# Study Notes: Derivation of Continuity Equation The **continuity equation** is a fundamental principle in fluid dynamics that expresses the conservation of mass in a flowing fluid. It states that the product of the cross-sectional area of a pipe and the velocity of the fluid at any point along the pipe remains constant. This principle is derived from the **law of conservation of mass**. --- ## Key Formula: Continuity Equation The continuity equation is mathematically expressed as: $$ R = A \cdot v = \text{constant} $$ Where: - $R$ = Volume flow rate (m³/s) - $A$ = Cross-sectional area of the pipe (m²) - $v$ = Velocity of the fluid (m/s) The continuity equation applies to both compressible and incompressible flows, but its form simplifies for incompressible fluids. --- ## Assumptions of the Continuity Equation For the derivation of the continuity equation, the following assumptions are made: - The tube has a **single entry and single exit**. - The fluid is **non-viscous** (no internal friction). - The flow is **incompressible** (density remains constant). - The fluid flow is **steady** (does not change with time). These assumptions simplify the derivation and ensure the equation is valid for idealized fluid flow scenarios. --- ## Derivation of the Continuity Equation ### Step-by-Step Derivation 1. **Consider a Pipe with Varying Cross-Section** Imagine a pipe where the cross-sectional area changes along its length. Let: - $A_1$ and $A_2$ be the cross-sectional areas at two points. - $v_1$ and $v_2$ be the fluid velocities at these points. - $\rho_1$ and $\rho_2$ be the fluid densities at these points. 2. **Fluid Movement Over a Small Time Interval ($\Delta t$)** - At the lower end of the pipe, the fluid travels a distance $\Delta x_1 = v_1 \cdot \Delta t$. - The volume of fluid entering the pipe at the lower end is: $$ V_1 = A_1 \cdot \Delta x_1 = A_1 \cdot v_1 \cdot \Delta t $$ - Similarly, at the upper end, the fluid travels a distance $\Delta x_2 = v_2 \cdot \Delta t$, and the volume of fluid exiting is: $$ V_2 = A_2 \cdot \Delta x_2 = A_2 \cdot v_2 \cdot \Delta t $$ 3. **Mass Conservation** The mass of the fluid entering and exiting the pipe must be equal (law of conservation of mass). Using the relationship $m = \rho \cdot V$: $$ \Delta m_1 = \rho_1 \cdot A_1 \cdot v_1 \cdot \Delta t $$ $$ \Delta m_2 = \rho_2 \cdot A_2 \cdot v_2 \cdot \Delta t $$ Since $\Delta m_1 = \Delta m_2$, we get: $$ \rho_1 \cdot A_1 \cdot v_1 = \rho_2 \cdot A_2 \cdot v_2 $$ 4. **General Form of the Continuity Equation** The above equation can be written as: $$ \rho \cdot A \cdot v = \text{constant} $$ For **incompressible flow** (where $\rho_1 = \rho_2$), the equation simplifies to: $$ A_1 \cdot v_1 = A_2 \cdot v_2 $$ **Example:** If water flows through a pipe with $A_1 = 0.05 \, \text{m}^2$ and $v_1 = 2 \, \text{m/s}$, and the pipe narrows to $A_2 = 0.02 \, \text{m}^2$, find $v_2$. Using $A_1 \cdot v_1 = A_2 \cdot v_2$: $$ 0.05 \cdot 2 = 0.02 \cdot v_2 $$ $$ v_2 = 5 \, \text{m/s} $$ --- ## Special Cases of the Continuity Equation ### 1. **Incompressible Flow** For incompressible fluids (constant density, $\rho$): $$ A \cdot v = \text{constant} $$ ### 2. **Steady Flow in Cylindrical Coordinates** In cylindrical coordinates, the continuity equation is expressed as: $$ \frac{\partial \rho}{\partial t} + \frac{1}{r} \frac{\partial}{\partial r} (r \rho u) + \frac{1}{r} \frac{\partial}{\partial \theta} (\rho v) + \frac{\partial}{\partial z} (\rho w) = 0 $$ ### 3. **Incompressible Flow in Vector Form** For incompressible flow, the continuity equation in vector form is: $$ \nabla \cdot \mathbf{v} = 0 $$ Where $\nabla \cdot \mathbf{v}$ is the divergence of the velocity field. This form is particularly useful in computational fluid dynamics (CFD) and advanced fluid mechanics. --- ## Key Takeaways - The **continuity equation** ensures mass conservation in fluid flow. - For **incompressible fluids**, the equation simplifies to $A \cdot v = \text{constant}$. - The equation applies to both steady and unsteady flows, but its form may vary depending on the assumptions (e.g., compressibility, coordinate system). **Common Mistake:** Forgetting to account for changes in density ($\rho$) when dealing with compressible fluids. Always check whether the fluid is compressible or incompressible before applying the simplified form of the equation. --- ## Practice Problems 1. Water flows through a pipe with a cross-sectional area of $0.1 \, \text{m}^2$ at a velocity of $3 \, \text{m/s}$. If the pipe narrows to $0.05 \, \text{m}^2$, what is the velocity of the water in the narrower section? 2. Air (compressible) flows through a duct where the density changes from $1.2 \, \text{kg/m}^3$ to $1.0 \, \text{kg/m}^3$. If the initial velocity and area are $5 \, \text{m/s}$ and $0.2 \, \text{m}^2$, respectively, find the velocity in the second section if the area is $0.15 \, \text{m}^2$. 3. Derive the continuity equation in cylindrical coordinates for steady, incompressible flow. Use these problems to test your understanding of the continuity equation and its applications!