A tiny temperature probe is fully immersed in a flowing fluid and is moving with zero relative velocity with respect to the fluid. The velocity field in the fluid is $\vec V = (2x) \hat i + (y + 3t) \hat j,$ and the temperature field in the fluid is T = 2x² + xy + 4t, where x and y are the spatial coordinates, and t is the time. The time rate of change of temperature recorded by the probe at (x = 1, y = 1, t = 1) is _______.

Consider the two-dimensional velocity field given by
$$\overrightarrow V = \left( {5 + {a_1}x + {b_1}y} \right)\widehat i + \left( {4 + {a_2}x + {b_2}y} \right)\widehat j,$$
where $${a_1},\,\,{b_1},\,\,{a_2}$$ and $${b^2}$$ are constants. Which one of the following conditions needs to be satisfied for the flow to be incompressible?

The volume tric flow rate (per unit depth) between two streamlines having stream functions $${\psi _1}$$ & $${\psi _2}$$ is

A channel of width $$450$$ $$mm$$ branches into two sub-channels having width $$300$$ $$mm$$ and $$200$$ $$mm$$ as shown in figure. If the volumetric flow rate (taking unit depth) of an incompressible flow through the main channel is $$0.9$$ $$3$$ $$m/s,$$ and the velocity in the sub-channel of width $$200$$ $$mm$$ is $$3$$ $$m/s,$$ the velocity in the sub-channel of width $$300$$ $$mm$$ is _____________ $$m/s$$.

Assume both inlet and outlet to be at the same elevation.