Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 9: Problem 2

Which of the following functions could represent the velocity potential for the two-dimensional flow of an inviscid fluid? (a) \(x+5 y ;\) (b) \(3 x^{2}-4 y^{2} ;\) (c) \(\cos (x-y)\); (d) \(\ln (x+y)\) (e) \(\arctan (x / y) ;\) (f) \(\operatorname{arccosec}(x / y)\)

Short Answer

Expert verified
The functions (a) \(x + 5y\) and (c) \(\cos(x - y)\) could represent the velocity potential for the two-dimensional flow of an inviscid fluid.

Step by step solution

01

Understand velocity potential functions

The velocity potential function, \(\phi\), of an inviscid, incompressible fluid must satisfy the Laplace equation: \(abla^2 \phi = 0\). For a function to be a potential function, \(\phi \), it must meet this criterion.
02

Consider option (a)

Evaluate \(\phi = x + 5y\). Compute \(abla^2 \phi\): \(\frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} = 0 + 0 = 0\). Option (a) satisfies the Laplace equation.
03

Consider option (b)

Evaluate \(\phi = 3x^2 - 4y^2\). Compute \(abla^2 \phi\): \(\frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} = 6 - 8 = -2 \e 0\). Option (b) does not satisfy the Laplace equation.
04

Consider option (c)

Evaluate \(\phi = \cos(x - y)\). Compute \(abla^2 \phi\): \(\frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} = -\cos(x - y) + \cos(x - y) = 0\). Option (c) satisfies the Laplace equation.
05

Consider option (d)

Evaluate \(\phi = \ln(x + y)\). Compute \(abla^2 \phi\): \(\frac{d}{dx}\left(\frac{1}{x+y}\right) + \frac{d}{dy}\left(\frac{1}{x+y}\right) = -\frac{1}{(x + y)^2} - \frac{1}{(x + y)^2} = -\frac{2}{(x + y)^2} \e 0\). Option (d) does not satisfy the Laplace equation.
06

Consider option (e)

Evaluate \(\phi = \arctan\left(\frac{x}{y}\right)\). Compute the second derivatives, which are not straightforward but confirm that the Laplace equation is not satisfied. Option (e) does not satisfy the Laplace equation.
07

Consider option (f)

Evaluate \(\phi = \operatorname{arccosec}(\frac{x}{y})\). Check second derivatives and confirm that the Laplace equation is not satisfied. Option (f) does not satisfy the Laplace equation.
08

Conclusion

From the calculations, the functions that satisfy the Laplace equation are (a) \(x + 5y\) and (c) \(\cos(x - y)\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Key Concepts

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

Laplace Equation
The Laplace equation is central to understanding many phenomena in physics and engineering, especially in fluid dynamics and electromagnetism. For a function \(\phi\), the Laplace equation is written as: \( \abla^2 \phi = 0 \).
This equation ensures that the potential function \(\phi\) produces a field with no sources or sinks.
In our context, it is applied to identify valid velocity potential functions for fluid flows.
Only those potential functions that satisfy \(\abla^2 \phi = 0\) can be considered valid.
This happens because in an ideal, inviscid, incompressible fluid, there can't be any net flux of fluid out or into any small volume, hence the Laplace condition.
Inviscid Fluid
An inviscid fluid is a theoretical fluid that has no viscosity; in other words, it experiences no internal friction.
This simplifies the equations of motion, making it easier to model fluid flows.
In reality, all fluids have some viscosity, but for high-speed flows or flows with very low viscosity, it's often a good approximation.
The flow of an inviscid fluid can be described more straightforwardly using potential flow theory.
That’s why we assume inviscid characteristics when working with velocity potentials.
Two-Dimensional Flow
Two-dimensional flow is a type of fluid flow where the fluid properties and flow parameters are functions of only two spatial coordinates, typically x and y.
This means there are no changes in the z-direction.
Such flows simplify analysis and computation significantly.
Examples include flow in a thin film or a wide, shallow river.
When dealing with two-dimensional flow, the velocity potential function, \(\phi\), is also a function of just x and y.
This allows us to apply the Laplace equation in two dimensions, examining second partial derivatives with respect to x and y.
Velocity Potential
Velocity potential is a scalar function denoted by \(\phi\), from which the velocity field of a fluid can be derived.
For a potential function, the fluid velocity components can be obtained as: \(\frac{\partial \phi}{\partial x}\) for the x-component and \(\frac{\partial \phi}{\partial y}\) for the y-component.
The potential \(\phi\) must satisfy the Laplace equation if it’s an inviscid, incompressible fluid's velocity potential.
This ensures the flow derived from \(\phi\) is irrotational.
In the problem, \(\phi\) functions like (a) \(x + 5y\) and (c) \(\cos(x - y)\) satisfy the Laplace Equation, making them valid velocity potential functions for a two-dimensional inviscid flow.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

A hollow cylindrical drum has an internal diameter of \(600 \mathrm{~mm}\) and is full of oil of relative density \(0.9\). At the centre of the upper face is a small hole open to atmosphere. Concentric with the axis of the drum (which is vertical) is a set of paddles \(300 \mathrm{~mm}\) in diameter. Assuming that all the oil in the central \(300 \mathrm{~mm}\), diameter rotates as a forced vortex with the paddles and that the oil outside this diameter moves as a free vortex, calculate the additional force exerted by the oil on the top of the drum when the paddles are steadily rotated at 8 revolutions per second.

A kite may be regarded as equivalent to a rectangular aerofoil of \(900 \mathrm{~mm}\) chord and \(1.8 \mathrm{~m}\) span. When it faces a horizontal wind of \(13.5 \mathrm{~m} \cdot \mathrm{s}^{-1}\) at \(12^{\circ}\) to the horizontal the tension in the guide rope is \(102 \mathrm{~N}\) and the rope is at \(7^{\circ}\) to the vertical. Calculate the lift and drag coefficients, assuming an air density of \(1.23 \mathrm{~kg} \cdot \mathrm{m}^{-3}\)

A rectangular aerofoil of \(100 \mathrm{~mm}\) chord and \(750 \mathrm{~mm}\) span is tested in a wind-tunnel. When the air velocity is \(30 \mathrm{~m} \cdot \mathrm{s}^{-1}\) and the angle of attack \(7^{\circ}\) the lift and drag are \(32.8 \mathrm{~N}\) and \(1.68 \mathrm{~N}\) respectively. Assuming an air density of \(1.23 \mathrm{~kg} \cdot \mathrm{m}^{-3}\) and an elliptical distribution of lift, calculate the coefficients of lift, drag and vortex drag, the corresponding angle of attack for an aerofoil of the same profile but aspect ratio \(5.0\), and the lift and drag coefficients at this aspect ratio.

A set of paddles of radius \(R\) is rotated with angular velocity \(\omega\) about a vertical axis in a liquid having an unlimited free surface. Assuming that the paddles are close to the free surface and that the fluid at radii greater than \(R\) moves as a free vortex, determine the difference in elevation between the surface at infinity and that at the axis of rotation.

An open cylindrical vessel, having its axis vertical, is \(100 \mathrm{~mm}\) diameter and \(150 \mathrm{~mm}\) deep and is exactly two-thirds full of water. If the vessel is rotated about its axis, determine at what steady angular velocity the water would just reach the rim of the vessel.
See all solutions

Recommended explanations on Physics Textbooks

Rotational Dynamics

Read Explanation

Thermodynamics

Read Explanation

Force

Read Explanation

Work Energy and Power

Read Explanation

Engineering Physics

Read Explanation

Physics of Motion

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free