Smoothed particle modelling of vapour-liquid coexistence Numerical - - PowerPoint PPT Presentation

Background Van der Waals Hydrodynamics SPH model Results Summary

Smoothed particle modelling of vapour-liquid coexistence

Numerical modelling of condensation in a quenched van der Waals fluid using a smoothed particle continuum method.

• A. Charles1

P . Daivis1

1Condensed Matter Theory Group

RMIT University

May 2008 / Applied Physics Seminar

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary

Outline

1

Background Smoothed Particles- the general idea Some mathematics A brief history of SPAM

2

Van der Waals Hydrodynamics Equilibrium - coexistence and interfaces Hydrodynamics

3

SPH Model

4

Results Droplets Films Spinodal Decomposition technology

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Smoothed Particles- the general idea Smooth particle interpolation A brief history of SPAM

Outline

1

Background Smoothed Particles- the general idea Some mathematics A brief history of SPAM

2

Van der Waals Hydrodynamics Equilibrium - coexistence and interfaces Hydrodynamics

3

SPH Model

4

Results Droplets Films Spinodal Decomposition technology

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Smoothed Particles- the general idea Smooth particle interpolation A brief history of SPAM

Smoothed particles

the general idea

SPH: smoothed particle hydrodynamics / SPAM: smoothed particle applied mechanics A numerical technique for solving problems in continuum mechanics. Occupies the same tool-space as Finite Element, Finite Difference Grid, Lattice Boltzmann, and Spectral Harmonic solvers.

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Smoothed Particles- the general idea Smooth particle interpolation A brief history of SPAM

Smoothed particles

the general idea

SPH: smoothed particle hydrodynamics / SPAM: smoothed particle applied mechanics A numerical technique for solving problems in continuum mechanics. Occupies the same tool-space as Finite Element, Finite Difference Grid, Lattice Boltzmann, and Spectral Harmonic solvers.

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Smoothed Particles- the general idea Smooth particle interpolation A brief history of SPAM

Smoothed particles

the general idea

SPH: smoothed particle hydrodynamics / SPAM: smoothed particle applied mechanics A numerical technique for solving problems in continuum mechanics. Occupies the same tool-space as Finite Element, Finite Difference Grid, Lattice Boltzmann, and Spectral Harmonic solvers.

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Smoothed Particles- the general idea Smooth particle interpolation A brief history of SPAM

Smoothed particles

the general idea

A continuous field is mapped by a set of arbitrarily distributed interpolation points which represent macroscopic elements.

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Smoothed Particles- the general idea Smooth particle interpolation A brief history of SPAM

Smoothed particles

the general idea

The field properties are mapped to the interpolation points (particles). Particles have mechanical and thermal properties (mass, temperature) The particles move according to equations of motion derived from an integral approximation of the continuum equations. Animation: little_expansion.avi, which shows a very small number of particles moving.

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Smoothed Particles- the general idea Smooth particle interpolation A brief history of SPAM

Smoothed particles

the general idea

The field properties are mapped to the interpolation points (particles). Particles have mechanical and thermal properties (mass, temperature) The particles move according to equations of motion derived from an integral approximation of the continuum equations. Animation: little_expansion.avi, which shows a very small number of particles moving.

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Smoothed Particles- the general idea Smooth particle interpolation A brief history of SPAM

Smoothed particles

the general idea

The field properties are mapped to the interpolation points (particles). Particles have mechanical and thermal properties (mass, temperature) The particles move according to equations of motion derived from an integral approximation of the continuum equations. Animation: little_expansion.avi, which shows a very small number of particles moving.

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Smoothed Particles- the general idea Smooth particle interpolation A brief history of SPAM

Smoothed particles

the general idea

The field properties are mapped to the interpolation points (particles). Particles have mechanical and thermal properties (mass, temperature) The particles move according to equations of motion derived from an integral approximation of the continuum equations. Animation: little_expansion.avi, which shows a very small number of particles moving.

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Smoothed Particles- the general idea Smooth particle interpolation A brief history of SPAM

Smoothed particles

the general idea

The density around each interpolation point (particle) is given by a kernel function.

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Smoothed Particles- the general idea Smooth particle interpolation A brief history of SPAM

Smoothed particles

the general idea

The value of any physical property at any point can be constructed by a summation over particles.

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Smoothed Particles- the general idea Smooth particle interpolation A brief history of SPAM

Smoothed particles

the general idea

A density dependent summation of over neighbouring particles gives us the field value at any position.

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Smoothed Particles- the general idea Smooth particle interpolation A brief history of SPAM

Smoothed particles

the general idea

Particles are not molecules Particles behave like classical particles in a molecular dynamics simulation. The effective inter-particle potential is soft, reasonably long ranged, and density dependent. volume. animation - gas_expansion_small: shows a compressed gas expanding to fill an available volume above the critical temperature.

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Smoothed Particles- the general idea Smooth particle interpolation A brief history of SPAM

Smoothed particles

the general idea

Particles are not molecules Particles behave like classical particles in a molecular dynamics simulation. The effective inter-particle potential is soft, reasonably long ranged, and density dependent. volume. animation - gas_expansion_small: shows a compressed gas expanding to fill an available volume above the critical temperature.

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Smoothed Particles- the general idea Smooth particle interpolation A brief history of SPAM

Smoothed particles

the general idea

Particles are not molecules Particles behave like classical particles in a molecular dynamics simulation. The effective inter-particle potential is soft, reasonably long ranged, and density dependent. volume. animation - gas_expansion_small: shows a compressed gas expanding to fill an available volume above the critical temperature.

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Smoothed Particles- the general idea Smooth particle interpolation A brief history of SPAM

Smoothed particles

the general idea

Particles are not molecules Particles behave like classical particles in a molecular dynamics simulation. The effective inter-particle potential is soft, reasonably long ranged, and density dependent. volume. animation - gas_expansion_small: shows a compressed gas expanding to fill an available volume above the critical temperature.

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Smoothed Particles- the general idea Smooth particle interpolation A brief history of SPAM

Smoothed particles

a most attractive numerical technique

the PDEs of continuum mechanics are transformed into a set of ODEs governing the motion of particles Particles as represent macroscopic elements, and (usually) behave intuitively Particles move with the streaming velocity of the ßuid - Lagrangian forumations of continuum mechanics are less complicated.

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Smoothed Particles- the general idea Smooth particle interpolation A brief history of SPAM

Smoothed particles

a most attractive numerical technique

the PDEs of continuum mechanics are transformed into a set of ODEs governing the motion of particles Particles as represent macroscopic elements, and (usually) behave intuitively Particles move with the streaming velocity of the ßuid - Lagrangian forumations of continuum mechanics are less complicated.

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Smoothed Particles- the general idea Smooth particle interpolation A brief history of SPAM

Smoothed particles

a most attractive numerical technique

the PDEs of continuum mechanics are transformed into a set of ODEs governing the motion of particles Particles as represent macroscopic elements, and (usually) behave intuitively Particles move with the streaming velocity of the ßuid - Lagrangian forumations of continuum mechanics are less complicated.

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Smoothed Particles- the general idea Smooth particle interpolation A brief history of SPAM

Smoothed particles

a very attractive numerical technique

Easy to code, a wealth of existing techniques for MD and

• ther particle methods can be applied.

Clear (though non-unique) relationship between continuum equations and particle equations. Good published results for a range of states - gases, liquids, solids.

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Smoothed Particles- the general idea Smooth particle interpolation A brief history of SPAM

Smoothed particles

a very attractive numerical technique

Easy to code, a wealth of existing techniques for MD and

• ther particle methods can be applied.

Clear (though non-unique) relationship between continuum equations and particle equations. Good published results for a range of states - gases, liquids, solids.

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Smoothed Particles- the general idea Smooth particle interpolation A brief history of SPAM

Smoothed particles

a very attractive numerical technique

Easy to code, a wealth of existing techniques for MD and

• ther particle methods can be applied.

Clear (though non-unique) relationship between continuum equations and particle equations. Good published results for a range of states - gases, liquids, solids.

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Smoothed Particles- the general idea Smooth particle interpolation A brief history of SPAM

Outline

1

Background Smoothed Particles- the general idea Some mathematics A brief history of SPAM

2

Van der Waals Hydrodynamics Equilibrium - coexistence and interfaces Hydrodynamics

3

SPH Model

4

Results Droplets Films Spinodal Decomposition technology

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Smoothed Particles- the general idea Smooth particle interpolation A brief history of SPAM

Smoothed particles

justification

A continuous field can be expressed as an integral of delta functions. f(r) =

• f(r′)δ(r − r′)dr′

(1) Substitute a Gaussian-like kernel W for the delta function to "smooth" and put dr′ = dxdydz = dV ′ f(r) =

• f(r′)W(r − r′, h)dV ′

(2)

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Smoothed Particles- the general idea Smooth particle interpolation A brief history of SPAM

Smoothed particles

justification

A continuous field can be expressed as an integral of delta functions. f(r) =

• f(r′)δ(r − r′)dr′

(1) Substitute a Gaussian-like kernel W for the delta function to "smooth" and put dr′ = dxdydz = dV ′ f(r) =

• f(r′)W(r − r′, h)dV ′

(2)

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Smoothed Particles- the general idea Smooth particle interpolation A brief history of SPAM

Smoothed particles

justification

Rewrite the volume integral dV ′ = dm(r′)

ρ(r′)

f(r) =

• f(r′)dm(r′)

ρ(r′) W(r − r′, h) (3) Divide the system into small finite elements such that dm = mi, and replace the integral with a sum. ˆ f(r) =

N

• i=1

fi mi ρi W(r − ri, h) (4)

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Smoothed Particles- the general idea Smooth particle interpolation A brief history of SPAM

Smoothed particles

justification

Rewrite the volume integral dV ′ = dm(r′)

ρ(r′)

f(r) =

• f(r′)dm(r′)

ρ(r′) W(r − r′, h) (3) Divide the system into small finite elements such that dm = mi, and replace the integral with a sum. ˆ f(r) =

N

• i=1

fi mi ρi W(r − ri, h) (4)

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Smoothed Particles- the general idea Smooth particle interpolation A brief history of SPAM

Summation Interpolant

Summation Interpolant

ˆ f(r) =

N

• i=1

fi mi ρi W(r − ri, h) ρ(r) =

N

• i=1

ρiW(r − ri, h)

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Smoothed Particles- the general idea Smooth particle interpolation A brief history of SPAM

The Smoothing Kernel

Smoothing Kernels

Kernels with gaussian shape but finite extent are most widely

• used. In theory any even, normalised function will do.

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Smoothed Particles- the general idea Smooth particle interpolation A brief history of SPAM

Mathematical Justification

Derivatives

ˆ ∇f(r) =

N

• i=1

mi ρi fi∇W(r − ri, h) (5) Most forces depend on spatial and temporal gradients - the relation above is needed to construct SP versions of the equations of motion.

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Smoothed Particles- the general idea Smooth particle interpolation A brief history of SPAM

Outline

1

Background Smoothed Particles- the general idea Some mathematics A brief history of SPAM

2

Van der Waals Hydrodynamics Equilibrium - coexistence and interfaces Hydrodynamics

3

SPH Model

4

Results Droplets Films Spinodal Decomposition technology

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Smoothed Particles- the general idea Smooth particle interpolation A brief history of SPAM

history

Developed for astrophysical simulations by Joe Monaghan, and Lucy in the 1970s Applied to non-equilibrium thermodynamics by William Hoover and co-workers in the 1980s: renamed Smoothed Particle Applied Mechanics. Refinined for studying incompressible fluids with viscosity and surface tension by JP Morris in the 90s

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Smoothed Particles- the general idea Smooth particle interpolation A brief history of SPAM

history

Developed for astrophysical simulations by Joe Monaghan, and Lucy in the 1970s Applied to non-equilibrium thermodynamics by William Hoover and co-workers in the 1980s: renamed Smoothed Particle Applied Mechanics. Refinined for studying incompressible fluids with viscosity and surface tension by JP Morris in the 90s

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Smoothed Particles- the general idea Smooth particle interpolation A brief history of SPAM

history

Developed for astrophysical simulations by Joe Monaghan, and Lucy in the 1970s Applied to non-equilibrium thermodynamics by William Hoover and co-workers in the 1980s: renamed Smoothed Particle Applied Mechanics. Refinined for studying incompressible fluids with viscosity and surface tension by JP Morris in the 90s

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Smoothed Particles- the general idea Smooth particle interpolation A brief history of SPAM

history

Used for studies of entropy, stability, compressible flows and phase transitions and solid fracture by Posch, Kum, Hoover in the 90s. Developers of DPD method incorporate SPH style density dependence into their technique in the late 90s (Warren, Espanol) Astrophysics: numerous advances, the technique is standardised and stable, several high quality codes freely available.

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Smoothed Particles- the general idea Smooth particle interpolation A brief history of SPAM

history

Used for studies of entropy, stability, compressible flows and phase transitions and solid fracture by Posch, Kum, Hoover in the 90s. Developers of DPD method incorporate SPH style density dependence into their technique in the late 90s (Warren, Espanol) Astrophysics: numerous advances, the technique is standardised and stable, several high quality codes freely available.

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Smoothed Particles- the general idea Smooth particle interpolation A brief history of SPAM

history

Used for studies of entropy, stability, compressible flows and phase transitions and solid fracture by Posch, Kum, Hoover in the 90s. Developers of DPD method incorporate SPH style density dependence into their technique in the late 90s (Warren, Espanol) Astrophysics: numerous advances, the technique is standardised and stable, several high quality codes freely available.

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Smoothed Particles- the general idea Smooth particle interpolation A brief history of SPAM

applications

Binary systems (Thieulot, others). Arctic sea ice (Lindsay and Stern). Die casting (Paul Cleary’s group at CSIRO). Solidification (Monaghan). One component vapour-liquid coexistence. (Nugent and Posch) Special effects for the film industry(CSIRO). Swimming efficiency (CSIRO). Capillary phenomena (Tartatovsky, Meakin) Ocean wave modelling. (Various)

• Explosions. (Liu)

Solid fracture and penetration. (Hoover) Origin of the moon. (Benz)

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Smoothed Particles- the general idea Smooth particle interpolation A brief history of SPAM

applications

Binary systems (Thieulot, others). Arctic sea ice (Lindsay and Stern). Die casting (Paul Cleary’s group at CSIRO). Solidification (Monaghan). One component vapour-liquid coexistence. (Nugent and Posch) Special effects for the film industry(CSIRO). Swimming efficiency (CSIRO). Capillary phenomena (Tartatovsky, Meakin) Ocean wave modelling. (Various)

• Explosions. (Liu)

Solid fracture and penetration. (Hoover) Origin of the moon. (Benz)

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Smoothed Particles- the general idea Smooth particle interpolation A brief history of SPAM

applications

Binary systems (Thieulot, others). Arctic sea ice (Lindsay and Stern). Die casting (Paul Cleary’s group at CSIRO). Solidification (Monaghan). One component vapour-liquid coexistence. (Nugent and Posch) Special effects for the film industry(CSIRO). Swimming efficiency (CSIRO). Capillary phenomena (Tartatovsky, Meakin) Ocean wave modelling. (Various)

• Explosions. (Liu)

Solid fracture and penetration. (Hoover) Origin of the moon. (Benz)

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Smoothed Particles- the general idea Smooth particle interpolation A brief history of SPAM

applications

Binary systems (Thieulot, others). Arctic sea ice (Lindsay and Stern). Die casting (Paul Cleary’s group at CSIRO). Solidification (Monaghan). One component vapour-liquid coexistence. (Nugent and Posch) Special effects for the film industry(CSIRO). Swimming efficiency (CSIRO). Capillary phenomena (Tartatovsky, Meakin) Ocean wave modelling. (Various)

• Explosions. (Liu)

Solid fracture and penetration. (Hoover) Origin of the moon. (Benz)

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Smoothed Particles- the general idea Smooth particle interpolation A brief history of SPAM

applications

Binary systems (Thieulot, others). Arctic sea ice (Lindsay and Stern). Die casting (Paul Cleary’s group at CSIRO). Solidification (Monaghan). One component vapour-liquid coexistence. (Nugent and Posch) Special effects for the film industry(CSIRO). Swimming efficiency (CSIRO). Capillary phenomena (Tartatovsky, Meakin) Ocean wave modelling. (Various)

• Explosions. (Liu)

Solid fracture and penetration. (Hoover) Origin of the moon. (Benz)

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Smoothed Particles- the general idea Smooth particle interpolation A brief history of SPAM

applications

Binary systems (Thieulot, others). Arctic sea ice (Lindsay and Stern). Die casting (Paul Cleary’s group at CSIRO). Solidification (Monaghan). One component vapour-liquid coexistence. (Nugent and Posch) Special effects for the film industry(CSIRO). Swimming efficiency (CSIRO). Capillary phenomena (Tartatovsky, Meakin) Ocean wave modelling. (Various)

• Explosions. (Liu)

Solid fracture and penetration. (Hoover) Origin of the moon. (Benz)

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Smoothed Particles- the general idea Smooth particle interpolation A brief history of SPAM

applications

Binary systems (Thieulot, others). Arctic sea ice (Lindsay and Stern). Die casting (Paul Cleary’s group at CSIRO). Solidification (Monaghan). One component vapour-liquid coexistence. (Nugent and Posch) Special effects for the film industry(CSIRO). Swimming efficiency (CSIRO). Capillary phenomena (Tartatovsky, Meakin) Ocean wave modelling. (Various)

• Explosions. (Liu)

Solid fracture and penetration. (Hoover) Origin of the moon. (Benz)

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Smoothed Particles- the general idea Smooth particle interpolation A brief history of SPAM

applications

Binary systems (Thieulot, others). Arctic sea ice (Lindsay and Stern). Die casting (Paul Cleary’s group at CSIRO). Solidification (Monaghan). One component vapour-liquid coexistence. (Nugent and Posch) Special effects for the film industry(CSIRO). Swimming efficiency (CSIRO). Capillary phenomena (Tartatovsky, Meakin) Ocean wave modelling. (Various)

• Explosions. (Liu)

Solid fracture and penetration. (Hoover) Origin of the moon. (Benz)

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Smoothed Particles- the general idea Smooth particle interpolation A brief history of SPAM

applications

Binary systems (Thieulot, others). Arctic sea ice (Lindsay and Stern). Die casting (Paul Cleary’s group at CSIRO). Solidification (Monaghan). One component vapour-liquid coexistence. (Nugent and Posch) Special effects for the film industry(CSIRO). Swimming efficiency (CSIRO). Capillary phenomena (Tartatovsky, Meakin) Ocean wave modelling. (Various)

• Explosions. (Liu)

Solid fracture and penetration. (Hoover) Origin of the moon. (Benz)

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Smoothed Particles- the general idea Smooth particle interpolation A brief history of SPAM

applications

Binary systems (Thieulot, others). Arctic sea ice (Lindsay and Stern). Die casting (Paul Cleary’s group at CSIRO). Solidification (Monaghan). One component vapour-liquid coexistence. (Nugent and Posch) Special effects for the film industry(CSIRO). Swimming efficiency (CSIRO). Capillary phenomena (Tartatovsky, Meakin) Ocean wave modelling. (Various)

• Explosions. (Liu)

Solid fracture and penetration. (Hoover) Origin of the moon. (Benz)

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Smoothed Particles- the general idea Smooth particle interpolation A brief history of SPAM

applications

Binary systems (Thieulot, others). Arctic sea ice (Lindsay and Stern). Die casting (Paul Cleary’s group at CSIRO). Solidification (Monaghan). One component vapour-liquid coexistence. (Nugent and Posch) Special effects for the film industry(CSIRO). Swimming efficiency (CSIRO). Capillary phenomena (Tartatovsky, Meakin) Ocean wave modelling. (Various)

• Explosions. (Liu)

Solid fracture and penetration. (Hoover) Origin of the moon. (Benz)

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Smoothed Particles- the general idea Smooth particle interpolation A brief history of SPAM

applications

Binary systems (Thieulot, others). Arctic sea ice (Lindsay and Stern). Die casting (Paul Cleary’s group at CSIRO). Solidification (Monaghan). One component vapour-liquid coexistence. (Nugent and Posch) Special effects for the film industry(CSIRO). Swimming efficiency (CSIRO). Capillary phenomena (Tartatovsky, Meakin) Ocean wave modelling. (Various)

• Explosions. (Liu)

Solid fracture and penetration. (Hoover) Origin of the moon. (Benz)

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Equilibrium - coexistence and interfaces Hydrodynamics

Outline

1

Background Smoothed Particles- the general idea Some mathematics A brief history of SPAM

2

Van der Waals Hydrodynamics Equilibrium - coexistence and interfaces Hydrodynamics

3

SPH Model

4

Results Droplets Films Spinodal Decomposition technology

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Equilibrium - coexistence and interfaces Hydrodynamics

van der Waals hydrodynamics

Use the van der Waals equation of state to close the continuum momentum and energy equations. More or less the Navier-Stokes equations with the isotropic pressure obtained from the van der Waas equation of state.

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Equilibrium - coexistence and interfaces Hydrodynamics

van der Waals hydrodynamics

Use the van der Waals equation of state to close the continuum momentum and energy equations. More or less the Navier-Stokes equations with the isotropic pressure obtained from the van der Waas equation of state.

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Equilibrium - coexistence and interfaces Hydrodynamics

van der Waals’ equation of state

p = NkT V − Nb − N2a V 2 (6) Possibly the simplest equation of state to exhibit a phase transition. Mean field attraction and hard-core volume exclusion Region with negative compressibility defines the phase coexistence region.

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Equilibrium - coexistence and interfaces Hydrodynamics

van der Waals’ equation of state

p = NkT V − Nb − N2a V 2 (6) Possibly the simplest equation of state to exhibit a phase transition. Mean field attraction and hard-core volume exclusion Region with negative compressibility defines the phase coexistence region.

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Equilibrium - coexistence and interfaces Hydrodynamics

van der Waals’ equation of state

p = NkT V − Nb − N2a V 2 (6) Possibly the simplest equation of state to exhibit a phase transition. Mean field attraction and hard-core volume exclusion Region with negative compressibility defines the phase coexistence region.

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Equilibrium - coexistence and interfaces Hydrodynamics

Maxwell Construction

Employ the Maxwell construction to predict two-phase coexistence at constant pressure. Equal area/ free energy common tangent constructions are equivalent.

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Equilibrium - coexistence and interfaces Hydrodynamics

Maxwell Construction

Employ the Maxwell construction to predict two-phase coexistence at constant pressure. Equal area/ free energy common tangent constructions are equivalent.

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Equilibrium - coexistence and interfaces Hydrodynamics

Coexistence in T and ρ

Equilbirum two phase coexistence and stability is can be expressed graphically in the (T,ρ) plane. Right: the spinodal (unstable) region is shaded. Left: contour shading of the pressure.

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Equilibrium - coexistence and interfaces Hydrodynamics

van der Waals to scale

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Equilibrium - coexistence and interfaces Hydrodynamics

reduced units

p = ρ ¯ kbT 1 − ρ¯ b − ¯ aρ2 (7) Reduced units, with density as independent variable. The temperature is given by the caloric van der Waals equation of

• state. In this set of reduced units (Nugent and Posch)

¯ a = 2.0, ¯ b = 0.5, ¯ kb = 1 and Tc 1.2 T = u + ¯ aρ ¯ kb (8)

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Equilibrium - coexistence and interfaces Hydrodynamics

Outline

1

Background Smoothed Particles- the general idea Some mathematics A brief history of SPAM

2

Van der Waals Hydrodynamics Equilibrium - coexistence and interfaces Hydrodynamics

3

SPH Model

4

Results Droplets Films Spinodal Decomposition technology

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Equilibrium - coexistence and interfaces Hydrodynamics

Continuity Equation

Continuity Equation

We use the Lagrangian (co-moving) formulation of the equations of motion. Good derivation of this material in Evans’ book which is available for free online. dρ dt = −ρ∇ · v (9)

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Equilibrium - coexistence and interfaces Hydrodynamics

Continuity Equation

Continuity Equation

We use the Lagrangian (co-moving) formulation of the equations of motion. Good derivation of this material in Evans’ book which is available for free online. dρ dt = −ρ∇ · v (9)

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Equilibrium - coexistence and interfaces Hydrodynamics

Momentum Equation

momentum eq

dv dt = −1 ρ∇ · P All the detail is in the constitutive relations used for the pressure

• tensor. vdW equation of state, shear and bulk viscosity.

P = ρ ¯ kbT 1 − ρ¯ b − ¯ aρ2

• 1 − 2η∇vos − (ηv∇ · v) 1

In more recent work we’ve added a density gradient term, which is needed to account for the mechanical stability of diffuse interfaces.

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Equilibrium - coexistence and interfaces Hydrodynamics

Momentum Equation

momentum eq

dv dt = −1 ρ∇ · P All the detail is in the constitutive relations used for the pressure

• tensor. vdW equation of state, shear and bulk viscosity.

P = ρ ¯ kbT 1 − ρ¯ b − ¯ aρ2

• 1 − 2η∇vos − (ηv∇ · v) 1

In more recent work we’ve added a density gradient term, which is needed to account for the mechanical stability of diffuse interfaces.

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Equilibrium - coexistence and interfaces Hydrodynamics

Momentum Equation

momentum eq

dv dt = −1 ρ∇ · P All the detail is in the constitutive relations used for the pressure

• tensor. vdW equation of state, shear and bulk viscosity.

P = ρ ¯ kbT 1 − ρ¯ b − ¯ aρ2

• 1 − 2η∇vos − (ηv∇ · v) 1

In more recent work we’ve added a density gradient term, which is needed to account for the mechanical stability of diffuse interfaces.

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Equilibrium - coexistence and interfaces Hydrodynamics

Momentum Equation

momentum eq

dv dt = −1 ρ∇ · P All the detail is in the constitutive relations used for the pressure

• tensor. vdW equation of state, shear and bulk viscosity.

P = ρ ¯ kbT 1 − ρ¯ b − ¯ aρ2

• 1 − 2η∇vos − (ηv∇ · v) 1

In more recent work we’ve added a density gradient term, which is needed to account for the mechanical stability of diffuse interfaces.

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Equilibrium - coexistence and interfaces Hydrodynamics

Momentum Equation

momentum eq

dv dt = −1 ρ∇ · P All the detail is in the constitutive relations used for the pressure

• tensor. vdW equation of state, shear and bulk viscosity.

P = ρ ¯ kbT 1 − ρ¯ b − ¯ aρ2

• 1 − 2η∇vos − (ηv∇ · v) 1

In more recent work we’ve added a density gradient term, which is needed to account for the mechanical stability of diffuse interfaces.

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Equilibrium - coexistence and interfaces Hydrodynamics

Momentum Equation

momentum eq

dv dt = −1 ρ∇ · P All the detail is in the constitutive relations used for the pressure

• tensor. vdW equation of state, shear and bulk viscosity.

P = ρ ¯ kbT 1 − ρ¯ b − ¯ aρ2

• 1 − 2η∇vos − (ηv∇ · v) 1

In more recent work we’ve added a density gradient term, which is needed to account for the mechanical stability of diffuse interfaces.

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Equilibrium - coexistence and interfaces Hydrodynamics

Energy equation

momentum eq

du dt = 1 ρ

• −∇ · Jq − PT : ∇v
• (10)

This just says that we have PV work, viscous heating, and conduction from hot to cold.

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary

Physical Model

Equations of motion

The equations of continuum mechanics in Lagrangian form can be expressed as ODEs governing the motion of the smooth particles.

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary

Continuity Equation

momentum eq

dρ dt = −ρ∇ · v (11) Mass density is defined by a summation over particles which explicitly conserves mass. ρ(r) =

n

• j=1

mjW

• r − rj, h
• (12)

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary

Momentum Equation

momentum eq

dv dt = −1 ρ∇ · P The SPH approximation, symmetrised to ensure observance of Newton III: dvi dt =

N

• j=1

mj

• Pj

ρ2

i

+ Pi ρ2

j

• · ∇iWij

(13)

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary

Pressure tensor

pressure tensor

P = peq1 + −2η∇vos + −ηv∇ · v (14) (∇ · v)i = 1 ρi

• j=1

mj(vj − vi) · ∇Wij (15)

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary

Energy equation

momentum eq

du dt = 1 ρ

• −∇ · Jq − PT : ∇v
• (16)

dui dt = 1 2

N

• j

mj

• Pi

ρ2

i

+ Pj ρ2

j

• : vab∇Wij−
• j

mj

• Jqi

ρ2

i

+ Jqj ρ2

j

• ·∇Wij

(17) Jq = −λ∇T = λ

• j

mij ρij

• Tj − Ti
• ∇Wij

(18)

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary

Thermostat

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Droplets Films Spinodal Decomposition

Outline

1

Background Smoothed Particles- the general idea Some mathematics A brief history of SPAM

2

Van der Waals Hydrodynamics Equilibrium - coexistence and interfaces Hydrodynamics

3

SPH Model

4

Results Droplets Films Spinodal Decomposition technology

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Droplets Films Spinodal Decomposition

Droplets

• f the model against Nugent and Posch (2000)

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Droplets Films Spinodal Decomposition

Outline

1

Background Smoothed Particles- the general idea Some mathematics A brief history of SPAM

2

Van der Waals Hydrodynamics Equilibrium - coexistence and interfaces Hydrodynamics

3

SPH Model

4

Results Droplets Films Spinodal Decomposition technology

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Droplets Films Spinodal Decomposition

Films

Equilbrated a set of vapour-liquid films (stable due to pbcs) at various temperatures

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Droplets Films Spinodal Decomposition

Films

Under periodic boundary conditions a film often has a smaller surface area than a droplet. movie - drop_to_film.avi

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Droplets Films Spinodal Decomposition

Films

Final density profiles at reduced temperatures of 7 and 1.05. Hard to resolve the low density phase with so few particles.

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Droplets Films Spinodal Decomposition

Films

The final bulk vapour and liquid densities show good agreement with the predicted binodal curve. Near the critical point equilibration times are excessive, and density fluctuations approach the box dimensions.

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Droplets Films Spinodal Decomposition

Metastability

I was pulling my hair out trying to see why the system wasn’t phase seperating until I realised it’s not difficult to quench into the metastable region by accident. movie - metastable_nucleate.avi

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Droplets Films Spinodal Decomposition

Outline

1

Background Smoothed Particles- the general idea Some mathematics A brief history of SPAM

2

Van der Waals Hydrodynamics Equilibrium - coexistence and interfaces Hydrodynamics

3

SPH Model

4

Results Droplets Films Spinodal Decomposition technology

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Droplets Films Spinodal Decomposition

Spinodal Decomposition

3600 particles in a 70 by 70 box Equilibrate at reduced temperature of 1.5 [movie - gas_parti.avi] Shock waves present in the equilibration are more apparent when pixels are shaded by density [movie - gas_eq.avi]. Note the scale is not constant - density fluctuations in our final ’near equilibrium’ gas. Quench the system into the spinodal part of the phase

• diagram. [movie-first_spinodal.avi]

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Droplets Films Spinodal Decomposition

Spinodal Decomposition

3600 particles in a 70 by 70 box Equilibrate at reduced temperature of 1.5 [movie - gas_parti.avi] Shock waves present in the equilibration are more apparent when pixels are shaded by density [movie - gas_eq.avi]. Note the scale is not constant - density fluctuations in our final ’near equilibrium’ gas. Quench the system into the spinodal part of the phase

• diagram. [movie-first_spinodal.avi]

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Droplets Films Spinodal Decomposition

Spinodal Decomposition

3600 particles in a 70 by 70 box Equilibrate at reduced temperature of 1.5 [movie - gas_parti.avi] Shock waves present in the equilibration are more apparent when pixels are shaded by density [movie - gas_eq.avi]. Note the scale is not constant - density fluctuations in our final ’near equilibrium’ gas. Quench the system into the spinodal part of the phase

• diagram. [movie-first_spinodal.avi]

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Droplets Films Spinodal Decomposition

Spinodal Decomposition

3600 particles in a 70 by 70 box Equilibrate at reduced temperature of 1.5 [movie - gas_parti.avi] Shock waves present in the equilibration are more apparent when pixels are shaded by density [movie - gas_eq.avi]. Note the scale is not constant - density fluctuations in our final ’near equilibrium’ gas. Quench the system into the spinodal part of the phase

• diagram. [movie-first_spinodal.avi]

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Droplets Films Spinodal Decomposition

Outline

1

Background Smoothed Particles- the general idea Some mathematics A brief history of SPAM

2

Van der Waals Hydrodynamics Equilibrium - coexistence and interfaces Hydrodynamics

3

SPH Model

4

Results Droplets Films Spinodal Decomposition technology

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Droplets Films Spinodal Decomposition

Fortran code

The main smoothed particle code is written in Fortran 90 Design issues are similar to an MD code Neighbour list and simulation box components straight from Peter Daivis WMD code Potential for efficiency improvements (at cost of simplicity)

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Droplets Films Spinodal Decomposition

Fortran code

The main smoothed particle code is written in Fortran 90 Design issues are similar to an MD code Neighbour list and simulation box components straight from Peter Daivis WMD code Potential for efficiency improvements (at cost of simplicity)

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Droplets Films Spinodal Decomposition

Fortran code

The main smoothed particle code is written in Fortran 90 Design issues are similar to an MD code Neighbour list and simulation box components straight from Peter Daivis WMD code Potential for efficiency improvements (at cost of simplicity)

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Droplets Films Spinodal Decomposition

Fortran code

The main smoothed particle code is written in Fortran 90 Design issues are similar to an MD code Neighbour list and simulation box components straight from Peter Daivis WMD code Potential for efficiency improvements (at cost of simplicity)

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Droplets Films Spinodal Decomposition

Workflow

Simulations run on VPAC’s edda about 16 seconds per timestep for 3600 particles ASCII output files converted to binary netCDF format.

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Droplets Films Spinodal Decomposition

Workflow

Simulations run on VPAC’s edda about 16 seconds per timestep for 3600 particles ASCII output files converted to binary netCDF format.

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Droplets Films Spinodal Decomposition

Workflow

Simulations run on VPAC’s edda about 16 seconds per timestep for 3600 particles ASCII output files converted to binary netCDF format.

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Droplets Films Spinodal Decomposition

Rendering

Post-processing handled by a set of Python scripts Rendering algorithm is simply the smoothed particle sum applied to regular grid points Heavy use of SciPy, NumPy, Matplotlib libraries. Daniel Price from Cambridge Astro SPLASH SPH rendering code

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Droplets Films Spinodal Decomposition

Rendering

Post-processing handled by a set of Python scripts Rendering algorithm is simply the smoothed particle sum applied to regular grid points Heavy use of SciPy, NumPy, Matplotlib libraries. Daniel Price from Cambridge Astro SPLASH SPH rendering code

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Droplets Films Spinodal Decomposition

Rendering

Post-processing handled by a set of Python scripts Rendering algorithm is simply the smoothed particle sum applied to regular grid points Heavy use of SciPy, NumPy, Matplotlib libraries. Daniel Price from Cambridge Astro SPLASH SPH rendering code

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Droplets Films Spinodal Decomposition

Rendering

Post-processing handled by a set of Python scripts Rendering algorithm is simply the smoothed particle sum applied to regular grid points Heavy use of SciPy, NumPy, Matplotlib libraries. Daniel Price from Cambridge Astro SPLASH SPH rendering code

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Droplets Films Spinodal Decomposition

Numerical Technique

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Droplets Films Spinodal Decomposition

Smoothing length

Simply plugging the van der Waals equation of state into the SPH machinery leads to odd looking results [movie hequalsH.avi]:

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Droplets Films Spinodal Decomposition

interparticle potential

purely attractive force is unstable.

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Droplets Films Spinodal Decomposition

Smoothing length

The method has stability problems when treating attractive forces. Modelling surface tension, Bill Hoover first used a longer smoothing length for an ad-hoc attractive density gradient force. Nugent and Posch also used the longer smoothing length applied to the attractive part of the pressure in their 2000 work Other particle techniques based on ’particles’ , including DPD and other formulations of SPH with ad-hoc additional forces also need to use this technique to obtain phase seperation, coexistence, and surface tension. (Tartatovsky, Meakin, Warren)

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Droplets Films Spinodal Decomposition

Smoothing length

The method has stability problems when treating attractive forces. Modelling surface tension, Bill Hoover first used a longer smoothing length for an ad-hoc attractive density gradient force. Nugent and Posch also used the longer smoothing length applied to the attractive part of the pressure in their 2000 work Other particle techniques based on ’particles’ , including DPD and other formulations of SPH with ad-hoc additional forces also need to use this technique to obtain phase seperation, coexistence, and surface tension. (Tartatovsky, Meakin, Warren)

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Droplets Films Spinodal Decomposition

Smoothing length

The method has stability problems when treating attractive forces. Modelling surface tension, Bill Hoover first used a longer smoothing length for an ad-hoc attractive density gradient force. Nugent and Posch also used the longer smoothing length applied to the attractive part of the pressure in their 2000 work Other particle techniques based on ’particles’ , including DPD and other formulations of SPH with ad-hoc additional forces also need to use this technique to obtain phase seperation, coexistence, and surface tension. (Tartatovsky, Meakin, Warren)

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Droplets Films Spinodal Decomposition

Smoothing length

The method has stability problems when treating attractive forces. Modelling surface tension, Bill Hoover first used a longer smoothing length for an ad-hoc attractive density gradient force. Nugent and Posch also used the longer smoothing length applied to the attractive part of the pressure in their 2000 work Other particle techniques based on ’particles’ , including DPD and other formulations of SPH with ad-hoc additional forces also need to use this technique to obtain phase seperation, coexistence, and surface tension. (Tartatovsky, Meakin, Warren)

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Droplets Films Spinodal Decomposition

Smoothing length

With a longer cohesive resolution, the interparticle force mimics the shape of the Lennard-Jones force.

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Droplets Films Spinodal Decomposition

Diffuse interfaces

SPH interfaces in simulations are of the order of the smoothing width, which is unsurprising. If we were using a grid method, any interface would appear to be at minimum one cell wide. It’s highly unlikely the density profile is accurate. (Why - resolution is too low, too many potential numerical artifacts.) if it is accurate it’s probably for the wrong reasons... To explain the diffuse interface, van der Waals Included the density gradient contribution to the free energy.

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Droplets Films Spinodal Decomposition

Diffuse interfaces

SPH interfaces in simulations are of the order of the smoothing width, which is unsurprising. If we were using a grid method, any interface would appear to be at minimum one cell wide. It’s highly unlikely the density profile is accurate. (Why - resolution is too low, too many potential numerical artifacts.) if it is accurate it’s probably for the wrong reasons... To explain the diffuse interface, van der Waals Included the density gradient contribution to the free energy.

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Droplets Films Spinodal Decomposition

Diffuse interfaces

SPH interfaces in simulations are of the order of the smoothing width, which is unsurprising. If we were using a grid method, any interface would appear to be at minimum one cell wide. It’s highly unlikely the density profile is accurate. (Why - resolution is too low, too many potential numerical artifacts.) if it is accurate it’s probably for the wrong reasons... To explain the diffuse interface, van der Waals Included the density gradient contribution to the free energy.

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Droplets Films Spinodal Decomposition

Diffuse interfaces

SPH interfaces in simulations are of the order of the smoothing width, which is unsurprising. If we were using a grid method, any interface would appear to be at minimum one cell wide. It’s highly unlikely the density profile is accurate. (Why - resolution is too low, too many potential numerical artifacts.) if it is accurate it’s probably for the wrong reasons... To explain the diffuse interface, van der Waals Included the density gradient contribution to the free energy.

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Droplets Films Spinodal Decomposition

Diffuse interfaces

SPH interfaces in simulations are of the order of the smoothing width, which is unsurprising. If we were using a grid method, any interface would appear to be at minimum one cell wide. It’s highly unlikely the density profile is accurate. (Why - resolution is too low, too many potential numerical artifacts.) if it is accurate it’s probably for the wrong reasons... To explain the diffuse interface, van der Waals Included the density gradient contribution to the free energy.

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Droplets Films Spinodal Decomposition

The gradient term

No book or paper explains it better that his original paper (translated by Rowlinson). Only matters for interfaces (inhomogeneous density). Diffuse interfaces are largest near the critical point. Similar method used by Cahn and Hilliard for two-component systems

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Droplets Films Spinodal Decomposition

The gradient term

No book or paper explains it better that his original paper (translated by Rowlinson). Only matters for interfaces (inhomogeneous density). Diffuse interfaces are largest near the critical point. Similar method used by Cahn and Hilliard for two-component systems

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary Droplets Films Spinodal Decomposition

The gradient term

No book or paper explains it better that his original paper (translated by Rowlinson). Only matters for interfaces (inhomogeneous density). Diffuse interfaces are largest near the critical point. Similar method used by Cahn and Hilliard for two-component systems

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary

Summary

Our SPAM model obeys the equation of state pretty well in the coexistence region We do not know whether the interface is represented accurately or artifactually Future Work Publish the code Study the effect of the density gradient term on the interface. More complex systems - solutions, boundary driven flow, different equations of state.

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary

Summary

Our SPAM model obeys the equation of state pretty well in the coexistence region We do not know whether the interface is represented accurately or artifactually Future Work Publish the code Study the effect of the density gradient term on the interface. More complex systems - solutions, boundary driven flow, different equations of state.

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary

Summary

Our SPAM model obeys the equation of state pretty well in the coexistence region We do not know whether the interface is represented accurately or artifactually Future Work Publish the code Study the effect of the density gradient term on the interface. More complex systems - solutions, boundary driven flow, different equations of state.

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary

Summary

Our SPAM model obeys the equation of state pretty well in the coexistence region We do not know whether the interface is represented accurately or artifactually Future Work Publish the code Study the effect of the density gradient term on the interface. More complex systems - solutions, boundary driven flow, different equations of state.

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary

Summary

Our SPAM model obeys the equation of state pretty well in the coexistence region We do not know whether the interface is represented accurately or artifactually Future Work Publish the code Study the effect of the density gradient term on the interface. More complex systems - solutions, boundary driven flow, different equations of state.

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary

Acknowledgements

RMIT: Peter Daivis Daniel Price (University of Exeter): SPLASH rendering code William and Carol Hoover (Ruby Valley Research Institute): support and encouragement Jeff Whitaker (NOAA): netCDF python libraries CAWCR have generously supported my studies.

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary

Acknowledgements

RMIT: Peter Daivis Daniel Price (University of Exeter): SPLASH rendering code William and Carol Hoover (Ruby Valley Research Institute): support and encouragement Jeff Whitaker (NOAA): netCDF python libraries CAWCR have generously supported my studies.

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary

Acknowledgements

RMIT: Peter Daivis Daniel Price (University of Exeter): SPLASH rendering code William and Carol Hoover (Ruby Valley Research Institute): support and encouragement Jeff Whitaker (NOAA): netCDF python libraries CAWCR have generously supported my studies.

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary

Acknowledgements

RMIT: Peter Daivis Daniel Price (University of Exeter): SPLASH rendering code William and Carol Hoover (Ruby Valley Research Institute): support and encouragement Jeff Whitaker (NOAA): netCDF python libraries CAWCR have generously supported my studies.

Andrew Charles, Peter Daivis

Background Van der Waals Hydrodynamics SPH model Results Summary

Acknowledgements

RMIT: Peter Daivis Daniel Price (University of Exeter): SPLASH rendering code William and Carol Hoover (Ruby Valley Research Institute): support and encouragement Jeff Whitaker (NOAA): netCDF python libraries CAWCR have generously supported my studies.

Andrew Charles, Peter Daivis