M1E508 - Fluids of Biological Systems

UNIVERSITY OF TORONTO

DEPT. MECHANICAL & INDUSTRIAL ENGINEERING

Final Exam

Tuesday, December 18, 2018, 2 pm - 4:30 pm, BA3116

Examiner: Edmond W.K. Young

NOTES:

This is a "Type D" examination. One personal double-sided 8.5" x 11" aid sheet

is allowed.

Calculators are permitted.

Attempt all parts of all questions - each part has the value indicated.

Time limit is 2.5 hours.

Maximum 100 marks.

1

Question 1

[30 marks] Answer the following short-answer questions.

(a) Name 5 types of biofluid that were studied in the course, and name the organ

system or tissue in which they can be found. Each answer should be written as a

pair: fluid and system. (5 marks)

(b) Name 3 types of biofluid that were not studied in this course, and name the organ

system or tissue in which they can be found. (S marks)

(c) Define or describe the following terminology:

Rouleaux formation (2 marks)

Hematocrit (2 marks)

Womersley number (2 marks)

Casson fluid (2 marks)

Windkessel model for arterial flow (2 marks)

(d) The pleural cavity of the lung contains intrapleural fluid, which has intrapleu-

ral pressure. What is the magnitude of this intrapleural pressure relative to

atmospheric pressure (positive or negative), and what is the biomechanical reason

why this pressure is necessary for respiratory health? (6 marks)

(e) A liquid plug is formed between two parallel plates that converge from left to right,

as shown in Figure 1 below. There are two air-liquid interfaces, one on the left with

radius of curvature R1 , and one on the right with radius of curvature R2. Use the

Young-Laplace equation for surface tension to determine whether the plug moves

left or right. (6 marks)

Does the plug move left or right?

liquid plug

Figure 1: Plug moving in a converging channel. Does the plug move left or right?

91

Question 2

[24 marks] A viscous liquid flows through the sectional cylindrical vessel shown in

Figure 2 below. The vessel has three sections with different lengths and radii. The

second (middle) section is filled with a porous hydrogel with permeability K, while the

first and last sections are simply filled with the fluid.

= 20 mm

L2 = 4.0 mm

L3 = 36 mm

= 1.2 mm

= 0.8 mm

= 0.4 mm

= 0.001 Pa-s

Q = 1.6uL/s

P3 = 40 Pa

Po = 0

K=?

r1 '2

- - -- - - - - -r-I

P3 =40 Pa P2 P1 P0 =0

Figure 2: A porous hydrogel is positioned in the middle section.

What must be the permeability K of the hydrogel, in units of [mm2]? (12 marks)

'What must be the pressure before and after the hydrogel, P2 and Pi, respectively,

in units of [Pa]? (12 marks)

ri

Question 3

[30 marks] A biomaterial is found to have viscoelastic properties that can be approxi-

mated by a Zener model, consisting of two elastic "spring" components and one viscous

"damper" component (Figure 3). Note that the left side by itself is essentially the fa-

miliar Voigt body, with one spring and one damper in parallel. The springs have shear

moduli C1 and C2, respectively, and the damper has viscosity , as shown in the figure.

At t = 0+, the Zener biomaterial is exposed to constant deformation 'yo, leading to stress

relaxation over time (see Figure below).

lo, t<o -

1'Yo, t>0

Derive a single differential equation relating total strain of the Zener body 'y to

total stress of the Zener body T. Your solution should include C1, C2, ji, 'y and r

only, with the component strains and stresses eliminated. (12 marks)

Solve the differential equation above for the shear stress response (t) for t > 0

(i.e., relaxation). Include 'ye, C1, C2, and i in your solution. (12 marks)

State the time constant of the response in terms of the variables given, and draw

the basic shape of the stress relaxation graph. (6 marks)

Note: Just as a reminder, the stress-strain relationship for the Voigt model in Assignment

#2 was:

TtotaJ = G(t) + p d'y(t)

dt

5

Gi

constant deformation T(t)

YO - 7 S

t=O t=O

Figure 3: Viscoelastic material represented by a Zener model exposed to constant defor-

mation, leading to stress relaxation.

Question 4

[16 marks] A simple 2D model is desired for studying the velocity profile u(y) of lymph

in a lymphatic vessel, where there is leakage into the vessel from one side of the wall,

and leakage out of the vessel from the other side of the wall.

It is desirable to simplify the geometry to look at flow between two flat porous plates,

as shown in Figure 4. In this way, the bottom porous plate allows lymph to leak in,

modelling the bottom half of the lymph vessel, and the top porous plate allows lymph

to leak out, modelling the top half of the lymph vessel.

For simplicity, consider that the flow is driven by a constant applied pressure gradient

of —dp/dx. Flow through both plates is upward such that the y-component velocity is

constant, v = V,. Assume that the plates are infinitely large in the x-z plane (into the

page), and assume the flow is fully developed everywhere. The goal is to solve for the

x-component velocity profile, n = u(y), representing the velocity profile of lymph as it

flows between these two porous walls.

v = V (leakage out)

y y = +h

4:= 4= 4= f =~ f

/ui

(centreline)

appiie pressure v = V (leakage in) gradient

Figure 4: Flow between two flat porous plates with bottom injection and top suction.

ro

For V,, = 0, what type of flow does this problem become? Derive the velocity

profile u = u(y) for V = 0. (8 marks)

For V > 0, use the Navier-Stokes equations to derive the appropriate ordinary

differential equation (ODE), and write the boundary conditions (BCs) needed

to solve u = u(y). (8 marks)

(Don't try to solve the ODE! Just show how you arrive at the ODE and the boundary

conditions.)

Note: For your interest, the solution to the ODE is:

u(y) -

2

I

y - 1

e - em1

Umax - Re h +

sinh Re ]

(ö]

AID SHEET

Governing Equations

Navier-Stokes Equations, Cartesian coordinates:

Coordinates F = (x,y,z)

Velocity = (u,v,w)

(Note: for constant density and constant viscosity fluids)

Du Dv 3w —+—+— ax ay Dz

Iwo

Du Du DuI p

[Du

at x ay Dzj

Dv Dv Dy 1 p

[Dv

at x ay Dzj

Dw 3w 3w1 p [aw

x Dy Dzj

Dp D2u D2u D2u = --+

ax Dx +—+-- +f Dy Dz

Dp 32v (92V D2v = —-- +i +f

Dp D2w 92w D2w = --+i

Dz Dx —+----+-- +f

Dy Dz

Navier-Stokes Equations, cylindrical coordinates:

Coordinates i? = (r, 0, z)

Velocity '7 = (Ur, u9, u)

(Note: for constant density and constant viscosity fluids)

1 D 1Du0 Du --(rur)+—+ = rDr r 9 Dz

au, au, u9 Dur u Dur =

at Dr r DO r Dz

Du9 Du9 u9 Du9 urue NO

Dr rDO r Dz

Du Du u0 Du Du =

at Dr rDO Dz

1DP 1V _ur 2Du0 ' --- +1-i• Ur j+fr pDr r

rpDO llDP _uo 2Dur

----- +1' U8 r2 r2 ao

1Dp 2 + vV u2 + f p Dz

where

V2 1D(D\ 1D2 32

Reynold's Transport Theorem:

Drn

JVW

ap Dt DtS

Stress Tensor

The shear stress tensor for a Newtonian fluid of constant density is defined as follows:

i'a ni = ( +

\0Xi 3x

Trigonometric ]Identities

sin(A + B) = sin A cos B + cos A sin B

cos (A + B) = cos A cos B sin A sin B

sin 2 X + c0s2 x = 1

Calculus

Chain Rule:

f(g(x)) = f(g(x))g'(x) dx

Product Rule:

xx = f'xgx + f(xg'(x) dx

Quotient Rule: d (f(x)) f'(x)g(x) - f(x)g(x)

dx g(x) - (g(x))2

Porous Media Flows

Kozeny-Carman relationship for permeability:

E3 K

2n2S2

where e is the porosity, 'r is the tortuosity, and S is the surface area-to-volume ratio.

Flow rate through a single cylindrical straight pore:

-

Qpore nr4 /p

8p L

Darcy's Law: Q KLp

AJ1L

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