Consider the steady-state flow of an incompressible Newtonian fluid (viscosity µ, density p) through a horizontal, circular tube made from an elastic material. Since the tube is elastic, its radius depends on the local pressure in the fluid. Since the pressure decreases for increasing z, the radius of the tube also decreases with the z-coordinate. The relationship between the radius (R) and local fluid pressure (P) is given by R= R where the exponent r is a coefficient that represents the compliance (i.e., reciprocal of stiffness) of the elastic and Ro is the radius at z = 0. The elastic is this particular tube is fairly stiff, such that r«1 and so the tube radius does not vary much between z= 0 and z = L. R(2) As shown in the sketch, the pressure measured at two positions L apart (with L>» Ro) are Po and Pt respectively. P= P, P- P. The flow rate within the tube is such that Re = pR1 v+/µ <« 1 (where v, is the centerline velocity and R1, is the radius at z = L). Since the fluid is incompressible, the volumetric flow in this tube is the same for all z. However, note that the pressure gradient ƏPləz cannot be a constant since the tube radius changes with z. a) By analyzing the flow and pressure in this tube (e.g., by scaling and solving the appropriate balance equations), determine R(2) for 0 S2SL. b) Determine the volumetric flow rate within this tube in terms of Po, Pt, and other appropriate variables. Please provide analytical solutions (equations), not just scaling estimates for both parts (a) and (b).
Consider the steady-state flow of an incompressible Newtonian fluid (viscosity µ, density p) through a horizontal, circular tube made from an elastic material. Since the tube is elastic, its radius depends on the local pressure in the fluid. Since the pressure decreases for increasing z, the radius of the tube also decreases with the z-coordinate. The relationship between the radius (R) and local fluid pressure (P) is given by R= R where the exponent r is a coefficient that represents the compliance (i.e., reciprocal of stiffness) of the elastic and Ro is the radius at z = 0. The elastic is this particular tube is fairly stiff, such that r«1 and so the tube radius does not vary much between z= 0 and z = L. R(2) As shown in the sketch, the pressure measured at two positions L apart (with L>» Ro) are Po and Pt respectively. P= P, P- P. The flow rate within the tube is such that Re = pR1 v+/µ <« 1 (where v, is the centerline velocity and R1, is the radius at z = L). Since the fluid is incompressible, the volumetric flow in this tube is the same for all z. However, note that the pressure gradient ƏPləz cannot be a constant since the tube radius changes with z. a) By analyzing the flow and pressure in this tube (e.g., by scaling and solving the appropriate balance equations), determine R(2) for 0 S2SL. b) Determine the volumetric flow rate within this tube in terms of Po, Pt, and other appropriate variables. Please provide analytical solutions (equations), not just scaling estimates for both parts (a) and (b).
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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Transcribed Image Text:Consider the steady-state flow of an incompressible Newtonian fluid (viscosity u, density p) through a
horizontal, circular tube made from an elastic material. Since the tube is elastic, its radius depends on
the local pressure in the fluid. Since the pressure decreases for increasing z, the radius of the tube also
decreases with the z-coordinate.
The relationship between the radius (R) and local fluid pressure (P) is given by
P
R= R
P.
where the exponent r is a coefficient that represents
the compliance (ie., reciprocal of stiffness) of the
elastic and Ro is the radius at z = 0. The elastic is
this particular tube is fairly stiff, such that I << 1
and so the tube radius does not vary much between
z= 0 and z = L.
R(:)
As shown in the sketch, the pressure measured at
two positions L apart (with L> Ro) are Po and Pi
respectively.
z=0
z=L
P= Po
P= P.
The flow rate within the tube is such that Re = pR, v/u << 1 (where v is the centerline velocity and
R, is the radius at z= L). Since the fluid is incompressible, the volumetric flow in this tube is the same
for all z. However, note that the pressure gradient aPlaz cannot be a constant since the tube radius
changes with z.
a) By analyzing the flow and pressure in this tube (e.g., by scaling and solving the
appropriate balance equations), determine R(2) for 0 <zSL.
b) Detemine the volumetric flow rate within this tube in terms of Po, Pt, and other
appropriate variables.
Please provide analytical solutions (equations), not just scaling estimates for both parts (a) and (b).
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