The steady velocity field in an inviscid fluid of density 1.5 is given to be $\vec{V}=(y^2-x^2)\hat{i}+(2xy)\hat{j} $. Neglecting body forces, the pressure gradient at (x =1, y = 1) is ______.

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.

Consider steady, one-dimensional compressible flow of a gas in a pipe of diameter 1 m. At one location in the pipe, the density and velocity are 1 kg/m³ and 100 m/s, respectively. At a downstream location in the pipe, the velocity is 170 m/s. If the pressure drop between these two locations is 10 kPa, the force exerted by the gas on the pipe between these two locations is _______ N.

A $$60$$ $$mm$$ $$-$$ diameter water jet strikes a plate containing a hole of $$40mm$$ diameter as shown in the figure. Part of the jet passes through the hole horizontally, and the remaining is deflected vertically. The density of water is $$1000$$ $$kg/{m^3}$$ . If velocities are as indicated in the figure, the magnitude of horizontal force (in $$N$$) required to hold the plate is _______________