1. The flow is one-dimensional.2. The density of the fluid is constant.3. The flow is laminar, which means that there are no rotational effects in the flow.Mathematically, this is equivalent to saying that the curl of the flow velocity is zero.4. The flow is sufficiently slow such that all nonlinear terms of the unknown variables (p, v) andtheir derivatives can be neglected.5. The resistance to the flow is assumed to be proportional to the velocity at a constant rate r.6. The leakage through the wall is assumed to be proportional to the pressure at a constant rateg.7. The cross-sectional area of the vessel, S, is a function of the pressure only.(S++at - S)Ax = (vS);At – (vS)x+axAt – gpSAxAt,(E1)Volume atVolumeVolumeVolumeVolume coming outthe endat startcoming in coming out through the wallpsAx(Av)(pS)x – (pS)x+ax – vrSAx.(E2)Ata. Show and explain that when Ax and At approach zero, equation (E1) can be rewrittenas:as a(v$)+ gpS = 0.(E3)atb. Showand explain that when Ax and At approach zero, equation (E2) can be rewrittenas:a(ps)axvrS.(E4)c. Let the ability to stretch c of the vessel wall be defined as the change in S per unitarea per unit change in p, or1 dsS dpGiven that S is only a function of the pressure p as previously assumed, and p is afunction of time and space, show and explain that equations (E3) and (E4) can beexpressed as follows:ap+cv+gp = 0.(ES.1)avParp+rv = 0.(ES.2)

Answered: Image /qna-images/answer/3fbbb8ce-ed6a-4885-9b0d-fd1856406ac3.jpg

(Sz+at – S;)Ax = (vS),At – (vS)x+axAt – gpSAxAt, (E1) Volume at Volume Volume Volume Volume coming out the end at start coming in coming out through the wall psax(Av) At (pS)x – (pS)x+Ax – vrSAx. (E2) a. Showand explain that when Ax and At approach zero, equation (E1) can be rewritten as: as , a(vs) + gpS = 0. (E3) at ax b. Show and explain that when Ax and At approach zero, equation (E2) can be rewritten as: a(ps) ps (E4) vrS. c. Let the ability to stretch c of the vessel wall be defined as the change in S per unit area per unit change in p, or 1 ds S dp Given that S is only a function of the pressure p as previously assumed, and p is a function of time and space, show and explain that equations (E3) and (E4) can be expressed as follows:

(Sz+at – S;)Ax = (vS),At – (vS)x+axAt – gpSAxAt, (E1) Volume at Volume Volume Volume Volume coming out the end at start coming in coming out through the wall psax(Av) At (pS)x – (pS)x+Ax – vrSAx. (E2) a. Showand explain that when Ax and At approach zero, equation (E1) can be rewritten as: as , a(vs) + gpS = 0. (E3) at ax b. Show and explain that when Ax and At approach zero, equation (E2) can be rewritten as: a(ps) ps (E4) vrS. c. Let the ability to stretch c of the vessel wall be defined as the change in S per unit area per unit change in p, or 1 ds S dp Given that S is only a function of the pressure p as previously assumed, and p is a function of time and space, show and explain that equations (E3) and (E4) can be expressed as follows:

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Question

1. The flow is one-dimensional.
2. The density of the fluid is constant.
3. The flow is laminar, which means that there are no rotational effects in the flow.
Mathematically, this is equivalent to saying that the curl of the flow velocity is zero.
4. The flow is sufficiently slow such that all nonlinear terms of the unknown variables (p, v) and
their derivatives can be neglected.
5. The resistance to the flow is assumed to be proportional to the velocity at a constant rate r.
6. The leakage through the wall is assumed to be proportional to the pressure at a constant rate
g.
7. The cross-sectional area of the vessel, S, is a function of the pressure only.
(S++at - S)Ax = (vS);At – (vS)x+axAt – gpSAxAt,
(E1)
Volume at
Volume
Volume
Volume
Volume coming out
the end
at start
coming in coming out through the wall
psAx(Av)
(pS)x – (pS)x+ax – vrSAx.
(E2)
At
a. Show and explain that when Ax and At approach zero, equation (E1) can be rewritten
as:
as a(v$)
+ gpS = 0.
(E3)
at
b. Showand explain that when Ax and At approach zero, equation (E2) can be rewritten
as:
a(ps)
ax
vrS.
(E4)
c. Let the ability to stretch c of the vessel wall be defined as the change in S per unit
area per unit change in p, or
1 ds
S dp
Given that S is only a function of the pressure p as previously assumed, and p is a
function of time and space, show and explain that equations (E3) and (E4) can be
expressed as follows:
ap
+cv+gp = 0.
(ES.1)
av
Par
p+rv = 0.
(ES.2)

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