chapter 36 - Vestibular Mechanics - KEMT FEI TUKE

The gel layer is treated as a Kelvin-Voight viscoelastic material where the gel shear stress has both an
elastic component and a viscous component acting in parallel. This viscoelastic material model is substituted
into the momentum equation, and the resulting gel layer equation of motion is
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(36.2)
with boundary and initial conditions: w(b,t) = v(t); w(0,t) = 0; w(y g,0) = 0; δ g (y g,0) = 0. The elastic term
in the equation is written in terms of the integral of velocity with respect to time, instead of displacement,
so the equation is in terms of a single dependent variable, the velocity.
The otoconial layer equation was developed using Newton’s second law of motion, equating the forces
that act on the otoconial layer—fluid shear, gel shear, buoyancy, and weight—to the product of mass
and inertial acceleration. The resulting otoconial layer equation is
( f ) ∂
ρobρo ρ
v ∂
+ −
∂ t
with the initial condition v(0) = 0.
ρ g
w
t G
∂
∂ =
⎛
∂ w
⎞
w
⎜ dt
⎜
⎟ +µ g
⎝ ∂y
⎟
g ⎠ y g
∂
2
2
2
2
∂
36.3 Nondimensionalization of the Motion Equations
t
∫0
Vs
u
g x f
∂t y f
−
⎛
⎡ ⎤
⎢ ⎥ =µ ⎜ ∂
⎣⎢
⎦⎥
⎜ ∂
⎝
⎞
⎟ G
⎟
⎠
−
⎛
⎜ ∂w
⎜ ∂y
⎝
⎞ ⎛
⎟ w
dt +µ ⎜ ∂
g ⎟ ⎜ ∂y
⎠ ⎝
0
y
g
y
g
f= 0
g= b y g= b
(36.3)
The equations of motion are then nondimensionalized to reduce the number of physical and dimensional
parameters and combine them into some useful nondimensional numbers. The following nondimensional
variables, which are indicated by overbars, are introduced into the motion equations:
y
f
y y
u v
y t t u v w
b b b V V
w
f
g f
= g = =
V
µ ⎛ ⎞
⎜ ⎟ = = =
2
⎝ ⎠
ρ o
(36.4)
Several nondimensional parameters occur naturally as a part of the nondimensionalization process. These
parameters are
ρ
R =
ρ
f
o
2
Gb µ ⎛ 2
ρ g b ⎞
o ρo
= M = gx= ⎜ ⎟ g
2
µ µ f ⎝ Vµ
f ⎠
f
(36.5)
These parameters represent the following: R is the density ratio, ε is a nondimensional elastic parameter,
M is the viscosity ratio and represents a major portion of the system damping, and – g x is the nondimensional
gravity.
The governing equations of motion in nondimensional form are then as follows. For the endolymph
fluid layer
R u
2
∂ ∂ u
=
∂ t ∂
(36.6)
with boundary conditions of – u (0, – t) = – v ( – t ) and – u (∞, – t) = 0 and initial conditions of – u ( – y f, 0) = 0. For
the otoconial layer
2
y f
∫
t
x
⎞
⎟
⎟
⎠