The Small Deborah Number Limit of the Doi-Onsager Equation to the Ericksen-Leslie Equation

AbstractLarge scale computation by molecular dynamics (MD) method is often challenging or even impractical due to its computational cost, in spite of its wide applications in a variety of fields. Although the recent advancement in parallel computing and introduction of coarse-graining methods have enabled large scale calculations, macroscopic analyses are still not realizable. Here, we present renormalized molecular dynamics (RMD), a renormalization group of MD in thermal equilibrium derived by using the Migdal–Kadanoff approximation. The RMD method improves the computational efficiency drastically while retaining the advantage of MD. The computational efficiency is improved by a factor of $$2^{n(D+1)}$$ 2 n ( D + 1 ) over conventional MD where D is the spatial dimension and n is the number of applied renormalization transforms. We verify RMD by conducting two simulations; melting of an aluminum slab and collision of aluminum spheres. Both problems show that the expectation values of physical quantities are in good agreement after the renormalization, whereas the consumption time is reduced as expected. To observe behavior of RMD near the critical point, the critical exponent of the Lennard-Jones potential is extracted by calculating specific heat on the mesoscale. The critical exponent is obtained as $$\nu =0.63\pm 0.01$$ ν = 0.63 ± 0.01 . In addition, the renormalization group of dissipative particle dynamics (DPD) is derived. Renormalized DPD is equivalent to RMD in isothermal systems under the condition such that Deborah number $$De\ll 1$$ D e ≪ 1 .

Download Full-text

A novel formulation of 3D Jeffery fluid flow near an irregular permeable surface having chemical reactive species

Advances in Mechanical Engineering ◽

10.1177/16878140211013609 ◽

2021 ◽

Vol 13 (5) ◽

pp. 168781402110136

Author(s):

Mumtaz Khan ◽

Amer Rasheed ◽

Shafqat Ali ◽

Qurat-ul-Ain Azim

Keyword(s):

Fluid Flow ◽

Variable Thickness ◽

Mass Transfer Rate ◽

Skin Friction Coefficient ◽

Deborah Number ◽

Pictorial Representation ◽

Fluid Temperature ◽

Flow Parameters ◽

Opposite Behavior ◽

Thickness Parameter

The main objective of this paper is to offer a comprehensive study regarding solar radiation and MHD effects on 3D boundary layer Jeffery fluid flow over a non-uniform stretched sheet along with variable thickness, porous medium and chemical reaction of first order are assumed. The system of equations representing temperature, velocity and concentration fields are converted into dimensionless form by introducing dimensionless variables. Thereafter, the aforesaid equations are solved with the help of BVP4C in MATLAB. The numerical results obtained through this scheme are more accurate when compared with those in the existing literature. In order to have a pictorial representation, the effects of material and flow parameters on velocity, temperature and concentration profiles are presented through graphs. Moreover, the numerical values of heat and mass transfer rate and skin friction coefficient are given in tabular form. It is evident from the acquired results, that the velocity offers two fold behavior for variable thickness parameter that is, n < 1 close and away from the non-uniform surface. It is also noted that the axial and transverse velocities show an increasing behavior for Deborah number while the fluid temperature and concentration shows opposite behavior at the same time.

Download Full-text

On the Measurement of Dynamic Properties of Lubricants under E. H. L. Conditions Using a Disc Machine

Journal of Mechanical Engineering Science ◽

10.1243/jmes_jour_1974_016_060_02 ◽

1974 ◽

Vol 16 (5) ◽

pp. 331-338 ◽

Cited By ~ 3

Author(s):

J. M. Davies ◽

K. Walters

Keyword(s):

Viscoelastic Properties ◽

Dynamic Properties ◽

Elastohydrodynamic Lubrication ◽

Deborah Number ◽

Theoretical Treatment

Johnson and Roberts (1)‡ have recently shown how a simple adaptation of the conventional disc machine can be used to measure the viscoelastic properties of lubricants under conditions of elastohydrodynamic lubrication. The present paper contains a theoretical treatment of the problem and derives operating formulae pertinent to conditions of low, finite and high Deborah number.

Download Full-text

Analysis of the Motion and Deformation of a Bubble Rising in a Viscoelastic Fluid

Volume 2: Fora ◽

10.1115/fedsm2006-98363 ◽

2006 ◽

Author(s):

Shriram Pillapakkam ◽

Pushpendra Singh ◽

Denis L. Blackmore ◽

Nadine Aubry

Keyword(s):

Velocity Field ◽

Newtonian Fluid ◽

Viscoelastic Fluid ◽

Bubble Radius ◽

Polymer Concentration ◽

Deborah Number ◽

Two Phase ◽

Bubble Volume ◽

Recirculation Zones ◽

Bubble Rising

A finite element code based on the level set method is developed for performing two and three dimensional direct numerical simulations (DNS) of viscoelastic two-phase flow problems. The Oldroyd-B constitutive equation is used to model the viscoelastic liquid and both transient and steady state shapes of bubbles in viscoelastic buoyancy driven flows are studied. The influence of the governing dimensionless parameters, namely the Capillary number (Ca), the Deborah Number (De) and the polymer concentration parameter c, on the deformation of the bubble is also analyzed. Our simulations demonstrate that the rise velocity oscillates before reaching a steady value. The shape of the bubble, the magnitude of velocity overshoot and the amount of damping depend mainly on the parameter c and the bubble radius. Simulations also show that there is a critical bubble volume at which there is a sharp increase in the bubble terminal velocity as the increasing bubble volume increases, similar to the behavior observed in experiments. The structure of the wake of a bubble rising in a Newtonian fluid is strikingly different from that of a bubble rising in a viscoelastic fluid. In addition to the two recirculation zones at the equator of the bubble rising in a Newtonian fluid, two more recirculation zones exist in the wake of a bubble rising in viscoelastic fluids which influence the shape of a rising bubble. Interestingly, the direction of motion of the fluid a short distance below the trailing edge of a bubble rising in a viscoelastic fluid is in the opposite direction to the direction of the motion of the bubble, thus creating a “negative wake”. In this paper, the velocity field in the wake of the bubble, the effect of the parameters on the velocity field and their influence on the shape of the bubble are also investigated.

Download Full-text

Numerical Study of Viscoelastic Micropolar Heat Transfer from a Vertical Cone for Thermal Polymer Coating

Nonlinear Engineering ◽

10.1515/nleng-2018-0064 ◽

2019 ◽

Vol 8 (1) ◽

pp. 449-460 ◽

Cited By ~ 1

Author(s):

K. Madhavi ◽

V. Ramachandra Prasad ◽

A. Subba Rao ◽

O. Anwar Bég ◽

A. Kadir

Keyword(s):

Heat Transfer ◽

Boundary Layer ◽

Angular Velocity ◽

Numerical Study ◽

Relaxation Times ◽

Viscoelastic Model ◽

Deborah Number ◽

Convection Boundary Layer ◽

Density Parameter ◽

R Ratio

Abstract A mathematical model is developed to study laminar, nonlinear, non-isothermal, steady-state free convection boundary layer flow and heat transfer of a micropolar viscoelastic fluid from a vertical isothermal cone. The Eringen model and Jeffery’s viscoelastic model are combined to simulate the non-Newtonian characteristics of polymers, which constitutes a novelty of the present work. The transformed conservation equations for linear momentum, angular momentum and energy are solved numerically under physically viable boundary conditions using a finite difference scheme (Keller Box method). The effects of Deborah number (De), Eringen vortex viscosity parameter (R), ratio of relaxation to retardation times (λ), micro-inertia density parameter (B), Prandtl number (Pr) and dimensionless stream wise coordinate (ξ) on velocity, surface temperature and angular velocity in the boundary layer regime are evaluated. The computations show that with greater ratio of retardation to relaxation times, the linear and angular velocity are enhanced whereas temperature (and also thermal boundary layer thickness) is reduced. Greater values of the Eringen parameter decelerate both the linear velocity and micro-rotation values and enhance temperatures. Increasing Deborah number decelerates the linear flow and Nusselt number whereas it increases temperatures and boosts micro-rotation magnitudes. The study is relevant to non-Newtonian polymeric thermal coating processes.

Download Full-text

A MATHEMATICAL MODEL OF NONSTATIONARY MOTION OF A VISCOELASTIC FLUID IN ROLLER BEARINGS

Problems of Strength and Plasticity ◽

10.32326/1814-9146-2019-81-4-501-512 ◽

2019 ◽

Vol 81 (4) ◽

pp. 501-512

Author(s):

I.A. Zhurba Eremeeva ◽

D. Scerrato ◽

C. Cardillo ◽

A. Tran

Keyword(s):

Mathematical Model ◽

Viscoelastic Fluid ◽

Viscoelastic Properties ◽

Fluid Model ◽

Maxwell Fluid ◽

Initial Boundary ◽

Deborah Number ◽

Nonstationary Motion ◽

Friction Forces ◽

Roller Bearings

Nowadays, the emergence of new lubricants requires an enhancement of the rheological models and methods used for solution of corresponding initial boundary-value problems. In particular, models that take into account viscoelastic properties are of great interest. In the present paper we consider the mathematical model of nonstationary motion of a viscoelastic fluid in roller bearings. We used the Maxwell fluid model for the modeling of fluid properties. The viscoelastic properties are exhibited by many lubricants that use polymer additives. In addition, viscoelastic properties can be essential at high fluid speeds. Also, viscoelastic properties can be significant in the case of thin gaps. Maxwell's model is one of the most common models of viscoelastic materials. It combines the relative simplicity of constitutive equations with the ability to describe a stress relaxation. In addition, viscoelastic fluids also allow us to describe some effects that are missing in the case of viscous fluid. An example it is worth to mention the Weissenberg effect and a number of others. In particular, such effects can be used to increase the efficiency of the film carrier in the sliding bearings. Here we introduced characteristic assumptions on the form of the flow, allowing to significantly simplify the solution of the problem. We consider so-called self-similar solutions, which allows us to get a solution in an analytical form. As a result these assumptions, the formulae for pressure and friction forces are derived. Their dependency on time and Deborah number is analyzed. The limiting values of the flow characteristics were obtained. The latter can be used for steady state of the flow regime. Differences from the case of Newtonian fluid are discussed. It is shown that viscoelastic properties are most evident at the initial stage of flow, when the effects of non-stationarity are most important.

Download Full-text

Influence of Melting Heat Transfer in the Stagnation-Point Flow of a Jeffrey Fluid in the Presence of Viscous Dissipation

Journal of Applied Mechanics ◽

10.1115/1.4005560 ◽

2012 ◽

Vol 79 (2) ◽

Cited By ~ 23

Author(s):

M. Mustafa ◽

T. Hayat ◽

Awatif A. Hendi

Keyword(s):

Heat Transfer ◽

Stagnation Point ◽

Viscous Dissipation ◽

Melting Process ◽

Deborah Number ◽

Stagnation Point Flow ◽

Heat Transfer Analysis ◽

Jeffrey Fluid ◽

Melting Heat ◽

Melting Heat Transfer

This communication studies the effect of melting heat transfer on the stagnation-point flow of a Jeffrey fluid over a stretching sheet. Heat transfer analysis is carried out in the presence of viscous dissipation. The arising differential system has been solved by the homotopy analysis method (HAM). The results indicate an increase in the velocity and the boundary layer thickness with an increase in the values of the elastic parameter (Deborah number) for a Jeffrey fluid which are opposite to those accounted for in the literature for the other subclasses of rate type fluids. Furthermore, an increase in the melting process corresponds to an increase in the velocity and a decrease in the temperature. A comparative study between the current computations and the previous studies is also presented in a limiting sense.

Download Full-text

Nematic-Smectic a Phase-Transition Under Shear Flow

MRS Proceedings ◽

10.1557/proc-177-153 ◽

1989 ◽

Vol 177 ◽

Cited By ~ 1

Author(s):

R. F. Bruinsma ◽

C. R. Safinya

Keyword(s):

Phase Transition ◽

Critical Temperature ◽

Shear Flow ◽

Structure Factor ◽

Deborah Number ◽

Flow Properties ◽

Newtonian Flow ◽

One Dimensional ◽

Smectic A ◽

Smectic A Phase

ABSTRACTWe discuss the effect of shear flow on the nematic to smectic A phase transition. The non-Newtonian flow properties (shear thinning and normal stress) ire correlated with the distortions of the structure-factor S(q). As a function of the Deborah number, we find first a deformation of S(q) and, beyond a critical Deborah number, highly distorted, one-dimensional fluctuations. The suppression of fluctuations by shear flow raises the critical temperature.

Download Full-text

Determination of the dipole moments of some polar liquids dissolved in polar solvents

Canadian Journal of Chemistry ◽

10.1139/v69-628 ◽

1969 ◽

Vol 47 (20) ◽

pp. 3767-3771 ◽

Cited By ~ 3

Author(s):

H. A. Rizk ◽

N. Youssef ◽

H. Grace

Keyword(s):

Dipole Moments ◽

Butyl Alcohol ◽

Polar Solvents ◽

Tert Butyl ◽

Tert Butyl Alcohol ◽

Initial Decrease ◽

Interacting Systems ◽

Solvent Molecules ◽

Onsager Equation

The application of a modified form of the Onsager equation at the condition of infinite dilution of a polar solute in a polar solvent leads to reasonable dipole moments for water, pyridine, acetone, tert-butyl alcohol, n-butyl alcohol, and β-octyl alcohol, except in the case of water in tert-butyl alcohol at 30 and 40 °C and the case of acetone in n-butyl alcohol at 30 to 50 °C. The initial decrease of the dielectric constant of solvent by addition of solute in each of these two cases is associated with a reduction in the Kirkwood g-factor of solute. In all 12 systems investigated, strong hydrogen bonding occurs between solute and solvent molecules and often between solvent molecules themselves. It is thought that this equation must fail when short-range interactions assume predominant importance, but why it works so well for those cases which are also strongly interacting systems is not clear.

Download Full-text