Let's break down the handwritten notes and provide a detailed explanation for each part: --- ### (a) **Newton's Law of Viscosity and Units** **Equation:** \[ \tau_{yx} = -\mu \frac{dv_x}{dy} \] Where: - \(\tau_{yx}\) = Shear stress - \(\mu\) = Viscosity - \(v_x\) = Velocity in x-direction - \(y\) = Distance perpendicular to flow #### **Units Table** | Symbol | SI Units | Conversion Factor | British Units | |--------|---------------------|--------------------------|---------------| | \(\tau\) | N/m² (Pa) | 1 N/m² = 0.0209 lbf/ft² | lbf/ft² | | \(\mu\) | N·s/m² = Pa·s | 1 N·s/m² = 0.0209 lbf·s/ft² | lbf·s/ft² | | \(v\) | m/s | 1 m/s = 3.281 ft/s | ft/s | | \(y\) | m | 1 m = 3.281 ft | ft | #### **Dimensional Consistency Check (in SI):** \[ \tau_{yx} = \mu \frac{dv_x}{dy} \] - \(\tau_{yx}\): \(\frac{N}{m^2}\) - \(\mu\): \(\frac{N \cdot s}{m^2}\) - \(\frac{dv_x}{dy}\): \(\frac{m/s}{m} = \frac{1}{s}\) So, \[ \frac{N}{m^2} = \frac{N \cdot s}{m^2} \times \frac{1}{s} = \frac{N}{m^2} \] **Conclusion:** The equation is dimensionally consistent. --- ### (b) **Viscosity Units** \[ [\text{Pa} \cdot \text{s}] = \frac{N \cdot s}{m^2} = \frac{kg \cdot m}{s^2} \cdot \frac{s}{m^2} = \frac{kg}{m \cdot s} \] #### **Conversion to Centipoise (cP):** \[ 1 \text{Pa} \cdot \text{s} = 1000 \text{cP} \] **Derivation:** \[ 1 \text{N} \cdot \text{s} / \text{m}^2 = 1 \text{N} \cdot \frac{10^5 \text{dyne}}{1 \text{N}} \cdot \frac{1}{1 \text{m}^2} \cdot \left(\frac{1 \text{m}}{100 \text{cm}}\right)^2 = 10 \text{Poise} \times 100 \text{cP} = 1000 \text{cP} \] --- ### (c) **Viscosity and Kinematic Viscosity of Water and Air** #### **Typical Values (at ~20°C):** | Fluid | Dynamic Viscosity (\(\mu\)) | Kinematic Viscosity (\(\nu\)) | |----------|-----------------------------|-------------------------------| | Water | \(1.0 \times 10^{-3}\) Pa·s (1 cP) | \(1.0 \times 10^{-6}\) m²/s | | Air | \(1.8 \times 10^{-5}\) Pa·s | \(1.5 \times 10^{-5}\) m²/s | **Sources:** - [Engineering Toolbox - Water Viscosity](https://www.engineeringtoolbox.com/water-dynamic-kinematic-viscosity-d_596.html) - [Engineering Toolbox - Air Viscosity](https://www.engineeringtoolbox.com/air-absolute-kinematic-viscosity-d_601.html) #### **Comparison and Everyday Effects:** - **Water** is much more viscous than air, so it resists flow more (e.g., swimming is harder than running). - **Kinematic viscosity** of air is higher than that of water due to its much lower density. - **Examples:** - **Water**: Oil flows more slowly than water; syrup even slower, due to higher viscosity. - **Air**: Air resistance (drag) is lower, so objects fall faster in air than in liquids. --- If you need further breakdowns or have more questions, let me know!

Certainly! Here's a detailed, point-by-point explanation addressing each part of your question, with calculations checked for accuracy and clarity, and avoiding tables as requested. --- ### (a) Newton's Law of Viscosity and Its Units **Equation:** \[ \tau_{yx} = -\mu \frac{dv_x}{dy} \] **Symbols and their units:** - **Shear stress \(\tau_{yx}\):** - *SI units:* Newton per square meter (N/m²), also called Pascal (Pa) - *British units:* pound-force per square foot (lbf/ft²) - **Viscosity \(\mu\):** - *SI units:* Newton-seconds per square meter (N·s/m²), which is equivalent to Pa·s - *British units:* pound-force seconds per square foot (lbf·s/ft²) - **Velocity \(v_x\):** - *SI units:* meters per second (m/s) - *British units:* feet per second (ft/s) - **Distance \(y\):** - *SI units:* meters (m) - *British units:* feet (ft) **Dimensional analysis for consistency:** - The shear stress \(\tau_{yx}\) has units of N/m². - The viscosity \(\mu\) has units of N·s/m². - The velocity gradient \(\frac{dv_x}{dy}\) has units of (m/s)/m = 1/s. Check units on RHS: \[ \text{Units of RHS} = (\text{N·s/m}^2) \times (1/s) = \text{N/m}^2, \] which matches the units of \(\tau_{yx}\). **Conclusion:** The equation is dimensionally consistent in SI units. --- ### (b) Units of Viscosity (\(\text{Pa}\cdot \text{s}\)) and Conversion to Centipoise (cP) - **Expressing \(\text{Pa}\cdot \text{s}\) in fundamental SI units:** \[ 1\, \text{Pa} = 1\, \text{N/m}^2 = \frac{kg \cdot m}{s^2 \cdot m^2} = \frac{kg}{m \cdot s^2} \] Therefore: \[ 1\, \text{Pa} \cdot \text{s} = \frac{kg}{m \cdot s^2} \times s = \frac{kg}{m \cdot s} \] - **Relation to centipoise (cP):** - 1 Pa·s = 1000 cP - *Derivation:* 1 cP = 1 mPa·s = \(1 \times 10^{-3}\) Pa·s Hence, \[ 1\, \text{Pa} \cdot \text{s} = 1000\, \text{cP} \] **Summary:** Viscosity in Pa·s can be converted to cP by multiplying by 1000. --- ### (c) Viscosity and Kinematic Viscosity of Water and Air **Values at approximately 20°C:** - **Water:** - Dynamic viscosity (\(\mu\)): approximately \(1.0 \times 10^{-3}\) Pa·s (which is 1 cP) - Kinematic viscosity (\(\nu\)): approximately \(1.0 \times 10^{-6}\) m²/s - **Air:** - Dynamic viscosity: approximately \(1.8 \times 10^{-5}\) Pa·s - Kinematic viscosity: approximately \(1.5 \times 10^{-5}\) m²/s **Sources:** - Engineering Toolbox and standard fluid property references. **Qualitative effects:** - **Water's higher viscosity** means it resists flow more strongly than air, which is why objects fall faster through air than through liquids. - **Kinematic viscosity** indicates how fluids flow under gravity; higher for air relative to water because of its lower density. - **Everyday examples:** - Moving through water feels much more resistant than moving through air. - Oil (more viscous than water) flows slowly, influencing lubrication and cooking. - Air's low viscosity facilitates efficient airflow over wings and around objects, impacting aerodynamics. --- ### Final notes: - No mistakes found in the approach or calculations. - The units are consistent and correctly derived. - The qualitative explanations align with standard fluid mechanics principles. If you'd like this information formatted differently or further clarified, please let me know!