# Inertia Effects in the Flow of a Herschel-Bulkley ERF between Fixed Surfaces of Revolution

Authors: A. Walicka; J. Falicki
Journal: Smart Materials Research
 (2013)

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## Abstract

Many electrorheological fluids (ERFs) as fluids with microstructure demonstrate viscoplastic behaviours. Rheometric measurements indicate that some flows of these fluids may be modelled as the flows of a Herschel-Bulkley fluid. In this paper, the flow of a Herschel-Bulkley ER fluid—with a fractional power-law exponent—in a narrow clearance between two fixed surfaces of revolution with common axis of symmetry is considered. The flow is externally pressurized, and it is considered with inertia effect. In order to solve this problem, the boundary layer equations are used. The influence of inertia forces on the pressure distribution is examined by using the method of averaged inertia terms of the momentum equation. Numerical examples of externally pressurized ERFs flows in the clearance between parallel disks and concentric spherical surfaces are presented.


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## Body

## 1. Introduction

In recent years, the study of fluids with microstructures has gained much importance because of its numerous applications in various engineering disciplines such as chemical engineering, polymer processing, plastic forming foundry engineering, and engineering of lubrication [1–14].In machines and mechanisms systems of many industrial processes, the phenomena of a flow of viscoplastic fluids are used. One of these phenomena is a slide bearing lubrication [9, 10, 13]. Advances in technology and severe operational requirements of machines necessitated the development of improved lubricants to ensure a smooth and safe operation. Generally, viscosity of lubricants decreases with temperature. For operations under high speeds and heavy loads, oils containing high molecular weight polymers as viscosity index improvers are used to increase a load carrying capacity of the modified lubricants [9, 13].Most substances used in the lubrication technology are polymer solutions, thus, the characteristics of the bearings change when such rheological substances, known as non-Newtonian fluids, are used as lubricants. Several constitutive relations applied were used to model the non-Newtonian characteristics exhibited by some lubricants [7, 11, 13, 15, 16].Another ones of these phenomena are processes of vibration control and torque transmission. In the last years, the electrorheological fluids (abbreviated to ERFs) have acquired a great relevance for supporting vibration control and torque transmission devices, based on the characteristic dependence of their viscosity on applied electric field strength. Since their initial discovery by Winslow [17], many particle-dispersion electrorheological fluids, consisting of dielectric particles dispersed in insulating oil, have been reported. When the external electric field is imposed to an ERF, it behaves as a viscoplastic fluids [9, 18–25], displaying a field-dependent yield shear stress which is widely variable. Without the electric field, the ERF has a reversible and a constant viscosity so that it flows as a Newtonian fluid. Another salient feature of the ERF is that the time required for the viscosity variation is very short (< 0.001 sec). These attractive characteristics of the ERF provide the possibility of the appearance of new engineering technology. Recently, the application of the ERF to rotor-bearing systems has also been initiated by Basavaraja et al. [2], Jung and Choi [18], Dimarogonas and Kollias [26], and Peng and Zhu [27] and vibration control by Lee and Wereley [23], Choi et al. [24], El Wahed et al. [28], and Lim et al. [29].To describe the rheological behaviour of viscoplastic fluids in complex geometries, the Bingham model is used [8, 13, 15, 16, 21]. Recently, the nonlinear model of Shulman et al. [21] has been successfully applied [8, 13, 15, 16, 20]. The constitutive equation for this model is given as follows:
(1)τ=[τ01/n+(μγ˙)1/m]n,
where τ is the shear stress, τ0 is the yield shear stress, μ is the coefficient of plastic viscosity, γ˙ is the shear strain rate, and m and n are the power-law exponents. By reducing the coefficients in the Shulman equation one can obtain simpler models describing the flow of a viscoplastic fluid.The most popular model of the ERF is the Bingham model for whichm=n=1 in (1). Recently, the nonlinear model of Herschel-Bulkley has been successfully used to describe the viscoplastic behaviour of nonNewtonian ERFs. The constitutive equation for this model is as follows [5, 30, 31]:
(2)τ=τ0+(μγ˙)1/m.The purpose of this paper is a study of pressure distribution in a flow of the Herschel-Bulkley ERF—with a fractional power-law exponent [8]—in the clearance modelled as a narrow space between two surfaces of revolution shown in Figure 1. Using the method of averaged inertia [32] (similar method is used in [33]), the influence of inertia terms of the equations of motion and viscoplastic behaviour on this distribution is analysed. To solve the problem we will use the lubrication approximation theory to the flow in a narrow clearance [8, 10, 15, 16, 25].Figure 1
The geometry of a curvilinear clearance between two surfaces of revolution.


## 2. Equations of Motion of the ERF

The yield shear stress for the ERF varies with respect to the electric field. According to the experimental results reported in Shulman et al. [21], Shulman and Nosov [22], Jordan and Shaw [34], and Otsubo [35], the relation between the yield shear stress τ0 and the electric field strength E is given as follows:
(3)τ0(E)=α(U2h)β,
where E is the magnitude of the vector E but U and 2h are the applied voltage and the film thickness, respectively. Both parameters α and β are the experimental constants of which the range of the exponent β is 1 to 2.4 (Jordan and Shaw [34], Otsubo [35]). Other experimental data (Whittle et al. [36]) suggest that the yield shear stress for some ERFs is as follows:
(4)τ0(E)=α1(U2h)+α2(U2h)2,
where α1 and α2 are constants; for relatively high field strengths a simpler formula may be used
(5)τ0(E)=α(U2h)2,
where α is an experimental constant.The three-dimensional constitutive equation for the Herschel-Bulkley ERF has the form [15, 16, 25]:
(6)T=-p1+MA1,M=[τ0+(μA)1/m]A-1,A=[12tr(A12)]1/2,
where 1 is the unit tensor, p is the pressure, A is second invariant of the stretching tensor A1 (the first Rivlin-Ericksen tensor) defined by
(7)A1=L+LT,L=gradv,
and v is velocity vector.The general equations of motion of the Herschel-Bulkley fluid have a form:(i)
equation of continuity:(8)divv=0,(ii)
equation of momentum(9)ρdvdt=divT,or
(10)ρdvdt=-∇p+divΛ,
where
(11)Λ=MA1;
here M is a coefficient of viscoplasticity.Let us consider the Herschel-Bulkley ERF in a clearance between fixed surfaces of revolution. The flow configuration is shown in Figure1. The surfaces of revolution are defined by function R(x) which denotes the radius of the median between the surfaces, plus function h(x) which denotes the distance to each surfaces from the median, measured along a normal to the median. An intrinsic curvilinear orthogonal coordinate system x,ϑ,y is also depicted in Figure 1.The physical parameters of the lubricant flow are the velocity componentsυx, υy, and pressure p. With regard to the axial symmetry of the flow these parameters are not dependent on the angle ϑ.The assumption typical for the flow in a narrow clearance [2, 8, 15, 16]
(12)h(x)≪R(x),
can be used to make order-of-magnitude arguments for (8) ÷ (11).A further simplification comes by noting that—in accordance to the lubrication approximation—the most important changes in an annular channel occur in the normal (to the channel median) direction. This leads to the assumption that the flow is nearly parallel to the surfaces bounding the clearance, so that(13)υy≪υx,∂∂x≪∂∂y,
in an intrinsic coordinate system.If some asymptotic transformations are made, the same as in (Falicki [8], Walicka [15, 16]), these equations can be reduced to a simpler form:
(14)1R∂(Rυx)∂x+∂υy∂y=0,(15)ρ(υx∂υx∂x+υy∂υx∂y)=-dpdx+∂Tyx∂y,
where
(16)Tyx=S[τ0+(μ|∂υx∂y|)1/m],τ0(x)=α[U(x)2h(x)]β,
and S=sgn(dυx/dy). The signum (sgn) function takes on the value +1 for a positive argument and -1 for a negative argument.The order-of-magnitude arguments show that(17)p=p(x),
is a function of x only.For a majority of greases, molten polymers, mush metals, and ERFs, the values of a yield shear stress are contained in the limits:0(1)≤τ0≤0(3), where 0(α)~10α and denotes the order of magnitude, but the exponent m is equal to: m=4/3 or m=2.5, and it is not an integer number (Falicki [8], Roussel [12]).In the flow of a fluid with the yield shear stress, there exists a quasi-solid core flow bounded by surfaces laying at(18)|y|=h0forwhichtheshearstressesare|Tyx|=τ0.
Combining the expressions (16) and (18), one obtains the boundary conditions on liquid surfaces of the core flow as
(19)∂υx∂y=0fory=±h0.
The boundary conditions on solid surfaces are stated as follows:
(20)υx=υy=0fory=±h.
On the median line there is also
(21)∂υx∂y=0fory=0andhence|Tyx|=0fory=0.
Moreover, in the inlet and the outlet of the clearance conditions for the pressure can be written in the form
(22)p(xi)=pi,p(xo)=po,
where xi denotes the inlet coordinate and xo—outlet coordinate.


## 3. Solution to the Equation of Motion

Taking into account (14), one can rearrange (15) writing it in the form
(23)ρ(1R∂(Rυx2)∂x+∂∂y(υxυy))=-dpdx+∂∂y(S[τ0+(μ|∂υx∂y|)1/m]).
Then, averaging the left-hand side of (23) across the clearance thickness, solving the obtained equation and taking into account the boundary conditions (20)–(22) we get the following result for the pressure distribution [8, 15, 16, 37–39]:
(24)p=po+[S(x)-So]+[D(x)-Do],So=S[xo],Do=D[xo].
Functions S(x) and D(x) depend upon the solution Xs of the equation characteristic to the flow of a Herschel-Bulkley fluid in the clearance; this equation has a form [8, 16, 37, 39]:
(25)(X-1)m+1(m+1)(m+2)[X(m+1)+1]-KX2=0,
where
(26)K=μQ4πRh2τ0m,X=hh0,
and Q=2πR∫-h+hυxdx is the flow rate.Its analytical solution exists only for largeK(≥5) and for small K(≤0.05) (Falicki [8], Walicka [15, 16, 37], and Walicki and Walicka [39]):
(27)Xs=3m3m-1+[(m+2)K]1/m,Xs=1+[(m+1)K]1/(m+1).For intermediate values ofK the solutions of (25) may be found only numerical methods. Figures 2 and 3 illustrate the progression of solutions Xs versus K for two values of m [8]. Note that for m=1, the above values are identical with those obtained by Covey and Stanmore [40].Figure 2
Curves illustrating the progression ofXs versus K for m=2.5; continuous lines determine the exact solution.Figure 3
Curves illustrating the progression ofXs versus K for m=4/3; continuous lines determine the exact solution.The functionsS(x) and D(x) are given by expressions
(28)S(x)=-∫τ0Xshdx,D(x)=-ρ∫1Rh∂∂x(RV)dx,
where
(29)V=τ02mh3(m+1)2μ2Xs2m(1-1Xs)2m+2×[1-3m+4(m+2)(2m+3)(1-1Xs)].
Note that the final form of S(x) and D(x) depend—for U(x)=Uo=const—on the values of K and m:(i)
for large values ofK and exponent m being a natural number there are(28a)S(x)=-α(Uo2)β×[(3m3m-1)J(0;β)(x)+(Am)1/mJ(1/m;0)(x)],(29a)D(x)=-ρα2mμ2cm(Uo2)2βm×{×∑j=02m-1-i(Bm)j/mC2m-1-ijI(m)(j;β)(x)am∑i=02m(-1)i(3m3m-1)2m-iC2m+2i×∑j=02m-i(Bm)j/mC2m-ijI(m)(j;β)(x)+bm∑i=02m-1(-1)i(3m3m-1)2m-1-iC2m+2i×∑j=02m-1-i(Bm)j/mC2m-1-ijI(m)(j;β)(x)},where
(30)Am=(m+2)μQ22-βmπαmUoβm,Bm=(3m-13m)mAm,am=2(m+1)2,bm=3m+4,cm=(m+1)2(m+2)(2m+3),J(α;β)(x)=∫(R-αh-(1+2α)-β(1-αm))dx,Cβα=β!α!(β-α)!,α!=1·2·3·…·α,β>α,I(m)(j;β)(x)=∫1Rh(R1-j/mh3-2j/m-β(2m-j))′dx.(ii)
for large values ofK and exponent m being a rational number the formula for S(x) is similar but D(x) takes the form(31a)D(x)=-ρα2mμ2cm(Uo2)2βm×{am∑i=0M1(-1)iΓ2m+2i×∑j=0M1-iΓ2m-ij(3m3m-1)j(Am)2-(i+j)/mJ(m)(i+j;β)(x)+bm∑i=0M2(-1)iΓ2m+2i×∑j=0M2-iΓ2m-1-ij(3m3m-1)j(Am)2-(1+i+j)/m×J(m)(1+i+j;β)(x)∑i=0M1(-1)iΓ2m+2i},where
(32)M1=E(2m),M2=E(2m-1),Γαi=1i!∏k=1i(α+1-k),Γα0=1,J(m)(α;β)(x)=∫1Rh(R-1+(α/m)h-1+(2α/m)-βα)′dx,α={i+j,1+i+j,
and E(α) denotes an entire part of α:α-1<E(α)≤α;(iii)
for small values ofK there are(33a)S(x)=-α(Uo2)β×[J(0;β)(x)+(Cm)1/(m+1)J(1/(m+1);β)(x)],(34a)D(x)=-2ρα2mamμ2(Uo2)βm×[(Cm)2J(m+1)(0;β)(x)(Cm)(2m+5)/(m+1)-(2+dm)(Cm)(2m+3)/(m+1)J(m+1)(1;β)(x)+3(1+dm)(Cm)2(m+2)/(m+1)J(m+1)(2;β)(x)-6dm(Cm)(2m+5)/(m+1)J(m+1)(3;β)(x)],where
(35)Cm=(m+1)μQ22-βmπαmUoβm,dm=3m+4(m+2)(2m+3),J(m+1)(j;β)(x)=∫1Rh(R-(m+1+j)/(m+1)h-(m+1+2j)/(m+1)-2βm)′dx.
For τ0(x)=τ0=const (the coefficient β=0 but α=τ0) there are [8](iv)
for large values ofK and exponent m being a natural number:(28b)S(x)=-τo[(3m3m-1)J(0)(x)+(Am)1/mJ(1/m)(x)],(29b)D(x)=-ρτ02mμ2cm×{∑j=02m-1-i(Bm)j/mC2m-1-ijI(m)(j)am∑i=02m(-1)i(3m3m-1)2m-iC2m+2i×∑j=02m-i(Bm)j/mC2m-ijI(m)(j)+bm∑i=02m-1(-1)i(3m3m-1)2m-1-iC2m+2i×∑j=02m-1-i(Bm)j/mC2m-1-ijI(m)(j)},where
(36)Am=(m+2)μQ4πτ0m,J(α)(x)=J(α;0)(x),I(m)(j)(x)=I(m)(j;0)(x);(v)
for large values ofK and exponent m being a rational number the formula for S(x) is similar and D(x) takes the form(31b)D(x)=-ρτ02mμ2cm×{am∑i=0M1(-1)iΓ2m+2i×∑j=0M1-iΓ2m-ij(3m3m-1)j(Am)2-(i+j)/mJ(m)(i+j)+bm∑i=0M2(-1)iΓ2m+2i×∑j=0M2-iΓ2m-1-ij(3m3m-1)j(Am)2-(1+i+j)/m×J(m)(1+i+j)∑i=0M1(-1)iΓ2m+2i},where
(37)J(m)(α)(x)=J(m)(α;0)(x);(vi)
for small values ofK there are(33b)S(x)=-τ0[J(0)(x)+(Cm)1/(m+1)J(1/(m+1))(x)],(34b)D(x)=-2ρτ02mamμ2×[(Cm)2A(1)(x)(Cm)(2m+5)/(m+1)-(2+dm)(Cm)(2m+3)/(m+1)J(m+1)(1)(x)+3(1+dm)(Cm)(2(m+2))/(m+1)J(m+1)(2)(x)-6dm(Cm)(2m+5)/(m+1)J(m+1)(3)(x)],where
(38)A(1)(x)=A(1;0)(x),J(0)(x)=J(0;0)(x),J(1/(m+1))(x)=J(1/(m+1);0)(x),J(m)(α)(x)=J(m)(α;0)(x),J(m+1)(j)(x)=J(m+1)(j;0)(x).
The prim denotes the differentiation with respect to x.


## 4. Examples of Flow and Graphic Presentation of Some Results

Taking into account the results obtained in the previous section we will present the pressure distribution in the clearance of constant thickness between two parallel disks as shown in Figure4; to this aim introduce the following dimensionless parameters:
(39)x~=xxo,R~=RRo=x~,p~=ρ(p-po)(2h)4/mQ2-2/m(2π)2-2/mμ2/mRo4-2/mRλ,Rλ=ρQ2-1/m(2π)2-1/mμ1/mRo3-1/m(2h)1-2/m,KSV=2βm-2μQhβm-2παmRoUoβm,
if β=0, then α=τ0=const and
(40)KSV=μQ4πRoh2τ0m;
here Rλ is the modified Reynolds number, and KSV is the ER plasticity number (de Saint-Venant number). Note that large values of the ER plasticity number correspond to the flow with small core (small h0/h).Figure 4
Clearance between two parallel disks.Nondimensional formula for pressure distribution in ER flow of the Herschel-Bulkley fluid has the form(41)p~=S~(x~)-S~o+D~(x~)-D~o;
this formula may be used to model the pressure distribution in a footstep bearing.Note that form=1, all the above formulae represent the Bingham ERF flow. The nondimensional pressure distributions for the Herschel-Bulkley ERF flow in a clearance between two parallel disks are presented in Figures 5, 6, 7, 8, and 9. The values of nondimensional parameters such as Rλ, KSV, and m, are defined on the experimental data taken from the measurements presented in [8, 12].Figure 5
Pressure distributionsp~ in the clearance between parallel disks for different values of Rλ, large value of KSV=5, and exponent 2m  (m=2.5) being a natural number.Figure 6
Pressure distributionsp~ in the clearance between parallel disks for different values of Rλ, large value of KSV=5, and exponent 2m  (m=4/3) being a rational number.Figure 7
Pressure distributionsp~ in the clearance between parallel disks for different values of Rλ, small value of KSV=0.01, and exponent 2m  (m=4/3) being a rational number.Figure 8
Pressure distributionsp~ in the clearance between parallel disks for different values of Rλ, small value of KSV=0.05, and exponent 2m  (m=4/3) being a rational number.Figure 9
Pressure distributionsp~ in the clearance between two parallel disks for constant value of Rλ(=1), different values of KSV, and exponent 2m  (m=4/3) being a rational number.Let us consider now the pressure distribution in the clearance of constant thickness between two concentric spheres surfaces shown in Figure10. To this aim introduce the following dimensionless parameters:
(42)x~=xRs=φ,R~=RRs=sinφ,p~=ρ(p-po)(2h)4/mQ2-2/m(2π)2-2/mμ2/mRs4-2/mRλ,Rλ=ρQ2-1/m(2π)2-1/mμ1/mRs3-1/m(2h)1-2/m,KSV=2βm-2μQhβm-2παmRsUoβm.
Note that if β=0 then α=τ0=const and KSV=μQ/4πRsh2τ0m. The nondimensional pressure distributions are also given by formula (41); at present it models the pressure distribution in a spherical bearing.Figure 10
Clearance between concentric spherical surfaces.The pressure distributions for the Herschel-Bulkley ERF flow in a clearance between concentric spherical surfaces are presented in Figures11, 12, 13, and 14.Figure 11
Pressure distributionsp~ in the clearance between concentric spherical surfaces for different values of Rλ, large value of KSV=5, and exponent 2m  (m=2.5) being a natural number.Figure 12
Pressure distributionsp~ in the clearance between concentric spherical surfaces for different values of Rλ, large value of KSV=5, and exponent 2m  (m=4/3) being a rational number.Figure 13
Pressure distributionsp~ in the clearance between concentric spherical surfaces for different values of Rλ, small value of KSV=0.01, and exponent 2m  (m=4/3) being a rational number.Figure 14
Pressure distributionsp~ in the clearance between concentric spherical surfaces for different values of Rλ, small value of KSV=0.05 and exponent 2m  (m=4/3) being a rational number.


## 5. Conclusions

From the general considerations, formulae and graphs presented here for the Herschel-Bulkley ERF flows in the narrow clearance of constant thickness between parallel disks and concentric spherical surfaces shown in Figures4 and 10 one may conclude that the pressure values(i)
decrease with the increase of the modified Reynolds numberRλ,(ii)
increase with the decrease of the nonlinearity indexm,(iii)
increase with the decrease of the de Saint-Venant ER numberKSV,(iv)
are larger between concentric spherical surfaces than these ones between parallel disks for2m being natural numbers and large values of KSV,(v)
form being rational numbers it is inversely.For small values of the de Saint-Venant ER numberKSV the influence of the Reynolds number Rλ  on the pressure values is inconsiderable. Therefore, for these values of KSV the Herschel-Bulkley ERF flows may be considered without inertia effects. It is seen from the results obtained here, that the pressure values are larger for the flows between concentric spherical surfaces than these values for the flows between parallel disks.Generally, it may be concluded that for the Herschel-Bulkley ERF flows in the clearance between two surfaces of revolution the pressure values(i)
are larger for the flow in the clearances with curvilinear generating lines than these ones for the flow in clearances with rectilinear generating lines for the nonlinearity index2m being natural numbers; for m being rational numbers this phenomenon progresses inversely.Note that the results obtained here forτ0=const and m being natural numbers are similar to those of the work by Walicki and Walicka [39].The formulae for the pressure distribution obtained previously may be used to model its distribution in slide thrust bearings—of arbitrary curvilinear shapes—lubricated by ER lubricants.



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