# BENG130 Biotransport: Flow Fundamentals Problems — Step-by-Step Solutions --- ## 1. **Steadiness of the Flow** **Given:** \[ \vec{v} = (x^2 - y^2)\hat{i} + (x - y^2)\hat{j} \] **To Determine:** Is the flow steady? ### **Solution:** A flow is **steady** if its velocity field does **not** change with **time** (i.e., \(\partial \vec{v}/\partial t = \)). - Here, \(\vec{v}\) depends only on \(x\) and \(y\). There is **no explicit dependence on time** (\(t\)). - **Therefore:** \[ \frac{\partial \vec{v}}{\partial t} = \] The flow is **steady**. **Final Answer:** \[ \boxed{\text{The flow is steady.}} \] --- ## 2. **Velocity Profile in a Channel** Given: \[ v_x = k\left(1 - \left(\frac{y}{W/2}\right)^2\right) \] where \(y = \) at the center, and \(W\) is the channel width. --- ### (a) **Sketch the Velocity Distribution (\(k=1, W=10\))** - \(W = 10\), so \(W/2 = 5\). - \(v_x = 1 - \left(\frac{y}{5}\right)^2\) - \(y\) ranges from \(-5\) to \(+5\). **Key Points:** - At \(y = \): \(v_x = 1\) (center, max velocity) - At \(y = \pm 5\): \(v_x = 1 - (1)^2 = \) (walls, zero velocity) **Sketch Description:** - The plot is a **parabola** opening downwards, centered at \(y = \), touching the \(y\)-axis at \(y = \pm 5\). --- ### (b) **Shear Stress at the Wall (\(y = \pm W/2\)), Newtonian Fluid** **Shear stress:** For a Newtonian fluid, \[ \tau = \mu \left. \frac{dv_x}{dy} \right|_{\text{wall}} \] **Calculate derivative:** \[ v_x = k\left[1 - \left(\frac{y}{W/2}\right)^2\right] \] Let \(a = W/2\): \[ v_x = k \left( 1 - \left(\frac{y}{a}\right)^2 \right) \] \[ \frac{dv_x}{dy} = k \cdot \left( -2\frac{y}{a^2} \right) = -\frac{2k}{a^2}y \] At \(y = a = W/2\): \[ \left. \frac{dv_x}{dy} \right|_{y = W/2} = -\frac{2k}{a^2}a = -\frac{2k}{a} = -\frac{2k}{W/2} = -\frac{4k}{W} \] **So,** \[ \tau_{\text{wall}} = \mu \left( -\frac{4k}{W} \right) \] **Final Answer:** \[ \boxed{ \tau_{\text{wall}} = -\frac{4\mu k}{W} } \] (Negative sign indicates direction; magnitude is \(\frac{4\mu k}{W}\).) --- ## 3. **Experimental Data: Fluid Type Identification** **Data Table:** | Shear Stress (dyne/cm\(^2\)) | 5 | 9 | 23 | 48 | 95 | |------------------------------|---|---|----|----|----| | Shear Rate (\(s^{-1}\)) |.1| 1 | 10 |100 |100| **To Do:** Plot shear stress vs. shear rate and determine if the fluid is Newtonian, shear-thinning, or shear-thickening. ### **Analysis:** - For a **Newtonian fluid**, \(\tau = \mu \dot{\gamma}\) (linear relationship). - **Shear-thinning:** Curve flattens (concave down); slope decreases with increasing rate. - **Shear-thickening:** Curve steepens (concave up); slope increases with increasing rate. #### **Calculate Apparent Viscosity (\(\tau/\dot{\gamma}\)):** | Shear Rate (\(s^{-1}\)) | .1 | 1 | 10 | 100 | 100 | |-------------------------|-----|---|----|-----|------| | Shear Stress (dyne/cm\(^2\)) | 5 | 9 | 23 | 48 | 95 | | Apparent Viscosity (\(\tau/\dot{\gamma}\)) | 50 | 9 | 2.3 | .48 | .095 | - **Viscosity decreases with increasing shear rate**. ### **Conclusion:** - **Shear-thinning** fluid (pseudoplastic behavior). **Final Answer:** \[ \boxed{\text{The fluid is shear-thinning.}} \] --- ## **Summary Table** | Part | Final Answer | |------|-------------| | 1 | The flow is steady. | | 2(a) | Parabolic velocity profile, max at center, zero at walls. | | 2(b) | \(\tau_{\text{wall}} = -\frac{4\mu k}{W}\) | | 3 | The fluid is shear-thinning. | --- If you need the plot for question 3, please let me know!