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Chapter 6: Problem 12

The velocity components of an incompressible, two-dimensional velocity field are given by the equations $$\begin{array}{l}u=y^{2}-x(1+x) \\\v=y(2 x+1)\end{array}$$ Show that the flow is irrotational and satisfies conservation of mass.

Short Answer

Expert verified
The given velocity field is indeed irrotational and satisfies the conservation of mass as both its curl and divergence are zero.

Step by step solution

01

Calculate the Curl of the Velocity Field

The curl of a two-dimensional velocity field \(F = (u, v)\) in Cartesian coordinates is defined as \(curl(F) = \frac{\partial v}{\partial x} - \frac{\partial u}{\partial y}\). For our given field, \(u=y^{2}-x(1+x)\) and \(v=y(2x+1)\), the curl is thus given by \(curl(F)= \frac{\partial v}{\partial x} - \frac{\partial u}{\partial y}= 2y - 2y = 0\). Thus, the velocity field is irrotational.
02

Calculate the Divergence of the Velocity Field

The divergence of a two-dimensional velocity field \(F = (u, v)\) in Cartesian coordinates is defined as \(div(F) = \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y}\). For our given field, \(u=y^{2}-x(1+x)\) and \(v=y(2x+1)\), the divergence is thus given by \(div(F) = \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y}= -1 - x + 2x = 0\). Thus, the velocity field obeys the principle of conservation of mass.

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Most popular questions from this chapter

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