# Series Solutions of Lifting and Drainage Problems of a Nonisothermal Modified Second Grade Fluid Using a Vertical Cylinder

Authors: M. Farooq; M. T. Rahim; S. Islam; M. Arif
Journal: Journal of Applied Mathematics
 (2014)

---

## Abstract

We studied the thin film flow for lifting and drainage problems using an incompressible, nonisothermal modified second grade
fluid. We developed nonlinear differential equations from momentum and energy equations, respectively. Series solutions for both lifting and drainage
problems are obtained. Expressions for the velocity profile, temperature distribution, volume flux, average film velocity, and shear stress on cylinder for both cases are obtained. Effects of different parameters on the flow problems are presented graphically.


---

## Body

## 1. Introduction

Thin film flows of linearly viscous fluids have been the subject of deep investigation for the last few decades. This is because of the widespread applications of such flows, both industrial and natural. The literature on thin film flows for Newtonian fluids is far-reaching but no proper attention has been given to these flows concerning non-Newtonian fluids. Siddiqui et al. [1–4] and Hayat et al. [5–7] made their valuable contributions to the thin film flows of non-Newtonian fluids. The most common examples of the non-Newtonian fluids are paints, clay coatings and other suspensions, drilling mud, blood, shampoo, ketchup, polymer melts, certain oils and greases, clastomers, and many other emulsions. Because of the fact that a single constitutive equation cannot exhibit all properties of non-Newtonian fluids, many constitutive equations or models have been proposed. Among these, the differential type fluids [8, 9] have received exceptional consideration. The modified second grade fluids, a subclass of the fluids of differential type, have been studied effectively in various types of flow situations [10–13].The flow of Newtonian fluids is described by the Navier-Stokes equations which are nonlinear partial differential equations and the exact solutions of these equations are very rare. The importance of exact solutions is not only because of the fact that they are the solutions of some fundamental flows but also because they are used as accuracy check for experimental, asymptotic, and numerical methods. The governing equations that describe the flow of non-Newtonian fluids are more complex and nonlinear than the Navier-Stokes equations and so the exact solutions obtained for these equations are considered as milestone.Many researchers have been attracted by the flow and heat transfer inside thin films [14–16]. This is due to their vast applications in engineering and industry such as food stuff processing, fiber and wire coating, reactor fluidization, transpiration cooling, polymer processing, gaseous diffusion, heat pipes, and fluidic cells of many biological and chemical detection systems. Lavrik et al. [17] considered the problem of chambers for chemical and biological detection systems such as fluidic cells for biological and chemical microcantilever. In majority of the problems related to flow and heat transfer studies, the power-law fluid model is taken as the non-Newtonian fluid. Only unpretentious interest has been devoted to the studies where the effects of viscous dissipation are incorporated, although its importance has been shown in many cases such as polymer processing.The aim of the present paper is to discuss two different flow problems for the modified second grade fluid motion on a vertically upward moving cylinder and down a stationary vertical cylinder. Series solutions subject to appropriate boundary conditions of the modeled highly nonlinear differential equations are obtained in both cases. Because of their practical significance, expressions for velocity profile, temperature distribution, volume flux, average film velocity, and shear stress on cylinder are also calculated. To the best of our knowledge, these kinds of problems with heat transfer have not yet been reported. At the end, graphical results are presented for different physical parameters appearing in the solution. We observed that modified second grade fluid shows the power law model results for steady case.This paper is organized as follows. In Section2, basic equations for the modified second grade fluid model are given. Section 3 provides formulation and solution of the lifting problem. Section 4 calculates volume flux, average film velocity, and shear stress on cylinder for lifting problem. Section 5 consists of equations and their solutions for drainage problem. Volume flux, average film velocity, and shear stress on cylinder for drainage problem are calculated in Section 6. Section 7 contains results and discussion, while concluding remarks are given in Section 8.


## 2. Basic Equations

The basic equations governing the flow of an incompressible, non-Newtonian fluid including thermal effects are [18]
(1)div⁡V=0,(2)ρDVDt=ρf-gradp+div⁡T,(3)ρCpDΘDt=κ∇2Θ+T·L,
where V is velocity vector and D/Dt is the material time derivative defined as
(4)DDt(*)=(∂∂t+V·∇)(*),
where ρ is constant density, f is body force, p is dynamic pressure, Θ is temperature, Cp is specific heat constant, κ is thermal conductivity, and T is extra stress tensor which is defined differently for different fluids. We have chosen the modified second grade fluid for our study. The extra stress tensor for such a fluid is given by
(5)T=μeffA1+α1A2+α2A12,
where α1 and α2 are normal stress coefficients, μeff is an effective viscosity for modified second grade fluid as a function of the shear rate and is defined as
(6)μeff=η(12tr⁡A12)m/2,
where η is flow consistency index and m is flow behavior index. The Rivlin Ericksen tensors, A1 and A2, are defined as
(7)A1=L+LT,L=gradV,A2=DA1Dt+A1L+LTA1.
It is worthwhile to mention here that, for m<0, the fluid is pseudoplastic or shear thinning; for m>0, the fluid is dilatant or shear thickening; for m=0, we obtain second grade fluid model. On the other hand, if α1=α2=0, (5) reduces to the power-law model. Furthermore, if m=α1=α2=0, we obtain the classical Newtonian model. It is important to note that the flow behavior index m has the limits -1<m<1 [19].


## 3. Lifting Problem

A modified second grade fluid is flowing on the outer surface of an infinitely long vertical cylinder of radiusR which moves vertically upward with constant speed V0 as illustrated in Figure 1(a). The flow is in the form of a thin uniform axisymmetric film of thickness δ in contact with stationary air. Gravity effects are in downward direction. We choose z-axis in the middle of the cylinder and r is normal to it. Furthermore, we suppose that motion is steady and there is no variation with respect to the component Θ. Thus, velocity field is chosen as
(8)V=[0,0,w(r)],Θ=Θ(r).
Using profile (8), the continuity equation (1) is identically satisfied and the momentum equation (2) reduces to
(9)0=-∂p∂r+(2α1+α2)1rddr[r(dwdr)2],(10)0=-1r∂p∂θ,(11)0=-ρg-∂p∂z+ηrddr[r(dwdr)m+1].
The velocity profile is obtained from (11). If we consider p atmospheric pressure and therefore a constant, then we can take ∂p/∂z=0; therefore, (11) becomes
(12)1rddr[r(dwdr)m+1]=ρgη,
which is highly nonlinear ordinary differential equation. Using profile (8) in the energy equation (3), we obtain
(13)κ[d2Θdr2+1rdΘdr]+η[dwdr]m+2=0.
The corresponding boundary conditions are
(14)atr=R+δ,dwdr=0,dΘdr=0,(15)atr=R,w=U,Θ=Θ0.
Integrating (12) with respect to r and applying the free space boundary condition (14), we get
(16)dwdr=-(ρg2η)1/(m+1)(R+δ)2/(m+1)r1/(m+1)[1-(rR+δ)2]1/(m+1).
In this expression, the sign of dw/dr is always opposite to that of r, since velocity decreases with the increase of r. The above equation can also be written as
(17)dwdr=-(ρg2η)1/(m+1)(R+δ)2/(m+1)×∑i=0∞‍(1m+1i)(-1)i(R+δ)2ir2i-1/(m+1).
For m≠0, solutions of (17) and (13), when boundary conditions (14) and (15) are applied, are
(18)w=U-(ρg2η)1/(m+1)×∑i=0∞‍(1m+1i)fffff×(-1)i+1R(2i+m/(m+1))(2i+m/(m+1))(R+δ)2i-2/(m+1)fffff×[1-(rR)2i+m/(m+1)],(19)Θ=Θ0+ηκ(ρg2η)(m+2)/(m+1)×∑i=0∞‍(m+2m+1i)(-1)i(2i+m/(m+1))(R+δ)2i-2((m+2)/(m+1))dddd×[R2i+m/(m+1)(2i+m/(m+1)){1-(rR)2i+m/(m+1)}ffifffff+(R+δ)2i+m/(m+1)ln⁡(rR)R2i+m/(m+1)(2i+m/(m+1))].
Introducing the nondimensional parameters,
(20)r*=rR,w*=wU,δ*=δR,Θ*=Θ-Θ0Θ1-Θ0,St=ρgR2Uμeff,Br=U2μeffκ(Θ1-Θ0),
where Θ1 is reference temperature, St is Stokes number, and Br is Brinkman number, (18) and (19), after dropping “*”, become
(21)w=1+(St2)1/(m+1)×∑i=0∞‍(1m+1i)(-1)i(2i+m/(m+1))(1+δ)2i-2/(m+1)gggg×[1-r2i+m/(m+1)],Θ=Br(St2)(m+2)/(m+1)×∑i=0∞‍(m+2m+1i)(-1)i(2i+m/(m+1))(1+δ)2idddd×[1(2i+m/(m+1)){1-r2i+m/(m+1)}ddddddd+(1+δ)2i+m/(m+1)ln⁡r1(2i+(m/m+1))],
which are the dimensionless velocity profile and temperature distribution for modified second grade fluid, respectively.Geometry of the problems.
(a)
(b)


## 4. Volume Flux, Average Film Velocity, and Shear Stress on Cylinder for Lifting Problem

Volume flux,Q, in cylindrical coordinates is given by
(22)Q=∫RR+δ‍∫02π‍rw(r)dθdr.
Using profile (18), (22) becomes
(23)Q=πU(R+δ)2[1-(RR+δ)2]-2π(ρg2η)1/(m+1)×∑i=0∞‍(1m+1i)(-1)i(2i+m/(m+1))(R+δ)2i-2/(m+1)ddid×[(R+δ)2i+m/(m+1)+2(2i+m/(m+1)+2)dddddd×{1-(RR+δ)2i+m/(m+1)+2}dddddd-(R+δ)22R2i+m/(m+1){1-(RR+δ)2}].
The average film velocity, V-, is defined as
(24)V-=Qπ[(R+δ)2-R2].
Therefore, V- for the upward moving cylinder is given by
(25)V-=U-(ρg2η)1/(m+1)×∑i=0∞‍(1m+1i)(-1)i(2i+m/(m+1))(R+δ)2i-2/(m+1)fffff×[(R+δ)2i+m/(m+1)(2i+m/(m+1)+2)fffffifff×{1-(R/(R+δ))2i+m/(m+1)+22[1-(R/(R+δ))2]}ffffiffff-R2i+m/(m+1)(R+δ)2i+(m/m+1)(2i+(m/m+1)+2)].
Shear stress on cylinder is
(26)Trz|r=R=-ρg(R+δ)22R[1-(RR+δ)2].


## 5. Drainage Problem

Again, we consider a modified second grade fluid now falling on the outer surface of a stationary infinitely long vertical cylinder of radiusR (Figure 1(b)). The flow is in the downward direction due to gravity. The governing equations (2) and (3) become
(27)1rddr[r(dwdr)m+1]=-ρgη,κ[d2Θdr2+1rdΘdr]+η[dwdr]m+2=0,
along with the corresponding boundary conditions
(28)atr=R+δ,dwdr=0,dΘdr=0,(29)atr=R,w=0,Θ=Θ0.
Solving (27) for m≠0, with boundary conditions (28) and (29), we obtain
(30)w=(ρg2η)1/(m+1)×∑i=0∞‍(1m+1i)(-1)i+1R(2i+m/(m+1))(2i+m/(m+1))(R+δ)2i-2/(m+1)fffff×[1-(rR)2i+m/(m+1)],(31)Θ=Θ0+ηκ(ρg2η)(m+2)/(m+1)×∑i=0∞‍(m+2m+1i)(-1)i(2i+m/(m+1))(R+δ)2i-2((m+2)/(m+1))ddd×[R2i+m/(m+1)(2i+m/(m+1)){1-(rR)2i+m/(m+1)}fffffff+(R+δ)2i+m/(m+1)ln⁡(rR)R2i+(m/m+1)(2i+(m/m+1))].
Using the nondimensional parameters, (20), (30), and (31) in the dimensionless forms, when the “*” is dropped, become
(32)w=(St2)1/(m+1)×∑i=0∞‍(1m+1i)(-1)i+1(2i+m/(m+1))(1+δ)2i-2/(m+1)dddd×[1-r2i+m/(m+1)],Θ=Br(St2)(m+2)/(m+1)×∑i=0∞‍(m+2m+1i)(-1)i(2i+m/(m+1))(1+δ)2iffff×[1(2i+m/(m+1)){1-r2i+m/(m+1)}fffffff+(1+δ)2i+m/(m+1)ln⁡r1(2i+(m/m+1))].


## 6. Volume Flux, Average Film Velocity, and Shear Stress on Cylinder for Drainage Problem

Volume flux,Q, is calculated from (22) by using (30), which is
(33)Q=2π(ρg2η)1/(m+1)×∑i=0∞‍(1m+1i)(-1)i(2i+m/(m+1))(R+δ)2i-2/(m+1)dddd×[(R+δ)2i+m/(m+1)+2(2i+m/(m+1)+2)ddddddd×{1-(RR+δ)2i+m/(m+1)+2}ddddddd-(R+δ)22R2i+m/(m+1){1-(RR+δ)2}].
The average film velocity, V-, is then given by
(34)V-=(ρg2η)1/(m+1)×∑i=0∞‍(1m+1i)(-1)i(2i+m/(m+1))(R+δ)2i-2/(m+1)dddd×[(R+δ)2i+m/(m+1)(2i+m/(m+1)+2)ddddddd×{1-(R/(R+δ))2i+m/(m+1)+22[1-(R/(R+δ))2]}dddddddd-R2i+m/(m+1)(R+δ)2i+(m/m+1)(2i+(m/m+1)+2)].
Shear stress on cylinder is
(35)Trz|r=R=ρg(R+δ)22R[1-(RR+δ)2].


## 7. Results and Discussion

In this paper, we studied thin film flows for lifting and drainage problems using vertical cylinders for an incompressible, nonisothermal modified second grade fluid. The series solutions of differential equations for both cases are obtained. Dependence of the flow behavior index,m, Stokes number, St, and Brinkman number, Br, on velocity profile and temperature distribution is investigated. The effect of flow behavior index, m, on velocity field and temperature distribution for both problems is illustrated graphically through Figures 2 and 5. Figures 3 and 6 show the effect of St and Br numbers on velocity and temperature for dilatant fluids. Graphs for pseudoplastic fluids are given in Figures 4 and 7. For the case of lifting, Figure 2(a) shows that as the fluid is becoming thicker, the magnitude of velocity decreases and vice versa, while the drainage problem is given in Figure 2(b). In Figure 5, we see that as the fluid is becoming thicker, the temperature increases. Figure 3(a) is showing the effect of Stokes number, St, on velocity profile for lifting problem. We see that as St is increasing, the gradient of velocity decreases, where as in the case of drainage problem Figure 3(b) shows that as St increases, velocity also increases. The effect of Stokes number, St, and Brinkman number, Br, on heat transfer for both problems can be observed in Figures 6(a) and 6(b), respectively. It is evident from these figures that increasing St and Br numbers results in the rise of temperature. Respective graphs for shear thinning fluids are shown in Figures 4 and 7.Effect ofm on velocity for lifting problem (a) and drainage problem (b).
(a)
(b)Effect on shear thickening fluid ofSt number on velocity for lifting problem (a) and drainage problem (b).
(a)
(b)Effect on shear thinning fluid ofSt number on velocity for lifting problem (a) and drainage problem (b).
(a)
(b)Figure 5
Effect ofm on temperature for both problems.Effect on shear thickening fluid ofSt and Br numbers on temperature for both problems.
(a)
(b)Effect on shear thinning fluid ofSt and Br numbers on temperature for both problems.
(a)
(b)


## 8. Conclusion

We have considered equations for steady, nonisothermal thin film flows for lifting and drainage problems in cylindrical coordinates of modified second grade fluid. In both cases, series solutions are obtained. Explicit expressions are given, in both problems, for the velocity field, temperature distribution, volume flux, average film velocity, and shear stress on cylinder. It is worth mentioning that normal stresses have no contribution for steady modified second grade fluid flow. We do not see any contribution of modified second grade fluid model verses power law model as solutions (18) and (29) for velocity profiles and also (19) and (30) for temperature distributions are the same as those of the power law fluid.



---

*Source: 842650-2014-02-24.xml*