- #1

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## Main Question or Discussion Point

In Batchelor's text (2000) on page 76, the stream function is defined as

[tex]

\psi - \psi_0 = \int\left(u dy - v dx\right)

[/tex]

where [itex] \psi_0 [/itex] is a constant

Now I begin with a simple function for [itex]u[/itex] where

[tex]

u = x^3

[/tex]

From mass conservation,

[tex]

\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0

[/tex]

[tex]

3x^2 + \frac{\partial v}{\partial y} = 0

[/tex]

[tex]

v = -3x^2y

[/tex]

Plugging this into the equation for the stream function

[tex]

\psi - \psi_0 = \int\left(u dy - v dx\right)

[/tex]

[tex]

\psi - \psi_0 = \int\left(x^3 dy + 3x^2y dx\right)

[/tex]

[tex]

\psi - \psi_0 = \int\left(x^3 dy\right) + \int\left(3x^2y dx\right)

[/tex]

[tex]

\psi - \psi_0 = x^3y + x^3y + C

[/tex]

[tex]

\psi - \psi_0 = 2x^3y + C

[/tex]

Now using the equations for [itex] u [/itex] and [itex] v [/itex],

[tex]

u = \frac{\partial \psi}{\partial y}

[/tex]

[tex]

u = \frac{\partial (2x^3y + C - \psi_0)}{\partial y}

[/tex]

[tex]

u = 2x^3

[/tex]

[tex]

v = -\frac{\partial \psi}{\partial x}

[/tex]

[tex]

v = -\frac{\partial (2x^3y + C - \psi_0)}{\partial x}

[/tex]

[tex]

v = -6x^2y

[/tex]

I seems like the initial [itex]v=-3x^2y[/itex] and [itex]u=x^3[/itex] are off from the recalculated [itex]v = -6x^2y[/itex] and [itex]u = 2x^3[/itex] by a factor of two. Am I doing something wrong? Thanks.

[tex]

\psi - \psi_0 = \int\left(u dy - v dx\right)

[/tex]

where [itex] \psi_0 [/itex] is a constant

Now I begin with a simple function for [itex]u[/itex] where

[tex]

u = x^3

[/tex]

From mass conservation,

[tex]

\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0

[/tex]

[tex]

3x^2 + \frac{\partial v}{\partial y} = 0

[/tex]

[tex]

v = -3x^2y

[/tex]

Plugging this into the equation for the stream function

[tex]

\psi - \psi_0 = \int\left(u dy - v dx\right)

[/tex]

[tex]

\psi - \psi_0 = \int\left(x^3 dy + 3x^2y dx\right)

[/tex]

[tex]

\psi - \psi_0 = \int\left(x^3 dy\right) + \int\left(3x^2y dx\right)

[/tex]

[tex]

\psi - \psi_0 = x^3y + x^3y + C

[/tex]

[tex]

\psi - \psi_0 = 2x^3y + C

[/tex]

Now using the equations for [itex] u [/itex] and [itex] v [/itex],

[tex]

u = \frac{\partial \psi}{\partial y}

[/tex]

[tex]

u = \frac{\partial (2x^3y + C - \psi_0)}{\partial y}

[/tex]

[tex]

u = 2x^3

[/tex]

[tex]

v = -\frac{\partial \psi}{\partial x}

[/tex]

[tex]

v = -\frac{\partial (2x^3y + C - \psi_0)}{\partial x}

[/tex]

[tex]

v = -6x^2y

[/tex]

I seems like the initial [itex]v=-3x^2y[/itex] and [itex]u=x^3[/itex] are off from the recalculated [itex]v = -6x^2y[/itex] and [itex]u = 2x^3[/itex] by a factor of two. Am I doing something wrong? Thanks.