GATE 2022 ME (ME1) – Question 63
A steady two-dimensional flow field is specified by the stream function $\psi = kx^3y$, where $x$ and $y$ are in meter and the constant $k = 1$ m$^{-2}$s$^{-1}$. The magnitude of acceleration at a point $(x, y) = (1$ m, $1$ m$)$ is ________ m/s$^2$ (round off to 2 decimal places).
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Correct answer: 4.22 to 4.26
Explanation
The velocity components are $u = \frac{\partial\psi}{\partial y} = x^3$ and $v = -\frac{\partial\psi}{\partial x} = -3x^2y$. The flow is steady, so $a_x = u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y} = x^3 \times 3x^2 + 0 = 3x^5$ and $a_y = u\frac{\partial v}{\partial x} + v\frac{\partial v}{\partial y} = x^3(-6xy) + (-3x^2y)(-3x^2) = 3x^4y$. At (1, 1): $a_x = 3$ and $a_y = 3$, so $|a| = \sqrt{18} = 4.24$ m/s².