Browsing by Author "Rana, Anirudh"

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Analytical and Numerical Solutions of Boundary Value Problems for the Regularized 13 Moment Equations
(AIP Conference Proceedings, 2011-05) Rana, Anirudh
Classical 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.
Applications of Nano-Biotechnological Approaches in Diagnosis and Protection of Wheat Diseases
(Springer, 2022-10) Rana, Anirudh
Wheat (Triticum aestivum) is a major staple food crop, plays a crucial role in food security, and is grown on an area of 221.6 million hectares (Mha) in multi-environments throughout the globe. Annual wheat production was recorded at 778.6 million metric tons in the years 2020–2021. Regardless of the abundant growth of wheat, people are facing food crises in some parts of the world because of the unavailability of food grains. The ever-growing population of the world is creating a new challenge for farmers and researchers. By the year 2050, the global need for agricultural products will have risen by 50%. To make it more challenging, biotic and abiotic factors become constant reasons for wheat yield losses. Continuously, the wheat crop suffers from a plethora of diseases (pests, insects, fungi, and bacteria). To deal with the challenges given above and meet future food needs, there is a strong need for new and cutting-edge technologies that can keep wheat farming sustainable and boost wheat production from current cropping systems and changing climates.
Coupled constitutive relations: a second law based higher-order closure for hydrodynamics
(RSC, 2018-10) Rana, Anirudh
In 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.
Coupled constitutive relations: a second lawbased higher-order closure for hydrodynamics
(RSC, 2018-10) Rana, Anirudh
In 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.
DSMC and R13 modeling of the adiabatic surface
(Elsevier, 2016-03) Rana, Anirudh
Adiabatic 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.
Efficient moment method for modeling nanoporous evaporation
(APS, 2022-02) Rana, Anirudh
Thin-film-based nanoporous membrane technologies exploit evaporation to efficiently cool microscale and nanoscale electronic devices. At these scales, when domain sizes become comparable to the mean-free path in the vapor, traditional macroscopic approaches such as the Navier-Stokes-Fourier (NSF) equations become less accurate, and the use of higher-order moment methods is called for. Two higher-order moment equations are considered; the linearized versions of the Grad 13 and Regularized 13 equations. These are applied to the problem of nanoporous evaporation, and results are compared to the NSF method and the method of direct simulation Monte Carlo (i.e., solutions to the Boltzmann equations). Linear and nonlinear versions of the boundary conditions are examined, with the latter providing improved results, at little additional computational expense, compared to the linear form. The outcome is a simultaneously accurate and computationally efficient method, which can provide simulation-for-design capabilities at the nanoscale.
Efficient simulation of non-classical liquid–vapour phase-transition flows: a method of fundamental solutions
(CUP, 2021-06) Rana, Anirudh
Classical continuum-based liquid–vapour phase-change models typically assume continuity of temperature at phase interfaces along with a relation which describes the rate of evaporation at the interface (Hertz–Knudsen–Schrage, for example). However, for phase-transition processes at small scales, such as the evaporation of nanodroplets, the assumption that the temperature is continuous across the liquid–vapour interface leads to significant inaccuracies (McGaughey et al., J. Appl. Phys., vol. 91, issue 10, pp. 6406–6415; Rana et al., Phys. Rev. Lett., vol. 123, 154501), as might the adoption of classical constitutive relations that lead to the Navier–Stokes–Fourier (NSF) equations. In this paper, to capture the notable effects of rarefaction at small scales, we adopt an extended continuum-based approach utilising the coupled constitutive relations (CCRs). In CCR theory, additional terms are invoked in the constitutive relations of the NSF equations originating from the arguments of irreversible thermodynamics as well as being consistent with the kinetic theory of gases. The modelling approach allows us to derive new fundamental solutions for the linearised CCR model, to develop a numerical framework based upon the method of fundamental solutions (MFS) and enables three-dimensional multiphase micro-flow simulations to be performed at remarkably low computational cost. The new framework is benchmarked against classical results and then explored as an efficient tool for solving three-dimensional phase-change events involving droplets.
Evaporation Boundary Conditions for the Linear R13 Equations Based on the Onsager Theory
(MDPI, 2018-09) Rana, Anirudh
Due 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.
Evaporation boundary conditions for the R13 equations of rarefied gas dynamics
(AIP, 2017-09) Rana, Anirudh
The 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.
Evaporation-driven vapour microflows: analytical solutions from moment methods
(CUP, 2018-03) Rana, Anirudh
Macroscopic 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.
A finite difference scheme for non-Cartesian mesh: Applications to rarefied gas flows
(AIP, 2022-07) Rana, Anirudh
A novel numerical scheme based on the finite-difference framework is developed, which allows us to model moderately rarefied gas flows in irregular geometries. The major hurdle in constructing numerical methods for rarefied gas flows is the prescription of the velocity-slip and temperature-jump boundary conditions as well as the discretization of an intricate set of partial differential equations. The proposed scheme is demonstrated to solve the non-linear coupled constitutive relations model along with the corresponding non-linear slip and jump boundary conditions. The computation of the discretized weights is proposed using two approaches: (i) polynomial shape functions and (ii) a generalized inverse distance approach. The non-linear terms are discretized using the fixed-point iteration method. The numerical method is validated for the Laplace equation over an annulus, and results are presented for a lid-driven curved cavity and a triangular lid-driven cavity, which delineates its performance on a skewed non-Cartesian grid. The results are validated with direct simulation Monte Carlo data from the literature, and a robust convergence for the solutions is demonstrated.
Fundamental solutions of an extended hydrodynamic model in two dimensions: Derivation, theory, and applications
(APS, 2023-07) Rana, Anirudh
The inability of the Navier-Stokes-Fourier equations to capture rarefaction effects motivates us to adopt the extended hydrodynamic equations. In the present work, a hydrodynamic model, which consists of the conservation laws closed with the recently propounded coupled constitutive relations (CCR), is utilized. This model is referred to as the CCR model and is adequate for describing moderately rarefied gas flows. A numerical framework based on the method of fundamental solutions is developed to solve the CCR model for rarefied gas flow problems in quasi two dimensions. To this end, the fundamental solutions of the linearized CCR model are derived in two dimensions. The significance of deriving the two-dimensional fundamental solutions is that they cannot be deduced from their three-dimensional counterparts that do exist in literature. As applications, the developed numerical framework based on the derived fundamental solutions is used to simulate (i) a rarefied gas flow between two coaxial cylinders with evaporating walls and (ii) a temperature-driven rarefied gas flow between two noncoaxial cylinders. The results for both problems have been validated against those obtained with the other classical approaches. Through this, it is shown that the method of fundamental solutions is an efficient tool for addressing quasi-two-dimensional multiphase microscale gas flow problems at a low computational cost. Moreover, the findings also show that the CCR model solved with the method of fundamental solutions is able to describe rarefaction effects, like transpiration flows and thermal stress, generally well.
Fundamental solutions to the regularised 13-moment equations: efficient computation of three-dimensional kinetic effects
(CUP, 2017-11) Rana, Anirudh
Fundamental 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.
H -theorem and boundary conditions for the linear R26 equations: application to flow past an evaporating droplet
(CUP, 2021-08) Rana, Anirudh
Determining physically admissible boundary conditions for higher moments in an extended continuum model is recognised as a major obstacle. Boundary conditions for the regularised 26-moment (R26) equations obtained using Maxwell's accommodation model do exist in the literature; however, we show in this article that these boundary conditions violate the second law of thermodynamics and the Onsager reciprocity relations for certain boundary value problems, and, hence, are not physically admissible. We further prove that the linearised R26 (LR26) equations possess a proper H-theorem (second-law inequality) by determining a quadratic form without cross-product terms for the entropy density. The establishment of the H-theorem for the LR26 equations in turn leads to a complete set of boundary conditions that are physically admissible for all processes and comply with the Onsager reciprocity relations. As an application, the problem of a slow rarefied gas flow past a spherical droplet with and without evaporation is considered and solved analytically. The results are compared with the numerical solution of the linearised Boltzmann equation, experimental results from the literature and/or other macroscopic theories to show that the LR26 theory with the physically admissible boundary conditions provides an excellent prediction up to Knudsen number ≲1 and, consequently, provides transpicuous insights into intriguing effects, such as thermal polarisation. In particular, the analytic results for the drag force obtained in the present work are in an excellent agreement with experimental results even for very large values of the Knudsen number.
Heat transfer in micro devices packaged in partial vacuum
(IOP, 2012) Rana, Anirudh
The 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.
Lifetime of a Nanodroplet: Kinetic Effects and Regime Transitions
(APS, 2019-10) Rana, Anirudh
A 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.
Macroscopic description of steady and unsteady rarefaction effects in boundary value problems of gas dynamics
(Springer, 2009-10) Rana, Anirudh
Four basic flow configurations are employed to investigate steady and unsteady rarefaction effects in monatomic ideal gas flows. Internal and external flows in planar geometry, namely, viscous slip (Kramer’s problem), thermal creep, oscillatory Couette, and pulsating Poiseuille flows are considered. A characteristic feature of the selected problems is the formation of the Knudsen boundary layers, where non-Newtonian stress and non-Fourier heat conduction exist. The linearized Navier–Stokes–Fourier and regularized 13-moment equations are utilized to analytically represent the rarefaction effects in these boundary-value problems. It is shown that the regularized 13-moment system correctly estimates the structure of Knudsen layers, compared to the linearized Boltzmann equation data.
Microscopic molecular dynamics characterization of the second-order non-Navier-Fourier constitutive laws in the Poiseuille gas flow
(AIP, 2016-08) Rana, Anirudh
The second-order non-Navier-Fourier constitutive laws, expressed in a compact algebraic mathematical form, were validated for the force-driven Poiseuille gas flow by the deterministic atomic-level microscopic molecular dynamics (MD). Emphasis is placed on how completely different methods (a second-order continuum macroscopic theory based on the kinetic Boltzmann equation, the probabilistic mesoscopic direct simulation Monte Carlo, and, in particular, the deterministic microscopic MD) describe the non-classical physics, and whether the second-order non-Navier-Fourier constitutive laws derived from the continuum theory can be validated using MD solutions for the viscous stress and heat flux calculated directly from the molecular data using the statistical method. Peculiar behaviors (non-uniform tangent pressure profile and exotic instantaneous heat conduction from cold to hot [R. S. Myong, “A full analytical solution for the force-driven compressible Poiseuille gas flow based on a nonlinear coupled constitutive relation,” Phys. Fluids 23(1), 012002 (2011)]) were re-examined using atomic-level MD results. It was shown that all three results were in strong qualitative agreement with each other, implying that the second-order non-Navier-Fourier laws are indeed physically legitimate in the transition regime. Furthermore, it was shown that the non-Navier-Fourier constitutive laws are essential for describing non-zero normal stress and tangential heat flux, while the classical and non-classical laws remain similar for shear stress and normal heat flux.
Modeling of Phase Change in Nanoconfinement Using Moment Methods
(ASME, 2023-01) Rana, Anirudh; Aneesh, A.M.
Accurate prediction of liquid–vapor phase change phenomena is critical in the design of thin vapor chambers and microheat pipes for the thermal management of miniaturized electronic systems. In view of this, we have considered the heat and mass transfer between two-liquid meniscuses separated by a thin gap of its own vapor. Assuming the heat and mass flow are to be steady and one-dimensional, analytic solutions are obtained to the linearized equations from the regularized 26-moment framework. Our analytic solutions provide excellent predictions for the effective heat conductivity of a dilute gas with those from the molecular dynamics (MD) and Boltzmann equation where Fourier's law fails. We also verified that the predicted heat and mass flow rates over the whole range of the Knudsen number are consistent with the kinetic theory of gases. Further, the model has been used to predict the effect of evaporation and accommodation coefficients on the heat and mass transfer between the liquid layers
A numerical study of the heat transfer through a rarefied gas confined in a microcavity
(Springer, 2014-07) Rana, Anirudh
Flow and heat transfer in a bottom-heated square cavity in a moderately rarefied gas is investigated using the R13 equations and the Navier–Stokes–Fourier equations. The results obtained are compared with those from the direct simulation Monte Carlo (DSMC) method with emphasis on understanding thermal flow characteristics from the slip flow to the early transition regime. The R13 theory gives satisfying results—including flow patterns in fair agreement with DSMC—in the transition regime, which the conventional Navier–Stokes–Fourier equations are not able to capture.