Faucet problem with Body Force.

  Ferrofluids are fluids that respond to magnetic fields. So, if you impose a magnetic field on a ferrofluid, you can make the body force be whatever magnitude and direction you like, depending on how the magnetic field is applied. Suppose that you have the “faucet” problem that we did (are doing?) for homework. Now, however, you have a body force that is gravity plus an additional downward force that changes with vertical distance. Suppose you have the following body force Fbody = ⇢g ↵zk Now, assume that you are given • the flow rate Q • the diameter d0 at “section 0” • the length L • the value of potential function . Use the combination of (a) the macroscopic mass balance on water, and (b) Bernoulli’s principle to predict the diameter, d1 and the velocity, v1 of the stream at the downstream (“section 1”) end with the information given? You should be able to generate an explicit formula; please do not put any numerical values in at this juncture. To be extra clear, here are the equations I would like you to start with the equations listed below. Conservation of mass Z V @⇢ @t + Z A ⇢v · n = 0 Bernoulli’s principle 1 2 ⇢v2 0 + (z0) + p0 = 1 2 ⇢v2 1 + (z1) + p1 Here, for Bernoulli’s principle, you need to use a more general potential energy. Instead of = ⇢gz, you will have to use (z) = Z z 0 ⇢g + ↵z k dz