Let's break down the problem step by step: --- ## **Given Equation** \[ \frac{dP}{dz} = -\frac{\mu}{K} V_z + \mu \frac{d^2 V_z}{dy^2} \] Where: - \( P \): Pressure - \( \mu \): Fluid viscosity - \( K \): Fiber matrix effect (permeability) - \( V_z \): Velocity in \( z \) direction (function of \( y \) only) - \( y \): Coordinate perpendicular to the plates (\(-B \leq y \leq B\)) - \( \frac{dP}{dz} = C \): Constant No-slip boundary conditions: \( V_z(\pm B) = \). --- ## **1. Case I: \( K/B^2 \gg 1 \) (Permeability is large)** ### **Simplification** If \( K/B^2 \gg 1 \), then \( \frac{\mu}{K} V_z \) is much **smaller** than \( \mu \frac{d^2 V_z}{dy^2} \). So, neglect \( \frac{\mu}{K} V_z \): \[ \frac{dP}{dz} = \mu \frac{d^2 V_z}{dy^2} \] \[ C = \mu \frac{d^2 V_z}{dy^2} \] \[ \frac{d^2 V_z}{dy^2} = \frac{C}{\mu} \] ### **Integrate Twice** 1st integration: \[ \frac{d V_z}{dy} = \frac{C}{\mu} y + A \] 2nd integration: \[ V_z(y) = \frac{C}{2\mu} y^2 + A y + B \] ### **Apply Boundary Conditions** \( V_z(\pm B) = \): At \( y = B \): \[ = \frac{C}{2\mu} B^2 + A B + B \] At \( y = -B \): \[ = \frac{C}{2\mu} B^2 - A B + B \] Subtract the two equations: \[ = [\frac{C}{2\mu} B^2 + A B ] - [\frac{C}{2\mu} B^2 - A B ] = 2AB \implies A = \] Plug back in: \[ = \frac{C}{2\mu} B^2 + B \implies B = -\frac{C}{2\mu} B^2 \] So: \[ V_z(y) = \frac{C}{2\mu} (y^2 - B^2) \] ### **Final Answer for Case 1:** \[ \boxed{V_z(y) = \frac{C}{2\mu} (y^2 - B^2)} \] --- ## **2. Case II: \( K/B^2 \ll 1 \) (Permeability is small)** Here, \( \frac{\mu}{K} V_z \) dominates, except near the boundaries where the no-slip condition must be satisfied. This gives rise to a **boundary layer** of thickness \( \delta \). ### **Outer Region (Away from Boundaries):** Neglect the second derivative term: \[ \frac{dP}{dz} = -\frac{\mu}{K} V_z \implies V_z = -\frac{K}{\mu} \frac{dP}{dz} = -\frac{K}{\mu} C \] So, in the bulk: \[ V_z = V_{z,} = -\frac{K}{\mu} C \] ### **Boundary Layer Near Plates (\( y \approx \pm B \)):** Here, the second derivative term is important to match the no-slip condition. - Let \( y = B - \eta \) where \( \eta \ll B \) (near upper plate). - The boundary layer thickness \( \delta \) is found by balancing the two terms: \[ \frac{\mu}{K} V_z \sim \mu \frac{d^2 V_z}{dy^2} \] Estimate \( \frac{d^2 V_z}{dy^2} \sim \frac{V_z}{\delta^2} \): \[ \frac{\mu}{K} V_z \sim \mu \frac{V_z}{\delta^2} \implies \frac{1}{K} \sim \frac{1}{\delta^2} \implies \delta \sim \sqrt{K} \] Since \( K/B^2 \ll 1 \implies \delta \ll B \). ### **Boundary Layer Equation:** Rescale close to the wall: \( \eta = B - y \), \( \eta \sim \delta \): \[ C = -\frac{\mu}{K} V_z + \mu \frac{d^2 V_z}{d\eta^2} \] Or, \[ \frac{d^2 V_z}{d\eta^2} - \frac{1}{K} V_z = -\frac{C}{\mu} \] This is a nonhomogeneous linear ODE. #### **General Solution:** Homogeneous part: \( \frac{d^2 V_z}{d\eta^2} - \frac{1}{K} V_z = \) Solution: \( V_z^h = A e^{\eta/\sqrt{K}} + B e^{-\eta/\sqrt{K}} \) Particular solution: \( V_z^p = \text{constant} = V_{z,} = -\frac{K}{\mu} C \) So, \[ V_z(\eta) = A e^{\eta/\sqrt{K}} + B e^{-\eta/\sqrt{K}} - \frac{K}{\mu} C \] ### **Boundary Conditions:** At the wall (\( \eta = \)), \( V_z = \): \[ = A + B - \frac{K}{\mu} C \] Far from the wall (\( \eta \rightarrow \infty \)), \( V_z \rightarrow V_{z,} = -\frac{K}{\mu} C \). But as \( \eta \rightarrow \infty \), \( e^{\eta/\sqrt{K}} \) diverges, so set \( A = \): So, \[ V_z(\eta) = B e^{-\eta/\sqrt{K}} - \frac{K}{\mu} C \] As \( \eta \to \infty \), \( V_z \to -\frac{K}{\mu} C \). At \( \eta = \): \[ = B - \frac{K}{\mu} C \implies B = \frac{K}{\mu} C \] So, \[ V_z(\eta) = \frac{K}{\mu} C \left[e^{-\eta/\sqrt{K}} - 1\right] \] Or, in terms of \( y \): \[ \boxed{ V_z(y) = -\frac{K}{\mu} C \left[1 - \exp\left(-\frac{B-y}{\sqrt{K}}\right)\right] \quad \text{(near } y = B\text{)} } \] And similarly near \( y = -B \): \[ \boxed{ V_z(y) = -\frac{K}{\mu} C \left[1 - \exp\left(-\frac{B+y}{\sqrt{K}}\right)\right] \quad \text{(near } y = -B\text{)} } \] --- ### **Summary** #### **(1) If \( K/B^2 \gg 1 \):** \[ \boxed{V_z(y) = \frac{C}{2\mu}(y^2 - B^2)} \] #### **(2) If \( K/B^2 \ll 1 \):** - **Thickness of boundary layer:** \( \boxed{\delta \sim \sqrt{K}} \ll B \) - **Velocity in the boundary layer near the wall:** \[ \boxed{ V_z(y) = -\frac{K}{\mu} C \left[1 - \exp\left(-\frac{B-y}{\sqrt{K}}\right)\right] } \] - **Outside boundary layer:** \( V_z = -\frac{K}{\mu} C \) --- Let me know if you need a plot or further explanation!