Lubricating motion of a sphere in a conical vessel

Masmoudi, K.; Masséi, Nicolas; Anthore, R.; May, Sandra; Feuillebois, François

doi:10.1063/1.869790

Cited by 4 publications

(7 citation statements)

References 3 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The discrepancy between experiments and theory (least-squares method completed with BM solution) also does not exceed 1%. In the lower part of the cone, there is a gap between the present numerical solution and the lubrication solution of Masmoudi et al (1998). More work is required here.…”

Section: Experimental Results Comparison With Calculations and Discumentioning

confidence: 91%

“…The BM theoretical solution for a sphere moving towards a single plane wall (dashed lines) is superimposed on the experimental results. The circle near the origin of coordinates indicates the domain of validity of the lubrication solution obtained by Masmoudi et al (1998).…”

Section: Experimental Results Comparison With Calculations and Discumentioning

confidence: 99%

“…The experimental technique and numerical scheme could be applied, for instance, to a slightly deformed sphere. Finally, the flow field around a sphere near the apex of a cone and matching with the Masmoudi et al (1998) lubrication solution are under consideration (see Lecoq & Feuillebois 2007).…”

Section: Resultsmentioning

confidence: 99%

“…Fortunately, there is an overlap domain in which both calculations are in agreement (see figure 11). Now, when the sphere comes close to the vertex of the cone, no analytical solution exists, except for the lubrication solution of Masmoudi et al (1998). This solution is only valid for a gap that is very small compared with the sphere radius.…”

Section: Mathematical Formulation and Numerical Proceduresmentioning

confidence: 99%

See 3 more Smart Citations

Creeping motion of a sphere along the axis of a closed axisymmetric container

et al. 2007

Self Cite

View full text Add to dashboard Cite

The creeping flow around a sphere settling along the axis of a closed axisymmetric container is obtained both theoretically and experimentally. The numerical technique for solving the Stokes equations uses the classical Sampson expansion; the boundary conditions on the sphere are satisfied exactly and those on the container walls are applied in the sense of least squares. This is an extension to the axisymmetric case of the technique for solving various two-dimensional flow problems. Two types of axisymmetric container are considered here as examples: circular cylinders closed by planes at both ends, and cones closed by a base plane. Calculated streamlines patterns show various sets of eddies, depending upon the geometry and the sphere position. Results are in agreement with earlier Stokes flow calculations of eddies in corners and in closed containers. Experiments use laser interferometry to measure the vertical displacement of a steel bead a few millimetres in diameter settling in a container filled with a very viscous silicone oil. The Reynolds number based on the sphere radius is typically of the order of 10−5. The accuracy on the vertical displacement is 50nm. Experiments show that the motion towards the apex of a cone is much slower than that towards a plane; the bead takes hours to reach the micrometre size roughness asperities on a conical wall, as compared with minutes to reach those on a plane wall. The numerical results for the drag force are in excellent agreement with experiments both for the cylindrical and the conical containers. With standard computer accuracy, the present numerical technique applies when the gap between the sphere and the nearby wall is larger than about one radius. For a sphere in the vicinity of any plane horizontal wall, these results also match with a previous analytical solution. That solution is in excellent agreement with our experimental results at small distances from the wall (typically less than a diameter, depending on the sphere size).

show abstract

Section: Experimental Results Comparison With Calculations and Discumentioning

confidence: 91%

Section: Experimental Results Comparison With Calculations and Discumentioning

confidence: 99%

Section: Resultsmentioning

confidence: 99%

Section: Mathematical Formulation and Numerical Proceduresmentioning

confidence: 99%

See 2 more Smart Citations

Creeping motion of a sphere along the axis of a closed axisymmetric container

et al. 2007

Self Cite

View full text Add to dashboard Cite

show abstract

“…It provides an accuracy for the sphere displacement of 0.1 µm, a significant improvement over other experiments using video. This technique was developed for this particular application, but has also been applied to other creepingflow hydrodynamics problems (see Lecoq et al 1993Lecoq et al , 1995Masmoudi et al 1998Masmoudi et al , 2002; details on the experimental setup may be found therein).…”

Section: Introductionmentioning

confidence: 99%

Drag force on a sphere moving towards a corrugated wall

et al. 2004

Self Cite

View full text Add to dashboard Cite

From the solution of the creeping-flow equations, the drag force on a sphere becomes infinite when the gap between the sphere and a smooth wall vanishes at constant velocity, so that if the sphere is displaced towards the wall with a constant applied force, contact theoretically may not occur. Physically, the drag is finite for various reasons, one being the particle and wall roughness. Then, for vanishing gap, even though some layers of fluid molecules may be left between the particle and wall roughness peaks, conventionally it may be said that contact occurs. In this paper, we consider the example of a smooth sphere moving towards a rough wall. The roughness considered here consists of parallel periodic wedges, the wavelength of which is small compared with the sphere radius. This problem is considered both experimentally and theoretically. The motion of a millimetre size bead settling towards a corrugated horizontal wall in a viscous oil is measured with laser interferometry giving an accuracy on the displacement of 0.1 $\mu$m. Several wedge-shaped walls were used, with various wavelengths and wedge angles.From the results, it is observed that the velocity of the sphere is, except for small gaps, similar to that towards a smooth plane that is shifted down from the top of corrugations. Indeed, earlier theories for a shear flow along a corrugated wall found such an equivalent smooth plane. These theories are revisited here. The creeping flow is calculated as a series in the slope of the roughness grooves. The cases of a flow along and across the grooves are considered separately. The shift is larger in the former case. Slightly flattened tops of the wedges used in experiments are also considered in the calculations. It is then demonstrated that the effective shift for the sphere motion is the average of the shifts for shear flows in the two perpendicular directions. A good agreement is found between theory and experiment.

show abstract

Lubrication in a flat cone: The transition from a cone to a plane wall

Masséi

Feuillebois

2007

Physics of Fluids

View full text Add to dashboard Cite

A lubrication formula is derived for the drag on a sphere settling in a viscous fluid on the axis of a flat cone vessel. This formula provides the link between the classical −1 result for the plane wall and our earlier −5/2 result for the cone ͓Masmoudi et al., Phys. Fluids 10, 1231 ͑1998͔͒, where is the gap between the sphere and wall. An excellent agreement is obtained with experimental results based on the measurement of the displacement of a sphere with laser interferometry. Moreover, this formula practically applies for a large range of values of cone angles; it then proves better than the earlier formula ͓Masmoudi et al. ͑1998͔͒ in describing experimental results.In the paper by Masmoudi et al., 1 denoted here as ͑I͒, the final stage of sedimentation of a sphere along the axis of a conical vessel containing a viscous liquid was considered both theoretically using lubrication theory and experimentally using laser interferometry. The main result for the sphere velocity V iswhere ␣ is the cone vertex angle, a denotes the sphere radius, d is the smallest distance between the sphere and the cone and U s is the Stokes velocity of the settling sphere when in an unbounded fluid. This lubrication result, valid for = d / a 1, was shown to be in excellent agreement with the experimental results for values of ranging from the order of the normalized roughness up to O͑4 ϫ 10 −2 ͒. Cones with angle ␣ = 31.6°and 45°were used in the experiment.Formula ͑1͒ is not valid in the limit ␣ → / 2. Indeed, for ␣ = / 2, that is for a sphere moving towards a plane, it is well known from lubrication theory that V / U s = . Thus for a cone with ␣ → / 2, the transition from the 5/2 variation to the variation is still unresolved. This question is addressed here using the same tools as in ͑I͒, viz theoretically using lubrication theory and experimentally using laser interferometry.The notation is shown in Fig. 1. Let ␤ = /2−␣. We assumeThe fluid domain is between z = z c = t ͑cone͒ and z = z s ͑sphere͒. The vertical distance D = z s − z c between the sphere and cone is also assumed to be much smaller than the sphere radius for lubrication theory to apply. Thus, the approximation of the sphere surface is: z s Ӎ D 0 + 2 / ͑2a͒. At contact, the distance between the sphere and the cone apex is a͑1 / cos ␤ −1͒Ӎat 2 / 2. Thus, we writewhere denotes the dimensionless distance along z between the sphere and its contact position on the cone. Our lubrication assumption is 1. The minimum vertical distance D is at position = at and its value is D m = a. Note that d in ͑1͒ is d = D m cos ␤, that is simply d = D m = a after neglecting terms O͑␤ 2 ͒. Let u and v be the components of the fluid velocity along and z, respectively. The lubrication calculation is done in a standard way. The Stokes momentum equation for u is first integrated along z giving a parabolic profile for u͑z͒. The continuity equation then is integrated from z = z c where u = v =0 to z = z s where u =0,v = V, the result being the Reynolds equation for the pressure p: a͒ Electronic mai...

show abstract

Lubricating motion of a sphere in a conical vessel

Cited by 4 publications

References 3 publications

Creeping motion of a sphere along the axis of a closed axisymmetric container

Creeping motion of a sphere along the axis of a closed axisymmetric container

Drag force on a sphere moving towards a corrugated wall

Lubrication in a flat cone: The transition from a cone to a plane wall

Contact Info

Product

Resources

About