chapter 36 - Vestibular Mechanics - KEMT FEI TUKE

FIGURE 36.5 Schematic structure of the semicircular canal showing a cross-section through the canal duct and
utricle. Also shown in the upper right corner is a cross-section of the duct. R is the radius of curvature of the
semicircular canal, a is the inside radius to the duct wall, and r is the spacial coordinate in the radial direction of
the duct.
We are interested in the flow of endolymph fluid with respect to the duct wall, and this requires that
the inertial motion of the duct wall RΩ be added to the fluid velocity u measured with respect to the
duct wall. The curvature of the duct can be shown to be negligible since a R, and no secondary flow
is induced; thus the curve duct can be treated as straight. Pressure gradients arise from two sources in
the duct: (1) the utricle, (2) the cupula. The cupula when deflected exerts a restoring force on the
endolymph. The cupula can be modeled as a membrane with linear stiffness K = ∆p/∆V, where ∆p is the
pressure difference across the cupula and ∆V is the volumetric displacement, where
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t a
= π ( )
0 0
∆V 2 u r, t rdr dt
(36.11)
If the angle subtended by the membranous duct is denoted by β, the pressure gradient in the duct
produced by the cupula is
The utricle pressure gradient can be approximated [see Van Buskirk, 1977] by
∫
∫
∂
∂ =
p K∆V z βR
∂
∂ =
p ( 2π−β) ρRα z β
(36.12)
(36.13)
When this information is substituted into the Navier-Stokes equation, the following governing equation
for endolymph flow relative to the duct wall is obtained:
( ) +
∂
∂ +
⎛ π⎞
⎜ ⎟ =−
⎝ ⎠
π
u
K
∂ ⎛ ∂ ⎞
R
rdr dt v
t ∫ ∫
∂
⎜r
R
r r⎝
∂
⎟
⎠
u
t a
2 2 1
α
u
β ρβ 0 0
r
(36.14)