BMES 444 CH06 Problem Set 20221207 - Tagged (3)

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BMES 444 PROBLEM SET – CHAPTER 6. FLUID RHEOLOGY Contents Problem 6.1: Fluid Rheology ......................................................................................................................................... 2 Problem 6.2: Fluid Rheology ......................................................................................................................................... 3 Problem 6.3: Fluid Rheology ......................................................................................................................................... 4 Problem 6.4: Blood Rheology ....................................................................................................................................... 5 Problem 6.5: Blood Rheology ....................................................................................................................................... 6 Problem 6.6: Blood Rheology ....................................................................................................................................... 7 [Solution] Problem 6.1 .................................................................................................................................................... 8 [Solution] Problem 6.2 .................................................................................................................................................... 9 [Solution] Problem 6.3 ................................................................................................................................................... 10 [Solution] Problem 6.4 ................................................................................................................................................... 11 [Solution] Problem 6.5 ................................................................................................................................................... 12 [Solution] Problem 6.6 ................................................................................................................................................... 13 1

Problem 6.1: Fluid Rheology Draw shear stress vs. shear rate curves for Newtonian, shear thinning, shear thickening, and Bingham plastic fluids (draw the curves on the same set of axes for comparison). 2

Problem 6.2: Fluid Rheology Draw effective viscosity vs. shear rate curves for Newtonian, shear thinning, shear thickening, and Bingham plastic fluids (draw the curves on the same set of axes for comparison). 3

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Problem 6.3: Fluid Rheology Using the rheology data posted in an Excel spreadsheet (ch6_rheology.xlsx), determine the rheological behavior and parameters for each of the four (4) fluids. Depending on the type of fluid, find the following parameters:  Newtonian fluid: viscosity  Bingham plastic: yield shear stress  Power law fluids: n Briefly describe how you determined the rheological behavior of each fluid, and how you analyzed the data to find the aforementioned parameters. 4

Problem 6.4: Blood Rheology Do blood thinners (e.g., warfarin, heparin, aspirin) affect blood viscosity? If so, how? If not, what do they do? 5

Problem 6.5: Blood Rheology Identify three diseases or conditions where blood viscosity changes. In each case, describe what causes the change in blood viscosity (and if possible, how much it changes). 6

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Problem 6.6: Blood Rheology One relatively simple model of non-Newtonian fluid rheology is the Herschel-Bulkley fluid, which combines the yield stress behavior of viscoplastic fluids with the power law description for shear thinning and shear thickening fluids: τ = τ 0 + k ˙ γ n ,τ ≥τ 0 In the Herschel-Bulkley model, τ 0 is the yield shear stress, k is the consistency index, and n is the flow index (where n < 1 models shear thinning and n > 1 models shear thickening). As you can see, this is basically a combination of the equation for a Bingham plastic and the Ostwald-de Waele relationship. The Herschel-Bulkley model can be applied to the behavior of blood, which exhibits a (small) yield stress and generally shear thinning behavior (that becomes pseudo-Newtonian at higher shear rates). Using the data provided in ch6_herschel.xlsx, determine n, k , and τ 0 for blood. 7

[Solution] Problem 6.1 Draw shear stress vs. shear rate curves for Newtonian, shear thinning, shear thickening, and Bingham plastic fluids (draw the curves on the same set of axes for comparison). 8

[Solution] Problem 6.2 Draw effective viscosity vs. shear rate curves for Newtonian, shear thinning, shear thickening, and Bingham plastic fluids (draw the curves on the same set of axes for comparison). 9

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[Solution] Problem 6.3 Using the rheology data posted in an Excel spreadsheet (ch6_rheology.xlsx), determine the rheological behavior and parameters for each of the four (4) fluids. Depending on the type of fluid, find the following parameters:  Newtonian fluid: viscosity  Bingham plastic: yield shear stress  Power law fluids: n Briefly describe how you determined the rheological behavior of each fluid, and how you analyzed the data to find the aforementioned parameters. You can use curve fitting and data transformation to find the behavior of the different fluids. One of the more effective approaches is to plot the data, and then either fit a linear model or a power law model to the data. Please see the posted Excel spreadsheet for details. 10

[Solution] Problem 6.4 Do blood thinners (e.g., warfarin, heparin, aspirin) affect blood viscosity? If so, how? If not, what do they do? “Blood thinners” generally do not directly affect viscosity. They are anticoagulants, i.e., they interfere with the coagulation cascade to prevent the formation of thrombi or blood clots. They are typically prescribed when thrombosis or embolism (when a blood clot detaches and lodges in another organ / vessel, e.g., pulmonary artery) is a serious risk, such as when you implant an artificial heart valve or when someone is diagnosed with deep vein thrombosis. However, there is some evidence (though not clear cut) that specific anticoagulants may have some effect on blood viscosity. 11

[Solution] Problem 6.5 Identify three diseases or conditions where blood viscosity changes. In each case, describe what causes the change in blood viscosity (and if possible, how much it changes). There are a number of possible diseases or conditions you could identify. Broadly speaking, any disease or condition that causes one or more of the following could alter blood viscosity:  Red blood cell deformability, e.g., sickle cell anemia, malaria  Red blood cell adhesion, e.g., sickle cell anemia, malaria  Hematocrit, e.g., dehydration, polycythemia, anemia  Plasma protein concentration, e.g., monoclonal gammopathies, multiple myeloma The conditions that cause increased blood viscosity are sometimes grouped together as hyperviscosity syndromes . 12

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[Solution] Problem 6.6 One relatively simple model of non-Newtonian fluid rheology is the Herschel-Bulkley fluid, which combines the yield stress behavior of viscoplastic fluids with the power law description for shear thinning and shear thickening fluids: τ = τ 0 + k ˙ γ n ,τ ≥τ 0 In the Herschel-Bulkley model, τ 0 is the yield shear stress, k is the consistency index, and n is the flow index (where n < 1 models shear thinning and n > 1 models shear thickening). As you can see, this is basically a combination of the equation for a Bingham plastic and the Ostwald-de Waele relationship. The Herschel-Bulkley model can be applied to the behavior of blood, which exhibits a (small) yield stress and generally shear thinning behavior (that becomes pseudo-Newtonian at higher shear rates). Using the data provided in ch6_herschel.xlsx, determine n, k , and τ 0 for blood. If we plot the data from ch6_herschel.xlsx, we can see a typical shear stress versus shear rate curve for blood: 0 1 2 3 4 5 6 7 0 50 100 150 200 250 300 Shear Stress (dynes/cm2) Shear Rate (1/s) Blood has a very small yield stress, and should exhibit shear thinning behavior, so τ 0 ≤ 0.1 dynes/cm 2 and n < 1. We then do curve fitting of some sort using the Herschel-Bulkley model (you can do this in Excel in one of several ways, or in MATLAB, or in some other software). One example curve fit process is to use Excel’s simple Trendline option, which includes a power fit. However, the Herschel-Bulkley model is not a true power law equation because of the yield stress. If we estimate a yield stress and subtract it from the shear stress values, and plot the re-zeroed shear stress versus the shear rate, we can do a power law fit in Excel. The resultant values are as follows: 13

τ 0 = 0.1 dynescm − 2 n = 0.89 k = 0.04 dynes cm − 2 s 0.89 We can plot the results of the fit: 0 1 2 3 4 5 6 7 0 50 100 150 200 250 300 Shear Stress (dynes/cm 2 ) Shear Rate (1/s) 14