Equations related to Bernoulli's equation?

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SUMMARY

This discussion focuses on the application of Bernoulli's equation in predicting soap incorporation in a shower head design. The user is developing a soap-dispensing shower head that utilizes the viscosity of soap and water flow dynamics to determine soap flow rates. Key insights reveal that while Bernoulli's principle is relevant, it primarily applies to inviscid fluids, necessitating additional considerations for viscous fluids. The user seeks equations that can provide a prediction accuracy of 20-25% for soap flow based on water flow rates.

PREREQUISITES
  • Understanding of Bernoulli's equation and its applications
  • Knowledge of fluid dynamics, particularly viscosity effects
  • Familiarity with engineering principles related to fluid flow
  • Basic mathematical skills for equation manipulation and analysis
NEXT STEPS
  • Research the Navier-Stokes equations for viscous fluid flow
  • Explore empirical models for soap and water mixing ratios
  • Investigate flow rate measurement techniques for shower heads
  • Learn about the effects of mesh design on fluid dynamics
USEFUL FOR

Engineers, product designers, and researchers involved in fluid dynamics and product development, particularly those focusing on innovative shower head designs and fluid mixing technologies.

DyslexicHobo
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I'm writing an engineering white paper on a soap-dispensing shower head. The product will have a canister of soap on top of a basic generic shower head separated by a mesh so that due to the viscosity of the soap, the flow of the water rushing by will be the main determining factor on how much soap comes through (rather than gravity pulling it at a constant rate). From what I have researched, the pulling of the soap into the water would be due to Bernoulli's principle.

I'm just wondering if there's any equations that I can use to accurately (within 20-25%, even) predict how much soap will be incorporated into the water at a given rate of flow for the water.
 
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You mentioned that the main mechanism is viscosity, while the Bernoulli equation is based on an inviscid fluid. Thoughts?
 

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