Example slides from a few Prosper-based seminar presentations

...with lots of graphics

Matthias Heil & Andrew L. Hazel

                               .
  M.Heil@maths.man.ac.uk     &     ahazel@maths.man.ac.uk
             http://www.maths.man.ac.uk/˜ mheil

Department of Mathematics University of Manchester

Lagrangian wall mechanics

Matthias Heil & Andrew L. Hazel, Department of Mathematics, University of Manchester, UK http://www.maths.man.ac.uk/˜ mheil – p.2/4

Lagrangian wall mechanics

• Principle of virtual displacements for a linearly elastic ring of undeformed radius

R_{0} and thickness h, subject to load f:

∫_{0}^{2π} [γ δγ + \frac{1}{12} ( \frac{h}{R_{0}} )^{2} κ δκ − \frac{1}{12} ( \frac{h}{R_{0}} )^{3} (( \frac{R_{0}}{h} ) f − λ_{T}^{2} \frac{∂^{2}R_{w}}{∂t^{2}} ) · δR_{w}] dζ = 0.

Matthias Heil & Andrew L. Hazel, Department of Mathematics, University of Manchester, UK http://www.maths.man.ac.uk/˜ mheil – p.2/4

Lagrangian wall mechanics

• Principle of virtual displacements for a linearly elastic ring of undeformed radius

R_{0} and thickness h, subject to load f:

∫_{0}^{2π} [γ δγ + \frac{1}{12} ( \frac{h}{R_{0}} )^{2} κ δκ − \frac{1}{12} ( \frac{h}{R_{0}} )^{3} (( \frac{R_{0}}{h} ) f − λ_{T}^{2} \frac{∂^{2}R_{w}}{∂t^{2}} ) · δR_{w}] dζ = 0.

• γ and κ represent the ring’s mid-plane strain and change of curvature, respectively.

Matthias Heil & Andrew L. Hazel, Department of Mathematics, University of Manchester, UK http://www.maths.man.ac.uk/˜ mheil – p.2/4

Lagrangian wall mechanics

• Principle of virtual displacements for a linearly elastic ring of undeformed radius

R_{0} and thickness h, subject to load f:

∫_{0}^{2π} [γ δγ + \frac{1}{12} ( \frac{h}{R_{0}} )^{2} κ δκ − \frac{1}{12} ( \frac{h}{R_{0}} )^{3} (( \frac{R_{0}}{h} ) f − λ_{T}^{2} \frac{∂^{2}R_{w}}{∂t^{2}} ) · δR_{w}] dζ = 0.

• Load non-dimensionalised by bending stiffness K, i.e.

f^{∗} = K f  where  K = \frac{E}{12(1 − ν^{2})} ( \frac{h}{R_{0}} )^{3} .

Matthias Heil & Andrew L. Hazel, Department of Mathematics, University of Manchester, UK http://www.maths.man.ac.uk/˜ mheil – p.2/4

Fluids: Weak form of equations

Momentum:

∫∫∫ [− \frac{∂p}{∂x_{i}} + \frac{∂}{∂x_{j}} ( \frac{∂u_{i}}{∂x_{j}} + \frac{∂u_{j}}{∂x_{i}} )] ψ^{(F)} dV = 0

Fluids: Weak form of equations

Momentum:

∫∫∫ [− \frac{∂p}{∂x_{i}} + \frac{∂}{∂x_{j}} ( \frac{∂u_{i}}{∂x_{j}} + \frac{∂u_{j}}{∂x_{i}} )] ψ^{(F)} dV = 0

[...integrate by parts.]

Fluids: Weak form of equations

Momentum:

∫∫∫ [p \frac{∂ψ^{(F)}}{∂x_{i}} − ( \frac{∂u_{i}}{∂x_{j}} + \frac{∂u_{j}}{∂x_{i}} ) \frac{∂ψ^{(F)}}{∂x_{j}} ] dV

+ ∫∫ [−pn_{i} + ( \frac{∂u_{i}}{∂x_{j}} + \frac{∂u_{j}}{∂x_{i}} ) n_{j}] ψ^{(F)} dS = 0

[...split the surface integral into dS = dS_{f} + dS_{\S_{f}}]

Fluids: Weak form of equations

Momentum:

∫∫∫ [p \frac{∂ψ^{(F)}}{∂x_{i}} − ( \frac{∂u_{i}}{∂x_{j}} + \frac{∂u_{j}}{∂x_{i}} ) \frac{∂ψ^{(F)}}{∂x_{j}} ] dV

+ ∫∫ [−pn_{i} + ( \frac{∂u_{i}}{∂x_{j}} + \frac{∂u_{j}}{∂x_{i}} ) n_{j}] ψ^{(F)} dS_{\S_{f}}

+ ∫∫ [−pn_{i} + ( \frac{∂u_{i}}{∂x_{j}} + \frac{∂u_{j}}{∂x_{i}} ) n_{j}] ψ^{(F)} dS_{f} = 0

[Note: ψ^{(F)} = 0 on S_{\S_{f}}]

Fluids: Weak form of equations

Momentum:

∫∫∫ [p \frac{∂ψ^{(F)}}{∂x_{i}} − ( \frac{∂u_{i}}{∂x_{j}} + \frac{∂u_{j}}{∂x_{i}} ) \frac{∂ψ^{(F)}}{∂x_{j}} ] dV

+ ∫∫ [−pn_{i} + ( \frac{∂u_{i}}{∂x_{j}} + \frac{∂u_{j}}{∂x_{i}} ) n_{j}] ψ^{(F)} dS_{f} = 0

[...use the traction boundary condition on the free surface S_{f}.]

Fluids: Weak form of equations

Momentum:

∫∫∫ [p \frac{∂ψ^{(F)}}{∂x_{i}} − ( \frac{∂u_{i}}{∂x_{j}} + \frac{∂u_{j}}{∂x_{i}} ) \frac{∂ψ^{(F)}}{∂x_{j}} ] dV

− ∫∫ [p_{b} + \frac{1}{Ca} κ] ψ^{(F)}n_{i} dS_{f} = 0

[...apply Weatherburn’s surface divergence theorem to the surface integral.]

Fluids: Weak form of equations

Momentum:

∫∫∫ [p \frac{∂ψ^{(F)}}{∂x_{i}} − ( \frac{∂u_{i}}{∂x_{j}} + \frac{∂u_{j}}{∂x_{i}} ) \frac{∂ψ^{(F)}}{∂x_{j}} ] dV

+ \frac{1}{Ca} ∫∫ \frac{1}{g} [g_{1}]_{i} (g_{22} \frac{∂ψ^{(F)}}{∂ζ_{1}} − g_{12} \frac{∂ψ^{(F)}}{∂ζ_{2}} ) + \frac{1}{g} [g_{2}]_{i} (g_{11} \frac{∂ψ^{(F)}}{∂ζ_{2}} − g_{12} \frac{∂ψ^{(F)}}{∂ζ_{1}} ) dS_{f}

− ∫∫ p_{b}ψ^{(F)}n_{i} dS_{f} − \frac{1}{Ca} ∮ ψ^{(F)}m_{i} ds = 0

Fluids: Weak form of equations

Momentum:

∫∫∫ [p \frac{∂ψ^{(F)}}{∂x_{i}} − ( \frac{∂u_{i}}{∂x_{j}} + \frac{∂u_{j}}{∂x_{i}} ) \frac{∂ψ^{(F)}}{∂x_{j}} ] dV

+ \frac{1}{Ca} ∫∫ \frac{1}{g} [g_{1}]_{i} (g_{22} \frac{∂ψ^{(F)}}{∂ζ_{1}} − g_{12} \frac{∂ψ^{(F)}}{∂ζ_{2}} ) + \frac{1}{g} [g_{2}]_{i} (g_{11} \frac{∂ψ^{(F)}}{∂ζ_{2}} − g_{12} \frac{∂ψ^{(F)}}{∂ζ_{1}} ) dS_{f}

− ∫∫ p_{b}ψ^{(F)}n_{i} dS_{f} − \frac{1}{Ca} ∮ ψ^{(F)}m_{i} ds = 0

m  m mS_{f} s  ^{g^{2}}  ^{g^{1}}

Matthias Heil & Andrew L. Hazel, Department of Mathematics, University of Manchester, UK http://www.maths.man.ac.uk/˜ mheil – p.3/4

Fluids: Weak form of equations

Momentum:

∫∫∫ [p \frac{∂ψ^{(F)}}{∂x_{i}} − ( \frac{∂u_{i}}{∂x_{j}} + \frac{∂u_{j}}{∂x_{i}} ) \frac{∂ψ^{(F)}}{∂x_{j}} ] dV

+ \frac{1}{Ca} ∫∫ \frac{1}{g} [g_{1}]_{i} (g_{22} \frac{∂ψ^{(F)}}{∂ζ_{1}} − g_{12} \frac{∂ψ^{(F)}}{∂ζ_{2}} ) + \frac{1}{g} [g_{2}]_{i} (g_{11} \frac{∂ψ^{(F)}}{∂ζ_{2}} − g_{12} \frac{∂ψ^{(F)}}{∂ζ_{1}} ) dS_{f}

− ∫∫ p_{b}ψ^{(F)}n_{i} dS_{f} − \frac{1}{Ca} ∮ ψ^{(F)}m_{i} ds = 0

Conservation of mass:

∫∫∫ \frac{∂u_{i}}{∂x_{i}} ψ^{(P)} dV = 0

Non-penetration on free surface:

∫∫ u_{i}n_{i}ψ^{(H)} dS_{f} = 0

• Discretise using Taylor–Hood elements

• Solve matrix equations directly by frontal method (HSL 2000)

Results (displayed as a pseudo-animation)

Surface-tension-driven collapse of a liquid-lined elastic ring:

• Fully coupled discretisation of the free-surface Navier-Stokes equations and the equations of large-displacement shell theory.

• Solution by the Newton-Raphson method.

Matthias Heil & Andrew L. Hazel, Department of Mathematics, University of Manchester, UK http://www.maths.man.ac.uk/˜ mheil – p.4/4

• Fully coupled discretisation of the free-surface Navier-Stokes equations and the equations of large-displacement shell theory.

• Solution by the Newton-Raphson method.

Matthias Heil & Andrew L. Hazel, Department of Mathematics, University of Manchester, UK http://www.maths.man.ac.uk/˜ mheil – p.4/4

• Fully coupled discretisation of the free-surface Navier-Stokes equations and the equations of large-displacement shell theory.

• Solution by the Newton-Raphson method.

Matthias Heil & Andrew L. Hazel, Department of Mathematics, University of Manchester, UK http://www.maths.man.ac.uk/˜ mheil – p.4/4

• Fully coupled discretisation of the free-surface Navier-Stokes equations and the equations of large-displacement shell theory.

• Solution by the Newton-Raphson method.

Matthias Heil & Andrew L. Hazel, Department of Mathematics, University of Manchester, UK http://www.maths.man.ac.uk/˜ mheil – p.4/4

Results (displayed as real animation)

Self-excited oscillations during finite Reynolds number flow in a collapsible channel. Start external animation

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Self-excited oscillations during finite Reynolds number flow in a collapsible channel. Start external animation

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Self-excited oscillations during finite Reynolds number flow in a collapsible channel. Start external animation

If this doesn’t work, check that (i) the file Stream.avi is located in the same directory as your pdf file when you display these slides and (ii) that acroread knows how to display avi files (maybe download a later version...)

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Self-excited oscillations during finite Reynolds number flow in a collapsible channel. Start external animation

If this doesn’t work, check that (i) the file Stream.avi is located in the same directory as your pdf file when you display these slides and (ii) that acroread knows how to display avi files (maybe download a later version...)

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