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Question

# In a steady, two-dimensional flow field in the xy - plane, the x-component of velocity is $u=a x+b y+c x^{2}$ where a, b, and c are constants with appropriate dimensions. Generate a general expression for velocity component v such that the flow field is incompressible.

Solution

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Recall that the flow is incompressible if its volumetric strain rate is 0:

$\begin{equation} \dfrac{1}{\mathcal{V}} \dfrac{d \mathcal {V}}{dt} = \epsilon_{xx} + \epsilon_{yy} = 0 \end{equation}$

since the flow is 2-dimensional.

Calculate from the given $u$ function:

$\begin{equation} u(x, y) = ax + by + cx^2 \end{equation}$

the appropriate linear strain rate:

$\begin{equation} \epsilon_{xx} = \dfrac{\partial u}{\partial x} = 2cx + a \end{equation}$

From (1) now we conclude:

$\begin{equation} \epsilon_{yy} = \dfrac{\partial v}{\partial y} = -\epsilon_{xx} = - 2cx - a \end{equation}$

Now integrate (4) with respect to $y$:

$\begin{equation} v(x, y) = - (2cx+a)y + f(x) \end{equation}$

where $f(x)$ is $\textit{arbitrary function of $x$}$, which arrives because the derivative in (4) was only $\textit{partial}$.

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