Question

According to a mathematical model, the velocity v of the airstream in a snall circular pipe (here. the trachea) is related to the radius ro of the pipe by v(r)=k(r0​−r)r2,2r0​​≤r≤r0​ where k>0 and r is the radial distance from the center of the pipe. (The model is not good for r<π0​/2.)

(a) Slietch a graph of this cubic polynomial using solely the multiplicity of ench root and the limiting behaviout as r→±[infinity]. Sketch the polynomial for all r as a dasbed line, then hightight the restricted domain by a solid line.
(b) Find the absolute maximum valus of von the interval [2r0​​,r0​]. Add the corresponding point(s) into your graph in (a).

Answer 1

The question deals with the velocity of an airstream in the trachea based on its radius, the calculation of the Reynolds number, and determining whether the flow is laminar or turbulent.

Step-by-step explanation:

The question asks to analyze the velocity of an airstream within a trachea modeled by a mathematical relationship between velocity (v) and the radial distance (r) within a range, and to find the maximum value of v on the given interval. When dealing with a fluid flowing through a constriction such as in a pipe or Venturi tube, the speed of the fluid increases as the cross-sectional area decreases, due to the conservation of mass and the principle of continuity. This concept, along with calculations of the Reynolds number, determines whether the flow is laminar or turbulent.

To calculate the average air speed in the trachea, we use the equation v(r) = k(r0 - r)r2, taking into account the given radius r0 is approximately 10-2 m. For the Reynolds number, which is a dimensionless value used to predict flow patterns in different fluid flow situations, we would need the density of the air p, the viscosity of the air μ, as well as the average speed ū. The formula for the Reynolds number Rec is Rec = (pūr0)/μ).

As for whether the airflow would be laminar or turbulent, this would depend on the value of the Reynolds number calculated: a value below approximately 2000 suggests laminar flow, while a value above that suggests turbulence.