need question 3 and 4 completed

On frame indifference, consitutive equations: Slattery_on_Consist_Funct_Fall_16 (Sections §1.2, 2.3.1, 2.3.2, 5.4); Notes_Frame_Indifference_10 (see course website Supplemental Reading)

• On balance laws, frame indifference, constitutive equations, second law: Muller_ Thermodynamics_1987_on viscous fluids and Liu method (sections §1.1,1.2,1.3,5.4.3) (see course website Supplemental Reading)

• On scaling: Deen (2nd Edition) Chapter 3 (section §3.3) • On similarity solutions, perturbation analysis of conduction dominated transport problems: Deen (2nd Edition) Chapter 4

• On FFT method for conduction dominated transport problems: Deen (2nd Edition) Chapter 5 (sections §5.1-6)

2. Proof of an Identity: Prove using Cartesian tensor notation the identity shown below which was used to derive the internal energy balance for a binary mixture

1

2

2∑ i=1

ωiv 2 i =

1

2 v2 +

1

2

2∑ i=1

ωiu 2 i

.

3. Reiner-Rivlin model: Many fluids exhibit significant deviations from Newton’s law of viscosity, even in very simple flows. An early attempt at describing elastic effects exhibited by (nearly incompressible) polymeric fluids in simple shear flow was the Reiner-Rivlin model

τ = −ηγ̇ − ηcγ̇ · γ̇ γ̇ = Ov + Ov†

where η and ηc are the shear and “cross” viscosity, respectively, taken to be constants here.

(a) What are the “fundamental” units of η and ηc (i.e. the units expressed in mass, length and time) ?

(b) Show that the Reiner-Rivlin model is frame indifferent (note you may use any of the results appearing in the Supplemental Reading online, without derivation).

(c) One of the simplest homogeneous flows that can be implemented in the laboratory for nearly any fluid is steady simple shear, where

vx = γ̇yx y

vy = 0

vz = 0

1

in a Cartesian system. Here, γ̇yx is the (constant) shear rate. For the Reiner Rivlin model, find the apparent shear viscosity, η(γ̇) , and the first and second normal stress coeffi cients, Ψ1(γ̇) and Ψ2(γ̇) respectively, defined by

τ yx = −η(γ̇)γ̇yx τxx − τ yy = −Ψ1(γ̇)γ̇2yx τ yy − τ zz = −Ψ2(γ̇)γ̇2yx

γ̇ = ∣∣γ̇yx∣∣

in a steady simple shear flow.

(d) Why do you think the coeffi cient ηc is termed the “cross” viscosity ?

4. Constitutive equation development for motionless (rigid) media: In class we developed the constitutive laws for a simple “viscous” fluid using determinism, local action, frame indifference and the second law. Repeat the entire development step by step for motionless, rigid materials, which correspond to simple viscous fluids, but with the constraint v = 0 imposed. In this case, only two “primary” fields ρ(r, t) and T (r, t) are of interest.

(a) What balance equations govern these materials? (Hint: simplify the (Eulerian) balances developed in class for the materials in question).

(b) Add to the above, a definition of temperature, and a statement of the second law, and carry out a degrees of freedom analysis to determine what constitutive equations are needed.

(c) Write down weak-gradient representations of the required constitutive equations and find the restrictions on these needed to ensure the second law is satisfied. For the latter part of the question, use the method of Liu and insist that equilibrium corresponds to a minimum in th entropy production rate.

2

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