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

V(𝑥, 𝑦) = 𝑘(𝑥𝑖̂ − 𝑦𝑗̂)

where 𝑘 = 2 s^-1. The coordinates 𝑥 and 𝑦 are in meters. Assume gravitational effects to be negligible.

If the density of the fluid is 1000 kg/m³ and the pressure at the origin is 100 kPa, the pressure at the location (2 m, 2 m) is _____________ kPa.

(Answer in integer)

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.

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)

A tube of uniform diameter D is immersed in a steady flowing inviscid liquid stream of velocity V, as shown in the figure. Gravitational acceleration is represented by g. The volume flow rate through the tube is ______.