For small strain, small deformation physics the CBL and the constitutive theories presented in reference [3] modified for dissipation due to rate of strain (CCM) and rate of symmetric part of rotation gradient tensor (micropolar NCCT) are given by

ρ 0 D 2 u k D t 2 − ρ 0 F k b − ∂ σ j k ∂ x j = 0 (1)

I Θ 0 ρ 0 D 2 ( i Θ k ) D t 2 − ϵ i j k σ i j − ∂ m j k ∂ x j = 0 (2)

ϵ j k l ( I Θ 0 ρ 0 v j ( i ω k ) − m j k ) = 0 (3)

ρ 0 D e D t + ∂ q i ∂ x i − tr ( [ σ s ] [ ε ˙ ] ) − tr ( [ m s ] [ J ˙ s i Θ ] ) − tr ( [ m a ] [ J ˙ a i Θ ] ) = 0 (4)

Additivedecompositionof   σ :     σ = σ s + σ a ;     σ s = σ s d + σ s e (5)

σ s d = σ 0 | Ω _ I + 2 μ ε + λ ( tr ( ε ) ) I + 2 η ε ˙ + κ tr ( ε ˙ ) I (6)

m s = 2 μ m ( J s i Θ ) + 2 η m ( J ˙ s i Θ ) (7)

q = − κ g (8)

In the mathematical model we have a total of 25 equations BLM(3), BAM(3), BMM(3), FLT(1), constitutive theories for σ s ( 6 ) , m s ( 6 ) , q ( 3 ) in a total of 25 dependent variables u ( 3 ) , σ s d ( 6 ) , σ a ( 3 ) , m s ( 6 ) , m a ( 3 ) , q ( 3 ) , θ ( 1 ) hence the model has closure. The last term in the entropy inequality

q : g θ − σ s d : ε ˙ − m s : J ˙ s i Θ − m a : J ˙ a i Θ ≤ 0 (9)

must be set to zero to ensure that (9) is always satisfied. Then, in addition to (1)-(8). We must also satisfy

m a : J ˙ a i Θ = 0 (10)

Equation (10) serves as a constraint on the mathematical Model (1)-(8).

We consider incompressible micropolar thermoviscous fluid medium. The CBL and constitutive theories of reference [4] are considered. These naturally include dissipation mechanism due to classical as well as non-classical micropolar physics. Following reference [4] we have the following for CBL and constitutive theories.

D ρ ¯ D t + ρ ¯   div ( v ¯ ) = 0 ( CM ) (CM) (11)

ρ ¯ ∂ v ¯ ∂ t + ρ ¯ ( v ¯ : ∇ ¯ ) v ¯ − ρ ¯ F ¯ b − ∇ ¯ : σ ¯ ( 0 ) = 0 (BLM) (12)

I ¯ Θ ρ ¯ D D t ( i ω ¯ k ) − ϵ i j k σ ¯ i j ( 0 ) − m ¯ m k , m ( 0 ) = 0 (BAM) (13)

ϵ j k l ( I ¯ Θ ρ ¯ v ¯ j ( i ω ¯ k ) − m ¯ j k ( 0 ) ) = 0 (BMM) (14)

ρ ¯ D e ¯ D t + ∇ ¯ : q ¯ − σ ¯ s ( 0 ) : D ¯ − m ¯ s ( 0 ) : J ¯ s i r Θ − m ¯ a ( 0 ) : J ¯ a i r Θ = 0 (FLT) (15)

ρ ¯ ( D Φ ¯ D t + η ¯ D θ ¯ D t ) + q ¯ : g ¯ θ ¯ − σ ¯ s ( 0 ) : D ¯ − m ¯ s ( 0 ) : J ¯ s i r Θ − m ¯ a ( 0 ) : J ¯ a i r Θ ≤ 0 (SLT) (16)

σ ¯ s d ( 0 ) = s ¯ | Ω _ + 2 η D ¯ + κ ( tr ( D ¯ ) ) I (17)

Additive   decomposition   of   σ ¯ ( 0 ) :     σ ¯ ( 0 ) = σ ¯ s ( 0 ) + σ ¯ a ( 0 ) ;     σ ¯ s ( 0 ) = σ ¯ s d ( 0 ) + σ ¯ s e ( 0 ) (18)

m ¯ s ( 0 ) = 2 η m ( J ¯ s i r Θ ) (19)

q ¯ = − κ g ¯ (20)

where

ω ¯ i = Θ ¯ i r   ,   D ( ω ¯ i ) D t = Θ ¯ ˙ i r (21)

In this mathematical model we have CM(1), BLM(3), BAM(3), BMM(3), FLT(1) and the constitutive theories for σ ¯ s d ( 0 ) ( 6 ) , m ¯ s ( 0 ) ( 6 ) , q ¯ ( 3 ) , a total of 26 equations in 26 dependent variables ρ ¯ ( 1 ) , v ¯ ( 3 ) , σ ¯ s d ( 0 ) ( 6 ) , σ ¯ a ( 0 ) ( 3 ) , m ¯ s ( 0 ) ( 6 ) , m ¯ a ( 0 ) ( 3 ) , q ¯ ( 3 ) , θ ¯ ( 1 ) . The last term in the entropy inequality must be set to zero to ensure that (16) is not violated. Thus

m ¯ s ( 0 ) : J ¯ a i r Θ = 0 (22)

must serve as constraint, hence constitutes additional condition that must be satisfied to guarantee that the entropy inequality is not violated.

The objective of this study is to demonstrate existence of pure rotational waves in micropolar solids with rotational inertial physics. Just like in classical continuum mechanics (CCM) we can show existence, propagation, reflection and interaction of translational (stress) waves, in case of micropolar solids with rotational inertial physics we can additionally show existence, propagation, reflection and interaction of rotational (moment) waves. That is, in micropolar solids with rotational inertial physics, translational and rotational waves coexist. In the following study we consider purely one dimensional case of BLM and BAM. In the studies presented here we show that BLM permits translational waves where as BAM with rotational inertial physics permits pure rotational waves.

Mathematical models

One dimensional form of BLM and balance of angular momenta with rotational inertial physics (Equations (1) and (2)) for small strain small deformation and the associated constitutive theories for thermoviscoelastic (without memory) micropolar solid with dissipation mechanism due to rate of strain (CCM) and due to rate of symmetric part of the rotation gradient tensor are given by (in the absence of body forces, body moments, initial stress and equilibrium Cauchy stress, thus σ 11 = σ s 11 = σ s d 11 and using x for x₁)

ρ 0 ∂ 2 u 1 ∂ t 2 − ∂ ( σ s d 11 ) ∂ x = 0 (23)

I Θ 0 ρ 0 ∂ 2 ( i Θ 1 ) ∂ t 2 − ∂ s m 11 ∂ x = 0 (24)

σ s d 11 = ( 2 μ + λ ) ∂ u 1 ∂ x + ( 2 η + κ ) ∂ ∂ t ( ∂ u 1 ∂ x ) (25)

m s 11 = ( α 1 ) ∂ ( i Θ 1 ) ∂ x + ( α 2 ) ∂ ∂ t ( ∂ ( i Θ 1 ) ∂ x )     α 1 = μ m , α 2 = η m (26)

Substituting (25) in (23) and (26) in (24) we obtain

ρ 0 ∂ 2 u 1 ∂ t 2 − ( 2 μ + λ ) ∂ 2 u 1 ∂ x 2 − ( 2 η + κ ) ∂ ∂ t ( ∂ 2 u 1 ∂ x 2 ) = 0 (27)

I Θ 0 ρ 0 ∂ 2 ( i Θ 1 ) ∂ t 2 − ( α 1 ) ∂ 2 ( i Θ 1 ) ∂ x 2 − ( α 2 ) ∂ ∂ t ( ∂ 2 ( i Θ 1 ) ∂ x 2 ) = 0 (28)

for 1D case and for incompressible physics 2 μ + λ = E , modulus of elasticity, as Poison’s ratio ν = 0 and κ = 0 , hence (28) reduces to (but (28) remains same).

ρ 0 ∂ 2 u 1 ∂ t 2 − E ∂ 2 u 1 ∂ x 2 − 2 η ∂ ∂ t ( ∂ 2 u 1 ∂ x 2 ) = 0 (29)

I Θ 0 ρ 0 ∂ 2 ( i Θ 1 ) ∂ t 2 − ( α 1 ) ∂ 2 ( i Θ 1 ) ∂ x 2 − ( α 2 ) ∂ ∂ t ( ∂ 2 ( i Θ 1 ) ∂ x 2 ) = 0 (30)

Remarks

1) Equation (29) describes translational or σ s d 11 stress wave in a viscous elastic medium. In the absence of the last term in (29), (29) represents translational waves in inviscid elastic medium. Dissipation (last term in (29)) results in amplitude decay and base elongation.

2) Equation (30) is analogous to equation (29) and can be obtained b replacing u 1 , ρ 0 , E, and 2 η in (29) by i Θ 1 , I Θ 0 ρ 0 , α 1 and α 2 . Thus, (30) describes rotational waves i.e., moment ( m s 11 ) wave. The third term in (30) is due to micropolar dissipation physics which would result in amplitude decay and base elongation of m s 11 rotational wave.

3) We note the lack of coupling between (29) and (30) due to zero antisymmetric part of the deviatoric Cauchy stress. This is not the case in 2D and 3D applications.

4) Presence of (29) and (30) confirm coexistence of translational and rotational waves in micropolar elastic solids.

Dimensionless form of the mathematical model

We first nondimensionalize (29) and (30) before presenting their solutions. We write (29) and (30) with hat ( ∧ ) on all quantities indicating that all quantities have their usual dimensions (units)

ρ ^ 0 ∂ 2 u ^ 1 ∂ t ^ 2 − E ^ ∂ 2 u ^ 1 ∂ x ^ 2 − 2 η ^ ∂ ∂ t ^ ( ∂ 2 u ^ 1 ∂ x ^ 2 ) = 0 (31)

I Θ 0 ρ ^ 0 ∂ 2 ( i Θ ^ 1 ) ∂ t ^ 2 − ( α ^ 1 ) ∂ 2 ( i Θ ^ 1 ) ∂ x ^ 2 − ( α ^ 2 ) ∂ ∂ t ^ ( ∂ 2 ( i Θ ^ 1 ) ∂ x ^ 2 ) = 0 (32)

We note that in general the speed of translational wave is different than the speed of rotational wave. However, since in this case (31) and (32) are decoupled, it is possible to nondimensionalize (31) and (32) such that in the dimensionless domain the speed of prorogation of both translation and rotational waves is unity. Obviously in 2D and 3D cases it is not possible to do so. We present details in the following.

BLM: Model TW1

Choosing L 0 , η 0 , ( ρ 0 ) r e f , t 0 , v 0 = L 0 / t 0 = E 0 ρ 0 and E 0 = ( ρ 0 ) r e f v 0 2 as reference quantities in which v 0 is the speed of translational wave (speed of sound in CCM).

ρ 0 ∂ 2 u 1 ∂ t 2 − E E 0 ( ρ 0 ) r e f v 0 2 ∂ 2 u 1 ∂ x 2 − 2 η R e ∂ ∂ t ( ∂ 2 u 1 ∂ x 2 ) = 0 (33)

where 1 / R e = η 0 L 0 ( ρ 0 ) r e f v 0 , Reynolds number. If we choose ( ρ 0 ) r e f = ρ ^ 0 , E 0 = E ^ = ( ρ 0 ) r e f v 0 2 and C d = 2 η R e as damping coefficient, then (33) can be reduced to

∂ 2 u 1 ∂ t 2 − ∂ 2 u 1 ∂ x 2 − c d ∂ ∂ t ( ∂ 2 u 1 ∂ x 2 ) = 0 (34)

In (34), the dimensionless speed of sound (based on reference quantities) in an inviscid medium is unity. We use (34) in the numerical studies.

BAM: Using speed of rotational waves as reference velocity: Model RW1

In this case, when nondimensionalizing BAM we choose speed of rotational wave as reference velocity. We also choose some of the same reference quantities as in BLM except we choose ( I Θ 0 ) r e f , ( α 1 ) 0 , ( α 2 ) 0 , ω 0 = ( α 1 ) 0 / ( I Θ 0 ) r e f ( ρ 0 ) r e f and t 0 = L 0 / ω 0 , ( α 1 ) 0 = ( I Θ 0 ) r e f ( ρ 0 ) r e f ω 0 2 , then (32) can be written as

I Θ 0 ρ 0 ∂ 2 ( i Θ 1 ) ∂ t 2 − ( α 1 ) ( α 1 ) 0 ( I Θ 0 ) r e f ( ρ 0 ) r e f ω 0 2 ∂ 2 ( i Θ 1 ) ∂ x 2 − ( α 2 ) 1 R e * ∂ ∂ t ( ∂ 2 ( i Θ 1 ) ∂ x 2 ) = 0 (35)

If ( I Θ 0 ) r e f = I ^ Θ 0 , ( ρ 0 ) r e f = ρ ^ 0 , then (35) can be reduced to

∂ 2 ( i Θ 1 ) ∂ t 2 − ∂ 2 ( i Θ 1 ) ∂ x 2 − ( α 2 ) R e * ∂ ∂ t ( ∂ 2 ( i Θ 1 ) ∂ x 2 ) = 0 (36)

in which R e * = ( I Θ 0 ) r e f ( ρ 0 ) r e f ω 0 L 0 ( α 2 ) 0 is the Reynolds number associated with dissipation due to micropolar physics. If we define c d * = α 2 R e * as the dissipation coefficient due to micropolar physics, then (36) can be written as

∂ 2 ( i Θ 1 ) ∂ t 2 − ∂ 2 ( i Θ 1 ) ∂ x 2 − c d * ∂ ∂ t ( ∂ 2 ( i Θ 1 ) ∂ x 2 ) = 0 (37)

Equation (34) and (37) are the simplified dimensionless forms of BLM and BAM describing translational and rotational waves in the presence of dissipation mechanisms. In the numerical studies we use equations (34) and (37).

BAM: Using translation wave speed as reference velocity: Model RW2

In this case we choose

v 0 = E 0 ( ρ 0 ) r e f t 0 = L 0 v 0 } (38)

using these reference values we can nondimensionalize (32) as follows.

I Θ 0 ρ 0 ∂ 2 ( i Θ 1 ) ∂ t 2 − ( α 1 ) ( α 1 ) 0 ( I Θ 0 ) r e f ( ρ 0 ) r e f v 0 2 ∂ 2 ( i Θ 1 ) ∂ x 2 − ( α 2 ) ( α 2 ) 0 ( I Θ 0 ) r e f ( ρ 0 ) r e f L 0 v 0 ∂ ∂ t ( ∂ 2 ( i Θ 1 ) ∂ x 2 ) = 0 (39)

If we choose ( I Θ 0 ) r e f = I ^ Θ 0 , ( ρ 0 ) r e f = ρ ^ 0 and define

α 1 * = ( α 1 ) 0 / 2 ( I Θ 0 ) r e f ( ρ 0 ) r e f v 0 2 ,   c d * = α 2 R e * ,   1 / R e * = ( α 2 ) 0 ( I Θ 0 ) r e f ( ρ 0 ) r e f L 0 v 0

then (39) can be reduced to

∂ 2 ( i Θ 1 ) ∂ t 2 − α 1 * ∂ 2 ( i Θ 1 ) ∂ x 2 − c d * ∂ ∂ t ( ∂ 2 ( i Θ 1 ) ∂ x 2 ) = 0 (40)

In this case the speed of dimensionless rotational wave is α 1 * which may not be unity. We use (40) in the numerical studies.

Remarks

1) We consider (34), (37) and (40) for presenting numerical studies.

2) Numerical studies are presented using space-time coupled finite element method based on space-time residual functional for a space-time strip with time marching [5] in which space-time local approximation for a space-time element are p-version hierarchical with higher order global differentiability in space and time.

3) We recast (34), (37) and (40) as a system of first order PDEs for convenience of defining BCs and ICs.

First order system of PDEs

Translational (or stress) waves (Model TW1)

We recast (34) as follows

∂ v 1 ∂ t − ∂ ( σ s d 11 ) ∂ x = 0 σ s d 11 − ∂ u 1 ∂ x − c d ∂ v 1 ∂ x = 0 v 1 − ∂ u 1 ∂ t = 0 } (41)

Rotational (or moment) waves (Model RW1)

We recast (37) as

∂ ( i ω 1 ) ∂ t − ∂ ( s m 11 ) ∂ x = 0 s m 11 − ∂ ( i Θ 1 ) ∂ x − c d * ∂ ( i ω 1 ) ∂ x = 0 i ω 1 − ∂ ( i Θ 1 ) ∂ t = 0 } (42)

Rotational (or moment) waves (Model RW2)

∂ ( i ω 1 ) ∂ t − ∂ ( s m 11 ) ∂ x s m 11 − α 1 * ∂ ( i Θ 1 ) ∂ x − c d * ∂ ( i ω 1 ) ∂ x = 0 i ω 1 − ∂ ( i Θ 1 ) ∂ t = 0 } (43)

Numerical studies

Evolutions are computed for models TW1, RW1 and RW2 using space-time coupled finite element method for a space-time strip with time marching [7] . The space-time local approximation for a space-time element is p-version hierarchical with higher order global differentiability in space and time. Discretizations, p-levels and the minimally confirming spaces are chosen such that for each space-time strip the space-time residual functional for the discretization is O(10⁻⁸) or lower ensuring that PDEs are satisfied accurately by the computed solution. In the present studies we choose a uniform mesh of 30, nine node p-version hierarchical finite elements with p-levels of 9 in space and time for all elements of the space-time strip. We choose local approximations in space and time of class C¹¹.

A time increment of Δ t = 0.1 is used in all computations. The solution is computed for the first space-time strip ( 0 ≤ t ≤ Δ t ) and then time marched to obtain evolution for desired value of τ (final value of time). Since all evolutions considered here are smooth, choice of solutions of class C¹¹ in space and time suffices. For this choice of the order of the approximation space, the space-time integrals for the discretization of the space-time strip are Riemann. The 30 element uniform discretization with p-levels of 9 for space and time and k = 2 for both space and time, yield space-time residual functional values for each space-time strip of the order O(10⁻⁸) or lower during the entire evolution, confirming good accuracy of the computed solution. In the following, we present computed evolutions for: translation wave using Model TW1 and rotations wave using Models TR1 and TR2.

Figure 1 shows schematic, space-time domain Ω ¯ x t , discretization Ω ¯ x t T = ∪ e     Ω ¯ x t ( e ) of the space time domain into space-time strips, discretization ( Ω ¯ x t ( 1 ) ) T = ∪ e     Ω ¯ x t ( e ) of the first space-time strip Ω ¯ x t ( 1 ) and the details of stress σ s d 11 boundary conditions (for TW1) applied to the last space-time element face at x = 1 . In the numerical studies, we choose negative σ 0 (compressive) for the applied pulse at x = 1 . The details of applying BC at x = L for mathematical models RW1 and RW2 are similar.

Translational wave: Model TW1

We consider mathematical model TW1. Evolution is computed for 30 time steps for the undamped case, c d = 0 using space-time coupled finite element formulation based on space-time residual functional. Since in this case, the speed of wave propagation is one, in ten time steps with Δ t = 0.1 , the applied pulse at x = 1 reaches the impermeable boundary at x = 0 . Due to zero damping, we expect the applied pulse shape to be preserved during evolution. Figure 2(a) shows a plot of σ s d 11 versus x for various values of time. Time steps 2, 6, 10 show the incident compressive pulse entering the spatial domain at x = L and propagating without amplitude decay or base elongation confirming the absence of numerical dispersion. At the 10^th time step the compressive pulse is precisely at x = 0 as expected. Reflection of the compressive incident pulse at the impermeable boundary (at x = 0 ) results in reflected compressive pulse at the 11^th time step with double the peak value. Upon further evolution the reflected compressive pulse recovers its peak and base at the 12^th time step (not shown). At the 14^th time step, we see reflected pulse with the same amplitude and base as the original incident pulse propagating toward the right end. At the 20^th time step the reflected compressive pulse reaches the free boundary at x = L = 1 and

reflects as a tensile pulse of the same amplitude and base as the original incident pulse propagating towards the boundary at x = 0 (time steps 23 and 28 in Figure 2(a)).

In the next study for TW1, we consider dimensionless damping coefficient c d = 0.002 . Figure 2(b) shows evolution of σ s d 11 versus x for time steps 2, 6, 10; 11, 14, 18; 23 and 28 (same as in Figure 2(a) for the undamped case). In the incident pulse, amplitude decays and the base elongates during propagation (time steps 2, 6, 10). During reflection (time step 11), the magnitude of the pulse increases but recovers upon further evolution (time step 14). The reflected pulse reaches the free boundary and reflection from the free boundary results in a tensile pulse that propagates to the left. We observe the amplitude decay is more pronounced in the propagating incident pulse. The reflected pulses also shows some amplitude decay, but the base elongation in the reflected pulse is more pronounced. All evolutions are smooth i.e., free of oscillations. Integrated sum of squares of the space-time residual functional are of the order of O(10⁻⁸) or lower for each space-time strip ensuring that the computed solution is accurate. Choice of k = 2 (order of the approximation space in space and time) is minimally conforming for the first order system of PDEs, hence all space-time integrals for ( Ω ¯ x t ( i ) ) T , discretization of space-time strip Ω ¯ x t ( i ) are Riemann. The space-time residual functional of the order O(10⁻⁸) ensures that the PDEs are satisfied in point wise sense over each space-time strip discretization Ω ¯ x t ( i ) , hence good time accuracy of the evolution is ensured.

Rotational wave: Model RW1

We consider mathematical model RW1 defined by Equation (42). In this case, the equations are nondimensionalized using the speed of rotational wave, hence the dimensionless speed of the rotational wave is one. In the first study, we apply moment pulse i.e., m s 11 pulse of time duration 2 Δ t (similar to σ s d 11 shown in Figure 1(e)). We choose p-levels of 9 in space and time, Δ t = 0.1 and order k of the approximation space in space and time is chosen to be 2, which is minimally confirming for a first order system of PDEs. We choose peak values of m s 11 = − 1 . Evolution is computed using space-time strip with time marching, Integral form in space-time finite element process is based on space-time residual functional. Figure 3(a) shows the evolution for c d * = 0 (undamped case). Incident moment pulse propagates towards the impermeable boundary without amplitude decay and base elongation. Upon reflection at x = 0 , the peak amplitude doubles during reflection but the pulse recovers to the original shape in the 12^th time steps and continues to propagate toward the free boundary. Reflection from the free boundary results in tensile moment pulse of the same shape as original incident compressive moment pulse that continues to propagate toward the impermeable boundary at x = 0 .

Figure 3(b) shows the propagation, reflection and propagation upon reflection of the same compressive m s 11 pulse as in Figure 3(a) but in the presence of dissipation. We choose dimensionless dissipation coefficient c d * = 0.002 . From Figure 3(b) we observe almost the same behavior of the moment pulse as that of σ s d 11 pulse in Figure 2(b). Presence of dissipation resulting in continued amplitude decay and base elongation during the propagation. Amplitude decay is more pronounced for the incident pulse and the base elongation is more significant in the reflected pulse.

Rotational Wave: Model RW2

In this study, we consider mathematical model RW3, Equation (43). In this mathematical model rotational wave speed is α 1 * and the dissipation is controlled by the dimensionless dissipation coefficient c d * . We apply a moment ( m s 11 ) pulse of duration 2 Δ t at the boundary x = L = 1 .

In the first study, we choose Δ t = 0.1 , α 1 * = 2.25 and c d * = 0 (undamped case). The wave speed is 2.25 = 1.5 , thus the pulse would reach x = 0

boundary in ( 1 / 1.5 ) / 0.1 = 6.67 time steps. Figure 4(a) show propagation and reflection of the moment pulse. Figure 4(b) show propagation and reflection of the moment pulse for α 1 * = 2.25 i.e., at the wave speed of 1.5 when c d * = 0.002 . Diminished amplitudes and elongated base of the pulse is clearly observed during evolution. This is similar to RW1 ( c d * = 0.002 ). In the next study, we apply the same moment pulse as in the previous study but choose α 1 * = 0.49 and c d * = 0 , hence rotational wave speed of 0.49 = 0.7 , thus the moment pulse will reach the boundary at x = 0 in ( 1 / 0.7 ) / 0.1 = 14.285 time steps. Figure 5(a) and Figure 5(b) show propagation and reflection of the moment wave for c d * = 0 and c d * = 0.002 . The undamped pulse ( c d * = 0 ) maintains its base and amplitude during the entire evolution and when c d * = 0.002 progressive amplitude decay and base elongation of the pulse is observed for both values of α 1 * .

We remark that when the rotational wave speed is one, we are able to precisely locate the pulse at the impermeable boundary as the time to reach the impermeable boundary is integer multiple of Δ t . In this case, we see perfect reflection of the m s 11 pulse at the 11^th time step and the peak value of m s 11 doubles, same as in the case of the translational stress wave. However, when rotational wave speed is not an integer multiple of Δ t , precise arrival of m s 11 pulse at x = 0 cannot be simulated as elapsed time is always an integer multiple of Δ t . For this reason, we are not able to observe 2 ( m s 11 ) in the peak values during the reflection process of the undamped case (Figure 4(a) and Figure 5(a)) when the rotational speed is not one.

1) In the model problem studies presented here the translational and rotational waves are decoupled. In R 2 and R 3 this may not be the case. Since the stress tensor is a function of [ J s d ] and the moment tensor is a function of [ J a d ] , both stress and moment tensor waves are dependent on gradients of displacement. Thus, in R 2 and R 3 we expect interaction between the two waves. That is, as shown here, rotational waves depend upon α ^ 1 and I ¯ Θ which in turn influences [ J d ] , thus [ J s d ] , hence influencing translational waves. The model problem studies in R 2 and R 3 are needed to illustrate this physics.

2) The medium with and without microconstituents is always considered isotropic and homogeneous. Thus, in this micropolar theory there is absence of wave dispersion in the micropolar medium as it is considered to be isotropic and homogeneous. This micropolar theory can only simulate wave propagation physics of translational and rotational waves and their interaction (when it exists) in R 2 and R 3 (future work).

Rotational wave?

The objective of the model problem studies presented in this section is to demonstrate the influence of rotational inertial physics in micropolar fluids in which NCCT is based on internal rotation rate physics. It is well known that in classical or micropolar viscous fluids, translational deviatoric stress waves can not exist due to the absence of elasticity. In this study, we only consider balance of angular momenta in 1D and associated constitutive theories to investigate the influence of rotational inertial physics.

Mathematical model: rotational wave?

The one dimensional form of BAM and the constitutive theory for the Cauchy moment tensor are given by (using x ¯ for x ¯ 1 ).

I ¯ Θ ρ ¯ ∂ ( i ω ¯ 1 ) ∂ t − ∂ ( s m ¯ 11 ) ∂ x ¯ = 0 (44)

s m ¯ 11 = α ¯ 1 ( 1 2 ∂ ( i ω ¯ 1 ) ∂ x ¯ ) (45)

Substituting (45) in (44)

I ¯ Θ ρ ¯ ∂ ( i ω ¯ 1 ) ∂ t − α ¯ 1 2 ( ∂ 2 ( i ω ¯ 1 ) ∂ x ¯ 2 ) = 0 (46)

Remarks

1) We clearly see that this Equation (46) will not permit a moment ( s m ¯ 11 )wave as it is not a wave equation due to the absence of ∂ 2 ( i ω ¯ 1 ) ∂ t 2 in place of ∂ ( i ω ¯ 1 ) ∂ t .

2) However it is interesting to study its solution for varying I ¯ Θ values.

Dimensionless form of the mathematical model

Recasting (46) with hat ( ∧ ) on all quantities indicating that they have their usual dimensions (or units)

I ¯ ^ Θ ρ ¯ ^ ∂ ( i ω ¯ ^ 1 ) ∂ t ^ − α ¯ ^ 1 2 ( ∂ 2 ( i ω ¯ ^ 1 ) ∂ x ¯ ^ 2 ) = 0 (47)

and using

I ¯ Θ = I ¯ ^ Θ / I Θ 0 ,   ρ ¯ = ρ ¯ ^ / ρ 0 ,   x ¯ = x ¯ ^ / L 0 ω ¯ i 1 = ω ¯ ^ i 1 / ω 0 ,   t 0 = L 0 / ω 0 , and   ω 0 = ( α 1 ) 0 / I Θ 0 ρ 0 ,   α ¯ 1 = α ¯ ^ 1 / ( α 1 ) 0 } (48)

We can write (47) as

I ¯ Θ ρ ¯ ∂ ( i ω ¯ 1 ) ∂ t − ( α ¯ 1 2 ) ( α 1 ) 0 L 0 I Θ 0 ρ 0 ( ∂ 2 ( i ω ¯ 1 ) ∂ x ¯ 2 ) = 0 (49)

If we choose I Θ 0 = I ¯ ^ Θ , ρ 0 = ρ ¯ ^ and ( α 1 ) 0 = α ¯ ^ 1 then (49) can be reduced to

∂ ( i ω ¯ 1 ) ∂ t − c ¯ 2 ( ∂ 2 ( i ω ¯ 1 ) ∂ x ¯ 2 ) = 0 (50)

in which c ¯ 2 = ( α ¯ ^ 1 ) / ( 2 L 0 I ¯ ^ Θ ρ ¯ ^ ) , dimensionless dissipation coefficient. PDE (50) can be cast as a system of two first order PDEs by using moment s m ¯ 11 as auxiliary variable and we have

∂ ( i ω ¯ 1 ) ∂ t − ∂ ( s m ¯ 11 ) ∂ x ¯ = 0 s m ¯ 11 = c ¯ 2 ∂ ( i ω ¯ 1 ) ∂ x ¯ } (51)

PDEs in (51) are helpful in defining BCs and ICs, hence are used in the computations.

Numerical Studies

Rotational wave?

We consider mathematical model (50) or (51). It is evident (50) is not a wave equation as a wave equation must have second order spatial and second order time derivatives of the dependent variable. Thus, (50) can not describe a wave, which is not a surprise, because classical as well as micropolar fluids do not possess elasticity, hence do not have stiffness, thus they cannot support deviatoric Cauchy stress wave or Cauchy moment wave.

Equation (50) is in fact a time dependent diffusion equation in which c ¯ 2 ∝ 1 I ¯ Θ is the diffusion coefficient. Thus low values of I ¯ Θ correspond to high values of c ¯ 2 and vice versa. Details of the schematic, space-time strips and discretization of a space-time strip remain the same as shown in Figure 1. We apply a negative s m ¯ 11 moment pulse of duration 2 Δ t on the boundary at x ¯ = 1 such that s m ¯ 11 ∈ [ 0, − 1 ] ∀     t ∈ [ 0, Δ t ] ; s m ¯ 11 ∈ [ − 1,0 ] ∀     t ∈ [ Δ t ,2 Δ t ] and s m ¯ 11 = 0 for t ≥ 2 Δ t (similar to BC shown in Figure 1(e)). We choose a discretization of 30 nine node p-version hierarchical space-time elements with p-levels of nine in space and time. Evolution is computed for 30 time steps using Δ t = 0.1 with local approximation of class C¹¹ in space and time, We choose two values of c ¯ 2 = 0.0001 , 0.001 . Evolutions for different values of time for the two choices of c ¯ 2 are shown in Figure 6(a) and Figure 6(b). For both values of c ¯ 2 , the applied pulse progressively diffuses as time elapses. We observe significantly high diffusion of the applied moment pulse for c ¯ 2 = 0.001 compared to c ¯ 2 = 0.0001 as expected. We clearly observe lack of existence and propagation of rotational or moment waves due to lack of elasticity, demonstrating that rotational waves can not exist in micropolar fluids with rotational inertial physics.