- #1

rafisondi

- 11

- 1

I am new here I hope it is okay to ask this question here.

As part of a project I am trying to examine the effect of a pulsating laminar duct flow on the heat transfer behavior for constant heat flux at the wall for a Newtonian incompressible fluid.

As such I am assuming an oscillating pressure gradient dp/dz = A*exp( i*ω*t), A being a real constant denoting the pressure amplitude, ω being the radial velocity and time t > 0. Using conservation of mass and the Navier Stokes equations, I get a quasi steady state flow profile that is only dependent on time and the radius w(r,t) in axial direction x which only depends on time and radius r. I attached a plot graph of my velocity profile vs radius (Radius R of duct = 0.01m). The different colored graphes are ploted for different times t_i.

I essentially only evaluated the profile 1dimensional over the radius since we are hydrodynamically developed and can neglect the azimutal direction bc of symmetry arguments.

My next step would have been to evaluate the temperature profile T(r,x,t) using the energy conservation assuming constant heat flux q over the duct wall area. Assuming no dissipation of energy and neglecting the thermal diffusion in axial direction (∂^2T/∂x^2) to be small compared to radial diffusion I get

ρ*c_p (∂T/∂t + u * ∂T/∂x ) = k (1/r *∂/∂r(r*∂T/∂r ) )

My biggest issue right now lies in the red term ∂T/∂x. I don't know how to "deal" with it... I can't really neglect it since that would knock out my fluid velocity. I somehow need to add some kind of "value" or expression to it, such that I can later numerically evaluate this PDE. I somehow feel like this should be actually very simple to solve but I can't seem can't seem to wrap my head around this problem.

Hopefully I framed my problem clearly. I am still a student and this is my first time working on such a problem so please excuse me possibly not being very exact with my formulations.

Anyways thanks in advance and have a nice day.