Department of Mathematics
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Item Coupled constitutive relations: a second lawbased higher-order closure for hydrodynamics(RSC, 2018-10) Rana, AnirudhIn theclassicalframework,theNavier–Stokes–Fourier equations areobtainedthroughthelinearuncoupled thermodynamic force-fluxrelationswhichguarantee the non-negativityoftheentropyproduction. However,theconventionalthermodynamicdescrip- tion isonlyvalidwhentheKnudsennumberis sufficientlysmall.Here,itisshownthattherangeof validity oftheNavier–Stokes–Fourierequationscan be extendedbyincorporatingthenonlinearcoupling among thethermodynamicforcesandfluxes.The resultingsystemofconservationlawsclosedwith the coupledconstitutiverelationsisabletodescribe many interestingrarefactioneffects,suchasKnudsen paradox, transpirationflows,thermalstress,heat flux withouttemperaturegradients,etc.,which cannot bepredictedbytheclassicalNavier–Stokes– Fourier equations.Forthissystemofequations, a setofphenomenologicalboundaryconditions, which respectthesecondlawofthermodynamics, is alsoderived.Someofthebenchmarkproblems in fluidmechanicsarestudiedtoshowthe applicability ofthederivedequationsandboundary conditions.Item Analytical and Numerical Solutions of Boundary Value Problems for the Regularized 13 Moment Equations(AIP Conference Proceedings, 2011-05) Rana, AnirudhClassical hydrodynamics—the laws of Navier-Stokes and Fourier—fails in the description of processes in rarefied gases. For not too large Knudsen numbers, extended macroscopic models offer an alternative to the solution of the Boltzmann equations. Anlytical and numerical solutions show that the regularized 13 moment equations can capture all important linear and non-linear rarefaction effects with good accuracy.Item Evaporation-driven vapour microflows: analytical solutions from moment methods(CUP, 2018-03) Rana, AnirudhMacroscopic models based on moment equations are developed to describe the transport of mass and energy near the phase boundary between a liquid and its rarefied vapour due to evaporation and hence, in this study, condensation. For evaporation from a spherical droplet, analytic solutions are obtained to the linearised equations from the Navier–Stokes–Fourier, regularised 13-moment and regularised 26-moment frameworks. Results are shown to approach computational solutions to the Boltzmann equation as the number of moments are increased, with good agreement for Knudsen number ≲1, whilst providing clear insight into non-equilibrium phenomena occurring adjacent to the interface.Item Evaporation Boundary Conditions for the Linear R13 Equations Based on the Onsager Theory(MDPI, 2018-09) Rana, AnirudhDue to the failure of the continuum hypothesis for higher Knudsen numbers, rarefied gases and microflows of gases are particularly difficult to model. Macroscopic transport equations compete with particle methods, such as the Direct Simulation Monte Carlo method (DSMC), to find accurate solutions in the rarefied gas regime. Due to growing interest in micro flow applications, such as micro fuel cells, it is important to model and understand evaporation in this flow regime. Here, evaporation boundary conditions for the R13 equations, which are macroscopic transport equations with applicability in the rarefied gas regime, are derived. The new equations utilize Onsager relations, linear relations between thermodynamic fluxes and forces, with constant coefficients, that need to be determined. For this, the boundary conditions are fitted to DSMC data and compared to other R13 boundary conditions from kinetic theory and Navier–Stokes–Fourier (NSF) solutions for two one-dimensional steady-state problems. Overall, the suggested fittings of the new phenomenological boundary conditions show better agreement with DSMC than the alternative kinetic theory evaporation boundary conditions for R13. Furthermore, the new evaporation boundary conditions for R13 are implemented in a code for the numerical solution of complex, two-dimensional geometries and compared to NSF solutions. Different flow patterns between R13 and NSF for higher Knudsen numbers are observed.Item Coupled constitutive relations: a second law based higher-order closure for hydrodynamics(RSC, 2018-10) Rana, AnirudhIn the classical framework, the Navier–Stokes–Fourier equations are obtained through the linear uncoupled thermodynamic force-flux relations which guarantee the non-negativity of the entropy production. However, the conventional thermodynamic descrip- tion is only valid when the Knudsen number is sufficiently small. Here, it is shown that the range of validity of the Navier–Stokes–Fourier equations can be extended by incorporating the nonlinear coupling among the thermodynamic forces and fluxes. The resulting system of conservation laws closed with the coupled constitutive relations is able to describe many interesting rarefaction effects, such as Knudsen paradox, transpiration flows, thermal stress, heat flux without temperature gradients, etc., which cannot be predicted by the classical Navier–Stokes–Fourier equations. For this system of equations, a set of phenomenological boundary conditions, which respect the second law of thermodynamics, is also derived. Some of the benchmark problems in fluid mechanics are studied to show the applicability of the derived equations and boundary conditions.Item Fundamental solutions to the regularised 13-moment equations: efficient computation of three-dimensional kinetic effects(CUP, 2017-11) Rana, AnirudhFundamental solutions (Green’s functions) are derived for the regularised 13-moment system (R13) of rarefied gas dynamics, for small departures from equilibrium; these solutions show the presence of Knudsen layers, associated with exponential decay terms, that do not feature in the solution of lower-order systems (e.g. the Navier–Stokes–Fourier equations). Incorporation of these new fundamental solutions into a numerical framework based on the method of fundamental solutions (MFS) allows for efficient computation of three-dimensional gas microflows at remarkably low computational cost. The R13-MFS approach accurately recovers analytic solutions for low-speed flow around a stationary sphere and heat transfer from a hot sphere (for which a new analytic solution has been derived), capturing non-equilibrium flow phenomena missing from lower-order solutions. To demonstrate the potential of the new approach, the influence of kinetic effects on the hydrodynamic interaction between approaching solid microparticles is calculated. Finally, a programme of future work based on the initial steps taken in this article is outlined.Item Lifetime of a Nanodroplet: Kinetic Effects and Regime Transitions(APS, 2019-10) Rana, AnirudhA transition from a d2 to a d law is observed in molecular dynamics (MD) simulations when the diameter (d) of an evaporating droplet reduces to the order of the vapor’s mean free path; this cannot be explained by classical theory. This Letter shows that the d law can be predicted within the Navier-Stokes-Fourier (NSF) paradigm if a temperature-jump boundary condition derived from kinetic theory is utilized. The results from this model agree with those from MD in terms of the total lifetime, droplet radius, and temperature, while the classical d2 law underpredicts the lifetime of the droplet by a factor of 2. Theories beyond NSF are also employed in order to investigate vapor rarefaction effects within the Knudsen layer adjacent to the interface.Item Heat transfer in micro devices packaged in partial vacuum(IOP, 2012) Rana, AnirudhThe influence of rarefaction effects on technical processes is studied numerically for a heat transfer problem in a rarefied gas, a box with bottom heated plate. Solutions obtained from several macroscopic models, in particular the classical Navier-Stokes-Fourier equations with jump and slip boundary conditions, and the regularized 13 moment (R13) equations [Struchtrup & Torrilhon, Phys. Fluids 15, 2003] are compared. The R13 results show significant flow patterns which are not present in the classical hydrodynamic description.Item Evaporation boundary conditions for the R13 equations of rarefied gas dynamics(AIP, 2017-09) Rana, AnirudhThe regularized 13 moment (R13) equations are a macroscopic model for the description of rarefied gas flows in the transition regime. The equations have been shown to give meaningful results for Knudsen numbers up to about 0.5. Here, their range of applicability is extended by deriving and testing boundary conditions for evaporating and condensing interfaces. The macroscopic interface conditions are derived from the microscopic interface conditions of kinetic theory. Tests include evaporation into a half-space and evaporation/condensation of a vapor between two liquid surfaces of different temperatures. Comparison indicates that overall the R13 equations agree better with microscopic solutions than classical hydrodynamics.Item DSMC and R13 modeling of the adiabatic surface(Elsevier, 2016-03) Rana, AnirudhAdiabatic wall boundary conditions for rarefied gas flows are described with the isotropic scattering model. An appropriate sampling technique for the direct simulation Monte Carlo (DSMC) method is presented, and the corresponding macroscopic boundary equations for the regularized 13-moment system (R13) are obtained. DSMC simulation of a lid driven cavity shows slip at the wall, which, as a viscous effect, creates heat that enters the gas while there is no heat flux in the wall. Analysis with the macroscopic equations and their boundary conditions reveals that this heat flux is due to viscous slip heating, and is the product of slip velocity and shear stress at the adiabatic surface. DSMC simulations of the driven cavity with adiabatic walls are compared to R13 simulations, which both show this non-linear effect in good agreement for Kn < 0.3.