Consider a unidirectional fluid flow with the velocity field given by

V(𝑥, 𝑦, 𝑧, 𝑡) = 𝑢(𝑥, 𝑡) 𝑖̂

where 𝑢(0,𝑡) = 1. If the spatially homogeneous density field varies with time 𝑡 as

𝜌(𝑡) = 1 + 0.2𝑒^−𝑡

the value of 𝑢(2, 1) is ______________. (Rounded off to two decimal places) Assume all quantities to be dimensionless.

The figure shows two fluids held by a hinged gate. The atmospheric pressure is P_a = 100 kPa. The moment per unit width about the base of the hinge is ____________ kNm/m. (Rounded off to one decimal place)

Take the acceleration due to gravity to be g = 9.8 m/s².

An explosion at time t = 0 releases energy 𝐸 at the origin in a space filled with a gas of density ρ. Subsequently, a hemispherical blast wave propagates radially outwards as shown in the figure.

Let R denote the radius of the front of the hemispherical blast wave. The radius R follows the relationship 𝑅 = 𝑘 𝑡^𝑎 𝐸 ^𝑏 𝜌^𝑐, where k is a dimensionless constant. The value of exponent a is ___________.

(Rounded off to one decimal place)

Consider incompressible laminar flow of a constant property Newtonian fluid in an isothermal circular tube. The flow is steady with fully-developed temperature and velocity profiles. The Nusselt number for this flow depends on