Question

Under conditions of constant stress, Newtonian flow is observed for the Maxwell model Starting with the Maxwell’s equation for shear rate, prove that this is true.

Answer 1

• A Newtonian fluid is independent of both time and rate of shear.
• Newtonian model describes an idealized type of flow. In many systems, a material may exhibit Newtonian behaviour over a wide range of shear rate, but may, surprisingly, demonstrate non-Newtonian behaviour outside of that range.
• For Such a conditions equation using Maxwell Model are describe below.
• The Maxwell model is a two-element model consisting of a linear spring element and a linear viscous dashpot element connected in series, as shown in fig.

• The stress-strain relations of spring and dashpot respectively are:

  σ=Eɛ2

  σ=η x dɛ/dt

  Because both elements are connected in series, the total strain is:

  ɛ=ɛ1+ɛ2  

  By eliminating ɛ1 and ɛ2, the following constitutive equation for the Maxwell model is obtained:

  dɛ/dt=(1/E)(dσ/dt) +(σ/η)

  The response of the model to various stress or strain conditions can be obtained after applying integration together with appropriate initial conditions.

• applying a constant stress σ = σo at t = 0, the following response is obtained:

  ɛ(t) =(σ/E) +(σ/η)t

• From above equation it is prove that at constant stress for Newtonian flow is observed for the Maxwell Model starting with the Maxwell' equation for shear rate.