Question

The $y$ component of velocity in a steady incompressible flow field in the $x y$ plane is \[v=\frac{2 x y}{\left(x^{2}+y^{2}\right)^{2}}\] Show that the simplest expression for the $x$ component of velocity is \[u=\frac{1}{\left(x^{2}+y^{2}\right)}-\frac{2 y^{2}}{\left(x^{2}+y^{2}\right)^{2}}\]

Name: Problem 8, Chapter 5: Introduction to Fluid Mechanics
Uploaded: 2020-10-11T12:07:45-08:00
Duration: 3 min 58 s
Channel: Khoobchandra Agrawal
Description: The y component of velocity in a steady incompressible flow field in the x y plane is v=(2 x y)/((x^2+y^2)^2) Show that the simplest expression for the x component of velocity is u=(1)/((x^2+y^2))-(2 y^2)/((x^2+y^2)^2)

   The $y$ component of velocity in a steady incompressible flow field in the $x y$ plane is
\[v=\frac{2 x y}{\left(x^{2}+y^{2}\right)^{2}}\]
Show that the simplest expression for the $x$ component of velocity is
\[u=\frac{1}{\left(x^{2}+y^{2}\right)}-\frac{2 y^{2}}{\left(x^{2}+y^{2}\right)^{2}}\]

Introduction to Fluid Mechanics

Philip J. Pritchard,… 8th Edition

Chapter 5, Problem 8

Instant Answer

Solved by Expert Khoobchandra Agrawal

10/11/2020

Step 1

This means that the divergence of the velocity field is zero. Mathematically, this is expressed as: \[\nabla \cdot \mathbf{v} = 0\] where $\mathbf{v}$ is the velocity vector. In two dimensions, this equation becomes: \[\frac{\partial u}{\partial x} + Show more…

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The $y$ component of velocity in a steady incompressible flow field in the $x y$ plane is \[v=\frac{2 x y}{\left(x^{2}+y^{2}\right)^{2}}\] Show that the simplest expression for the $x$ component of velocity is \[u=\frac{1}{\left(x^{2}+y^{2}\right)}-\frac{2 y^{2}}{\left(x^{2}+y^{2}\right)^{2}}\]

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Key Concepts

Incompressible Flow and Continuity Equation

In a steady incompressible flow, the density remains constant, which leads to the continuity equation in two dimensions: ?u/?x + ?v/?y = 0. This equation ensures conservation of mass and imposes a relationship between the velocity components u and v, so that if one component is known, the other can be determined through differentiation and integration.

Application of Differentiation and Integration to Satisfy the Continuity Equation

When one component of the velocity field is provided—as in this case with v—the continuity equation can be used to find the other component u by differentiating v with respect to y and then integrating with respect to x (or vice versa). This process is essential to ensure that the resulting velocity components combine to produce a divergence-free flow, conforming to the condition of incompressibility.

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