Application Note 18

Shear Stress and Shear Rate Calculations
for the µ-Slide y-shaped Based on Computational
Fluid Dynamics (CFD)This Application Note provides instructions on how to calculate the wall shear stress (WSS) in theµ-Slide y-shaped. For simplicity reasons, the term "shear stress" used here will always refer towall shear stress. Further, the shear rate is calculated.
The µ-Slide y-shaped was designed for studies of non-uniform laminar shear stress. Under flow
conditions, the shear stress levels depend on the region on the slide: The prevalent shear stressin the branched regions is approximately half of the regions with only the single channel.
Due to the standardized Luer adapters, the µ-Slide y-shaped can easily be combined with any
flow system (e.g., the ibidi Pump System). The shear stress calculations apply equally for allsystems.
For a detailed explanation on how to calculate shear stress and shear rates in the ibidi µ-Slides,
including a glossary and considerations before setting up an experiment, please have a look at:
AN 11 “Shear Stress and Shear Rates for ibidi µ-Slides Based on Numerical Calculations”Table of Contents
1Flow Regions in the µ-Slide y-shaped ................................................................................... 22Shear Stress and Shear Rate Calculations in Uniform Laminar Flow Regions ..................... 2

2.1 Reference Table for Shear Stress Values in the Single Channel Area ................................. 3
2.2 Reference Table for Shear Stress Values in the Branched Channel Area ............................ 43Shear Stress and Shear Rate Calculations in Non-Uniform Laminar Flow Regions ............. 5____________________________________________________________________________________
Application Note 18© ibidi GmbH, Version 2.0, February 14th, 2022

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Application Note 18

1Flow Regions in the µ-Slide y-shaped

The µ-Slide y-shaped provides regions of uniform laminar flow and non-uniform laminar flow.
Laminar flow is defined as the movement of liquids without turbulences. The fluid flows in parallel
layers with no disruption between them. Furthermore, a uniform laminar shear stress has aconstant direction and velocity over time.
In the case of non-uniform laminar flow, the flow direction is constant, whereas the velocity
spatially varies across the cell layer. Non-uniform laminar flow is characterized by flow velocity
gradients in small sub millimeter regions.
Please note: The flow pattern in the µ-Slide y-shaped is always laminar! Therefore, it isexperimentally impossible to generate turbulences with aqueous solutions in the µ-Slidey-shaped.

2

Shear Stress and Shear Rate Calculations in Uniform Laminar FlowRegionsThe shear stress is uniform in the straight parts of the channel. This applies for the single channel
region as well as for the branched parts of the channel. The flow rate and shear stress in theorange regions are twice as high as in the green regions (Figure 1).

Figure 1: Bottom view of the µ-Slide y-shaped. The dashed rectangles indicate zones of uniform laminarflow and, therefore, also uniform laminar shear stress. Measures in mm indicate the size of the zones.

Nomenclature and units:

Φ flow rate
τ shear stressη dynamical viscosity

ml/mindyn/cm²dyn∙s/cm²To use the following equations, insert the flow rate, shear stress, and viscosity values in theindicated units. For simplicity reasons, the calculations include all unit conversions (not shown).
To calculate the wall shear stress correctly, you need to know the viscosity of the perfused
medium (for details, see Application Note 11).
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Application Note 18© ibidi GmbH, Version 2.0, February 14th, 2022

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Shear Stress

Shear Rate

Single channel area(100% flow velocity)τ = η ∙ 227.4 ∙ Φ𝛾𝛾 = 227.4 ∙ ΦBranched area (50%flow velocity)τ = η ∙ 113.7 ∙ Φ𝛾𝛾 = 113.7 ∙ Φ

2.1 Reference Table for Shear Stress Values in theSingle Channel Area
These tables are suitable for a quick determination of the needed
flow rate. The shear stress is calculated for medium at 37°C withviscosity η = 0.0072 dyn·s/cm².

τ = η ∙ 227.4 ∙ Φτ [dyn/cm²]

Φ [ml/min]τ [dyn/cm²]

Φ [ml/min]τ [dyn/cm²]Φ [ml/min]

0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
3

0.06
0.12
0.18
0.24
0.31
0.37
0.43
0.49
0.55
0.61
0.73
0.86
0.98
1.10
1.22
1.34
1.47
1.59
1.71
1.83

3.5
4
4.5
5
5.5
6
7
8
9
10
11
12
13
14
15
16
18
20
22
24

2.14
2.44
2.75
3.05
3.36
3.66
4.28
4.89
5.50
6.11
6.72
7.33
7.94
8.55
9.16
9.77
10.99
12.22
13.44
14.66

25
30
35
40
45
50
55
60
65
70
75
80
85
90
95
100105110
115
120

15.27
18.32
21.38
24.43
27.48
30.54
33.59
36.65
39.70
42.75
45.81
48.86
51.92
54.97
58.02
61.08
64.13
67.18
70.24
73.29

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Application Note 18© ibidi GmbH, Version 2.0, February 14th, 2022

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Application Note 18

2.2 Reference Table for Shear Stress Values in the
Branched Channel Area
These tables are suitable for a quick determination of the needed
flow rate. The shear stress is calculated for medium at 37°C withviscosity η = 0.0072 dyn·s/cm².

τ = η ∙ 113.7 ∙ Φτ [dyn/cm²]

Φ [ml/min]τ [dyn/cm²]

Φ [ml/min]τ [dyn/cm²]

Φ [ml/min]

0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
3

0.12
0.24
0.37
0.49
0.61
0.73
0.86
0.98
1.10
1.22
1.47
1.71
1.95
2.20
2.44
2.69
2.93
3.18
3.42
3.66

3.5
4
4.5
5
5.5
6
7
8
9
10
11
12
13
14
15
16
18
20
2224

4.284.89
5.50
6.11
6.72
7.33
8.55
9.77
10.99
12.22
13.44
14.66
15.88
17.10
18.32
19.54
21.99
24.43
26.87
29.32

25
30
35
40
45
50
55
60
65
70
75
80
85
90
95
100
105
110
115
120

30.54
36.65
42.75
48.86
54.97
61.08
67.18
73.29
79.40
85.51
91.62
97.72
103.83109.94116.05122.15128.26134.37140.48146.58____________________________________________________________________________________
Application Note 18© ibidi GmbH, Version 2.0, February 14th, 2022Page 4 of 7Application Note 18

3Shear Stress and Shear Rate Calculations in Non-Uniform Laminar
Flow Regions

In the kink and branching regions of the µ-Slide y-shaped (Figure 2), the laminar flow and thus
also the shear stress are non-uniform and need to be calculated based on the velocity distribution.

Figure 2: Bottom view of the µ-Slide y-shaped. The dashed rectangles indicate zones of non-uniform
laminar flow and, therefore, non-uniform laminar shear stress. The branching region on the left provides
an opening angle of 30°. The branching region on the right provides a 45° opening angle.

The shear stress is assumed to be proportional to the calculated velocity at 1 µm distance fromthe bottom. The value of 100 % velocity is defined as 0.00024 m/s, which is the velocity in theuniform flow region of the single channels. In this region, the shear stress is 2.274 dyn/cm² at a 1ml/min flow rate.
Further, for calculating the shear stress at a specific location within a non-uniform flow region, thecorrespondent flow velocity and the viscosity of the used medium are required. Detailed flow
velocities for every spot on the µ-Slide y-shaped were simulated by computational fluid dynamics(CFD) and can be looked up in Figures 3–5.
The shear stress in the non-uniform regions of the µ-Slide y-shaped is calculated as follows:𝜏𝜏 =𝑑𝑑𝑑𝑑𝑑𝑑 ∙ 𝑚𝑚𝑚𝑚𝑚𝑚
𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣
𝑣𝑣𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 (𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠)∙ 2.274∙Φ∙𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣 𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 𝑎𝑎𝑎𝑎 20°𝐶𝐶
100% 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣
𝑐𝑐𝑐𝑐² ∙ 𝑚𝑚𝑚𝑚____________________________________________________________________________________
Application Note 18© ibidi GmbH, Version 2.0, February 14th, 2022

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Application Note 18

Calculation Example
1. Define the spot where you want to calculate the shear stress. The spot used in thisexample is marked in Figure 3.
2. Determine the color at the desired position and look up the corresponding flow velocity inthe legend. The spot in this example lies in the yellow region of Figure 3, with acorresponding flow velocity of 0.00018 m/s.
3. In this example, the flow rate is assumed at 2 ml/min. The viscosity is assumed at 0.0072
dyn∙s/cm².
The shear stress at the spot used in this example is calculated as follows:
𝑑𝑑𝑑𝑑𝑑𝑑 ∙ 𝑠𝑠
𝑚𝑚
𝑑𝑑𝑑𝑑𝑑𝑑 ∙ 𝑚𝑚𝑚𝑚𝑚𝑚
𝑚𝑚𝑚𝑚 0.0072 𝑐𝑐𝑚𝑚2
𝑑𝑑𝑑𝑑𝑑𝑑𝑠𝑠
𝜏𝜏 == 2.46𝑚𝑚 ∙ 2.274 𝑐𝑐𝑚𝑚2 ∙ 𝑚𝑚𝑚𝑚 ∙ 2 𝑚𝑚𝑚𝑚𝑚𝑚 ∙𝑑𝑑𝑑𝑑𝑑𝑑 ∙ 𝑠𝑠
𝑐𝑐𝑐𝑐²
0.00024
0.01
𝑠𝑠
𝑐𝑐𝑚𝑚2
0.00018

The velocity distribution in non-uniform laminar flow regions shown in Figures 3–5 was determinedwith a computational fluid dynamic (CFD) simulation (ANSYS FLUENT Flow Modeling Software)with the following parameters and assumptions: stationary, isothermal, laminar, water assumed
to be incompressible, density of water 1000 kg/m³, viscosity 0.001 kg/(m∙s), and at given flow ofmass = 1.001 ml/min. Calculated is the velocity in 1 µm distance from the bottom.

Figure 3: Calculated data for the 30° branching region. The arrows indicate the flow direction. The color
scale gives the velocity [m/s]. The spot used in the calculation example is marked with a red cross.
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Application Note 18© ibidi GmbH, Version 2.0, February 14th, 2022

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Application Note 18

Figure 4: Calculated data for the 45° branching region. The arrows indicate the flow direction. The color
scale gives the velocity [m/s].

Figure 5: Calculated data for the kink region. The arrows indicate the flow direction. The color scale gives
the velocity [m/s].
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Application Note 18© ibidi GmbH, Version 2.0, February 14th, 2022

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