ME521 HW 3 October 16, 2019

1. The generlized Leibniz theorem states that (see, Kundu and Cohen)

D

Dt

∫V (t)

FdV =

∫V (t)

∂F

∂tdV +

∫A(t)

Fu · ndA, (1)

where F (x, t) is a tensor of any rank, V (t) is the time raying volume over whic theintegration is carried out and A(t) is the surface area of the volume. Using thistheorem and conservation of mass, show that

D

Dt

∫V (t)

ρfdV =

∫V (t)

ρDf

DtdV, (2)

where f(x, t) is a tensor of any rank. Note that this is the theorem used in the lecturein deriving the energy equation.

2. This problem is just for practice. Make sure that you understand every step ofderivation presented in the lecture.From our discussion in class, we have the following relation

TdS = dϵ− pdρ

ρ2(3)

and neglecting the volumetric heat addition, deduce the following equation for thespecific entropy S

ρDS

Dt= 2µ

eijeijT

+ λ∆2

T+ k

∇T∇T

T 2+∇ · (k∇ lnT ). (4)

(a) Observe thatDS

Dt= 0, (5)

when µ = k = λ = 0.(b) Show that the material properties µ, k and λ must be positive to satisfy the

second law of thermodynamics.

3. Constitutive Equations:1 In this problem, we consider the constitutive equation fora material with microstructure, such as liquid crystals which are now used in many

1This question might be little challenging. Give a try and compare your results with mine later.

different everyday components (e.g., digital displays, computer screens, etc.). Con-sider these materials to consists of a Newtonian solvent that has suspended in it smallrigid rod-shaped particles as illustrated in Fig. 1. The particles are orientable andwe denote the average orientation vector N (with N · N = 1), which is also calledthe director field (see Fig. 1); in general the director field is unknown and has to besolved simultaneously with the velocity field, which is a complicated problem! Herewe wish only to think about the constitutive equation relating stress to the rate ofstrain. We wish to treat this new “fluid” as incompressible continuum (velocity fieldu) since the suspended particles are very small, but we also wish to account in theconstitutive equation for the orientability of the microstructure. Hence, we expectN to enter the form of the stress tensor, τij, where we recall that the total stress isσij = −pδij + τij.Consider the form taken by the stress versus the rate-of-strain relation in the form

τ = A : ∇u or τij = Aijkl∂uk

∂xl

, (6)

where A depends on N as well, i.e., A(N). Assume that there are no applied externalfields so that τ is symmetric. Also, assume that Eq. (6) only depends on the rate ofstrain tensor e, i.e., τij = Aijklekl where ekl =

12

(∂uk

∂xl+ ∂ul

∂xk

).

(a) Determine the most general form of A(N) with the property that is invariant(i.e., unchanged when N → −N.Hint: The components of Aijkl involve of the identity tensor I, the orientationtensor N, and their products, which satisfy the symmetries required of A.

(b) What is the form of the corresponding constitutive equation for the stress tensorand how many material constants are there?

(c) What is the rate of energy dissipation in the material?

4. Poisson equation for the pressure field: Consider a Newtonian fluid with constantproperties, i.e., ρ = const, µ = const. By taking divergence of the momentum equa-tions and using the incompressibility condition, derive an equation of the pressurefield in the form

∇2p = f(e,Ω), (7)

where e and Ω are the rate-of-strain and rate-of-rotation tensors, respectively. Youshould determine the functional form of the right-hand side.

5. Some manipulations of the Navier-Stokes Equations: In some problems, notablyacoustics, there are small amplitude motions which are accompanied by small den-sity and pressure variations. It is useful to develop insighht for the mathematicaldescription of these problems by linearizing the governing equations, as describedhere.

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Figure 1: Schematic illustration of microstructure of a nematic fluid.

(a) Let ρ0 and p0 denote, respectively, the density and pressure in a static fluid.The account for small amplitude motions, let ρ = ρ0 + ρ, p = p0 + p with|ρ| ≪ ρ0 and |p| ≪ p0, and denote the small velocity of the fluid motion by u.Formally the linearization idea simply amounts to neglecting all quadratic (andhigher order) terms involving products of small quantities. Linearize the Navier-Stokes equations (accounting for both coefficients of viscosity) and continuityequations, allowing for compressibility, and assuming µ, λ and ρ0 are constants,to arrive at

ρ0∂u∂t

= −∇p+ µ∇2u +(λ+

µ

3

)∇ (∇ · u) , ∂ρ

∂t= −ρ0∇ · u. (8)

(b) Combine with the isentropic approximation, c2 =(

∂p∂ρ

)s, where c is the speed

of sound, to obtain a single wave-like equation for the density variations:

∂2ρ

∂t2− 1

ρ0

(λ+

4µ

3

)∇2∂ρ

∂t= c2∇2ρ. (9)

(c) In order to understand the sense in which the viscous terms produce ”damping”,consider a wave-like solution in one dimension. Let ρ(x, t) = Aeiωteikx and findthe dispersion relation, i.e., k = k(ω). Give both the real and imaginary parts.Note that if k has an imaginary part, Im(k) = ki > 0, then this corresponds tosolutions which are exponentially damped as e−kix.

(d) Finally, suppose that undamped (µ = λ = 0) acoustic waves have frequency ωand wavelength λa where λaω = c. Using Eq. (9), under what conditions doyou expect viscous effects to be small during the wave propagation? Make anorder of magnitude estimate.