Air flows through a venture and into atmosphere. Air density is $$\rho $$; atmospheric pressure is $$'\,{P_a}';$$ throat diameter is exit diameter is $$'{D_t}';$$ and exit velocity is $$U.$$ the throat is connected to a cylinder containing a frictionless piston attached to a spring. The spring constant is $$'k'.$$ the bottom surface of the piston is exposed to atmosphere. Due to the flow, the piston moves by distance $$x.$$ assuming incompressible frictionless flow, $$x$$ is

A syringe with a frictionless plunger contains water and has at its end a $$100$$ $$mm$$ long needle of $$1$$ $$mm$$ diameter. The internal diameter of the syringe is $$10$$ $$mm.$$ Water density is $$1000\,\,kg/{m^3}.$$ . The plunger is pushed in at $$10$$ $$mm/s$$ and the water comes out as a jet.

Neglect losses in the cylinder and assume fully developed laminar viscous flow throughout the needle; the Darcy friction factor is $${64/R_e}$$. Where $${R_e}$$ is the Reynolds number. Given that the viscosity of water is $$1.0 \times {10^{ - 3}}\,\,kg/m\,\,\,s,$$ the force $$F$$ in newtons required on the plunger is

A syringe with a frictionless plunger contains water and has at its end a $$100$$ $$mm$$ long needle of $$1$$ $$mm$$ diameter. The internal diameter of the syringe is $$10$$ $$mm.$$ Water density is $$1000\,\,kg/{m^3}.$$ . The plunger is pushed in at $$10$$ $$mm/s$$ and the water comes out as a jet.

Assuming ideal flow, the force $$F$$ in Newton’s required on the plunger to push out the water is

Heat is being transferred by convection from water at $${48^ \circ }C$$ to a glass plate whose surface that is exposed to the water is at $${40^0}C.$$ The thermal conductivity of water is $$0.6$$ $$W/mK$$ and the thermal conductivity of glass is $$1.2$$ $$W/mK.$$ The spatial gradient of temp in the water at the water glass interface is
$$dT/dy = 1 \times {10^4}\,\,K/m...$$

The value of the temperature gradient in the glass at the water-glass interface in $$K/m$$ is