Question

5.29 Consider the flow field given by V mine (a) the number of dimensions of the flow, (b) if it is a possible incompressible flow, and (c) the acceleration of a fluid particle at point (x, y, z) (2, 3, 4). хузі-4y+yk. Deter- 5.1 Which of the following sets of equations represent possible two- dimensional incompressible flow cases? (d) 11 = (2x+4y)st; u=3(x+y)yt

Answer 1

5.29

(a) No.of dimensions is 3.

(b) For incompressible flow

$\vec{\nabla}\cdot\vec{v}=0$

because

$\frac{\partial (xy^3)}{\partial x}+\frac{\partial (y^4)}{4\partial y}+\frac{\partial (xy)}{\partial z}=0$

hence the given field represents in-compressible flow.

5.1 The approach is same as above. Just check

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

I'll do only b and c

(b)

$\frac{\partial (2xy-x^2y)}{\partial x}+\frac{\partial (2xy-y^2+x^2)}{\partial y}=2y-2xy+2x-2y\neq0$

Not incompressible.

(c)

òy

Not incompressible.