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Chapter 4: Problem 16

Determine whether the following expressions satisfy the continuity equation: (a) \(u=10 x t, v=-10 y t, \rho=\) constant [Yes] (b) \(u=U(y / \delta)^{1 / 7}, v=0, \rho=\) constant. [Yes]

Short Answer

Expert verified
Both expressions satisfy the continuity equation.

Step by step solution

01

Understanding the Continuity Equation

The continuity equation for a two-dimensional flow, assuming incompressibility and constant density, can be written as: \(\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0\). We will apply this equation to the given expressions.
02

Checking Part (a)

For \(u=10xt\) and \(v=-10yt\), compute the partial derivatives: \(\frac{\partial u}{\partial x} = 10t\) and \(\frac{\partial v}{\partial y} = -10t\). Adding these derivatives gives: \(10t + (-10t) = 0\), which satisfies the continuity equation.
03

Checking Part (b)

For \(u=U(y / \delta)^{1 / 7}\) and \(v=0\), compute the partial derivatives: \(\frac{\partial u}{\partial x} = 0\) because \(u\) does not depend on \(x\), and \(\frac{\partial v}{\partial y} = 0\) because \(v\) is constant 0. Their sum is: \(0 + 0 = 0\), which also satisfies the continuity equation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Incompressible Flow
Incompressible flow is a type of fluid flow where the fluid density remains constant. Such flow is prevalent in liquids since their density hardly changes under pressure. Key aspects to understand about incompressible flow include:

  • Constant Density: The density of the fluid does not change with time or space. Thus, in any control volume, the mass of the fluid remains unchanged.
  • Equation Constraint: For incompressible flow, the continuity equation simplifies significantly. If the density \(\rho\) is constant, we use the form \(abla \cdot \mathbf{V} = 0\), where \(\mathbf{V}\) is the velocity vector of the fluid.
  • Applications: Engineers often assume incompressible flow for liquids. It's useful in hydraulic systems and when studying oceans, rivers, and even airflow at lower speeds.
Understanding these principles helps solve fluid dynamics problems effectively, ensuring adherence to the continuity equation in varied applications.
Partial Derivatives
Partial derivatives are a fundamental tool in calculus used to analyze functions of multiple variables. They involve differentiating a function concerning one variable while keeping others constant. Key points include:

  • Powers of Specification: Consider a function \(f(x, y)\); to find the partial derivative \(\frac{\partial f}{\partial x}\), treat \(y\) as constant and differentiate with respect to \(x\).
  • Applications in Fluid Dynamics: In the context of the continuity equation, partial derivatives help express changes in the velocity field along different axes, capturing the effect of incompressible flow dynamics.
  • Partial Derivative Notation: Denoted by different symbols, such as \(\frac{\partial}{\partial x}\), partial derivatives emphasize the focus on rate changes within a multi-dimensional function.
This conceptual understanding aids in extending calculus methods for predicting physical behaviors in two or three dimensions, particularly in contexts like the given problem.
Two-Dimensional Flow
Two-dimensional flow refers to a simplification in fluid dynamics where the flow properties are assumed to vary only in two dimensions, typically \(x\) and \(y\). Here are critical features of two-dimensional flow:

  • Simplification: By neglecting variations along the third axis, the complexity of the problem is reduced, allowing for more straightforward solutions.
  • Governing Equations: In 2-D flow, the continuity equation simplifies to \(\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0\) for incompressible flow, where \(u\) and \(v\) are velocity components.
  • Practical Examples: Many flows can be approximated as two-dimensional under certain conditions. Examples include airflow over a wing or water flow across a flat plate.
This approach forms the core of many fluid dynamics analyses since it offers a balance between analytical tractability and real-world applicability.

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Most popular questions from this chapter

Water flows through a pipe \(25 \mathrm{~mm}\) in diameter at a velocity of \(6 \mathrm{~m} \mathrm{~s}^{-1}\). Determine whether the flow will be laminar or turbulent assuming that the dynamic viscosity of water is \(1.30 \times 10^{-3} \mathrm{~kg} \mathrm{~m}^{-1} \mathrm{~s}^{-1}\) and its density \(1000 \mathrm{~kg} \mathrm{~m}^{-3}\). If oil of specific gravity \(0.9\) and dynamic viscosity \(9.6 \times\) \(10^{-2} \mathrm{~kg} \mathrm{~m}^{-1} \mathrm{~s}^{-1}\) is pumped through the same pipe, what type of flow will occur? [Turbulent, \(\operatorname{Re}=115\) 385; laminar, \(\operatorname{Re}=1406]\)

The air entering a compressor has a density of \(1.2 \mathrm{~kg} \mathrm{~m}^{-3}\) and a velocity of \(5 \mathrm{~m} \mathrm{~s}^{-1}\), the area of the intake being \(20 \mathrm{~cm}^{2}\). Calculate the mass flow rate. If air leaves the compressor through a \(25 \mathrm{~mm}\) diameter pipe with a velocity of \(4 \mathrm{~m} \mathrm{~s}^{-1}\), what will be its density? $$ \left[12 \times 10^{-3} \mathrm{~kg} \mathrm{~s}^{-1}, 6.11 \mathrm{~kg} \mathrm{~m}^{-3}\right] $$

An air duct is of rectangular cross-section \(300 \mathrm{~mm}\) wide by \(450 \mathrm{~mm}\) deep. Determine the mean velocity in the duct when the rate of flow is \(0.42 \mathrm{~m}^{3} \mathrm{~s}^{-1}\). If the duct tapers to a cross-section \(150 \mathrm{~mm}\) wide by \(400 \mathrm{~mm}\) deep, what will be the mean velocity in the reduced section assuming that the density remains unchanged? \(\left[3.11 \mathrm{~m} \mathrm{~s}^{-1}, 7.0 \mathrm{~m} \mathrm{~s}^{-1}\right]\)

The velocity along the centreline of a nozzle of length \(L\) is given by $$ u=2 t\left(1-0.5 \frac{x}{L}\right)^{2} $$ where \(u\) is the velocity in metres per second, \(t\) is the time in seconds from the commencement of flow, \(x\) is the distance from the inlet to the nozzle. Find the convective acceleration and the local acceleration when \(t=3 \mathrm{~s}, x=\frac{1}{2} L\) and \(L=0.8 \mathrm{~m}\). $$ \left[18.99 \mathrm{~m} \mathrm{~s}^{-2}, 1.125 \mathrm{~m} \mathrm{~s}^{-2}\right] $$

A sphere of diameter \(300 \mathrm{~mm}\) falls axially down a \(305 \mathrm{~mm}\) diameter vertical cylinder which is closed at its lower end and contains water. If the sphere falls at a speed of \(150 \mathrm{~mm} \mathrm{~s}^{-1}\), what is the mean velocity relative to the cylinder wall of the water in the gap surrounding the midsection of the sphere? \(\left[4.46 \mathrm{~m} \mathrm{~s}^{-1}\right]\)
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