Discussion

The velocity field of a two-dimensional, incompressible flow is given by

\(\vec{V} = 2\sinh x\,\hat{i} + v(x, y)\,\hat{j}\)

where \(\hat{i}\) and \(\hat{j}\) denote the unit vectors in \(x\) and \(y\) directions, respectively. If \(v(x, 0) = \cosh x\), then \(v(0, -1)\) is

Answer: C

The value of \(v(0, -1)\) is 3.

For two-dimensional incompressible flow, continuity requires \(\partial u/\partial x + \partial v/\partial y = 0\). With \(u = 2\sinh x\):

\[ \begin{aligned} \frac{\partial v}{\partial y} &= -2\cosh x \\ v &= -2y\cosh x + g(x) \end{aligned} \]

The condition \(v(x, 0) = \cosh x\) gives \(g(x) = \cosh x\), so \(v = (1 - 2y)\cosh x\). At \(x = 0\), \(y = -1\):

\[ v(0, -1) = (1 + 2)\cosh 0 = 3 \]

1 is the given value at y = 0, which ignores how v changes with y; 2 and 4 do not follow from the continuity equation.

No comment is present. Be the first to comment.

Loading…