runs we photograph the stream to capture its shape. We then isolated the silhouette of the stream and compare it to the predicted shape generated by the Bernoulli equation and an altered equation which includes an additional term that accounts for surface tension. We have found that the altered equation better predicts the shape of the stream and therefore should be considered in relevant applications of fluid dynamics.

#### Summary of research results to be presented

Our current data strongly suggests the accepted method for plotting the shape of a vertical laminarly flowing fluid with a free surface is incomplete without accounting for surface energy. We derived a term to account for the energy lost due to the formation of the free surface of the stream which leads to an altered version of the Bernoulli equation. When comparing the actual shape of the stream under specific conditions to the shape predicted by identical conditions by the Bernoulli equation and the altered equation, the altered equation better matches the real silhouette of the stream while the unmodified Bernoulli equation exhibits notable deviation. Data collected from numerous trials of this process additionally indicates that surface tension plays a more significant role in the shape of the stream when the flow rate is low. These results strongly support our claim that surface energy must be considered in certain applications of the study of fluid dynamics. Our presentation will include a report of the statistical tests evaluating the improved fit to the real silhouette of the stream, a detailed description of the conditions under which this concept must be considered, and a discussion of the potential applications of this new term.

Investigating the Absence of Surface Energy in the Bernoulli Equation

C308

Our work aims to show that under certain circumstances surface tension must be factored into the Bernoulli equation to accurately model the behavior of a fluid in motion. The Bernoulli equation is an expression of the conservation of energy principle applied to a fluid in motion. Therefore, we propose it must include the energy used in creating the surface of the fluid, or surface tension. In the absence of this term, the Bernoulli equation does not account for all energies and hence cannot accurately predict the behavior of a fluid in motion. To test this claim, we consider the scenario of a vertical laminarly flowing fluid with a free surface. As a fluid falls in the manner described, it takes a recognizable shape. It is currently accepted that this shape can be determined using the Bernoulli equation by plotting the radius at any point along the stream as a function of its vertical distance from a fixed point. To test the accuracy of the accepted method, we compare the resultant shape of a laminar vertical stream predicated with the Bernoulli equation to the actual shape of a real fluid flowing under the same conditions. In order to accomplish this, we built a device that puts out a constant laminar stream when a fluid runs through it. As the device runs we photograph the stream to capture its shape. We then isolated the silhouette of the stream and compare it to the predicted shape generated by the Bernoulli equation and an altered equation which includes an additional term that accounts for surface tension. We have found that the altered equation better predicts the shape of the stream and therefore should be considered in relevant applications of fluid dynamics.