^{1}, George Karapetsas

^{1}, Yannis Dimakopoulos

^{1,a)}and John Tsamopoulos

^{1,b)}

### Abstract

The injection of a viscoplasticmaterial, driven by a constant pressure drop, inside a pipe or between two parallel coaxial disks under creeping flow conditions is examined. The transient nature of both flow arrangements requires solving a time-dependent problem and fully accounting for the advancing liquid/air interface. Materialviscoplasticity is described by the Papanastasiou constitutive equation. A quasi-elliptic grid generation scheme is employed for the construction of the mesh, combined with local mesh refinement near the material front and, periodically, full mesh reconstruction. All equations are solved using the mixed finite element/Galerkin formulation coupled with the implicit Euler method. For a viscoplastic fluid, the flow field changes qualitatively from that of a Newtonian fluid because the material gets detached from the walls. For small Bingham numbers, the contact line moves in the flow direction, so that initially the flow resembles that of a Newtonian fluid, but even in that case detachment eventually occurs. The distance covered by the contact line, before detachment takes place, decreases as the Bingham number increases. For large enough Bingham numbers, the fluid may even detach from the wall without advancing appreciably. In pipe flow, when detachment occurs, unyielded material arises at the front and the flow changes into one under constant flow rate with pressure distribution that does not vary with time. In the flow between disks, it remains decelerating and the material keeps rearranging at its front because of the increased cross section through which it advances. The wall detachment we predict has been observed experimentally by Bates and Bridgwater [Chem. Eng. Sci. **55**, 3003–3012 (2000)] in radial flow of pastes between two disks.

I. INTRODUCTION

II. PROBLEM FORMULATION

A. Boundary and initial conditions

III. NUMERICAL IMPLEMENTATION

A. Elliptic grid generation

B. Global mesh reconstruction

C. Local mesh refinement

D. Mixed finite element method

E. Solution procedure

F. Yield surface determination

IV. RESULTS AND DISCUSSION

A. Flow inside a cylindrical pipe

B. Radial flow between two parallel disks

V. CONCLUDING REMARKS

### Key Topics

- Free surface
- 55.0
- Viscoplasticity
- 55.0
- Poiseuille flow
- 28.0
- Viscosity
- 26.0
- Yield stress
- 25.0

## Figures

Initial arrangement of a finite amount of viscoplastic fluid (a) in a semi-infinite pipe and (b) between parallel and coaxial disks.

Initial arrangement of a finite amount of viscoplastic fluid (a) in a semi-infinite pipe and (b) between parallel and coaxial disks.

Schematic representation of the remapping scheme of the interfacial nodes (a) before and (b) after the procedure.

Schematic representation of the remapping scheme of the interfacial nodes (a) before and (b) after the procedure.

Typical mesh with two levels of local refinement behind the advancing front for the flow between two disks with , , and at time

Typical mesh with two levels of local refinement behind the advancing front for the flow between two disks with , , and at time

Contours of the radial, upper half, and the axial, lower half, velocity component of a Newtonian fluid in a straight pipe at (a) , (b) , and (c) for . The interval between the max and min in each snapshot was divided by 16, 15, and 14 contour lines, respectively. The number of contour lines for in each of the three snapshots is 16, 14, and 14, respectively. The M2 mesh was used.

Contours of the radial, upper half, and the axial, lower half, velocity component of a Newtonian fluid in a straight pipe at (a) , (b) , and (c) for . The interval between the max and min in each snapshot was divided by 16, 15, and 14 contour lines, respectively. The number of contour lines for in each of the three snapshots is 16, 14, and 14, respectively. The M2 mesh was used.

Contours of the pressure field, upper half, and , lower half of a Newtonian fluid at time for . There are 31 contour lines for and 36 contour lines for the pressure.

Contours of the pressure field, upper half, and , lower half of a Newtonian fluid at time for . There are 31 contour lines for and 36 contour lines for the pressure.

Variation in the axial distance between the flow front tip and the triple contact point , with the axial position of the front tip for a Newtonian fluid in a straight pipe with . Comparison with experimental data provided by Behrens *et al.* (1987). The same symbols have been used for all the experimental data, although the indicated variation is caused because the experiments took place with different fluids, pipe diameters, and flow rates.

Variation in the axial distance between the flow front tip and the triple contact point , with the axial position of the front tip for a Newtonian fluid in a straight pipe with . Comparison with experimental data provided by Behrens *et al.* (1987). The same symbols have been used for all the experimental data, although the indicated variation is caused because the experiments took place with different fluids, pipe diameters, and flow rates.

Contours of the second invariant of the stresses, upper half, and the axial component of the velocity, lower half, of a viscoplastic fluid in a straight pipe at (a) , (b) , and (c) . The dimensionless parameters are . The range between the maximum and minimum value of in each of the three snapshots was divided by 13, 11, and 10 contour lines, respectively. The number of isolines for the second invariant of the stresses in each snapshot is 28. The M2 mesh was used.

Contours of the second invariant of the stresses, upper half, and the axial component of the velocity, lower half, of a viscoplastic fluid in a straight pipe at (a) , (b) , and (c) . The dimensionless parameters are . The range between the maximum and minimum value of in each of the three snapshots was divided by 13, 11, and 10 contour lines, respectively. The number of isolines for the second invariant of the stresses in each snapshot is 28. The M2 mesh was used.

Comparison of the analytical prediction for the axial velocity profile with numerical simulation along with at for .

Comparison of the analytical prediction for the axial velocity profile with numerical simulation along with at for .

Contours of the second invariant of the stresses, upper half, and the pressure field, lower half of a viscoplastic fluid at (a) , (b) , and (c) for . In each of the three snapshots, we have plotted 40 isolines for and 30 isolines for the pressure. The M3 mesh was used.

Contours of the second invariant of the stresses, upper half, and the pressure field, lower half of a viscoplastic fluid at (a) , (b) , and (c) for . In each of the three snapshots, we have plotted 40 isolines for and 30 isolines for the pressure. The M3 mesh was used.

Time evolution of the shape of the fluid/air interface for a viscoplastic material with (a) and (b) . The rest dimensionless parameters are (a) and (b) .

Time evolution of the shape of the fluid/air interface for a viscoplastic material with (a) and (b) . The rest dimensionless parameters are (a) and (b) .

Time evolution of the axial component of the velocity at the interface tip for various Bn. The rest of the dimensionless parameters are .

Time evolution of the axial component of the velocity at the interface tip for various Bn. The rest of the dimensionless parameters are .

Variation in the axial distance between the flow front tip and the triple contact point , with the axial position of the front tip for various Bn numbers in a straight pipe.

Variation in the axial distance between the flow front tip and the triple contact point , with the axial position of the front tip for various Bn numbers in a straight pipe.

Dependence of the location of the triple contact point at detachment on its initial axial position for two sets of Bn numbers, each corresponding to a single , for pipe flow. The slope of the line with is 0.915, whereas the slope for is 0.918.

Dependence of the location of the triple contact point at detachment on its initial axial position for two sets of Bn numbers, each corresponding to a single , for pipe flow. The slope of the line with is 0.915, whereas the slope for is 0.918.

Contours of the pressure field, upper half, and the radial velocity component, lower half, of a Newtonian fluid between two parallel disks at (a) , (b) , and (c) for . The number of contours for the pressure in each of the three snapshots is 24. The interval between these two extremes in is divided by 20, 26, and 24 contour lines in each of the three snapshots, respectively. The M1 mesh was used.

Contours of the pressure field, upper half, and the radial velocity component, lower half, of a Newtonian fluid between two parallel disks at (a) , (b) , and (c) for . The number of contours for the pressure in each of the three snapshots is 24. The interval between these two extremes in is divided by 20, 26, and 24 contour lines in each of the three snapshots, respectively. The M1 mesh was used.

Contours of the pressure field, upper half, and the radial velocity component, lower half, of a viscoplastic material between two parallel disks at (a) and (b) for . The number of contour lines for both variables in each of the two snapshots is 24. The M1 mesh was used.

Contours of the pressure field, upper half, and the radial velocity component, lower half, of a viscoplastic material between two parallel disks at (a) and (b) for . The number of contour lines for both variables in each of the two snapshots is 24. The M1 mesh was used.

Contours of the axial, upper half, and the radial, lower half, velocity component of a viscoplastic material between two parallel disks at (a) , (b) , and (c) for . There are 25 contour lines for and 18 for in each one of the three snapshots.

Contours of the axial, upper half, and the radial, lower half, velocity component of a viscoplastic material between two parallel disks at (a) , (b) , and (c) for . There are 25 contour lines for and 18 for in each one of the three snapshots.

Time evolution of the shape of the fluid/air interface for a viscoplastic material with (a) , (b) , (c) , and (d) .

Time evolution of the shape of the fluid/air interface for a viscoplastic material with (a) , (b) , (c) , and (d) .

Contours of the pressure field of a viscoplastic material with between two parallel disks at . The rest of the dimensionless parameters are . The number of contour lines shown is 24. The M1 mesh was used.

Contours of the pressure field of a viscoplastic material with between two parallel disks at . The rest of the dimensionless parameters are . The number of contour lines shown is 24. The M1 mesh was used.

Vectors of the velocity for a viscoplastic material with , relative to the tip velocity. (a) The entire domain where the area of minimum velocity is shown and (b) a close-up in the region of the triple contact point.

Vectors of the velocity for a viscoplastic material with , relative to the tip velocity. (a) The entire domain where the area of minimum velocity is shown and (b) a close-up in the region of the triple contact point.

Stresses of a viscoplastic material with between parallel disks. (a) Circumferential upper half and radial lower half. (b) Axial upper half and the shear stress lower half. The rest of the dimensionless parameters are . The number of contour lines is 12 for and 30 for each one of the other components of the stress tensor.

Stresses of a viscoplastic material with between parallel disks. (a) Circumferential upper half and radial lower half. (b) Axial upper half and the shear stress lower half. The rest of the dimensionless parameters are . The number of contour lines is 12 for and 30 for each one of the other components of the stress tensor.

Time evolution of the radial component of the velocity at the interface tip for various Bn. The rest of the dimensionless parameters are .

Time evolution of the radial component of the velocity at the interface tip for various Bn. The rest of the dimensionless parameters are .

Variation in the radial distance between the flow front tip and the triple contact point , with the radial position of the front tip for various Bn numbers between two parallel disks.

Variation in the radial distance between the flow front tip and the triple contact point , with the radial position of the front tip for various Bn numbers between two parallel disks.

Dependence of the location of the triple contact point at detachment on its initial radial position for three sets of Bn numbers, each corresponding to a single , for fluid injection between parallel disks. The slope of the line with is 0.871, the slope for is 0.959, and that for is 0.999.

Dependence of the location of the triple contact point at detachment on its initial radial position for three sets of Bn numbers, each corresponding to a single , for fluid injection between parallel disks. The slope of the line with is 0.871, the slope for is 0.959, and that for is 0.999.

Contours of the second invariant of the stress tensor of a viscoplastic material with . The rest of the dimensionless parameters are .

Contours of the second invariant of the stress tensor of a viscoplastic material with . The rest of the dimensionless parameters are .

## Tables

Properties of the finite element meshes used in this paper.

Properties of the finite element meshes used in this paper.

Extent of unyielded region of a viscoplastic in a pipe material with , , and .

Extent of unyielded region of a viscoplastic in a pipe material with , , and .

Comparison of the radius of unyielded domain in a pipe at various time instants between the analytically predicted value and the calculated one using the present code for a material with , , and .

Comparison of the radius of unyielded domain in a pipe at various time instants between the analytically predicted value and the calculated one using the present code for a material with , , and .

Comparison of the radius of unyielded domain in a pipe at various time instants between the analytically predicted value and the calculated one using the present code for a material with , , and .

Article metrics loading...

Full text loading...

Commenting has been disabled for this content