# Unsteady Helical Flows of a Size-Dependent Couple-Stress Fluid

Authors: Qammar Rubbab; Itrat Abbas Mirza; Imran Siddique; Saadia Irshad
Journal: Advances in Mathematical Physics
 (2017)

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## Abstract

The helical flows of couple-stress fluids in a straight circular cylinder are studied in the framework of the newly developed, fully determinate linear couple-stress theory. The fluid flow is generated by the helical motion of the cylinder with time-dependent velocity. Also, the couple-stress vector is given on the cylindrical surface and the nonslip condition is considered. Using the integral transform method, analytical solutions to the axial velocity, azimuthal velocity, nonsymmetric force-stress tensor, and couple-stress vector are obtained. The obtained solutions incorporate the characteristic material length scale, which is essential to understand the fluid behavior at microscales. If characteristic length of the couple-stress fluid is zero, the results to the classical fluid are recovered. The influence of the scale parameter on the fluid velocity, axial flow rate, force-stress tensor, and couple-stress vector is analyzed by numerical calculus and graphical illustrations. It is found that the small values of the scale parameter have a significant influence on the flow parameters.


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## Body

## 1. Introduction

The existence of couple stresses in continuum mechanics is a consequence of the discrete character of the matter at the finest scale. Also, the noncentral property of forces between elementary particles of matter leads to the appearing of couple stresses [1]. E. Cosserat and F. Cosserat [2] were the first to have incorporated the couple stresses by considering the oriented material point triads and independent microrotation in theory of continuum solids. Toupin [3], Mindlin and Tiersten [4], and Koiter [5] have developed theories with couple stresses, in which the rigid body motion of the infinitesimal element of matter at each point of the continuum is described by six degrees of freedom.Based on these theories with couple stresses, Stokes [6] has developed the theory of couple-stress fluids in order to study the size-dependent behavior of flows. Stokes’ theory represents the simplest generalization of the classical theory of fluids, which considers the presence of couple stresses. All the above models have various inconsistencies. In the context of Stokes’ theory, the indeterminacy of the spherical part of couple-stress tensor is a major shortcoming. Another problem appears in the theory of Mindlin and Tiersten; namely, in their theory, the constitutive relation for the force-stress tensor contains the body couples. The indeterminate spherical part of couple-stress tensor is ignored in Stokes’ theory, without justifications. Also, the presence of body couples in the constitutive relations was neglected by Stokes in his theory.Hadjesfandiari et al. [1, 7, 8] have solved these issues by using arguments based upon the energy equation along with kinematical considerations. They developed a fully consistent couple-stress theory for solids and fluids, which establishes the couple-stress tensor and mean curvature rate tensor as skew-symmetric energy conjugate measures. Their results could be useful for examining fluid flow problems influenced by the mechanics at small scales. Significant application areas for size-dependent fluid mechanics involve modeling of blood flows, lubrication problems, liquid crystals, and polymeric suspensions. This branch of fluid mechanics has attracted a growing interest from researchers in the field.Bakhti and Azrar [9] studied the steady flow of a couple-stress fluid through constricted tapered artery under influence of a transverse magnetic field, moving catheter, and slip velocity. Solutions to velocity and shear stress are expressed with Bessel’s functions. Pralhad and Schultz [10] used the couple-stress fluid model for the study of the steady flow of blood through stenosed artery. Blood velocity, the resistance to flow, and shear stress distribution have been obtained. Verma et al. [11] have studied the blood flow in a stenosed tube considering the blood as a couple-stress fluid. Effects of slip velocity in stenosed tube were highlighted. Shenoy and Pai [12] have made the static analysis of a misaligned externally adjustable fluid-film bearing including turbulence and couple-stress effects in lubricants blended with polymer additives. Naduvinamani and Patil [13] obtained a numerical solution to finite modified Reynolds equation for couple-stress squeeze film lubrication of porous journal bearings.The unsteady three-dimensional flow of couple-stress fluid over a stretched surface with mass transfer and chemical reaction was investigated by Hayat et al. [14]. Devakar and Iyengar [15] investigated the generalized Stokes’ problems for incompressible couple-stress fluids. The effects of kinetic helicity (velocity-vorticity correlation) on turbulent momentum transport were investigated by Yokoi et al. [16, 17]. Other interesting topics can be found in references [18–21].We must mention that the models considered in the above articles are based on the theory developed by Mindlin, Tiersten, and Stokes.In the present paper we consider the consistent theory of couple-stress fluids elaborated by Hadjesfandiari and his coworkers [1, 8] and, we study the helical flows of couple-stress fluids within a straight circular cylinder under general boundary conditions. In the studied problem, the fluid motion is generated by the helical motion of the cylindrical surface with the time-dependent axial and azimuthal velocities and by the time-dependent couple-stress vector on the cylinder surface. The nonslip conditions are, also, considered. Using suitable nondimensional variables, we determine analytical solutions to the axial and azimuthal velocities by means of the integral transform method. The axial angular velocity and the flow rate are also obtained. Components of the nonsymmetric force-stress tensor and the couple-stress vector are determined from the constitutive relations and velocity field. Obviously, if the characteristic length of couple-stress fluids is zero, we recover results for the classical fluid. The obtained analytical solutions are used in order to perform numerical calculations using the Mathcad software for particular external loadings. The results are graphically presented. It is found that all flow parameters are influenced by the fluid scale parameter. The significant influence is obtained for small values of the scale parameter.


## 2. Statement of the Problem

We consider a homogeneous, incompressible, viscous couple-stress fluid flowing inside a straight circular cylinder of radiusR. The cylindrical coordinate system (r,θ,z) has the z-axis identical with the cylinder axis. The governing equations of the couple-stress fluid are as follows [1, 8]:(i) Continuity equation:(1)∇·v→=∂vr∂r+1r∂vθ∂θ+∂vz∂z=0,where v→=vre→r+vθe→θ+vze→z is the fluid velocity.(ii) Equation of linear momentum (the body forces are neglected):(2)ρ∂v→∂t+∇v→·v→=-∇p+μ∇2v→-η∇2∇2v→,where ρ is the fluid density, μ is the dynamic viscosity, η is the viscosity coefficient of couple-stress fluid, and p is the thermodynamic pressure.(iii) The constitutive equations:The nonsymmetric force-stress tensor:(3)T=-pI+τ,τ=2μD+2ηΩ,where(4)D=12∇v→+∇v→T,(5)Ω=12∇v→-∇v→Trepresent the strain rate tensor and the angular velocity tensor, respectively.The polar couple-stress vector:(6)M→=2ηΔv→.In this paper, we consider that the velocity field and pressure are functions of the form(7)v→r,t=vθr,te→θ+v→zr,te→z,p=pr,t.Equation (1) is identically satisfied and (2)–(6) become(8)∂p∂r=ρrvθ2,ρ∂vθ∂t=μΔ-1r2vθ-ηΔ-1r2Δ-1r2vθ,ρ∂vz∂t=μΔvz-ηΔ2vz,τrr=0,τrθ=μr∂∂rvθr-ηΔ1r∂∂rrvθ,τrz=μ∂vz∂r-ηΔ∂vz∂r,τθr=μr∂∂rvθr+ηΔ1r∂∂rrvθ,τθθ=0,τθz=0,τzr=μ∂vz∂r+ηΔ∂vz∂r,τzθ=0,τzz=0,Mr=0,Mθ=2ηΔ-1r2vθ,Mz=2ηΔvz.We consider the following initial-boundary conditions:(9)vθr,0=0,vzr,0=0,(10)vθ0,t=0,Mθ0,t=0,∂vz∂rr=0=0,(11)vθR,t=V1f1t,MθR,t=2ηm1f2t,(12)vzR,t=V0g1t,MzR,t=2ηm2g2t.Functionsf1(t), f2(t), g1(t), and g2(t) are piecewise continuous functions on [0,t]; for every t>0, they have exponential order at infinity and f1(0)=f2(0)=g1(0)=g2(0).We define the characteristic material length(13)l=ημm,which is absent in classical fluid mechanics but is fundamental for couple-stress fluids.Introducing the nondimensional variables(14)r∗=rR,t∗=νtR2,u∗=Rvθν,v∗=Rvzν,p∗=R2pρν2,a=l2R2,τ∗=R2ρν2τ,M→∗=Rρν2M→,f1∗t∗=RV1νf1R2t∗ν,f2∗t∗=m1R3νf2R2t∗ν,g1∗t∗=RV0νg1R2t∗ν,g2∗t∗=m2R3νg2R2t∗νinto (8)–(10) and dropping the star notation, we obtain the following nondimensional problem:(15)∂p∂r=1ru2,(16)∂u∂t=Δ-1r2u-aΔ-1r2Δ-1r2u,(17)∂v∂t=Δv-aΔ2v,(18)τrr=0,τrθ=r∂∂rur-aΔ1r∂∂rru,τrz=∂v∂r-aΔ∂v∂r,(19)τθr=r∂∂rur+aΔ1r∂∂rru,τθθ=0,τθz=0,(20)τzr=∂v∂r+aΔ∂v∂r,τzθ=0,τzz=0,(21)Mr=0,Mθ=2aΔ-1r2u,Mz=2aΔv,(22)ur,0=0,vr,0=0,(23)u0,t=0,Mθ0,t=0,∂v∂rr=0=0,(24)u1,t=f1t,Mθ1,t=2af2t,(25)v1,t=g1t,Mz1,t=2ag2t.In the end of this section, we give two lemmas regarding some properties to the operators from (16) and (17). These properties will be used in order to find solutions of the above problem.Lemma 1.
Ifαn,n=1,2,… are the positive roots of Bessel function J0(x), vn(t)=∫01v(r,t)rJ0(αnr)dr, and ∂v(r,t)/∂rr=0=0, then,(i)
(26)∫01Δvr,trJ0αnrdr=αnJ1αnv1,t-αn2vnt;(ii)
(27)∫01Δ2vr,trJ0αnrdr=-αn3J1αnv1,t+αnJ1αnΔvr,tr=1+αn4vnt.Lemma 2.
Ifβn,n=1,2,… are the positive roots of Bessel function J1(x), un(t)=∫01u(r,t)rJ1(βnr)dr, and Δ-1/r2u(r,t)r=0=0, then,(i)
(28)∫01Δ-1r2ur,trJ0αnrdr=-βnJ0αnu1,t-βn2unt;(ii)
(29)∫01Δ-1r22ur,trJ1βnrdr=βn3J0βnu1,t+βnJ0βnΔ-1r2ur,tr=1+βn4unt.The demonstration of relations (26)–(29) is made easy using integration by parts and properties of Bessel functions [22, 23]. It is noted that vn(t) is zero-order finite Hankel transform of function v(r,t) and un(t) is the finite Hankel transform of the first order of function u(r,t), respectively.


## 3. Solution of the Problem

In order to find the solution of problem (15)–(25), we use the Laplace transform with respect to the variable time and finite Hankel transform with respect the radial coordinate [24, 25].


### 3.1. Velocity Field

Applying the Laplace and finite Hankel transforms to (16) and (17), using the initial and boundary conditions (22)–(25) and Lemmas 1 and 2, we obtain the following transformed equations:(30)u-ns=-βnJ0βn1+aβn2s+βn21+aβn2f-1s+aβnJ0βns+βn21+aβn2f-2s,v-ns=αnJ1αn1+aαn2s+αn21+aαn2g-1s-aαnJ1αns+αn21+aαn2g-2s,where u-n(s)=∫0∞un(t)e-stdt, v-n(s)=∫0∞vn(t)e-stdt, f-1(s), f-2(s), g-1(s), and g-2(s) are the Laplace transforms of functions un(t), vn(t), f1(t), f2(t), g1(t), and g2(t), respectively. Equations (30) can be written in the suitable forms(31)u-ns=-J0βnβnf-1s+J0βnβnsf-1ss+βn21+aβn2+J0βnβn3f-2s-J0βnβn31+aβn2f-2s-aJ0βnβn1+aβn2sf-2ss+βn21+aβn2,v-ns=J1αnαng-1s-J1αnαnsg-1ss+αn21+aαn2-4+αn2J1αn4αn3g-2s+4+αn21+aαn24αn31+aαn2J1αng-2s+aJ1αnαn1+aαn2sg-2ss+αn21+aαn2.Now, using the integrals(32)∫01r2J1βnrdr=-J0βnβn,∫01r3-r8rJ1βnrdr=J0βnβn3,∫01rJ0αnrdr=J1αnαn,∫01r2-14rJ0αnrdr=-4+αn24αn3J1αn,and applying the inverse Hankel and Laplace transforms, we obtain closed forms to azimuthal velocity and axial velocities as(33)ur,t=rf1t+r3-r8f2t-2f2t∑n=1∞J1βnrβn31+aβn2J0βn+2∑n=1∞J1βnrβnJ0βn∫0tf˙1τexp⁡-βn21+aβn2t-τdτ-2a∑n=1∞J1βnrβn1+aβn2J0βn∫0tf˙2τexp⁡-βn21+aβn2t-τdτ,vr,t=g1t+r2-14g2t+2g2t∑n=1∞4+αn21+aαn2J0αnr4αn31+aαn2J1αn-2∑n=1∞J0αnrαnJ1αn∫0tg˙1τexp⁡-αn21+aαn2t-τdτ+2a∑n=1∞J0αnrαn1+aαn2J1αn∫0tg˙2τexp⁡-αn21+aαn2t-τdτ.In the above relations we have used the notationf˙(t)=df(t)/dt.By introducing notations(34)Ant=-2f2tβn31+aβn2J0βn+2βnJ0βn∫0tf˙1τexp⁡-βn21+aβn2t-τdτ-2aβn1+aβn2J0βn∫0tf˙2τexp⁡-βn21+aβn2t-τdτ,Bnt=2g2t4+αn21+aαn24αn31+aαn2J1αn-2αnJ1αn∫0tg˙1τexp⁡-αn21+aαn2t-τdτ+2aαn1+aαn2J1αn∫0tg˙2τexp⁡-αn21+aαn2t-τdτ,(33) are written as(35)ur,t=rf1t+r3-r8f2t+∑n=1∞AntJ1βnr,vr,t=g1t+r2-14g2t+∑n=1∞BntJ0αnr.It is observed that functionsAn(t) and Bn(t) have the property An(0)=Bn(0)=0.The axial angular velocity (the spin vector) is given by(36)ω→=12∇×v→=-12∂v∂re→θ+∂u∂r+ure→z=12∑n=1∞αnBntJ1αnr-r2g2te→θ+122f1t+2r2-14f2t+∑n=1∞βnAntJ0βnre→z,and the flow rate is(37)Qt=∫012πrvr,tdr=πg1t-π8g2t+2π∑n=1∞BntαnJ1αn.


### 3.2. Force-Stress Tensor and Couple-Stress Vector

Replacing (35) in (18)–(21) and performing calculations, we get the following expressions for the force-stresses and couple-stress vector:(38)τrθ=r24-2af2t+∑n=1∞βnAnt1+2aβn2J0βnr-2r∑n=1∞AntJ1βnr,τθr=r24+2af2t+∑n=1∞βnAnt1-2aβn2J0βnr-2r∑n=1∞AntJ1βnr,τrz=r2-a2rg2t-∑n=1∞αnBnt1+aαn2-1r2J1αnr,τzr=r2+a2rg2t-∑n=1∞αnBnt1-aαn2-1r2J1αnr,τrr=τθθ=τθz=τzθ=τzz=0,Mr=0,Mθ=2arf2t-∑n=1∞βn2AntJ1βnr,Mz=2ag2t-∑n=1∞αn2BntJ0αnr.It is easy to see that, fora=0 (the ordinary Newtonian fluid), the force-stress tensor becomes a symmetric tensor and the couple-stress vector is zero.


### 3.3. Particular Case (Constant Velocity and Couple Stress on the Boundary)

Let us consider the following boundary conditions:(39)f1t=p1Ht,f2t=p2Ht,g1t=q1Ht,g2t=q2Ht,where p1≥0, p2≥0, q1≥0, and q2≥0 are constants and Ht=1/2sign(t)(1+sign(t)) is the Heaviside unit step function. In this case, the derivatives of functions given by (39) are (40)f˙1t=p1δt,f˙2t=p2δt,g˙1t=q1δt,g˙2t=q2δt,δ(t) being the Dirac distribution. Functions An(t) and Bn(t) are written in simpler forms, as(41)Ant=-2p2Htβn31+aβn2J0βn+2HtβnJ0βnp1-ap21+aβn2exp⁡-βn21+aβn2t,Bnt=2q2Ht4+αn21+aαn24αn31+aαn2J1αn-2HtαnJ1αnq1-aq21+aαn2exp⁡-αn21+aαn2t.


## 3.1. Velocity Field

Applying the Laplace and finite Hankel transforms to (16) and (17), using the initial and boundary conditions (22)–(25) and Lemmas 1 and 2, we obtain the following transformed equations:(30)u-ns=-βnJ0βn1+aβn2s+βn21+aβn2f-1s+aβnJ0βns+βn21+aβn2f-2s,v-ns=αnJ1αn1+aαn2s+αn21+aαn2g-1s-aαnJ1αns+αn21+aαn2g-2s,where u-n(s)=∫0∞un(t)e-stdt, v-n(s)=∫0∞vn(t)e-stdt, f-1(s), f-2(s), g-1(s), and g-2(s) are the Laplace transforms of functions un(t), vn(t), f1(t), f2(t), g1(t), and g2(t), respectively. Equations (30) can be written in the suitable forms(31)u-ns=-J0βnβnf-1s+J0βnβnsf-1ss+βn21+aβn2+J0βnβn3f-2s-J0βnβn31+aβn2f-2s-aJ0βnβn1+aβn2sf-2ss+βn21+aβn2,v-ns=J1αnαng-1s-J1αnαnsg-1ss+αn21+aαn2-4+αn2J1αn4αn3g-2s+4+αn21+aαn24αn31+aαn2J1αng-2s+aJ1αnαn1+aαn2sg-2ss+αn21+aαn2.Now, using the integrals(32)∫01r2J1βnrdr=-J0βnβn,∫01r3-r8rJ1βnrdr=J0βnβn3,∫01rJ0αnrdr=J1αnαn,∫01r2-14rJ0αnrdr=-4+αn24αn3J1αn,and applying the inverse Hankel and Laplace transforms, we obtain closed forms to azimuthal velocity and axial velocities as(33)ur,t=rf1t+r3-r8f2t-2f2t∑n=1∞J1βnrβn31+aβn2J0βn+2∑n=1∞J1βnrβnJ0βn∫0tf˙1τexp⁡-βn21+aβn2t-τdτ-2a∑n=1∞J1βnrβn1+aβn2J0βn∫0tf˙2τexp⁡-βn21+aβn2t-τdτ,vr,t=g1t+r2-14g2t+2g2t∑n=1∞4+αn21+aαn2J0αnr4αn31+aαn2J1αn-2∑n=1∞J0αnrαnJ1αn∫0tg˙1τexp⁡-αn21+aαn2t-τdτ+2a∑n=1∞J0αnrαn1+aαn2J1αn∫0tg˙2τexp⁡-αn21+aαn2t-τdτ.In the above relations we have used the notationf˙(t)=df(t)/dt.By introducing notations(34)Ant=-2f2tβn31+aβn2J0βn+2βnJ0βn∫0tf˙1τexp⁡-βn21+aβn2t-τdτ-2aβn1+aβn2J0βn∫0tf˙2τexp⁡-βn21+aβn2t-τdτ,Bnt=2g2t4+αn21+aαn24αn31+aαn2J1αn-2αnJ1αn∫0tg˙1τexp⁡-αn21+aαn2t-τdτ+2aαn1+aαn2J1αn∫0tg˙2τexp⁡-αn21+aαn2t-τdτ,(33) are written as(35)ur,t=rf1t+r3-r8f2t+∑n=1∞AntJ1βnr,vr,t=g1t+r2-14g2t+∑n=1∞BntJ0αnr.It is observed that functionsAn(t) and Bn(t) have the property An(0)=Bn(0)=0.The axial angular velocity (the spin vector) is given by(36)ω→=12∇×v→=-12∂v∂re→θ+∂u∂r+ure→z=12∑n=1∞αnBntJ1αnr-r2g2te→θ+122f1t+2r2-14f2t+∑n=1∞βnAntJ0βnre→z,and the flow rate is(37)Qt=∫012πrvr,tdr=πg1t-π8g2t+2π∑n=1∞BntαnJ1αn.


## 3.2. Force-Stress Tensor and Couple-Stress Vector

Replacing (35) in (18)–(21) and performing calculations, we get the following expressions for the force-stresses and couple-stress vector:(38)τrθ=r24-2af2t+∑n=1∞βnAnt1+2aβn2J0βnr-2r∑n=1∞AntJ1βnr,τθr=r24+2af2t+∑n=1∞βnAnt1-2aβn2J0βnr-2r∑n=1∞AntJ1βnr,τrz=r2-a2rg2t-∑n=1∞αnBnt1+aαn2-1r2J1αnr,τzr=r2+a2rg2t-∑n=1∞αnBnt1-aαn2-1r2J1αnr,τrr=τθθ=τθz=τzθ=τzz=0,Mr=0,Mθ=2arf2t-∑n=1∞βn2AntJ1βnr,Mz=2ag2t-∑n=1∞αn2BntJ0αnr.It is easy to see that, fora=0 (the ordinary Newtonian fluid), the force-stress tensor becomes a symmetric tensor and the couple-stress vector is zero.


## 3.3. Particular Case (Constant Velocity and Couple Stress on the Boundary)

Let us consider the following boundary conditions:(39)f1t=p1Ht,f2t=p2Ht,g1t=q1Ht,g2t=q2Ht,where p1≥0, p2≥0, q1≥0, and q2≥0 are constants and Ht=1/2sign(t)(1+sign(t)) is the Heaviside unit step function. In this case, the derivatives of functions given by (39) are (40)f˙1t=p1δt,f˙2t=p2δt,g˙1t=q1δt,g˙2t=q2δt,δ(t) being the Dirac distribution. Functions An(t) and Bn(t) are written in simpler forms, as(41)Ant=-2p2Htβn31+aβn2J0βn+2HtβnJ0βnp1-ap21+aβn2exp⁡-βn21+aβn2t,Bnt=2q2Ht4+αn21+aαn24αn31+aαn2J1αn-2HtαnJ1αnq1-aq21+aαn2exp⁡-αn21+aαn2t.


## 4. Numerical Results and Discussion

Unsteady helical flows in the consistent theory of couple-stress fluids were considered, under general boundary conditions. By using suitable nondimensional variables, the governing flow equations are obtained in the dimensionless form. It is important to note that these equations contain as parameter the dimensionless scale flow parametera, defined as the square of the rate between the characteristic material length and the radius of circular cylinder. As a result from the studied particular problems, the scale parameter has a significant influence on the fluid behavior. Obviously, if the scale parameter equals zero, results corresponding to the Newtonian fluid are obtained. Solutions for fluid velocity, nonsymmetric force-stress tensor, and couple-stress vector were obtained using integral transforms method (Laplace transform with respect to the time variable and finite Hankel transform with respect to the radial coordinate). The axial angular velocity and the flow rate are also determined.The obtained solutions contain in their expressions the positive roots of the Bessel functionsJ0(x) and J1(x), denoted by αn and βn, roots which were generated by means of Mathcad subroutine “root(f(x),x,a,b).” In the numerical simulations, we have used n∈[500,1000], values for which the numerical approximation accuracy is very good.In our study, the azimuthal velocity, axial velocity, and the components of the couple-stress vector are given on the cylinder surface as arbitrary functions of the timet; therefore, the obtained solutions can generate solutions to various problems with practical applications.The numerical results for Figures1–3 and for Table 1 were generated under conditions f1(t)=f2(t)=g1(t)=g2(t)=H(t)=1/2sign(t)(1+sign(t)).Table 1
The influence of time values on fluid velocity components.
(a) a = 0.1 r u ( r , t ) ,   t = 1 u ( r , t ) ,   t = 5 u ( r , t ) ,   t = 10 0 0 0 0 0.1 0.093499 0.093499 0.093499 0.2 0.187262 0.187262 0.187262 0.3 0.281577 0.281577 0.281577 0.4 0.376776 0.376776 0.376776 0.5 0.473268 0.473268 0.473268 0.6 0.57157 0.57157 0.57157 0.7 0.672355 0.672355 0.672355 0.8 0.776506 0.776506 0.776506 0.9 0.885198 0.885198 0.885198 1 1 1 1  (b) a = 0.1 r v ( r , t ) ,   t = 1 v ( r , t ) ,   t = 5 v ( r , t ) ,   t = 10 0 1.158949 1.159123 1.159112 0.1 1.167755 1.167927 1.167916 0.2 1.169256 1.169421 1.16941 0.3 1.171729 1.171882 1.171872 0.4 1.175404 1.17554 1.175532 0.5 1.180568 1.180685 1.180677 0.6 1.187616 1.187711 1.187705 0.7 1.197082 1.197153 1.197149 0.8 1.209657 1.209704 1.209701 0.9 1.226044 1.226067 1.226065 1 1 1 1Figure 1
Profiles of azimuthal velocityu(r,t) and axial velocity v(r,t) for small time and different values of scale parameter a.
(a) (b) (c)Figure 2
Profiles of azimuthal velocityu(r,t) and axial velocity v(r,t) for t=1 and different values of the scale parameter a.Figure 3
Variation of flow rate with the scale parametera, respectively, with the time t.
(a) (b)Figures4 and 5 were plotted under conditions (42)f1t=g1t=0.2Ht=0.1sign⁡t1+sign⁡t,f2t=0,g2t=0.Figure 1 shows profiles of both azimuthal and axial velocities, versus radial coordinate r for different values of the scale parameter a and for three small values of the time t.Figure 4
The variation of shear stresses with the scale parametera.Figure 5
Variation of the couple stress with the scale parametera.The influence of the scale parametera on rotational velocity is significant only for very small values of the time t. It is observed from Figure 1(a) that azimuthal velocity increases with the scale parameter a. The rotational velocity of couple-stress fluid is bigger than the velocity of classical fluid, except the case of very small values of time t. In this particular case, there are values of the scale parameter, for which the couple-stress fluid flows more slowly than the Newtonian fluid in the central area of the flow domain. The influence of the scale parameter on the axial velocity is more significant than on the azimuthal velocity. It is seen from Figure 1(b) that the axial velocity increases with the scale parameter.As is apparent from Figure2 and values of Table 1, the fluid azimutal velocity is almost steady and the axial velocity has a small time-variation, for t>1. These properties are due to the exponential terms in (34) which tend fast to zero, because,(43)αn21+aαn2≥5.7831+5.783a,n≥1,α1≃2.405,βn21+aβn2≥14.6821+14.682a,n≥1,β1≃3.832(For t>1 and a≥0 the exponential terms in (34) are negligible).The influence of the scale parametera on the flow rate Q(t) in axial direction is presented in Figure 3. Obviously, the significant variation of the flow rate is for small values of the time t. It is important to note that, for large values of the scale parameter a or for large values of the time t, the flow rate in the axial direction becomes constant.The influence of the scale parametera on the components τrθ, τθr, τrz, and τzr of the force-stress tensor and on the components Mθ and Mz of the couple-stress vector is analyzed in Figures 4 and 5.The significant influence occurs for small values of the scale parameter. For these values the shear stressesτrθ and τθr and the couple-stress component Mθ have an extreme value and tend to approach zero for large values of the scale parameter.Shear stressesτrz and τzr and the couple-stress component Mz are monotone increasing/decreasing for small values of the parameter a and tend to a constant value if the parameter values are large.



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*Source: 9724381-2017-02-08.xml*