Solution Transport Phenomena Second Edition
Karri Weston
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Using the combination of the kinetic theory of gases (KTG), Boltzmann transport equation (BTE), and molecular dynamics (MD) simulations, we study the transport phenomena in the Knudsen layer near a planar evaporating surface. The MD simulation is first used to validate the assumption regarding the anisotropic velocity distribution of vapor molecules in the Knudsen layer. Based on this assumption, we use the KTG to formulate the temperature and density of vapor at the evaporating surface as a function of the evaporation rate and the mass accommodation coefficient (MAC), and we use these vapor properties as the boundary conditions to find the solution to the BTE for the anisotropic vapor flow in the Knudsen layer. From the study of the evaporation into a vacuum, we show the ratio of the macroscopic speed of vapor to the most probable thermal speed of vapor molecules in the flow direction will always reach the maximum value of 1.5 at the vacuum boundary. The BTE solutions predict that the maximum evaporation flux from a liquid surface at a given temperature depends on both the MAC and the distance between the evaporating surface and the vacuum boundary. From the study of the evaporation and condensation between two parallel plates, we show the BTE solutions give good predictions of transport phenomena in both the anisotropic vapor flow within the Knudsen layer and the isotropic flow out of the Knudsen layer. All the predictions from the BTE are verified by the MD simulation results.
The dimensionless (a) density, (b) temperature, and (c) molar flux of vapor at the evaporating surface as a function of dimensionless macroscopic velocity of vapor at the evaporating surface for different values of MAC.
(Top panel) A snapshot of the model system for the MD study of evaporation of liquid Ar on an Au surface at Th=85K into a vacuum, and (bottom panel) the temperature profile in the vapor flow direction. The uncertainty of Tx and Ty is smaller than the size of symbols.
The (a) molar, (b) momentum, and (c) energy flux in each bin of the vapor region. The horizontal dashed lines show the average value of molar, momentum and energy flux in the vapor region. The uncertainties are smaller than the size of symbols.
(Top panel) A snapshot of the model system for the MD simulation of evaporation of liquid Ar on a hot Au surface at Th=85K and condensation of vapor Ar on a cold Au surface at Tl=35K, and (bottom panel) the temperature profile in the vapor flow direction. The uncertainty of Tx and Ty is smaller than the size of symbols.
(Top panel) A snapshot of the model system for the study of evaporation of liquid Ar at TL=82.8K into a vacuum. The vacuum boundary is 103 nm from the evaporating surface. (Bottom panels) The (a) temperature, (b) density, and (c) dimensionless macroscopic velocity in the vapor region. The scatters are MD simulation results. The lines are predictions from the BTE.
(Top panel) A snapshot of the model system for the study of evaporation and condensation of fluid Ar between two parallel plates. The separation between the evaporating and the condensing surfaces is 106 nm. (Bottom panels) The (a) temperature, (b) density, and (c) dimensionless macroscopic velocity in the vapor region. The scatters are MD simulation results. The lines are predictions from the BTE.
(a) The velocity distribution of vapor molecules in the second bin of the vapor region. The red circles and blue diamonds are MD simulation results for the velocity components along the vapor flow direction and perpendicular to the flow direction, respectively. The dashed line is the Maxwell velocity distribution (MVD) of Ar molecules at T=71K. The solid line is the shifted MVD (SMVD) of Ar molecules at T=61K. (b) RJME vs. Jnet at the evaporating and condensing surfaces.
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Transport phenomena still stand as one of the most challenging problems in computational physics. By exploiting the analogies between Dirac and lattice Boltzmann equations, we develop a quantum simulator based on pseudospin-boson quantum systems, which is suitable for encoding fluid dynamics transport phenomena within a lattice kinetic formalism. It is shown that both the streaming and collision processes of lattice Boltzmann dynamics can be implemented with controlled quantum operations, using a heralded quantum protocol to encode non-unitary scattering processes. The proposed simulator is amenable to realization in controlled quantum platforms, such as ion-trap quantum computers or circuit quantum electrodynamics processors.
Transport phenomena in fluid flows play a crucial role for many applications in science and engineering. Indeed, a large variety of natural and industrial processes depend critically on the transport of mass, momentum and energy of chemical species by means of fluid flows across material media of assorted nature1. The numerical simulation of such transport phenomena still presents a major challenge to modern computational fluid dynamics. Among the reasons for this complexity stand out the presence of strong heterogeneities and huge scale separation in the basic mechanisms, namely advection, diffusion and chemical reactions2,3. In the last two decades, a novel concept for the solution of transport phenomena in fluid flows has emerged in the form of a minimal lattice Boltzmann (LB) kinetic equation. This approach is based on the statistical viewpoint typical of kinetic theory4,5. LB is currently used across a broad range of problems in fluid dynamics, from fully developed turbulence in complex geometries to micro and nanofluidics6,7, all the way down to lattice gas automata8 and quark-gluon applications9.
Recent improvements in ion trap and superconducting circuit experiments make these platforms ideal for challenging quantum information and simulation tasks. Trapped-ion experiments have demonstrated quantum information and simulation capabilities10,11,12, including the quantum simulation of highly correlated fermionic systems13, fermionic-bosonic models14,15 and lattice gauge theories16. Superconducting circuit setups can host nowadays top-end quantum information protocols, such as quantum teleportation17 and topological phase transitions18. These quantum devices are approaching the complexity required to simulate both classical and quantum nontrivial problems, as proposed by Feynman some decades ago19. Efforts in designing quantum algorithms for the implementation of fluid dynamics make use of quantum computer networks20,21. In these works, the quantum degrees of freedom are used on the same ground as classical parameters and the exponential gain of quantum computers is not properly exploited. In contrast, systems described by pseudospins coupled to bosonic modes, such as the aforementioned ion-trap and superconducting circuit platforms, can enjoy quantum superposition and have advantages with respect to pure-qubit quantum computers in simulating fluids.
In this article, we propose a quantum simulation of lattice Boltzmann dynamics, using coupled pseudospin-boson quantum platforms. Based on previously established analogies between Dirac and LB equations, we define here a full quantum mapping of transport equations in fluid flows. The LB dynamics is simulated sequentially by performing particle streaming and collision steps. The non-unitary collision process can be implemented with an heralded protocol, by sequential collapses of an ancillary qubit. The proposed mapping is amenable to realization in trapped-ion and superconducting circuit platforms.
Here, is the ith component of the particle fluid density associated with the lattice site at the time t and with discrete velocity . The macroscopic fluid density at the site is retrieved as , while the fluid velocity is defined as the weighted sum of the discrete velocities, . The velocity components , with , satisfy mass-momentum-energy conservation laws and rotational symmetry. Typical lattices are D2Q9 or D3Q15 models, for the case of two dimensions with 9 speeds and three dimensions with 15 speeds, respectively22.
Collisional properties are here expressed in scattering-relaxation form, making use of the local equilibrium distribution . The LB approach to compute the dynamics associated with Eq. (1) uses sequential computational steps. One initially performs a displacement (free-streaming) of each distribution component towards the nearest-neighbor lattice site pointed at by the discrete velocity . From there, the equilibrium distribution function is computed and the outcome of the collisional process is retrieved. Further iterations of these calculations allow the propagation of the lattice dynamics in time. We address the question of whether all these steps can be performed in a quantum simulator with practical quantum computing protocols.
Notice that the streaming matrices of the LB equation are diagonal, while the αij, which generate a Clifford algebra, cannot be simultaneously diagonalized. Additionally, the mass matrix βij is Hermitian, while standard collision matrices come in real symmetric form in the LB equation. Therefore, a complete codification of the LB scheme in quantum language requires the implementation of diagonal streaming matrices and of purely imaginary symmetric scattering matrices.
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