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Derivation and solution of the equations for fluid
flow in a helical channel
Hayden TronnoloneSupervisor: Yvonne Stokes
The University of Adelaide
School of Mathematical Sciences
November 2, 2011
Overview Equations Thin-film approximation Solution Existence
Gravity separation spirals
c© 2004 Tiomin Resources INC. (TIO) All rights reserved.
Derivation and solution of the equations for fluid flow in a helical channel University of Adelaide
Overview Equations Thin-film approximation Solution Existence
Secondary flow
Derivation and solution of the equations for fluid flow in a helical channel University of Adelaide
Overview Equations Thin-film approximation Solution Existence
Water slides
www.kulin.wa.gov.au
commons.wikimedia.org www.copacabanaresort.com
Derivation and solution of the equations for fluid flow in a helical channel University of Adelaide
Overview Equations Thin-film approximation Solution Existence
Flow in a river
Deposited
material
Outer bank
eroded
Surface
Rotating
secondary
flow
Inner bank
Outer bank
Derivation and solution of the equations for fluid flow in a helical channel University of Adelaide
Overview Equations Thin-film approximation Solution Existence
◮ This can result in the development of a meander.
◮ Cauto River (Rıo Cauto), Cuba (from Wikimedia Commons)
Derivation and solution of the equations for fluid flow in a helical channel University of Adelaide
Overview Equations Thin-film approximation Solution Existence
Mathematical modelling
◮ It is very difficult to measure thin flows without interferingwith them.
◮ Aim to develop a mathematical model of the fluid flow; asystem of equations that describes the real-world process.
◮ The equations need to capture the important aspects of thesystem while remaining tractable.
Derivation and solution of the equations for fluid flow in a helical channel University of Adelaide
Overview Equations Thin-film approximation Solution Existence
◮ The governing equations are◮ the Navier–Stokes equations for an incompressible fluid
(Newton’s second law); and◮ the continuity equation (conservation of mass).
Derivation and solution of the equations for fluid flow in a helical channel University of Adelaide
Overview Equations Thin-film approximation Solution Existence
◮ The governing equations are◮ the Navier–Stokes equations for an incompressible fluid
(Newton’s second law); and◮ the continuity equation (conservation of mass).
◮ We seek steady-state solutions; we assume the velocity is notchanging with time.
Derivation and solution of the equations for fluid flow in a helical channel University of Adelaide
Overview Equations Thin-film approximation Solution Existence
◮ The governing equations are◮ the Navier–Stokes equations for an incompressible fluid
(Newton’s second law); and◮ the continuity equation (conservation of mass).
◮ We seek steady-state solutions; we assume the velocity is notchanging with time.
◮ We also seek helically-symmetric solutions; we assume thevelocity (and hence geometry) in any cross section along thechannel.
Derivation and solution of the equations for fluid flow in a helical channel University of Adelaide
Overview Equations Thin-film approximation Solution Existence
◮ The helix can be describedby its curvature and torsion.
◮ We assume both are small.
Derivation and solution of the equations for fluid flow in a helical channel University of Adelaide
Overview Equations Thin-film approximation Solution Existence
Boundary conditions
◮ To complete our description, we need to say what happens atthe boundaries.
◮ Friction causes the fluid to “stick” to the channel wall, so thevelocity is zero there.
Derivation and solution of the equations for fluid flow in a helical channel University of Adelaide
Overview Equations Thin-film approximation Solution Existence
Boundary conditions
◮ To complete our description, we need to say what happens atthe boundaries.
◮ Friction causes the fluid to “stick” to the channel wall, so thevelocity is zero there.
◮ On the free surface, we have two boundary conditions.◮ Fluid particles on the free surface stay on the free surface.◮ There is no stress on the free surface (we ignore the effects of
surface tension).
Derivation and solution of the equations for fluid flow in a helical channel University of Adelaide
Overview Equations Thin-film approximation Solution Existence
Governing equations
Continuity equation:∂v
∂y+
∂w
∂z= 0
Navier–Stokes equations:
v∂u
∂y+ w
∂u
∂z= ∇2u +
R sinα
F2
v∂v
∂y+ w
∂v
∂z−
1
2Ku2 = −
∂p
∂y+∇2v
v∂w
∂y+ w
∂w
∂z= −
∂p
∂z+∇2w −
R2 cosα
F2
R = Uaν, F = U
√ga, K = 2ǫR2
Derivation and solution of the equations for fluid flow in a helical channel University of Adelaide
Overview Equations Thin-film approximation Solution Existence
Thin-film approximation
Derivation and solution of the equations for fluid flow in a helical channel University of Adelaide
Overview Equations Thin-film approximation Solution Existence
Thin-film approximation
◮ Rescale the vertical co-ordinate. Let
z =z
δ,
where 0 < δ ≪ 1 is some small aspect ratio.
◮ We substitute into the previous system and determinecorresponding scales on v ,w and p. We keep terms of leadingorder in δ (the largest terms).
Derivation and solution of the equations for fluid flow in a helical channel University of Adelaide
Overview Equations Thin-film approximation Solution Existence
Thin-film equations
Continuity equation:∂v
∂y+
∂w
∂z= 0
Navier–Stokes equations:
∂2u
∂z2+ sinα = 0
−∂p
∂y+
∂2v
∂z2+ χu2 = 0
−∂p
∂z− cosα = 0
χ = δK2R
Derivation and solution of the equations for fluid flow in a helical channel University of Adelaide
Overview Equations Thin-film approximation Solution Existence
Thin-film solution
◮ The thin-film solution is (dropping checks on variables)
p(y , z) = cosα(H + h − z)
u(y , z) =sinα
2(z − H)(H + 2h − z)
v(y , z) =−χ sin2 α
120(z − H){(z − H)3[(H + 2h − z)
× (H + 4h − z) + 2h2]− 16h5}
−cosα
2(z − H)(H + 2h − z)
∂
∂y(H + h)
Derivation and solution of the equations for fluid flow in a helical channel University of Adelaide
Overview Equations Thin-film approximation Solution Existence
◮ To complete the solution we need to find the free surfaceshape, which is given by solving
d
dy(H + h) =
6η
35h4
where η = χ sin2 αcosα
.
Derivation and solution of the equations for fluid flow in a helical channel University of Adelaide
Overview Equations Thin-film approximation Solution Existence
◮ How “good” is the thin-film model? How well does itcompare to the more complicated model?
◮ To answer this, we must try to solve the more complicatedequations using a numerical method (aided by ComsolMultiphysics).
◮ We compare the thin-film and numerical solutions to assessthe accuracy of the thin-film model.
◮ Compare solutions in a channel of rectangular cross section aswe can find analytical solutions for this case.
Derivation and solution of the equations for fluid flow in a helical channel University of Adelaide
Overview Equations Thin-film approximation Solution Existence
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
y
z
Navier–Stokes model (numerical solution)Thin-film approximation ( δ = 0.08 )
Derivation and solution of the equations for fluid flow in a helical channel University of Adelaide
Overview Equations Thin-film approximation Solution Existence
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
y
z
Navier–Stokes model (numerical solution)Thin-film approximation ( δ = 0.1 )
Derivation and solution of the equations for fluid flow in a helical channel University of Adelaide
Overview Equations Thin-film approximation Solution Existence
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
y
z
Navier–Stokes model (numerical solution)Thin-film approximation ( δ = 0.12 )
Derivation and solution of the equations for fluid flow in a helical channel University of Adelaide
Overview Equations Thin-film approximation Solution Existence
Flux down the channel
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0
0.1
0.2
0.3
0.4
0.5
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0
0.1
0.2
0.3
0.4
0.5
Thin-film (left) and Navier–Stokes (right).
Derivation and solution of the equations for fluid flow in a helical channel University of Adelaide
Overview Equations Thin-film approximation Solution Existence
Existence
◮ We are interested in the existence of the thin-film solution asδ increases.
◮ The thin-film solution depends upon determining the shape ofthe free-surface, h, which is given by solving
dh
dy=
6η
35h4, η =
χ sin2 α
cosα, χ =
δK
2R.
◮ Whether we can solve this or not depends upon the quantity η.
Derivation and solution of the equations for fluid flow in a helical channel University of Adelaide
Overview Equations Thin-film approximation Solution Existence
◮ Integrating the differential equation for the free surface shapeyields
h(y) =
(
h−3ℓ −
18η
35(1 + y)
)
−13
,
where hℓ is a constant.
◮ Solving for this constant reduces to finding the roots of apolynomial.
Derivation and solution of the equations for fluid flow in a helical channel University of Adelaide
Overview Equations Thin-film approximation Solution Existence
0.5 1 1.5 2 2.5 3 3.5
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
η
Roots
We find no root, and hence no solutions, for η > 3.28129.
Derivation and solution of the equations for fluid flow in a helical channel University of Adelaide
Overview Equations Thin-film approximation Solution Existence
Conclusions
◮ We have developed a mathematical model of a fluid flow in ahelical channel and derived a simplified model for shallowflows.
◮ For a rectangular channel cross-section, the thin-film solutionagrees with the numerical solution, especially along the freesurface.
◮ In this geometry, the ability to find a solution depends on theparameter η.
◮ It is possible to extend this analysis to different channelgeometries.
Derivation and solution of the equations for fluid flow in a helical channel University of Adelaide