Quiescent Newtonian Fluid Research Articles

On the numerical simulation of the unsteady free fall of a solid in a fluid: I. The Newtonian case

In the last decade, extensive experiments have been carried out in order to gain a better insight in the phenomenology related to the motion of a falling body in a quiescent Newtonian fluid. This renewal of interest in that subject is due not only to the lack of full theoretical explanations of the intricate body motion but also to the high relevance of that issue in applications ranging from meteorology, sedimentology and aerospace engineering to biology. We present a new numerical method for the simulation of the instationary free fall of a unique solid in a fluid. A key ingredient of the proposed approach is the reformulation of the conservation and kinetic equations in the solid frame as well as the explicit treatment of the fluid-body coupling. The issue of non-smooth data in time resulting from our explicit approach is addressed. The time stepping is based on the implicit fractional-step- θ scheme and the spatial discretization relies on the finite element method. Numerical experiments for the steady-falling regime, for the periodic oscillating motion as well as for the tumbling motion are presented following existing experimental set-up. The proposed method is validated by comparison with experimental data.

Cooling of aPr<1 fluid in a rectangular container

The flow behaviour associated with the cooling of an initially quiescent isothermal Newtonian fluid with Prandtl numberPrless than unity in a rectangular container by unsteady natural convection with an imposed lower temperature on vertical sidewalls is investigated by scaling analysis and direct numerical simulation. The flow is dominated by two distinct stages of development. i.e the boundary-layer development stage adjacent to the sidewall and the subsequent cooling-down stage. The first stage can be further divided into a start-up stage, transitional stage, and steady-state stage. The parameters characterizing the flow behaviour are the boundary-layer thickness, the maximum vertical velocity within the boundary layer, the time for the boundary layer to reach the steady state, the Nusselt number across the sidewall at the boundary-layer development stage, the time for the fluid in the container to be fully cooled down, and the average fluid temperature over the whole volume of the container.

Stokes flow in ellipsoidal geometry

Particle-in-cell models for Stokes flow through a relatively homogeneous swarm of particles are of substantial practical interest, because they provide a relatively simple platform for the analytical or semianalytical solution of heat and mass transport problems. Despite the fact that many practical applications involve relatively small particles (inorganic, organic, biological) with axisymmetric shapes, the general consideration consists of rigid particles of arbitrary shape. The present work is concerned with some interesting aspects of the theoretical analysis of creeping flow in ellipsoidal, hence nonaxisymmetric domains. More specifically, the low Reynolds number flow of a swarm of ellipsoidal particles in an otherwise quiescent Newtonian fluid, that move with constant uniform velocity in an arbitrary direction and rotate with an arbitrary constant angular velocity, is analyzed with an ellipsoid-in-cell model. The solid internal ellipsoid represents a particle of the swarm. The external ellipsoid contains the ellipsoidal particle and the amount of fluid required to match the fluid volume fraction of the swarm. The nonslip flow condition on the surface of the solid ellipsoid is supplemented by the boundary conditions on the external ellipsoidal surface which are similar to those of the sphere-in-cell model of Happel (self-sufficient in mechanical energy). This model requires zero normal velocity component and shear stress. The boundary value problem is solved with the aim of the potential representation theory. In particular, the Papkovich–Neuber complete differential representation of Stokes flow, valid for nonaxisymmetric geometries, is considered here, which provides the velocity and total pressure fields in terms of harmonic ellipsoidal eigenfunctions. The flexibility of the particular representation is demonstrated by imposing some conditions, which made the calculations possible. It turns out that the velocity of first degree, which represents the leading term of the series, is sufficient for most engineering applications, so long as the aspect ratios of the ellipsoids remains within moderate bounds. Analytical expressions for the leading terms of the velocity, the total pressure, the angular velocity, and the stress tensor fields are obtained. Corresponding results for the prolate and the oblate spheroid, the needle and the disk, as well as for the sphere are recovered as degenerate cases. Novel relations concerning the ellipsoidal harmonics are included in the Appendix.

3D simulations of hydrodynamic drag forces on two porous spheres moving along their centerline

This paper numerically evaluates the hydrodynamic drag force exerted on two highly porous spheres moving steadily along their centerline through a quiescent Newtonian fluid over a Reynolds number ranging from 0.1 to 40. At creeping-flow limit, the drag forces exerted on both spheres were approximately identical. At higher Reynolds numbers the drag force on the leading sphere (sphere #1) was higher than the following sphere (sphere #2), revealing the shading effects produced by sphere #1 on sphere #2. At dimensionless diameter β < 2 ( β = d f / 2 k 0.5 , d f and k are sphere diameter and interior permeability, respectively), the spheres can be regarded as “no-spheres” limit. At increasing β for both spheres, the drag force on sphere #2 was increased because of the more difficult advective flow through its interior, and at the same time the drag was reduced owing to the stronger wake flow produced by the denser sphere #1. The competition between these two effects leads to complicated dependence of drag force on sphere #2 on β value. These effects were minimal when β became low.

Scaling laws for unsteady natural convection cooling of fluid with Prandtl number less than one in a vertical cylinder

The flow behavior associated with cooling an initially quiescent isothermal Newtonian fluid with Prandtl number (Pr) less than one in a vertical cylinder by unsteady natural convection with an imposed lower temperature on vertical sidewalls is investigated by scaling analysis and direct numerical simulation. The flow is dominated by three distinct stages of development, i.e., the boundary-layer development stage adjacent to the sidewall, the stratification stage, and the cooling-down stage, respectively. The first stage can be further divided into three distinct substages, i.e., the start-up stage, the transitional stage, and the boundary-layer steady-state stage, respectively. A scaling analysis is carried out to obtain scaling laws for the basic flow features in terms of the flow control parameters, i.e., the Rayleigh number Ra, Pr, and the aspect ratio of the cylinder A , respectively. A series of direct numerical simulations with selected values of A , Ra, and Pr in the ranges of 1/3< or = A< or =3, 10(6) < or =Ra < or = 10(10) , and 0.01< or =Pr< or =0.5 are carried out, and it is found that the numerical results agree well with the scaling laws. These numerical results are further used to quantify these scaling laws for Ra, A , and Pr in the above-mentioned ranges.

Long-term behavior of cooling fluid in a vertical cylinder

The long-term behavior of cooling an initially quiescent isothermal Newtonian fluid in a vertical cylinder by unsteady natural convection with a fixed wall temperature has been investigated in this study by scaling analysis and direct numerical simulation. Two specific cases are considered. Case 1 assumes that the fluid cooling is due to the imposed fixed temperature on the vertical sidewall whereas the top and bottom boundaries are adiabatic. Case 2 assumes that the cooling is due to that on both the vertical sidewall and the bottom boundary whereas the top boundary is adiabatic. The long-term behavior of the fluid cooling in the cylinder is well represented by T a( t), the average fluid temperature in the cylinder at time t, and the average Nusselt number on the cooling boundary. The scaling analysis shows that for both cases θ a( τ) scales as e - C ( A Ra ) - 1 / 4 τ - 1 , where θ a( τ) is the dimensionless form of T a( t), τ the dimensionless time, A the aspect ratio of the vertical cylinder, Ra the Rayleigh number, and C a proportionality constant. A series of direct numerical simulations with the selected values of A, Ra, and Pr ( Pr is the Prandtl number) in the ranges of 1/3 ⩽ A ⩽ 3, 6 × 10 6 ⩽ Ra ⩽ 6 × 10 10, and 1 ⩽ Pr ⩽ 1000 have been carried out for both cases to validate the developed scaling relations, and it is found that these numerical results agree well with the scaling relations. These numerical results have also been used to quantify the scaling relations and it is found that C = 1.287 and 1.357 respectively for Case 1 and 2 with Ra, A and Pr in the ranges of 1/3 ⩽ A ⩽ 3, 6 × 10 6 ⩽ Ra ⩽ 6 × 10 10, and 1 ⩽ Pr ⩽ 1000.

Long-term behavior of cooling fluid in a rectangular container.

In this study, the long-term behavior of cooling an initially quiescent isothermal Newtonian fluid in a rectangular container with an infinite length by unsteady natural convection due to a fixed wall temperature has been investigated by scaling analysis and direct numerical simulation. Two specific cases are considered. Case 1 assumes that the cooling of the fluid is caused by the imposed fixed temperature on the vertical sidewall while the top and bottom boundaries are adiabatic. Case 2 assumes that the cooling is caused by the imposed fixed temperature on both the vertical sidewall and the bottom boundary while the top boundary is adiabatic. The appropriate parameters to represent the long-term behavior of the fluid cooling in the container are the transient average fluid temperature T(a)(t) over the whole volume of the container per unit length (i.e., the transient area average fluid temperature, as used in the subsequent numerical simulations) at time t and the average Nusselt number on the cooling boundary. A scaling analysis has been carried out which shows that for both cases theta(a)(tau) scales as e(-C(ARa)(-1/4) tau), where theta(a)(tau) is the dimensionless form of T(a)(t), tau is the dimensionless time, A is the aspect ratio of the container, Ra is the Rayleigh number, and C is a proportionality constant. A series of direct numerical simulations with the selected values of A, Ra, and Pr (Pr is the Prandtl number) in the ranges of 1/3< or =A< or =3, 6 x 10(6) < or =Ra< or =6 x 10(10), and 1< or =Pr< or =1000 have been carried out for both cases to validate the developed scaling relations. It is found that these numerical results agree well with the scaling relations. The numerical results have also been used to quantify the scaling relations and it is found that C=0.645 and 0.705 respectively for Cases 1 and 2 with Ra, A and Pr in the above-mentioned ranges.

Effects of the history force on an oscillating rigid sphere at low Reynolds number

The present work presents an experimental study on a free-falling rigid sphere in a quiescent incompressible newtonian fluid, placed in an oscillating frame. The goal of this investigation is to examine the effect of the history force acting on the sphere at small Reynolds numbers (Re≤2.5) and finite Strouhal numbers (1≤Sl≤20). The particle trajectory is measured by using a high-speed video camera and modern techniques of image processing. The average terminal velocity, the oscillation magnitude, and the phase shift with the oscillating frame are measured and compared with those obtained from theoretical approaches. The comparison is made by solving the equation of motion of the sphere with and without the history force. In addition to the significant role that this force plays in the momentum balance, it was found that the correction of the added mass force and the history force by the empirical coefficients of Odar and Hamilton (J Fluid Mech 18:302–314, 1964; J Fluid Mech 25:591–592, 1966) are not necessary in our Re and Sl ranges. The added mass is the same as that obtained by the potential flow theory and the history force is well predicted by the Basset expression (Treatise on hydrodynamics, 1888).

Hydrodynamic drag forces on two porous spheres moving along their centerline

This paper numerically evaluates the hydrodynamic drag force exerted on two highly porous spheres moving steadily along their centerline (sphere #1 and sphere #2) through a quiescent Newtonian fluid over a Reynolds number ranging from 0.1 to 40. At creeping flow limit, the drag forces exerted on both spheres were identical. At higher Reynolds numbers the drag force on sphere #1 was higher than sphere #2, revealing the shading effects produced by sphere #1 on sphere #2. At dimensionless diameter ( β, = d f /2 k 0.5, d f and k are floc diameter and interior permeability, respectively) >20, the spheres can be regarded nonporous. At β<20, the drag forces dropped. At β<2, the drag forces approached “no-spheres” limit. An increased size ratio of two spheres ( d f1 / d f2 ) would increase the drag force on sphere #1 and reduce that on sphere #2. At increasing β for both spheres, the drag force on sphere #2 was increased because of the more difficult advective flow through its interior, and at the same time the drag was reduced owing to the stronger wake flow produced by the denser sphere #1. The competition between these two effects leads to complicated dependence of drag force on sphere #2 on β value. These effects were minimal when β became low. Two identical spheres could move steadily along their centerline. At higher Reynolds number, the two spheres would move closer because of the incorporation of inertia force. For spheres of different diameters, the sphere # 2 would move faster than sphere #1 regardless of their size ratio and β value. This occurrence yielded efficient coagulation when two porous spheres were moving in-line.

The 3D Happel model for complete isotropic Stokes flow

The creeping flow through a swarm of spherical particles that move with constant velocity in an arbitrary direction and rotate with an arbitrary constant angular velocity in a quiescent Newtonian fluid is analyzed with a 3D sphere‐in‐cell model. The mathematical treatment is based on the two‐concentric‐spheres model. The inner sphere comprises one of the particles in the swarm and the outer sphere consists of a fluid envelope. The appropriate boundary conditions of this non‐axisymmetric formulation are similar to those of the 2D sphere‐in‐cell Happel model, namely, nonslip flow condition on the surface of the solid sphere and nil normal velocity component and shear stress on the external spherical surface. The boundary value problem is solved with the aim of the complete Papkovich‐Neuber differential representation of the solutions for Stokes flow, which is valid in non‐axisymmetric geometries and provides us with the velocity and total pressure fields in terms of harmonic spherical eigenfunctions. The solution of this 3D model, which is self‐sufficient in mechanical energy, is obtained in closed form and analytical expressions for the velocity, the total pressure, the angular velocity, and the stress tensor fields are provided.

Motion and shape of a viscoelastic drop falling through a viscous fluid

The steady shape of a drop of dilute polymer solution falling through a quiescent viscous Newtonian fluid is considered. Experimentally, we find that an immiscible drop of 0.16% xanthan gum in 80:20 glycerol/water falling through 9.8 P polydimethylsiloxane oil may exhibit a stable dimple at its trailing edge. At higher volumes the dimple extends far into the interior of the drop, and pinches off via a Rayleigh-type instability, injecting oil droplets into the polymer drop. At even larger volumes, a toroidal shape develops. We show that the dimpled shape can be reproduced mathematically with axisymmetric solutions for Stokes flow past a non-Newtonian drop, using the constitutive equation for a Simple Fluid of Order Three.

Hydrodynamic drag on non-spherical floc and free-settling test

This work numerically investigates the hydrodynamic drag force exerted on a porous spheroid floc moving steadily through a quiescent Newtonian fluid over a wide range of the Reynolds number. The flow patterns for a highly porous spheroid moving at an elevated Reynolds number are basically the same as those at a low Reynolds number, which extends the applicable range of a creeping-flow based correlation to the higher Reynolds number regime. The shape effect becomes more prominent as the spheroid becomes more porous. Using the equivalent diameter, defined as the geometric mean diameter of the principal axes, leads to a universal correlation relating to the drag force, aspect ratio, and interior permeability. In addition, free-settling experiments are performed to estimate how the non-spherical shape affects the three sludge samples. The possible errors in data reduction for the free-settling test are attributed to the a/ b ratio and the internal permeability. The errors range from 16–34% for a/ b=0.6–2.0.

An analytical study of the motion of a sphere rolling down a smooth inclined plane in an incompressible Newtonian fluid

The accelerated motion of a sphere rolling down a smooth inclined plane immersed in a quiescent Newtonian fluid has been investigated theoretically using Jan and Chen's drag and added mass coefficient data [C.D. Jan, J.C. Chen, J. Hydraulic Res. 35 (1997) 689]. A modified expression for the drag coefficient was used to obtain an analytical solution to the equations of motion in the inclined plane. Closed form solutions for the instantaneous acceleration, time and distance required by the sphere to reach a given velocity are presented, and the sphere's settling velocity is determined. The relationship between the plane's angle of inclination and the time and distance required by the sphere to reach 99% of its terminal velocity is also investigated. The analysis reported herein is, however, restricted to dense spheres immersed in light liquids where additional effects arising from the Basset forces can be neglected.

Hydrodynamic drag force exerted on a moving floc and its implication to free-settling tests

This paper numerically evaluates the hydrodynamic drag force exerted on an individual floc moving steadily through an quiescent Newtonian fluid over a wide range of Reynolds numbers. The floc is modelled as a highly porous sphere, whose interior flow field is governed by the combined Darcy–Brinkman model; while the bulk fluid field is governed by Navier–Stokes equations. We used a computational fluid dynamics software, FIDAP 7.5, to solve the fluid field within and around the moving floc, from which the corresponding hydrodynamic drag force exerted on the floc is subsequently obtained. When the floc permeability is rather high, which is usually the case for activated sludge flocs, the drag force was found to be inversely proportional to the floc Reynolds number. The floc density versus size relationship obtained on the basis of the Stokes law-like correlation is thereby applicable to estimate the flocs' fractal dimension, however, is erroneous for estimating the floc density.

Stokes flow in spheroidal particle-in-cell models with rappel and kuwabara boundary conditions

Particle-in-cell models are useful in the development of simple but reliable analytical expressions for heat and mass transfer in swarms of particles. Most such models consider spherical particles. Here the creeping flow through a swarm of spheroidal particles, that move with constant uniform velocity in the axial direction through an otherwise quiescent Newtonian fluid, is analyzed with a spheroid-in-cell model. The solid internal spheroid represents a particle of the swarm. The external spheroid contains the spheroidal particle and the amount of fluid required to match the fluid volume fraction of the swarm. The boundary conditions on the (conceptual) external spheroidal surface are similar to those of the sphere-in-cell Happel model [1], namely, nil normal velocity component and shear stress. The stream function is obtained in series form using the recently developed method of semiseparation of variables. It turns out that the first term of the series is sufficient for most engineering applications, so long as the aspect ratio of the spheroids remains within moderate bounds, say ∼1/5<a3<∼5. Analytical expressions for the streamfunction, the velocity components, the vorticity, the drag force acting on each particle, and the permeability of the swarm are obtained. Representative results are presented in graph form and they are compared with those obtained using Kuwabara-type boundary conditions. The Happel formulation is slightly superior because it leads to a particle-in-cell that is self sufficient in mechanical energy.

Wall-induced lift on a sphere

The lateral migration of rigid spheres sedimenting near a large flat vertical wall, in a quiescent Newtonian fluid, has been studied experimentally. The migration velocities were measured by recording the trajectory of the spheres as they sedimented near a wall for the particle Re range 0.1–10.0. The measurements indicate that the expression derived by Vasseur & Cox [J. Fluid Mech. 80, 561–591 (1977)] predicts, fairly accurately, the migration velocity up to a particle Re of 3.0.

Sedimentation of Noncolloidal Particles at Low Reynolds Numbers

Sedimentation, wherein particles fall under the action of gravity through a fluid in which they are suspended, is commonly used in the chemical and petroleum industries as a way of separating particles from fluid, as well as a way of separating particles with different settling speeds from each other. Examples of such separations include dewatering of coal slurries, clarifi­ cation of waste water, and processing of drilling and mining fluids containing rock and mineral particles of various sizes. The separation of different particles by sedimentation is also the basis of some laboratory techniques for determining the distribution of particle sizes in a particulate dispersion. Owing to the significance of the subject, there have been numerous experimental and theoretical investigations of the sedimentation of par­ ticles in a fluid. One of the earliest of these is Stokes' analysis of the translation of a single rigid sphere through an unbounded quiescent Newtonian fluid at zero Reynolds number, which led to his well-known law (0) _ 2a2(p. p )g u 9J.l ' (1.1)