Let’s break down the problem step by step, explaining each part so it’s clear and thorough. --- ### **Step 1: Understanding the Problem** We are given: - The velocity profile for a Newtonian fluid under a plate is: \( u = (12y^{1/4}) \) mm/s, where \( y \) is in mm. - The fluid's viscosity, \( \mu = 0.5 \times 10^{-3} \) N·s/m². - We are asked to **determine the minimum shear stress** within the fluid. --- ### **Step 2: Shear Stress Formula** For a Newtonian fluid, **shear stress** is: \[ \tau = \mu \frac{du}{dy} \] --- ### **Step 3: Find the Velocity Gradient** Given \( u = 12y^{1/4} \) mm/s, Let’s find \(\frac{du}{dy}\): First, recall the power rule: \[ \frac{d}{dy}(y^n) = n y^{n-1} \] Apply this to our velocity profile: \[ \frac{du}{dy} = 12 \cdot \frac{d}{dy}\left(y^{1/4}\right) \] \[ \frac{du}{dy} = 12 \cdot \left( \frac{1}{4} y^{-3/4} \right) \] \[ \frac{du}{dy} = 3 y^{-3/4} \] --- ### **Step 4: Where is Shear Stress Minimum?** The shear stress is minimum where the velocity gradient is minimum. Since \(\frac{du}{dy} = 3y^{-3/4}\), This expression **decreases** as \(y\) increases. Given the plate's height is 16 mm, the minimum gradient is at \(y = 16\) mm. --- ### **Step 5: Plug Values into Shear Stress Formula** Let’s substitute: - \(\mu = 0.5 \times 10^{-3}\) N·s/m² - \(\frac{du}{dy} = 3(16)^{-3/4}\) (make sure units are consistent) \[ \tau_{min} = \mu \frac{du}{dy} \] \[ \tau_{min} = \left[0.5 \times 10^{-3} \text{ N·s/m}^2\right] \times \left[3 \times (16 \text{ mm})^{-3/4}\right] \] Evaluate \((16)^{-3/4}\): - \(16^{1/4} = 2\), so \(16^{-3/4} = (16^{1/4})^{-3} = 2^{-3} = 1/8\) Now calculate: \[ \tau_{min} = 0.5 \times 10^{-3} \times 3 \times \frac{1}{8} \] \[ \tau_{min} = 0.5 \times 10^{-3} \times 0.375 \] \[ \tau_{min} = 0.1875 \times 10^{-3} \text{ N/m}^2 \] \[ \tau_{min} = 0.1875 \text{ mPa} \] --- ### **Step 6: Final Answer** The **minimum shear stress** within the fluid is: \[ \boxed{0.1875 \ \text{mPa}} \] --- #### **Extra Note** At \(y = 0\), the velocity gradient becomes infinite, so the formula isn’t valid at the wall. --- Let me know if you have any questions or need more explanation!