Understanding Critical Flow and Minimum M in Fluid Mechanics

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SUMMARY

The discussion focuses on determining the minimum M (Mmin) in fluid mechanics, specifically when the Froude number (Fr) equals one, represented by the equation Fr = (Q^2)B/g(A^3). Participants express confusion regarding the transition from δA to B and the justification of dividing by ∂y in the derivation process. The need for clarity on the representation of B and its relationship to δA is emphasized, as well as the distinction between parameters on the left and right pages of the provided material.

PREREQUISITES
  • Understanding of fluid mechanics principles, specifically Froude number.
  • Familiarity with the concepts of flow parameters such as A, Q, and g.
  • Knowledge of calculus, particularly differentiation and its applications in fluid dynamics.
  • Ability to interpret mathematical derivations in the context of physical phenomena.
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  • Research the derivation of the Froude number in fluid mechanics.
  • Study the relationship between flow parameters A, Q, and g in fluid dynamics.
  • Learn about the significance of minimum M in various flow scenarios.
  • Examine the mathematical justification for dividing by parameters in fluid equations.
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This discussion is beneficial for students and professionals in fluid mechanics, particularly those studying flow dynamics and mathematical modeling in engineering contexts.

fonseh
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Homework Statement


I am not sure how to get the Mmin . Why δA will become B (refer to the pink circled part)

Homework Equations

The Attempt at a Solution


Is there anything wrong with the author's steps ? I know the author's intention is to show that the Mmin occur when the Fr = (Q^2)B/ g(A^3) = 1 , but we have (Q^2)δA / g(A^3) ... How to make the B=δA ?
 

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I'm afraid I cannot make sense of the development over those two pages.
On the left page, the parameter appears to be distance along the stream. At any such distance, there is an A, a ##\bar y## etc. The height, y, is not a parameter here.
On the right page we are concerned with the minimum M has a function of height, and yet the equations appear to be derived from those on the left page. The step "dividing by ∂y" is not adequately justified by that description.
I see no explanation of what B represents.
 

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