ABSTRACT
Consider the steady, laminar, axially symmetric viscous flow of an incompressible fluid induced by an infinite disk rotating steadily with angular velocity Ω about the z-axis in a cylindrical coordinate system (r, θ, z). The motion of the fluid is governed by the continuity equation
1 r
∂(rVr) ∂r
+ 1 r
∂Vθ ∂θ
+ ∂Vz ∂z
= 0 (17.1)
and the Navier-Stokes equations
Vr ∂Vr ∂r
+ Vz ∂Vr ∂z
− V 2 θ
r = ν
[ ∂2Vr ∂r2
+ 1 r
∂Vr ∂r
+ ∂2Vr ∂z2
− Vr r2
] − 1
ρ
∂p
∂r , (17.2)
Vr ∂Vθ ∂r
+ Vz ∂Vθ ∂z
+ VrVθ r
= ν [ ∂2Vθ ∂r2
+ 1 r
∂Vθ ∂r
+ ∂2Vθ ∂z2
− Vθ r2
] , (17.3)
Vr ∂Vz ∂r
+ Vz ∂Vz ∂z
= ν [ ∂2Vz ∂r2
+ 1 r
∂Vz ∂r
+ ∂2Vz ∂z2
] − 1
ρ
∂p
∂z , (17.4)
subject to the nonslip boundary conditions
Vθ = rΩ, Vr = Vz = 0, when z = 0, (17.5)
and the conditions at infinity
Vr = Vθ = 0, when z = +∞, (17.6) where ρ denotes the fluid density, ν the kinematic viscosity coefficient, p the pressure, and Vr, Vθ, Vz the velocity components in the radial, azimuthal, and axial directions, respectively. Defining the similarity variable
η = z
√ Ω ν
(17.7)
and using the similarity transformation
Vr = (rΩ) f(η), (17.8) Vθ = (rΩ) g(η), (17.9)
Vz = √
νΩ w(η), (17.10) p = −ρνΩ P (η), (17.11)
equations to (17.6) to a set of ordinary differential equations
f ′′ = f2 − g2 + f ′ w, (17.12) g′′ = g′ w + 2f g, (17.13) w w′ = P ′ + w′′, (17.14)
2f + w′ = 0, (17.15)
subject to the boundary conditions
f(0) = f(+∞) = 0, g(0) = 1, g(+∞) = 0, w(0) = 0, (17.16) where the prime denotes the derivative with respect to η. From (17.15),
f = −w ′
2 . (17.17)
Substituting it into Equations (17.12) and (17.13), we have
w′′′ − w′′ w + 1 2 w′ w′ − 2g2 = 0, (17.18)
g′′ − w g′ + w′ g = 0, (17.19) subject to the boundary conditions
w(0) = w′(0) = w′(+∞) = 0, g(0) = 1, g(+∞) = 0. (17.20) For details the reader is referred to Von Ka´rma´n [112] and Zandbergen and Dijkstra [113].
