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MATHEMATICAL TRIPOS Part III
Wednesday 4 June 2003 9 to 11
PAPER 75
PHYSIOLOGICAL FLUID DYNAMICS
Attempt TWO questions.
There are four questions in total. The questions carry equal weight.
Candidates may use their lecture notes, any material handed out during the course and examples classes, and any hand-written or typed notes, taken from sources outside the lectures, which they have prepared themselves.
You may not start to read the questions
printed on the subsequent pages until
instructed to do so by the Invigilator.
1 Measurements of pressure and flow-rate wave forms at fixed sites in arteries show that, within a single cardiac cycle, the time t 1 at which the pressure is maximum is later than the time t 2 at which the flow rate is maximum. Conventional measurements at peripheral sites show that the time difference t 1 − t 2 decreases with increasing age. However, recent measurements in the ascending aorta indicate that t 1 − t 2 increases with age.
You are invited to seek to explain all the above findings by modelling the propa- gation of the pulse wave and its reflection at the aortic bifurcation, which is known to be a site at which the net cross-sectional area of the vessels decreases. It is also known that arteries become stiffer with age, but the geometry of the aortic bifurcation is relatively unaffected by age. Be explicit about all assumptions and approximations in your model.
[Hint: it is suggested that the peripherally-travelling part of the pulse wave in the aorta is modelled as a cosine wave, in which the pressure is
p = PI cos [ω (t − x/c 1 )] ,
where x, t are longitudinal coordinate and time, PI is a constant amplitude, ω is angular frequency and c 1 is wave speed.]
Paper 75
3 Fully-developed steady flow along an annular channel of width ˆh has a velocity profile ˆu = U uˆ 0 (y), 0 6 y 6 1 ,
where the radial coordinate is ˆr = hˆ(R + y), and in which the magnitudes of the shear-rate on the two walls are different, i.e.
u′ 0 (0) = γ 0 , u′ 0 (1) = −γ 1 , γ 0 > γ 1 > 0
and u 0 (0) = u 0 (1) = 0. Here Uˆ is a velocity scale, and R, γ 0 , γ 1 are dimensionless constants. Do NOT calculate u 0 (y) explicitly.
Axisymmetric perturbations to this flow can be analysed in the same way as for a planar channel, apart from the fact that the continuity equation is ˆuxˆ + (^1) rˆ (ˆrυˆ)ˆr = 0 where (ˆx, rˆ) are cylindrical polar coordinates with corresponding velocity components (ˆu, ˆυ), and the viscous terms are also modified.
The inner wall of the channel, y = 0, is subjected to a time-dependent indentation, y = F (x, t),
where F (x, t) = 0 for x 6 0 and x > 1, x = ˆx/λh, t = ωtˆ (ˆt is dimensional time), ω is characteristic frequency and λ, are dimensionless quantities such that
λ 1 , 1.
The Reynolds number is Re = Uˆ ˆh/ν 1; the Strouhal number is St = ωh/ˆ Uˆ 1.
(i) Explain carefully the relative orders of magnitude of the parameters , λ, Re, St that
(a) permit the flow to be analysed as an inviscid core with two boundary layers on the walls, of dimensionless thickness δ ; and (b) allow the dimensionless longitudinal velocity in the core to be written as
u = u 0 (y) +
A(x, t) R + y
u′ 0 (y) + ^2 u 2 (x, y, t) +....
Show that A(x, t) satisfies the following partial differential equation, as long as the boundary layer thickness remains of O(δ) everywhere:
σAxxx − βα 1 At − α 2 AAx = βγ 0 Ft + γ 02 F Fx +
γ^20 R
(AF )x, (∗)
where
β = λSt−^1 , σ = λ−^2 −^1
0
u^20 (y) R + y
dy, α 1 =
γ 0 R
γ 1 R + 1
, α 2 =
γ 02 R^2
γ^21 (R + 1)^2
(ii) Deduce that small amplitude (linear) sinusoidal waves can propagate downstream (and not upstream), with group velocity equal to three times the phase velocity.
For regions in which F = 0, investigate nonlinear waves of permanent form, given by A(ξ) where ξ = x + ct, and such that A and all its derivatives tend to zero smoothly as ξ → ±∞. By integrating equation (*) twice in that case, show (for example graphically) that such waves can propagate upstream (but not downstream), with A < 0 and |A|max = 3βα 1 c/α 2.
[The above is a model of a cardiac assist device consisting of a balloon mounted axisymmetrically on a catheter in the aorta, and inflated periodically.]
Paper 75
4 A model for a red blood cell passing steadily down an otherwise plasma-filled cap- illary consists of an axisymmetric elastic body of unstressed radius r 0 (x), −L 6 x 6 L, where x is the longitudinal coordinate, surrounded by incompressible viscous fluid con- tained in a rigid cylinder of radius a. Near x = 0, r 0 (x) is approximately parabolic:
r 0 (x) ≈ r 00 −
κx^2 ,
where κ > 0 and r 00 may be assumed to be greater than a. The cell elasticity is modelled linearly, so that the pressure in the lubricating film of fluid around the cell is given by
p = p 0 + α[r 0 (x) − r(x)],
where p 0 , α are positive constants and r(x) is the actual cell radius. The “cell” moves in the +x direction with speed U , and the pressure in the plasma behind the cell exceeds that in front by ∆p. The goal is to find a relationship between ∆p and U , on the assumption that inertia is negligible.
Taking axes fixed in the cell, use lubrication theory to analyse the flow in the lubricating film, showing in particular that
dp dx
6 μU h^2
12 μQ h^3
where h(x) is the film thickness, μ is the fluid viscosity and − 2 πaQ is the (unknown) volume flow rate of fluid past the cell in the +x direction. Write down boundary conditions at x = ±L. What further conditions must be imposed to complete the formulation of the problem?
Setting h = (2Q/U )H and x =
( 2 Q/U
κ
X, show that the problem can be
reduced to: dH dx
H^2
H^3
= X (1)
with
H(− L˜) − H( L˜) = Cλ
∫ L˜
− L˜
3 H
H^2
dX, (2)
where
C =
2 Q
U a
, λ =
6 μU ακ^1 /^2 (2Q/U )^5 /^2
, L˜ = L
κ 2 Q/U
Then a) express ∆p as a multiple of the left hand side of equation (2);
b) explain why self-consistency of the model requires C 1; c) seek a solution in the limit of small λ and large L˜, of the form
H = H 0 (x) + λH 1 (x) +....
Show that H 0 =
b^2 + X^2
Paper 75 [TURN OVER
