For all supplemental material see 492_supplemental.zip.
SUPPLEMENTARY MOVIE
Visualization of the flow within the 3D printed model (Re = 0.376, see Table 1). Flow in the folds consists of horizontal bands of distinct red and blue color, indicating no adoral component to flow and no mixing within the folds, consistent with Hypothesis 2 (see text, Figure 2.2). The animation is sped up 16x. Figure 4 shows a single frame from the flow pattern observed, showing the steady-state flow pattern after nine minutes of flow. Click on image to activate animation.
SUPPLEMENTARY TABLE S1.
Full calculations for in vivo flow conditions.
| What is the Re for blastoid Pentremites rusticus in vivo? | ||
| Assumed values shown in blue along with source | ||
| Environmental data | ||
| nu, kinematic viscosity, m^2/s | 1.00E-006 | Use 1e-6 for 25C, 1.4 for 12C, 1.8 for 0C |
| Case 1: Cilia in pore canals | ||
| At the pore | ||
| Pore diameter, m | 2.54E-005 | Measured from Schmidtling and Marshall 2010 |
| Pore area, m^2 | 0.0000000005 | |
| Clia velocity, m/s | 6.00E-004 | Paul 1978 |
| Volumetric flow rate m^3/s | 0.0 | |
| Re,D pore | 0.01524 | Implies laminar flow |
| In the folds | ||
| Spacing between folds, m | 4.26E-005 | |
| Spacing between pores, m | 2.65E-004 | Schmidtling and Marshall 2010 |
| Number of folds per pore | 3 | From Schmidtling and Marshall, figure 6 |
| Fold area, m^2 | 0.0000000339 | 66.809793371 |
| Volumetric flow rate m^3/s | 0.0 | |
| Velocity in fold, m/s | 0.0000089807 | m/s |
| Dh fold, 4*FA/WP | 0.0000733744 | for finite channels |
| Dh fold, 2*spacing | 0.0000851648 | for infinite array of closely spaced plates |
| Re,Dh fold | 0.0006589551 | |
| Re,Dh2 fold | 0.0007648415 | Implies laminar flow |
| Case 2: Cilia in folds | ||
| At the pore | ||
| Pore diameter, m | 2.54E-005 | Measured from Schmidtling and Marshall 2010 |
| Pore area, m^2 | 0.0000000005 | |
| Volumetric flow rate m^3/s | 0.0 | |
| Pore velocity, m/s | 0.040085876 | |
| Re,D pore | 1.018181251 | Implies laminar flow |
| In the folds | ||
| Spacing between folds, m | 4.26E-005 | NOT measured in Schmidtling and Marshall; Tony measured in x-ray image |
| Spacing between pores, m | 2.65E-004 | 7x pore diameter |
| Number of folds per pore | 3 | From Schmidtling and Marshall, figure 6 |
| Fold area, m^2 | 0.0000000339 | |
| Cilia velocity, m/s | 6.00E-004 | Paul 1978 |
| Volumetric flow rate m^3/s | 0.0 | |
| Dh fold, 4*FA/WP | 0.0000733744 | For finite channels |
| Dh fold, 2*spacing | 0.0000851648 | For infinite array of closely spaced plates |
| Re,Dh fold | 0.0440246517 | |
| Re,Dh2 fold | 0.0510989011 | Implies laminar flow |
| Case 2 is the controlling case (higher Re, highest mixing expected there). | ||
SUPPLEMENTARY TABLE S2.
Full calculations for model flow conditions as tested.
| What is the Re for the model as tested? | |||
| Assumed values shown in blue along with source | |||
| Environmental data | |||
| nu, kinematic viscosity, m^2/s | 1.00E-006 | Use 1e-6 for freshwater at 25 C | |
| Use 27e-6 for mineral oil | |||
| At the faucet | |||
| Time, s | 4034 | Tony Huynh measured | |
| Level change, m | 0.0025 | Tony Huynh measured | |
| Area, m^2 | 0.1114302763 | Tony Huynh measured, for McMurdo portable tank with black screens | |
| Volumetric flow rate m^3/s | 0.0000000691 | ||
| At the pore | |||
| pPore diameter, m | 1.85E-003 | TH design, CM, DE measured | |
| Pore area, m^2 | 2.69E-006 | ||
| Number of pores connected | 8 | ||
| Volumetric flow rate m^3/s | 0.0000000086 | 1/8 of flow from tap | |
| Velocity at pore, m/s | 0.0032113231 | ||
| Re,D pore | 5.9409476461 | transition, less than 10 | |
| In the folds | |||
| Spacing between folds, m | 3.10E-003 | TH design, CM DE measured | |
| Spacing between pores, m | 0.0153 | TH designed, DE meas STL | |
| Number of folds per pore | 3 | Tony Huynh designed into model, Fig 3 | |
| Fold area, m^2 | 1.42E-004 | 52.9347713135 | |
| Volumetric flow rate m^3/s | 0.0000000086 | ||
| Velocity in fold, m/s | 0.0000606657 | m/s | |
| Dh fold, 4*FA/WP | 0.0051554348 | for finite channels | |
| Dh fold, 2*spacing | 0.0062 | for infinite array of closely spaced plates | |
| Re,Dh fold | 0.3127578744 | ||
| Re,Dh2 fold | 0.376127117 | laminar | |
| 7.4 | x in vivo Re | ||
SUPPLEMENTARY TABLE S3.
Full calculations for model operated to match lowest Re in vivo case.
| What speed would we have to run at to match in vivo | ||
| Assumed values shown in blue along with source | ||
| Environmental data | ||
| nu, kinematic viscosity, m^2/s | 1.00E-006 | Use 1e-6 for freshwater at 25 C |
| Use 30e-6 for mineral oil | ||
| At the pore | ||
| Pore diameter, m | 1.85E-003 | Tony Huynh designed |
| Pore area, m^2 | 0.000002688 | |
| Volumetric flow rate m^3/s | 0.0000000016 | |
| Velocity at pore, m/s | 5.85E-004 | |
| Re,D pore | 1.081961112 | Transition, less than 10 |
| In the folds | ||
| Spacing between folds, m | 3.10E-003 | Tony Huynh designed |
| Spacing between pores, m | 0.0153 | Tony Huynh designed |
| Number of folds per pore | 3 | Tony Huynh designed into model, Fig 3 |
| Fold area, m^2 | 0.00014229 | |
| Volumetric flow rate m^3/s | 0.0000000016 | |
| Velocity in fold, m/s | 0.0000110484 | m/s |
| Dh fold, 4*FA/WP | 0.0051554348 | For finite channels |
| Dh fold, 2*spacing | 0.0062 | For infinite array of closely spaced plates |
| Re,Dh fold | 0.0569592391 | |
| Re,Dh2 fold | 0.0685 | Implies laminar flow |
| Re in vivo, case 1 | 0.0007648415 | |
| Re in vivo, case 2 | 0.0510989011 | |
SUPPLEMENTARY FILE.
Stereolithography (STL) file of model of the distal end of the hydrospire in Pentremites rusticus (see palaeo-electronica.org/content/2015/1073-blastoid-hydrospire-fluid-flow) (see 492_supplemental.zip).
