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A brand of toothpaste is made using a mixture of white and red paste. During the manufacturing process, both pastes are pumped through a pipe, with the red paste lying in the centre of the pipe, as shown in Figure Q2. The pipe has an internal radius, R₂, and the segment of red paste has a radius R₁. The flow is found to follow the Poiseuille equation, i.e. u(r): 1 dp 4μ dx (R² - 1²) Where is the viscosity of the paste and toothpaste through the pipe is Qtot, and the flowrate of the red paste is Qred. is the pressure gradient. The total flow of The toothpaste has a density of p = 1200 kg/m³, and a viscosity of 0.03 Pa.s. The pipe has a radius R₂ = 5 cm and a length L = 2 m.

A brand of toothpaste is made using a mixture of white and red paste. During the manufacturing process, both pastes are pumped through a pipe, with the red paste lying in the centre of the pipe, as shown in Figure Q2. The pipe has an internal radius, R₂, and the segment of red paste has a radius R₁. The flow is found to follow the Poiseuille equation, i.e. u(r): 1 dp 4μ dx (R² - 1²) Where is the viscosity of the paste and toothpaste through the pipe is Qtot, and the flowrate of the red paste is Qred. is the pressure gradient. The total flow of The toothpaste has a density of p = 1200 kg/m³, and a viscosity of 0.03 Pa.s. The pipe has a radius R₂ = 5 cm and a length L = 2 m.

Elements Of Electromagnetics
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
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Engineers find that when the shear stress at the interface between the red and white paste exceeds 0.5 Pa the colours begin to mix. If ?1 = 2 cm, what is the maximum pressure drop along the pipe that can be achieved without the colours smudging?

R₁ R₂ u(y) Figure Q2: Flow of toothpaste of two different colours.
Transcribed Image Text:R₁ R₂ u(y) Figure Q2: Flow of toothpaste of two different colours.
A brand of toothpaste is made using a mixture of white and red paste. During the manufacturing process, both pastes are pumped through a pipe, with the red paste lying in the centre of the pipe, as shown in Figure Q2. The pipe has an internal radius, R₂, and the segment of red paste has a radius R₁. The flow is found to follow the Poiseuille equation, i.e. u(r): 1 dp 4u dx - (R² - r²) Where μ is the viscosity of the paste and is the pressure gradient. The total flow of dx toothpaste through the pipe is Qtot, and the flowrate of the red paste is Qred. The toothpaste has a density of p = 1200 kg/m³, and a viscosity of 0.03 Pa.s. The pipe has a radius R₂ = 5 cm and a length L = 2 m.
Transcribed Image Text:A brand of toothpaste is made using a mixture of white and red paste. During the manufacturing process, both pastes are pumped through a pipe, with the red paste lying in the centre of the pipe, as shown in Figure Q2. The pipe has an internal radius, R₂, and the segment of red paste has a radius R₁. The flow is found to follow the Poiseuille equation, i.e. u(r): 1 dp 4u dx - (R² - r²) Where μ is the viscosity of the paste and is the pressure gradient. The total flow of dx toothpaste through the pipe is Qtot, and the flowrate of the red paste is Qred. The toothpaste has a density of p = 1200 kg/m³, and a viscosity of 0.03 Pa.s. The pipe has a radius R₂ = 5 cm and a length L = 2 m.
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