--- title: "Control volume, continuum and viscosity" subject: "Fluid Mechanics" chapter: 1 key: "semester_3/fluid_mechanics:control-volume-continuum-viscosity" source: https://www.bctengineer.com/q/semester_3/fluid_mechanics/control-volume-continuum-viscosity site: BCT Engineering --- # Control volume, continuum and viscosity Fluid Mechanics · Chapter 1: Fluid and its physical properties - Asked in: 2075 Chaitra, 2+2 marks - Video: [Control Volume and Continuum Concept, Fluid Mechanics](https://www.youtube.com/watch?v=7mUTl4skNR4) — Civil - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_3/fluid_mechanics/control-volume-continuum-viscosity - Whole chapter: https://www.bctengineer.com/md/semester_3/fluid_mechanics/chapter-1 ## Question Explain the concept of control volume and continuum in fluid mechanics. Define viscosity with its expression. ## Short answer A control volume is a fixed region used in Eulerian analysis. The continuum assumption treats fluid as continuous matter, valid when length scales exceed mean free path. Viscosity is the fluid’s resistance to shear; Newton’s law τ = μ du/dy gives dynamic viscosity μ, and kinematic viscosity ν = μ/ρ. ## Answer #### Control volume A **control volume** is a fixed region of space, of chosen shape and size, through whose boundary (the control surface) fluid may flow in and out. Instead of following a particular mass (system approach), we study what enters and leaves the region. It is the basis of the Eulerian approach, used to write the continuity, momentum and energy equations for flow through pipes, nozzles, turbines and bends. #### Continuum Fluids are made of molecules with empty spaces between them. The **continuum concept** ignores this and treats the fluid as continuously distributed matter, so properties such as density, pressure and velocity are smooth functions of position and time and can be differentiated. It is valid when the smallest length of interest is much larger than the mean free path of the molecules (Knudsen number $\ll1$). It fails for rarefied gases at very high altitude. #### Viscosity **Viscosity** is the property of a fluid by which it resists relative motion between its layers. By Newton's law of viscosity, the shear stress is proportional to the velocity gradient: $$ \tau=\mu\frac{du}{dy} $$ where $\mu$ is the **dynamic viscosity** (unit Pa·s = N·s/m²; 1 poise = 0.1 Pa·s). The **kinematic viscosity** is $\nu=\mu/\rho$ (m²/s; 1 stokes = $10^{-4}\ \text{m}^2/\text{s}$). Viscosity of liquids decreases and that of gases increases with temperature. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What is a control volume? A. A fixed mass of fluid followed in motion B. A fixed region of space through whose boundary fluid can flow C. A container with closed walls only D. The volume of fluid at constant pressure Answer: B. A control volume is a fixed region with a control surface across which fluid flows; it is the Eulerian approach. (Key point: Control volume) 2. When is the continuum concept valid? A. When the smallest length of interest is much larger than the molecular mean free path B. When the fluid is a gas at very high altitude C. When the length of interest is smaller than the mean free path D. Only for incompressible fluids Answer: A. Knudsen number << 1 means molecules can be ignored and properties treated as smooth functions. (Key point: Continuum validity) 3. Which expression defines viscosity? A. tau = mu du/dy B. tau = mu dy/du C. p = mu du/dy D. tau = rho du/dy Answer: A. Shear stress is proportional to velocity gradient, with dynamic viscosity mu as the constant. (Key point: Newton's law of viscosity) 4. How does temperature affect viscosity? A. Both liquids and gases decrease B. Liquids increase, gases decrease C. Liquids decrease, gases increase D. Neither is affected Answer: C. Liquid viscosity falls as cohesion weakens with heating, while gas viscosity rises with more molecular momentum exchange. 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