Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 6: Problem 26

The stream function for a given two-dimensional flow field is $$\psi=5 x^{2} y-(5 / 3) y^{3}$$ Determine the corresponding velocity potential.

Short Answer

Expert verified
The velocity potential is given by \(φ=5xy^{2} - (5/3)x^{3}\)

Step by step solution

01

Compute the component velocities

The x-component of the velocity \(u\) is given by \(-\frac{∂ψ}{∂y} = -5x^{2} + 5y^{2}\) and the y-component of the velocity \(v\) is given by \(\frac{∂ψ}{∂x} = 10xy\)
02

Integrate the x-component

Now integrate \(u\) with respect to \(x\) to find a part of the velocity potential. \(∫u dx =∫(-5x^{2} + 5y^{2})dx=- (5/3)x^{3} + 5yx^{2} + G(y)\). \(G(y)\) is an arbitrary function of \(y\) that arises due to the integration.
03

Integrate the y-component

Integrate \(v\) with respect to \(y\) to find another part of the velocity potential. \(∫v dy = ∫10xy dy = 5x y^{2} + H(x)\), where \(H(x)\) is an arbitrary function of \(x\).
04

Equate the results from step 2 and step 3

The velocity potential arises from both \(x\) and \(y\) components of the velocity. Hence, equate the results from steps 2 and 3. \(-(5/3)x^{3} + 5yx^{2} + G(y) = 5xy^{2} + H(x)\). Now find an appropriate \(G(y)\) and \(H(x)\) such that these two expressions are identical for all \(x\) and \(y\). In here, we can see that \(H(x)=-(5/3)x^{3}\) and \(G(y) = 0\).
05

Combine results

The final velocity potential φ is a combination of the results of these functions: \(φ=5xy^{2} - (5/3)x^{3}\)

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!

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

The flow in the impeller of a centrifugal pump is modeled by the superposition of a source and a free vortex. The impeller has an outer diameter of \(0.5 \mathrm{m}\) and an inner diameter of \(0.3 \mathrm{m}\). At the outlet from the impeller, the flowing water has the following velocity componeats, relative to the impeller: radial component \(2 \mathrm{m} / \mathrm{s}\) and tangential comporent \(7 \mathrm{m} / \mathrm{s}\). (a) Find the strength of the source and the vortex required to model this flow. (b) Assume that the impeller blades are shaped like the streamlines and plot an impeller blade shape. (c) Find the radial and tangential components of velocity at the inlet to the impeller.

According to Eq. 6.134 , the \(x\) -velocity in fully developed laminar flow between parallel plates is given by $$u=\frac{1}{2 \mu}\left(\frac{\partial p}{\partial x}\right)\left(y^{2}-h^{2}\right)$$ The \(y\) -velocity is \(v=0\). Determine the volumetric strain rate, the vorticity, and the rate of angular deformation. What is the shear stress at the plate surface?

The velocity in a certain flow field is given by the equation $$\mathbf{v}=x \hat{\mathbf{i}}+x^{2} z \hat{\mathbf{j}}+y z \hat{\mathbf{k}}$$ Determine the expressions for the three rectangular components of acceleration.

Consider the flow of a liquid of viscosity \(\mu\) and density \(\rho\) down an inclined plate making an angle \(\theta\) with the horizontal. The film thickness is \(t\) and is constant. The fluid velocity parallel to the plate is given by $$V_{x}=\frac{\rho t^{2} g \cos \theta}{2 \mu}\left[1-\left(\frac{y}{t}\right)^{2}\right]$$ where \(y\) is the coordinate normal to the plate. Calculate \(\Phi\) and \(\Psi\) for this flow and show that neither satisfies Laplace's equation. Why not?

For a certain incompressible, two-dimensional flow field the velocity component in the \(y\) direction is given by the equation $$v=3 x y+x^{2} y$$ Determine the velocity component in the \(x\) direction so that the volumetric dilatation rate is zero.
See all solutions

Recommended explanations on Physics Textbooks

Fluids

Read Explanation

Conservation of Energy and Momentum

Read Explanation

Further Mechanics and Thermal Physics

Read Explanation

Space Physics

Read Explanation

Mechanics and Materials

Read Explanation

Kinematics Physics

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