Let's walk through the step-by-step solution to this Navier-Stokes simplification problem for a thin film (inclined plane flow): --- ## **Step 1: Understanding the Scenario** - **Thin film of fluid** flows down an **inclined plane** (angle θ = 15°). - Film thickness **h = 1 mm**, width **b = 1 m**. - **Assume steady, fully developed, 2D flow** (no change in z direction, no change in time, velocity profile does not change along x). --- ## **Step 2: Continuity Equation** The continuity equation is: \[ \frac{\partial \rho}{\partial t} + \frac{\partial (\rho u)}{\partial x} + \frac{\partial (\rho v)}{\partial y} + \frac{\partial (\rho w)}{\partial z} = \] For **incompressible flow** (\(\rho\) is constant) and no variation in z (2D flow, \(w = \)): \[ \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = \] --- ## **Step 3: Navier-Stokes Equations (Component Form)** Given in the image for x, y, z directions. Let's analyze each: ### **x-direction:** \[ \rho \left(\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} + w \frac{\partial u}{\partial z}\right) = -\frac{\partial p}{\partial x} + \rho g_x + \mu \left(\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2}\right) \] ### **y-direction:** \[ \rho \left(\frac{\partial v}{\partial t} + u \frac{\partial v}{\partial x} + v \frac{\partial v}{\partial y} + w \frac{\partial v}{\partial z}\right) = -\frac{\partial p}{\partial y} + \rho g_y + \mu \left(\frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial y^2} + \frac{\partial^2 v}{\partial z^2}\right) \] ### **z-direction:** \[ \rho \left(\frac{\partial w}{\partial t} + u \frac{\partial w}{\partial x} + v \frac{\partial w}{\partial y} + w \frac{\partial w}{\partial z}\right) = -\frac{\partial p}{\partial z} + \rho g_z + \mu \left(\frac{\partial^2 w}{\partial x^2} + \frac{\partial^2 w}{\partial y^2} + \frac{\partial^2 w}{\partial z^2}\right) \] --- ## **Step 4: Assumptions and Simplifications** - **Steady flow**: \(\frac{\partial}{\partial t} = \) - **Fully developed flow in x**: \(\frac{\partial}{\partial x} = \) - **2D flow**: \(w = , \frac{\partial}{\partial z} = \) - **Thin film (height much smaller than length)**: Variation in y direction dominates. Apply these to each term: #### **x-direction:** - \(\frac{\partial u}{\partial t} = \) - \(u \frac{\partial u}{\partial x} = \) - \(w = \), so \(w \frac{\partial u}{\partial z} = \) - \(\frac{\partial^2 u}{\partial x^2} = \) - \(\frac{\partial^2 u}{\partial z^2} = \) So, the equation simplifies to: \[ \rho v \frac{\partial u}{\partial y} = -\frac{\partial p}{\partial x} + \rho g_x + \mu \frac{\partial^2 u}{\partial y^2} \] But for fully developed flow, the velocity profile does not change along y for v, so \(v = \), so: \[ = -\frac{\partial p}{\partial x} + \rho g_x + \mu \frac{\partial^2 u}{\partial y^2} \] --- #### **y-direction:** - \(\frac{\partial v}{\partial t} = \) - \(u \frac{\partial v}{\partial x} = \) - \(v \frac{\partial v}{\partial y}\) is small for thin film and fully developed flow, so can be neglected. - \(w = \), so \(w \frac{\partial v}{\partial z} = \) - \(\frac{\partial^2 v}{\partial x^2} = \) - \(\frac{\partial^2 v}{\partial z^2} = \) - \(\frac{\partial^2 v}{\partial y^2}\) is typically small and can be neglected for thin films. So, the equation simplifies to: \[ = -\frac{\partial p}{\partial y} + \rho g_y \] --- ## **Step 5: Final Simplified Equations** ### **x-direction:** \[ = -\frac{\partial p}{\partial x} + \rho g_x + \mu \frac{\partial^2 u}{\partial y^2} \] ### **y-direction:** \[ = \rho g_y - \frac{\partial p}{\partial y} \] Or: \[ \frac{\partial p}{\partial y} = \rho g_y \] --- ## **Step 6: Justification for Crossing Out Terms** - **Time derivatives**: Zero for steady flow. - **x-derivatives**: Zero for fully developed flow (no change along x). - **z-derivatives**: Zero for 2D flow (no variation in z). - **w component**: Zero for 2D flow. - **Convective terms**: Most are zero due to above reasons; \(v \frac{\partial u}{\partial y}\) is very small compared to pressure gradient and viscous terms in thin films and is neglected. --- ## **Final Answer (as shown in your image)** **The simplified equations are:** \[ \boxed{ \begin{align*} &= \rho g_x + \mu \frac{\partial^2 u}{\partial y^2} - \frac{\partial p}{\partial x} \\ &= \rho g_y - \frac{\partial p}{\partial y} \end{align*} } \] Or as in the image: - \(\frac{\partial v}{\partial y} = \) - \( = \rho g_x + \mu \frac{\partial^2 u}{\partial y^2} - \frac{\partial p}{\partial x}\) - \( = \rho g_y - \frac{\partial p}{\partial y}\) --- ### **Summary Table** | Term | Reason for Crossing Out | |----------------------|----------------------------------------------------------------------------------------------------------| | \(\frac{\partial}{\partial t}\) | Steady flow: no time dependence | | \(\frac{\partial}{\partial x}\) | Fully developed flow: profile doesn't change in x | | \(\frac{\partial}{\partial z}\), \(w\) | 2D flow: no changes in z, no velocity in z direction | | Convective terms (\(u \frac{\partial}{\partial x}\), \(v \frac{\partial}{\partial y}\), etc.) | Most are zero/neglected due to steady, 2D, thin film, and fully developed flow assumptions | | Pressure in y (\(\frac{\partial p}{\partial y}\)) | Hydrostatic, from gravity only (can be simplified) | --- **If you need the physical interpretation or further help with boundary conditions, let me know!**