A fluid of uniform density carries a pressure field

PHYSICS · Multi-part · Fermi University Physics · Fluids & Continuum Physics · Kinematics, conservation laws and ideal flow · Kinematics, conservation laws and ideal flow

Fluids and Continuum Physics, Ch Euler's equation and Bernoulli's theorem

Problem

A fluid of uniform density ρ\rho carries a pressure field p(x,y,z)p(x,y,z), with zz measured vertically upward and g=−gz^\mathbf{g} = -g\hat{\mathbf{z}}. ((a)) Consider a small rectangular box of fluid with edges δx\delta x, δy\delta y, δz\delta z parallel to the axes and centred at (x,y,z)(x,y,z). By comparing the pressure forces on the two faces whose normals lie along xx, and expanding pp to first order, show that the net pressure force on the box is −∇p δx δy δz-\nabla p\,\delta x\,\delta y\,\delta z, so that the pressure force per unit volume is −∇p-\nabla p. State what this gives when the fluid is at rest and gravity is the only other force. ((b)) Starting from Euler's equation ρ Du/Dt=−∇p+ρg\rho\,D\mathbf{u}/Dt = -\nabla p + \rho\mathbf{g}, take the component along a streamline, using arc length ss measured along it, and integrate to obtain p+12ρ∣u∣2+ρgz=constant along the streamline.p + \tfrac12\rho|\mathbf{u}|^{2} + \rho g z = \text{constant along the streamline}. Say at which step each of the four conditions (steady, inviscid, incompressible with uniform ρ\rho, along one streamline) is used. You may use that for a unit tangent s^=u/∣u∣\hat{\mathbf{s}} = \mathbf{u}/|\mathbf{u}| and any scalar field ff, s^⋅∇f=df/ds\hat{\mathbf{s}}\cdot\nabla f = \mathrm{d}f/\mathrm{d}s along the streamline. ((c)) Apply the result to a wide open tank of water punctured 2.5 m2.5\ \mathrm{m} below the free surface, with the surface and the emerging jet both at atmospheric pressure and the surface speed negligible. Take g=9.8 m s−2g = 9.8\ \mathrm{m\,s^{-2}} and give the efflux speed.

Browse the problem library