arXiv subjects

Vinay Kumar Gupta

Publications and source records attributed to Vinay Kumar Gupta.

9 recordsLinked to original sources

arXiv2025

A generalized fundamental solution technique for the regularized 13-moment system in rarefied gas flows

In this work, we explore the method of fundamental solutions (MFS) for solving the regularized 13-moment (R13) equations for rarefied monatomic gases. While previous applications of the MFS in rarefied gas flows relied on problem-specific fundamental solutions, we propose a generic approach that systematically computes the fundamental solutions for any linear moment system without predefined source terms. The generalized framework is first introduced using a simple example involving the Stokes equations, and is then extended to the R13 equations. The results obtained from the generic MFS are validated against an analytical solution for the R13 equations. Following validation, the framework is applied to the case of thermally-induced flow between two non-coaxial cylinders. Since no analytical solution exists for this case, we compare the results obtained from the MFS with those obtained from the finite element method (FEM). To further assess computational efficiency, we analyze the runtimes of the FEM and MFS. The results indicate that the MFS converges faster than the FEM and serves as a promising alternative to conventional meshing-based techniques.

physics.flu-dyn↗

arXiv2025

Exploring external rarefied gas flows through the method of fundamental solutions

The well-known Navier--Stokes--Fourier equations of fluid dynamics are, in general, not adequate for describing rarefied gas flows. Moreover, while the Stokes equations -- a simplified version of the Navier--Stokes--Fourier equations -- are effective in modeling slow and steady liquid flow past a sphere, they fail to yield a non-trivial solution to the problem of slow and steady liquid flow past an infinitely long cylinder (a two-dimensional problem essentially); this is referred to as Stokes' paradox. The paradox also arises when studying these problems for gases. In this paper, we present a way to obtain meaningful solutions for two-dimensional flows of rarefied gases around objects by circumventing Stokes' paradox. To this end, we adopt an extended hydrodynamic model, referred to as the CCR model, consisting of the balance equations for the mass, momentum and energy and closed with the coupled constitutive relations. We determine an analytic solution of the CCR model for the problem and compare it with a numerical solution based on the method of fundamental solutions. Apart from addressing flow past a circular cylinder, we aim to showcase the capabilities of the method of fundamental solutions to predict the flow past other objects in two dimensions for which analytic solutions do not exist or are difficult to determine. For that, we investigate the problem of rarefied gas flow past an infinitely long semicircular cylinder.

physics.flu-dyn↗

arXiv2024

Rarefied gas flow past a liquid droplet: interplay between internal and external flows

Experimental and theoretical studies on millimetre-sized droplets suggest that at low Reynolds number the difference between the drag force on a circulating water droplet and that on a rigid sphere is very small (less than 1 %) (LeClair et al., J. Atmos. Sci., vol. 29, 1972, pp. 728-740). While the drag force on a spherical liquid droplet at high viscosity ratios (of the liquid to the gas), is approximately the same as that on a rigid sphere of the same size, the other quantities of interest (e.g. the temperature) in the case of a rarefied gas flow over a liquid droplet differ from the same quantities in the case of a rarefied gas flow over a rigid sphere. The goal of this article is to study the effects of internal motion within a spherical microdroplet/nanodroplet -- such that its diameter is comparable to the mean free path of the surrounding gas -- on the drag force and its overall dynamics. To this end, the problem of a slow rarefied gas flowing over an incompressible liquid droplet is investigated analytically by considering the internal motion of the liquid inside the droplet and also by accounting for kinetic effects in the gas. Detailed results for different values of the Knudsen number, the ratio of the thermal conductivities and the ratio of viscosities are presented for the pressure and temperature profiles inside and outside the liquid droplet. The results for the drag force obtained in the present work are in good agreement with the theoretical and experimental results existing in the literature.

physics.flu-dyn↗

arXiv2023

Fundamental solutions of an extended hydrodynamic model in two dimensions: Derivation, theory, and applications

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 non-coaxial 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.

physics.flu-dyn↗

arXiv2021

$H$-theorem and boundary conditions for the linear R26 equations: application to flow past an evaporating droplet

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 $\,\lesssim 1$ 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.

physics.flu-dyn↗

arXiv2020

Moment theories for a d-dimensional dilute granular gas of Maxwell molecules

Various systems of moment equations---consisting of up to (d+3)(d^2+6d+2)/6 moments---in a general dimension d for a dilute granular gas composed of Maxwell molecules are derived from the inelastic Boltzmann equation by employing the Grad moment method. The Navier--Stokes-level constitutive relations for the stress and heat flux appearing in the system of mass, momentum and energy balance equations are determined from the derived moment equations. It has been shown that the moment equations only for the hydrodynamic field variables (density, velocity and granular temperature), stress and heat flux---along with the time-independent value of the fourth cumulant---are sufficient for determining the Navier--Stokes-level constitutive relations in the case of inelastic Maxwell molecules, and that the other higher-order moment equations do not play any role in this case. The homogeneous cooling state of a freely cooling granular gas is investigated with the system of the Grad (d+3)(d^2+6d+2)/6-moment equations and its various subsystems. By performing a linear stability analysis in the vicinity of the homogeneous cooling state, the critical system size for the onset of instability is estimated through the considered Grad moment systems. The results on critical system size from the presented moment theories are found to be in reasonably good agreement with those from simulations.

cond-mat.stat-mech↗

arXiv2019

Shear-banding instability in arbitrarily inelastic granular shear flows

One prototypical instability in granular flows is the shear-banding instability, in which a uniform granular shear flow breaks into alternating bands of dense and dilute clusters of particles having low and high shear (shear stress or shear rate), respectively. In this work, the shear-banding instability in an arbitrarily inelastic granular shear flow is analyzed through the linear stability analysis of granular hydrodynamic equations closed with Navier--Stokes-level constitutive relations. It is shown that the choice of appropriate constitutive relations plays an important role in predicting the shear-banding instability. A parametric study is carried out to study the effect of the restitution coefficient, channel width and mean density. Two global criteria relating the control parameters are found for the onset of the shear-banding instability.

cond-mat.soft↗

arXiv2019

Coupled constitutive relations: a second law based higher order closure for hydrodynamics

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 description 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 can not 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.

physics.flu-dyn↗

arXiv2017

Higher-order moment theories for dilute granular gases of smooth hard-spheres

Grad's method of moments is employed to develop higher-order Grad's moment equations---up to first 26-moments---for granular gases within the framework of the (inelastic) Boltzmann equation. The homogeneous cooling state of a freely cooling granular gas is investigated with the Grad's 26-moment equations in a semi-linearized setting and it is shown that the granular temperature in the homogeneous cooling state still decays according to Haff's law while the other higher-order moments decay on a faster time scale. The constitutive relations for stress and heat flux (the Navier--Stokes and Fourier relations) are obtained by performing a Chapman--Enskog-like expansion on the Grad's 26-moment equations and compared with those existing in the literature. The linear stability of the homogeneous cooling state is analyzed through the Grad's 26-moment system and various sub-systems by decomposing them into longitudinal and transverse systems. It is found that one eigenmode in both longitudinal and transverse systems in case of inelastic gases is unstable. By comparing the eigenmodes from various theories, it is established that the 13-moment eigenmode theory predicts that the unstable eigenmode remains unstable for all wavenumbers below a certain coefficient of restitution while any other higher-order moment theory shows that this mode becomes stable above some critical wavenumber for all values of coefficient of restitution. In particular, the Grad's 26-moment theory leads to a smooth profile for the critical wavenumber in contrast to the other considered theories. Furthermore, the critical system size obtained through the Grad 26-moment and existing theories are also in excellent agreement.

cond-mat.soft↗