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 _______.

In the following two-dimensional momentum equation for natural convection over a surface immersed in a quiescent fluid at temperature T_∞ (g is the gravitational acceleration, β is the volumetric thermal expansion coefficient, ν is the kinematic viscosity, u and v are the velocities in x and y directions, respectively, and T is the temperature)

$\rm u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} = g β (T - T_∞) + \nu \frac{\partial^2 u}{\partial y^2} $

the term gβ(T - T_∞) represent

The figure shows a purely convergent nozzle with a steady, inviscid compressible flow of an ideal gas with constant thermophysical properties operating under choked condition. The exit plane shown in the figure is located within the nozzle. If the inlet pressure (P0) is increased while keeping the back pressure (Pback) unchanged, which of the following statements is/are true?

A solid spherical bead of lead (uniform density = 11000 kg/m³) of diameter d = 0.1 mm sinks with a constant velocity V in a large stagnant pool of a liquid (dynamic viscosity = 1.1 × 10^-3 kg∙m^-1∙s^-1). The coefficient of drag is given by $\rm C_D = \frac{24}{Re} $, where the Reynolds number (Re) is defined on the basis of the diameter of the bead. The drag force acting on the bead is expressed as $\rm D = (C_D) (0.5 \rho V^2) \left( \frac{\pi d^2}{4} \right), $ where ρ is the density of the liquid. Neglect the buoyancy force. Using g = 10 m/s², the velocity V is __________ m/s.