Background Van der Waals Hydrodynamics Smoothed Particles- the general idea SPH model Smooth particle interpolation Results A brief history of SPAM Summary 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 Smoothed Particles- the general idea SPH model Smooth particle interpolation Results A brief history of SPAM Summary 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 Smoothed Particles- the general idea SPH model Smooth particle interpolation Results A brief history of SPAM Summary Smoothed particles a very attractive numerical technique Easy to code, a wealth of existing techniques for MD and other 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 Smoothed Particles- the general idea SPH model Smooth particle interpolation Results A brief history of SPAM Summary Smoothed particles a very attractive numerical technique Easy to code, a wealth of existing techniques for MD and other 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 Smoothed Particles- the general idea SPH model Smooth particle interpolation Results A brief history of SPAM Summary Smoothed particles a very attractive numerical technique Easy to code, a wealth of existing techniques for MD and other 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 Smoothed Particles- the general idea SPH model Smooth particle interpolation Results A brief history of SPAM Summary Outline Background 1 Smoothed Particles- the general idea Some mathematics A brief history of SPAM Van der Waals Hydrodynamics 2 Equilibrium - coexistence and interfaces Hydrodynamics SPH Model 3 Results 4 Droplets Films Spinodal Decomposition technology Andrew Charles, Peter Daivis
Background Van der Waals Hydrodynamics Smoothed Particles- the general idea SPH model Smooth particle interpolation Results A brief history of SPAM Summary Smoothed particles justification A continuous field can be expressed as an integral of delta functions. � f ( r ′ ) δ ( r − r ′ ) dr ′ f ( r ) = (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 Smoothed Particles- the general idea SPH model Smooth particle interpolation Results A brief history of SPAM Summary Smoothed particles justification A continuous field can be expressed as an integral of delta functions. � f ( r ′ ) δ ( r − r ′ ) dr ′ f ( r ) = (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 Smoothed Particles- the general idea SPH model Smooth particle interpolation Results A brief history of SPAM Summary Smoothed particles justification Rewrite the volume integral dV ′ = dm ( r ′ ) ρ ( r ′ ) f ( r ′ ) dm ( r ′ ) � ρ ( r ′ ) W ( r − r ′ , h ) f ( r ) = (3) Divide the system into small finite elements such that dm = m i , and replace the integral with a sum. N m i ˆ � f ( r ) = f i W ( r − r i , h ) (4) ρ i i = 1 Andrew Charles, Peter Daivis
Background Van der Waals Hydrodynamics Smoothed Particles- the general idea SPH model Smooth particle interpolation Results A brief history of SPAM Summary Smoothed particles justification Rewrite the volume integral dV ′ = dm ( r ′ ) ρ ( r ′ ) f ( r ′ ) dm ( r ′ ) � ρ ( r ′ ) W ( r − r ′ , h ) f ( r ) = (3) Divide the system into small finite elements such that dm = m i , and replace the integral with a sum. N m i ˆ � f ( r ) = f i W ( r − r i , h ) (4) ρ i i = 1 Andrew Charles, Peter Daivis
Background Van der Waals Hydrodynamics Smoothed Particles- the general idea SPH model Smooth particle interpolation Results A brief history of SPAM Summary Summation Interpolant Summation Interpolant N N m i ˆ � � f ( r ) = f i W ( r − r i , h ) ρ ( r ) = ρ i W ( r − r i , h ) ρ i i = 1 i = 1 Andrew Charles, Peter Daivis
Background Van der Waals Hydrodynamics Smoothed Particles- the general idea SPH model Smooth particle interpolation Results A brief history of SPAM Summary 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 Smoothed Particles- the general idea SPH model Smooth particle interpolation Results A brief history of SPAM Summary Mathematical Justification Derivatives N m i ˆ � ∇ f ( r ) = f i ∇ W ( r − r i , h ) (5) ρ i i = 1 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 Smoothed Particles- the general idea SPH model Smooth particle interpolation Results A brief history of SPAM Summary Outline Background 1 Smoothed Particles- the general idea Some mathematics A brief history of SPAM Van der Waals Hydrodynamics 2 Equilibrium - coexistence and interfaces Hydrodynamics SPH Model 3 Results 4 Droplets Films Spinodal Decomposition technology Andrew Charles, Peter Daivis
Background Van der Waals Hydrodynamics Smoothed Particles- the general idea SPH model Smooth particle interpolation Results A brief history of SPAM Summary 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 Smoothed Particles- the general idea SPH model Smooth particle interpolation Results A brief history of SPAM Summary 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 Smoothed Particles- the general idea SPH model Smooth particle interpolation Results A brief history of SPAM Summary 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 Smoothed Particles- the general idea SPH model Smooth particle interpolation Results A brief history of SPAM Summary 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 Smoothed Particles- the general idea SPH model Smooth particle interpolation Results A brief history of SPAM Summary 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 Smoothed Particles- the general idea SPH model Smooth particle interpolation Results A brief history of SPAM Summary 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 Smoothed Particles- the general idea SPH model Smooth particle interpolation Results A brief history of SPAM Summary 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 Smoothed Particles- the general idea SPH model Smooth particle interpolation Results A brief history of SPAM Summary 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 Smoothed Particles- the general idea SPH model Smooth particle interpolation Results A brief history of SPAM Summary 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 Smoothed Particles- the general idea SPH model Smooth particle interpolation Results A brief history of SPAM Summary 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 Smoothed Particles- the general idea SPH model Smooth particle interpolation Results A brief history of SPAM Summary 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 Smoothed Particles- the general idea SPH model Smooth particle interpolation Results A brief history of SPAM Summary 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 Smoothed Particles- the general idea SPH model Smooth particle interpolation Results A brief history of SPAM Summary 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 Smoothed Particles- the general idea SPH model Smooth particle interpolation Results A brief history of SPAM Summary 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 Smoothed Particles- the general idea SPH model Smooth particle interpolation Results A brief history of SPAM Summary 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 Smoothed Particles- the general idea SPH model Smooth particle interpolation Results A brief history of SPAM Summary 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 Smoothed Particles- the general idea SPH model Smooth particle interpolation Results A brief history of SPAM Summary 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 Smoothed Particles- the general idea SPH model Smooth particle interpolation Results A brief history of SPAM Summary 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 Equilibrium - coexistence and interfaces SPH model Hydrodynamics Results Summary Outline Background 1 Smoothed Particles- the general idea Some mathematics A brief history of SPAM Van der Waals Hydrodynamics 2 Equilibrium - coexistence and interfaces Hydrodynamics SPH Model 3 Results 4 Droplets Films Spinodal Decomposition technology Andrew Charles, Peter Daivis
Background Van der Waals Hydrodynamics Equilibrium - coexistence and interfaces SPH model Hydrodynamics Results Summary 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 Equilibrium - coexistence and interfaces SPH model Hydrodynamics Results Summary 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 Equilibrium - coexistence and interfaces SPH model Hydrodynamics Results Summary van der Waals’ equation of state V − Nb − N 2 a NkT p = (6) V 2 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 Equilibrium - coexistence and interfaces SPH model Hydrodynamics Results Summary van der Waals’ equation of state V − Nb − N 2 a NkT p = (6) V 2 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 Equilibrium - coexistence and interfaces SPH model Hydrodynamics Results Summary van der Waals’ equation of state V − Nb − N 2 a NkT p = (6) V 2 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 Equilibrium - coexistence and interfaces SPH model Hydrodynamics Results Summary 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 Equilibrium - coexistence and interfaces SPH model Hydrodynamics Results Summary 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 Equilibrium - coexistence and interfaces SPH model Hydrodynamics Results Summary 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 Equilibrium - coexistence and interfaces SPH model Hydrodynamics Results Summary van der Waals to scale Andrew Charles, Peter Daivis
Background Van der Waals Hydrodynamics Equilibrium - coexistence and interfaces SPH model Hydrodynamics Results Summary reduced units p = ρ ¯ k b T a ρ 2 b − ¯ (7) 1 − ρ ¯ 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 , ¯ ¯ k b = 1 and T c 1 . 2 T = u + ¯ a ρ (8) ¯ k b Andrew Charles, Peter Daivis
Background Van der Waals Hydrodynamics Equilibrium - coexistence and interfaces SPH model Hydrodynamics Results Summary Outline Background 1 Smoothed Particles- the general idea Some mathematics A brief history of SPAM Van der Waals Hydrodynamics 2 Equilibrium - coexistence and interfaces Hydrodynamics SPH Model 3 Results 4 Droplets Films Spinodal Decomposition technology Andrew Charles, Peter Daivis
Background Van der Waals Hydrodynamics Equilibrium - coexistence and interfaces SPH model Hydrodynamics Results Summary 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. d ρ dt = − ρ ∇ · v (9) Andrew Charles, Peter Daivis
Background Van der Waals Hydrodynamics Equilibrium - coexistence and interfaces SPH model Hydrodynamics Results Summary 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. d ρ dt = − ρ ∇ · v (9) Andrew Charles, Peter Daivis
Background Van der Waals Hydrodynamics Equilibrium - coexistence and interfaces SPH model Hydrodynamics Results Summary Momentum Equation momentum eq d v 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. � ρ ¯ k b T � 1 − 2 η ∇ v os − ( η v ∇ · v ) 1 a ρ 2 P = b − ¯ 1 − ρ ¯ 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 Equilibrium - coexistence and interfaces SPH model Hydrodynamics Results Summary Momentum Equation momentum eq d v 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. � ρ ¯ k b T � 1 − 2 η ∇ v os − ( η v ∇ · v ) 1 a ρ 2 P = b − ¯ 1 − ρ ¯ 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 Equilibrium - coexistence and interfaces SPH model Hydrodynamics Results Summary Momentum Equation momentum eq d v 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. � ρ ¯ k b T � 1 − 2 η ∇ v os − ( η v ∇ · v ) 1 a ρ 2 P = b − ¯ 1 − ρ ¯ 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 Equilibrium - coexistence and interfaces SPH model Hydrodynamics Results Summary Momentum Equation momentum eq d v 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. � ρ ¯ k b T � 1 − 2 η ∇ v os − ( η v ∇ · v ) 1 a ρ 2 P = b − ¯ 1 − ρ ¯ 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 Equilibrium - coexistence and interfaces SPH model Hydrodynamics Results Summary Momentum Equation momentum eq d v 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. � ρ ¯ k b T � 1 − 2 η ∇ v os − ( η v ∇ · v ) 1 a ρ 2 P = b − ¯ 1 − ρ ¯ 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 Equilibrium - coexistence and interfaces SPH model Hydrodynamics Results Summary Momentum Equation momentum eq d v 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. � ρ ¯ k b T � 1 − 2 η ∇ v os − ( η v ∇ · v ) 1 a ρ 2 P = b − ¯ 1 − ρ ¯ 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 Equilibrium - coexistence and interfaces SPH model Hydrodynamics Results Summary Energy equation momentum eq du dt = 1 � − ∇ · J q − P T : ∇ v � (10) ρ 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 d ρ (11) dt = − ρ ∇ · v Mass density is defined by a summation over particles which explicitly conserves mass. n � � � ρ ( r ) = m j W r − r j , h (12) j = 1 Andrew Charles, Peter Daivis
Background Van der Waals Hydrodynamics SPH model Results Summary Momentum Equation momentum eq d v dt = − 1 ρ ∇ · P The SPH approximation, symmetrised to ensure observance of Newton III: N � � d v i P j + P i � = m j · ∇ i Wij (13) ρ 2 ρ 2 dt i j j = 1 Andrew Charles, Peter Daivis
Background Van der Waals Hydrodynamics SPH model Results Summary Pressure tensor pressure tensor P = p eq 1 + − 2 η ∇ v os + − η v ∇ · v (14) ( ∇ · v ) i = 1 � m j ( v j − v i ) · ∇ W ij (15) ρ i j = 1 Andrew Charles, Peter Daivis
Background Van der Waals Hydrodynamics SPH model Results Summary Energy equation momentum eq du dt = 1 � − ∇ · J q − P T : ∇ v � (16) ρ N � � � � du i dt = 1 + P j J qi + J qj P i � � m j : v ab ∇ W ij − m j ·∇ W i j ρ 2 ρ 2 ρ 2 ρ 2 2 i j i j j j (17) m ij � � � J q = − λ ∇ T = λ T j − T i ∇ W ij (18) ρ ij j Andrew Charles, Peter Daivis
Background Van der Waals Hydrodynamics SPH model Results Summary Thermostat Andrew Charles, Peter Daivis
Background Droplets Van der Waals Hydrodynamics Films SPH model Spinodal Decomposition Results Summary Outline Background 1 Smoothed Particles- the general idea Some mathematics A brief history of SPAM Van der Waals Hydrodynamics 2 Equilibrium - coexistence and interfaces Hydrodynamics SPH Model 3 Results 4 Droplets Films Spinodal Decomposition technology Andrew Charles, Peter Daivis
Background Droplets Van der Waals Hydrodynamics Films SPH model Spinodal Decomposition Results Summary Droplets of the model against Nugent and Posch (2000) Andrew Charles, Peter Daivis
Background Droplets Van der Waals Hydrodynamics Films SPH model Spinodal Decomposition Results Summary Outline Background 1 Smoothed Particles- the general idea Some mathematics A brief history of SPAM Van der Waals Hydrodynamics 2 Equilibrium - coexistence and interfaces Hydrodynamics SPH Model 3 Results 4 Droplets Films Spinodal Decomposition technology Andrew Charles, Peter Daivis
Background Droplets Van der Waals Hydrodynamics Films SPH model Spinodal Decomposition Results Summary Films Equilbrated a set of vapour-liquid films (stable due to pbcs) at various temperatures Andrew Charles, Peter Daivis
Background Droplets Van der Waals Hydrodynamics Films SPH model Spinodal Decomposition Results Summary 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 Droplets Van der Waals Hydrodynamics Films SPH model Spinodal Decomposition Results Summary 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 Droplets Van der Waals Hydrodynamics Films SPH model Spinodal Decomposition Results Summary 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 Droplets Van der Waals Hydrodynamics Films SPH model Spinodal Decomposition Results Summary 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 Droplets Van der Waals Hydrodynamics Films SPH model Spinodal Decomposition Results Summary Outline Background 1 Smoothed Particles- the general idea Some mathematics A brief history of SPAM Van der Waals Hydrodynamics 2 Equilibrium - coexistence and interfaces Hydrodynamics SPH Model 3 Results 4 Droplets Films Spinodal Decomposition technology Andrew Charles, Peter Daivis
Background Droplets Van der Waals Hydrodynamics Films SPH model Spinodal Decomposition Results Summary 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 Droplets Van der Waals Hydrodynamics Films SPH model Spinodal Decomposition Results Summary 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 Droplets Van der Waals Hydrodynamics Films SPH model Spinodal Decomposition Results Summary 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 Droplets Van der Waals Hydrodynamics Films SPH model Spinodal Decomposition Results Summary 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 Droplets Van der Waals Hydrodynamics Films SPH model Spinodal Decomposition Results Summary Outline Background 1 Smoothed Particles- the general idea Some mathematics A brief history of SPAM Van der Waals Hydrodynamics 2 Equilibrium - coexistence and interfaces Hydrodynamics SPH Model 3 Results 4 Droplets Films Spinodal Decomposition technology Andrew Charles, Peter Daivis
Background Droplets Van der Waals Hydrodynamics Films SPH model Spinodal Decomposition Results Summary 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 Droplets Van der Waals Hydrodynamics Films SPH model Spinodal Decomposition Results Summary 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 Droplets Van der Waals Hydrodynamics Films SPH model Spinodal Decomposition Results Summary 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 Droplets Van der Waals Hydrodynamics Films SPH model Spinodal Decomposition Results Summary 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 Droplets Van der Waals Hydrodynamics Films SPH model Spinodal Decomposition Results Summary 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 Droplets Van der Waals Hydrodynamics Films SPH model Spinodal Decomposition Results Summary 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 Droplets Van der Waals Hydrodynamics Films SPH model Spinodal Decomposition Results Summary 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 Droplets Van der Waals Hydrodynamics Films SPH model Spinodal Decomposition Results Summary 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
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