# Derivation: Equation of Motion in Rectangular Coordinates (x-component) for Newtonian Fluid Let's derive the **Navier-Stokes equation** for the x-component in Cartesian coordinates, assuming **constant density** (ρ) and **constant viscosity** (μ) for a Newtonian fluid. --- ## **Step 1: Consider a Fluid Element** - Take a small rectangular fluid element of size \( dx \times dy \times dz \). - Let the velocity components be \( u, v, w \) in the \( x, y, z \) directions respectively. --- ## **Step 2: Apply Newton's Second Law (Conservation of Momentum)** **Sum of Forces = Rate of Change of Momentum** \[ \text{Net force in } x = \text{mass} \times \text{acceleration in } x \] --- ## **Step 3: Forces Acting on the Fluid Element** ### **A. Body Force (Gravity)** Let \( X \) be the body force per unit mass in the x-direction (e.g., gravity). \[ \text{Body force} = \rho\, X\, dx\, dy\, dz \] ### **B. Pressure Forces** - At face \( x \): \( P \cdot dy\, dz \) - At face \( x + dx \): \( [P + \frac{\partial P}{\partial x} dx] \cdot dy\, dz \) Net pressure force in x-direction: \[ F_P = -\frac{\partial P}{\partial x} dx\, dy\, dz \] ### **C. Viscous Forces** For a Newtonian fluid, the viscous (shear) stress in the x-direction: \[ \tau_{xx} = 2\mu \frac{\partial u}{\partial x} \] \[ \tau_{yx} = \mu \left( \frac{\partial u}{\partial y} + \frac{\partial v}{\partial x} \right) \] \[ \tau_{zx} = \mu \left( \frac{\partial u}{\partial z} + \frac{\partial w}{\partial x} \right) \] The sum of viscous forces per unit volume in the x-direction: \[ F_{\text{viscous, x}} = \frac{\partial}{\partial x} \left( 2\mu \frac{\partial u}{\partial x} \right) + \frac{\partial}{\partial y} \left[ \mu \left( \frac{\partial u}{\partial y} + \frac{\partial v}{\partial x} \right) \right] + \frac{\partial}{\partial z} \left[ \mu \left( \frac{\partial u}{\partial z} + \frac{\partial w}{\partial x} \right) \right] \] For **constant viscosity** \( \mu \): \[ = \mu \left[ \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2} \right] = \mu \nabla^2 u \] --- ## **Step 4: Acceleration (Substantial Derivative)** The acceleration in the x-direction for fluid motion is given by the **material derivative**: \[ a_x = \frac{Du}{Dt} = \frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} + w \frac{\partial u}{\partial z} \] --- ## **Step 5: Assemble the Equation** Combine all forces and acceleration per unit volume: \[ \rho \frac{Du}{Dt} = -\frac{\partial P}{\partial x} + \mu \nabla^2 u + \rho X \] \[ \boxed{ \rho \left( \frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} + w \frac{\partial u}{\partial z} \right) = -\frac{\partial P}{\partial x} + \mu \left( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2} \right) + \rho X } \] --- ## **Final Answer: x-component of the Navier-Stokes Equation** \[ \boxed{ \rho \left( \frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} + w \frac{\partial u}{\partial z} \right) = -\frac{\partial P}{\partial x} + \mu \left( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2} \right) + \rho X } \] --- ### **Explanation** - **Left side:** Rate of change of momentum (acceleration of fluid). - **First right term:** Pressure gradient force. - **Second right term:** Viscous force (diffusion of momentum). - **Third right term:** Body force (e.g., gravity). This is the **x-component of the Navier-Stokes equation** for a Newtonian fluid with constant density and viscosity in Cartesian coordinates.