Fluid Mechanics 2 - Continuity Equation
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What does the continuity equation represent in fluid dynamics?

  • The relationship between pressure and density in a fluid
  • The calculation of fluid viscosity under different conditions
  • The change in velocity of a fluid over time
  • The conservation of mass within a fluid flow (correct)

    • Which equation relates to Newton's second law of motion in fluid dynamics?

  • Bernoulli's equation
  • Continuity equation
  • Navier-Stokes equation (correct)
  • Energy conservation equation

    • In the cylindrical polar coordinate form, what is the structure of the continuity equation?

  • $\frac{\partial v_z}{\partial z} + r \frac{\partial (rv_\theta)}{\partial \theta} + \frac{\partial (rv_r)}{\partial r} = 0$
  • $1/r \frac{\partial (rv_r)}{\partial r} + \frac{1}{r} \frac{\partial v_\theta}{\partial \theta} + \frac{\partial v_z}{\partial z} = 0$ (correct)
  • $\frac{1}{r} \frac{\partial (rv_r)}{\partial r} + \frac{\partial v_\theta}{\partial \theta} + r \frac{\partial v_z}{\partial z} = 0$
  • $\frac{\partial v_z}{\partial z} + \frac{\partial v_r}{\partial r} + \frac{\partial v_\theta}{\partial \theta} = 1$

    • What does the term $\nabla (\rho v)$ in the continuity equation signify?

    <p>The divergence of mass flow rate</p> Signup and view all the answers

    Which term in the Navier-Stokes equation accounts for external forces acting on the fluid?

    <p>$\rho g$</p> Signup and view all the answers

    What does the variable $r$ represent in the cylindrical coordinate system?

    <p>The distance from the origin to a point in the xy-plane</p> Signup and view all the answers

    Which equation represents the continuity equation in cylindrical coordinates?

    <p>$\frac{\partial \rho}{\partial t} + \frac{1}{r}\frac{\partial (r\rho v_r)}{\partial r} + \frac{1}{r}\frac{\partial (\rho v_\theta)}{\partial \theta} + \frac{\partial (\rho v_z)}{\partial z} = 0$</p> Signup and view all the answers

    What is true about steady flow in the context of the continuity equation?

    <p>Steady flow means that all properties are constant over time.</p> Signup and view all the answers

    Which of the following describes the scenario of steady and incompressible flow?

    <p>Flow where the density remains constant and does not change with time.</p> Signup and view all the answers

    What does the variable $\theta$ represent in the cylindrical coordinate system?

    <p>The angular coordinate measured in the xy-plane</p> Signup and view all the answers

    In the context of velocity components in cylindrical coordinates, what does $v_z$ represent?

    <p>The velocity component in the vertical direction</p> Signup and view all the answers

    Which term describes the transformation from rectangular coordinates to cylindrical coordinates?

    <p>$r = \sqrt{x^2 + y^2}$</p> Signup and view all the answers

    Which condition must be satisfied for mass outflow in a cylindrical control volume?

    <p>The rate of mass accumulation equals the sum of mass inflows and the net outflows.</p> Signup and view all the answers

    What does the continuity equation mathematically express?

    <p>The conservation of mass in a flow system</p> Signup and view all the answers

    Which term represents the mass flow rate in the x-direction?

    <p>$\rho v_x (x + dx) dy dz$</p> Signup and view all the answers

    What is derived from the general form of law of mass conservation for a control volume?

    <p>The differential form of mass conservation</p> Signup and view all the answers

    In the differential form of mass conservation, what does $\frac{\partial \rho}{\partial t}$ represent?

    <p>The change in density over time</p> Signup and view all the answers

    When evaluating mass flow rates, which dimensions are relevant for volumetric analysis?

    <p>Length, width, and height</p> Signup and view all the answers

    What happens to the continuity equation when we take the limit as $dx$, $dy$, and $dz$ approach zero?

    <p>It results in a differential form reflecting local mass balance</p> Signup and view all the answers

    How does mass flow rate relate to density, velocity, and area?

    <p>Mass flow rate is density multiplied by velocity times area</p> Signup and view all the answers

    In the context of the continuity equation, what does the term $\nabla (\rho v)$ specify?

    <p>The divergence of mass in a control volume</p> Signup and view all the answers

    Which of the following is a part of the differential form of mass conservation?

    <p>$\frac{\partial \rho}{\partial t} + \frac{\partial \rho v_x}{\partial x} = 0$</p> Signup and view all the answers

    What does the term $\rho v_x (x + dx)$ indicate in the continuity equation?

    <p>Outflow mass at the x-direction</p> Signup and view all the answers

    The term $\rho v_y (y + dy)$ in the continuity equation signifies what?

    <p>Mass inflow from the positive y-direction</p> Signup and view all the answers

    What physical principle does the continuity equation illustrate in fluid mechanics?

    <p>The conservation of mass</p> Signup and view all the answers

    In the context of fluid flow, the term $\rho$ generally refers to which property?

    <p>Density of the fluid</p> Signup and view all the answers

    When examining a control volume in three dimensions, what aspect of the mass is primarily represented?

    <p>Mass flow rate</p> Signup and view all the answers

    What does applying the divergence operator ($\nabla$) on the mass flow density vector imply?

    <p>It assesses local mass transfer</p> Signup and view all the answers

    Study Notes

    Fluid Mechanics 2 - Continuity Equation

    • Course: KIL 3002, Fluid Mechanics 2, Department of Chemical Engineering, Universiti Malaya
    • Chapter Focus: Continuity Equation in Differential Form
    • Key Concept: Mass conservation in a fluid flow system
    • Mathematical Formulation: The continuity equation in differential form is ∂ρ/∂t + ∇ • (ρν) = 0. This describes the transport of mass.
    • Derivation: Derived from an infinitesimal control volume. The equation represents the balance between the rate of mass accumulation within a control volume and the net rate of mass flow across its boundaries.
    • Infinitesimal Control Volume: A tiny volume element used to analyze mass conservation in the equation. Mass flow into and out of this small control volume is factored.
    • Mass Flow Rate: density × velocity × area, denoted as ρ ν A
    • Net Outflow of Mass Calculation: In x-direction: ((ρ ν)x+dx - (ρ ν)x) dy dz, and similar expressions for y and z directions, describing mass flow in and out of the infinitesimal volume.
    • Differential Form of Mass Conservation: The rate of mass accumulation within a control volume plus the net outflow of mass from the control volume is equal to zero.
    • Final Differential Form of Continuity Equation: ∂ρ/∂t + (∂(ρνx))/∂x + (∂(ρνy))/∂y + (∂(ρνz))/∂z = 0

    Cylindrical Coordinate System

    • Transformation: The continuity equation is reformulated using cylindrical coordinates (r, θ, z) for more relevant applications.
    • Derivation: In cylindrical coordinates, the equation is ∂ρ/∂t + 1/r ∂(rρνr)/∂r + 1/r ∂(ρνθ)/∂θ + ∂(ρνz)/∂z = 0
    • Significance: This version is useful in situations with axisymmetric or cylindrical geometry.
    • General Compact Form: ∂ρ/∂t + ∇ • (ρν) = 0, emphasizing the fundamental concept.

    Common Flow Cases

    • Steady Flow: Properties like density do not change with time; ∂ρ/∂t = 0.
    • Steady and Incompressible Flow: Incompressible fluids have constant density; ρ is unchanging. Using this principle, the terms simplify to a form like ∂(ρ νx)/∂x + ∂(ρ νy)/∂y + ∂(ρ νz)/∂z = 0.
    • Cylindrical Polar Coordinate Applications: There are specific cylindrical and polar representations of these flow types and their resulting formulations.

    Summary

    • Overall Concept: The continuity equation expresses the principle of mass conservation in fluid mechanics.
    • Derivation/Methodology: Mathematical derivation is crucial for understanding the relationship of mass accumulation within a control volume. This approach calculates the inflow and outflow rates, and setting this balance to zero forms the equation.

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    Description

    This quiz focuses on the Continuity Equation in Differential Form as part of the Fluid Mechanics 2 course (KIL 3002) at Universiti Malaya. It covers the key concepts of mass conservation in fluid flow and includes mathematical formulations and derivations related to the transport of mass. Test your understanding of the theoretical and practical aspects of the continuity equation.

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