OURNAL

Alexandria Engineering Journal (2015) 54, 935-939

HOSTED BY

Alexandria University Alexandria Engineering Journal

www.elsevier.com/locate/aej www.sciencedirect.com

SHORT COMMUNICATION

Exact solution of unsteady flow generated by c^Ma*

sinusoidal pressure gradient in a capillary tube

M. Abdulhameed a,% D. Vierub, R. Roslana, S. Shafiec

a Centre for Research in Computational Mathematics, Universiti Tun Hussein Onn Malaysia, 86400 Parit Raja, Batu Pahat, Johor, Malaysia

b Department of Theoretical Mechanics, Technical University of Iasi, Iasi R-6600, Romania c Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi, 81310 Skudai, Malaysia

Received 9 October 2014; revised 22 June 2015; accepted 22 July 2015 Available online 8 September 2015

KEYWORDS

Oscillating flow; Second grade fluid; Capillary tube; Exact solution

Abstract In this paper, the mathematical modeling of unsteady second grade fluid in a capillary tube with sinusoidal pressure gradient is developed with non-homogenous boundary conditions. Exact analytical solutions for the velocity profiles have been obtained in explicit forms. These solutions are written as the sum of the steady and transient solutions for small and large times. For growing times, the starting solution reduces to the well-known periodic solution that coincides with the corresponding solution of a Newtonian fluid. Graphs representing the solutions are discussed. © 2015 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

Generally, many engineering fluids, e.g. dilute polymer, pastes, slurries, synovial, paints exhibit numerous strange features, e.g. shear loss/thickening and display of elastic effects which cannot be well described by the Navier-Stokes equations. Various rheological models have been proposed to portray their non-Newtonian flow behavior. The fluids of a differential type have acquired special status due to their elegance, Dunn and Rajagopal [1]. One such type of rheological model is the differential type fluid model and second grade fluid is one of the subclasses of these differential type fluid models. Due to its ability

* Corresponding author. Tel.: +60 146183613.

E-mail address: moallahyidi@gmail.com (M. Abdulhameed).

Peer review under responsibility of Faculty of Engineering, Alexandria

University.

in successfully capturing various non-Newtonian effects, it has been the subject of many investigations [2-7], etc.

Recently, some of the newly developed approximate analytical tools have been employed by various researchers to solve several basic flow problems of second grade fluid in cylindrical geometry, and the approximate solutions were found for the velocity profiles [8-11]. All these quoted analyses of the fluid flow take place due to the drag of boundary in a bath of fluid. However, no solution expressions were obtained for the flow rate that is solely due to the oscillating pressure gradient. The task of the present paper is to venture further in this regime. For what we are interested to examine the unsteady second grade fluid in a capillary round tube driving by a sinusoidal pressure gradient. Due to the complexity of the governing equations, finding accurate solutions is not easy. Therefore, we made an attempt to obtain an exact solution to the differential equation. A solution for the velocity field is derived as the sum of steady and transient solutions, describing the

http://dx.doi.org/10.1016/j.aej.2015.07.014

1110-0168 © 2015 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

motion of the fluid for small and large times using exact analysis. This review would serve as an important reference for researchers in this area.

2. Formulation of the problem

Consider an incompressible, laminar, viscoelastic fluid pulsating flow in a capillary tube with a radius of r0 driven by a pressure gradient that varies sinusoidally with time as

Vp = ez(B0 + Bi exp (imt)).

where the pressure gradient contains a steady and a pulsating part, of amplitudes B0 and B1, respectively. The unit vector ez is in the z-direction parallel to the flow, m is the frequency of the pressure gradient, t is the time and i — is the imaginary constant. Using the pressure gradient given in Eq. (1), the cosine and sine oscillations can be treated by taking the real and imaginary parts of the pressure gradient Vp. Fig. 1 shows the physical configuration.

The Cauchy stress tensor T for an incompressible homogeneous second grade fluid is given by the constitutive equations

T — —pI + S, S — iA1 + a1 A2 + a2 Aj,

where I is the identity tensor, p is the pressure, S is the extrastress tensor, i is the dynamic viscosity, and a1, a2 are normal stress moduli and A1 and A2 are the kinematic tensors defined as

Ai = (grad u) + (grad u)T,

Aj — dt Ai + Ai (grad u) + (grad u) Ai,

where d is the material time derivative and u is velocity vector.

The fluid velocity through capillary tube is moving with velocity of the form

u — u(r, t) — u(r, t)ez, (5)

where ez is the unit vector along z-axis.

Introducing Eq. (5) into Eq. (2), we find that

Tr,z = l + ai

d\ du(r, t)

'Ft dr '

By considering the pressure gradient in the axial direction, the balance of the linear momentum in the absence of body forces leads to the following equation

du ( d A

q — — (Bo + Bi exp (imt)) + i — + H T-,z

Eliminating Tr,z between Eqs. (6) and (7), we obtain

du (du i du\

q -Qt— (Bo+ Bi exp(ixt)) + H +- d~r)+ ai

d d2 u i du

dt dr2 r dr The initial and boundary conditions are

u — 0 at t = 0' for 0 6 r 6 r0,

(9) (i0)

u — 0 at r = r0, for all t p 0. (ii)

Consider the following dimensionless quantities

du — 0 at r — 0, for all t p 0,

u „ ai u = —, a

It „ r „ xrj . , t ——^ ' r —— ' x —-. (i2)

we obtain the dimensionless initial-boundary values problem (dropping * the notation)

du du 1 du

- — r(Bo + B1exp (imt))+ dpr + 7 @r

d d2 u i du dt V d r2 r dr

u — 0 at t — 0, for 0 6 r 6 i,

du — 0 at r — 0, for all t p 0,

u = 0 at r— i, for all t p 0.

where c — -rL is a constant that controls the amplitude of the

' ium f

pressure fluctuation and um — is the cross-sectional

mean velocity for the time-averaged flow.

3. Solution technique

3.1. Steady solution

Assume that the solution to Eq. (13) is of the form

u(r, t) — us(r) + ut(r, t); (17)

where us is a steady solution and ut is the transient solution component. Note that, if we allow t !i, we obtain the steady solution.

Substituting Eq. (17) into (13) we have

dus dut . ^ .. ,, df + di — C(Bo + B1 exp (ixt))

d us d ut i dus i dut dr dr r dr r dr

d d ut i dut dt dr r dr

Considering — 0, Eq. (i8) can be separated into two equations

Figure 1 The physical configuration.

d us i dus

dr r dr

d ut i dut d d ut i dut dr r dr dt dr r dr dut

— "dt — — cBi exp (ixt),

with initial and boundary conditions

us(r) — -Ut(r, 0), dus @ut

— = — = 0 at r — 0, or or

us — ut — 0 at r — 1. The solution of Eq. (19) is obtain as 1

= 4 CM1 - r1) •

(21) (22)

We realize that the solution us in Eq. (24) provides a parabolic velocity profile that coincides with the corresponding solution of a Newtonian problem and is independent of the second-grade parameter a.

3.2. Transient solution

Consider that the transient solution can be written as a function

ut(r, t) — yßius(r, t).

Substituting Eq. (25) into Eq. (20), gives

@2ud 1 @ud 0 /@2ud 1 dus\ dus or2 r or ot or2 r or ot — — exp (ixt),

applying finite Hankel transform technique, Eq. (26) can be expressed as

rô2ud 1 dud d icPus 1 dus @r2 r dr dt\ dr2 r dr

dus ~dt

— Jo[-exp (iat)},

where J0 is the first class Bessel function with the following properties

rd2g(r) , 1 dg(r)

c r2 r c r

'dg(r)

— -knJ0[g(r)},

@Jo[g(r)} dt ;

— exp (ixt) Jo(-exp (iat)) —---Ji (kn

and kn are the positive roots of J0(kn). Eq. (27) reduce to

-k2nJo[us(r, t)} — ak;

2 dJo[us(r, t)} dJo[us(r, t)}

— exp (iat)

J1(kn).

Take Jo[us(r, t)} — F(kn, t), Eq. (31) gives

F(kn, t)

A dF(kn, t) exp (iat) * + k2 ) -dt- —-^-J1(kn

Eq. (32) is a non-homogeneous differential equation with constant coefficients. The general solution is given by

F(kn, t) — C exp

J1 (kn) exp (iat)

ak2 + 1 kn [k2n + iX(akn + 1)] '

where C is a constant. The constant can be determined through the initial conditions.

Using Eqs. (21) and (25) we have

(r, 0) —— Ä- (1 — r2)

Applying the finite Hankel transform with J0

F(kn, 0)= J0{us(r, 0)] = - r{ 1 - r2) J0(knr)dr, (35)

4B1 J0

The following simplification is obtained

F(kn, 0)= J(){us(r, 0)]

B fkn B fkn

= zJ0(z)dz + z3J0(z)dZ; (36)

4B1 kn 0 4B1 kn 0

where z = knr. Using the following properties

J zJ0(z)dz — zJ1(z), J z3Jo(z)dz — z3 Jx (z) — 2z1Jl(z),

we obtain

F(kn, 0) —

(26) But, using

J2(kn).

Jn— 1 (z) + Jn+1 (z) —— Jn(z)

with n — 1 and z — kn, implies J0(kn) + J2(kn) — j~Ji(k„) with J0(kn) — 0 (by hypothesis) and

B0 Jl(kn)

F(kn, 0) — —-

Making t — 0 into Eq. (33) and using condition (40), we get

B0 J1 (kn) J1 [kn] exp (iat)

B kn kn K + ix{akl + 0]

The solution of Eq. (32) with initial condition (40) corresponding to (21) could immediately follow

A B0 J1(kn) I klt \

F(kn; t) = -B—73 exp —

B1 kn \ akn + 1J

kn [kl + m(akn +1)]

exp (iat) — exp —

akn +1

B0 J1(kn) n

F(kn,t) —— B0—expl—*£+

(—-ft-) \ akkn + V

[kn — ^K + ^J1(k„) J kit \ ^ *k2n + 1)

kn [k4 + a.k2n + 1)2]

[k2n — im(xk2n + 1)]J1(krl kn [k4 + a2(ak2n + 1)2]

[cos (at) + i sin (at)\,

Further,

F(k„, t) — —

B0 Ji(k„)

Bi k exp| —.. ,2

V akn + V

J i (k„)k:

■ exp

V akn + V

k. [k4 + x2( akj + i)2]

Ji(kn)kjjcos (xt) + Ji(kn)XakJj + i) sin (xt) k. [k. + x2 (ak. + i)2]

< Ji(kn)x(akjj + i) exp / k.t !

k [k. + x2(ak2 + i)2] V ak. + i )

Ji(kn)k2 sin (xt) — Ji (kjx^k. + i) cos (xt)

k. [k4 + x2( ak2 + i)2]

Implies

F(k„, t) — Fi(k„, t) + iF2(k„, t). (45)

Observations for t ! 1 (The ''permanent solution" or solution for large time)

Fi (k«' t)

Ji(kn)kj; cos (xt) + Ji (kjx^k2 + i) sin (xt)

k. [k. + x2(ak2 + i)2]

A Ji(k»)k;;sin (xt) — -MkM^ + ^cos (xt)

F2 (k.' t) ! -^-(----ri-. (4')

k. [k. + x2(ak2 + i)2]

To observe that we have obtained only these solutions of Eq. (32). Since for an experiment, these solutions are important. Inverse Hankel transform of Eq. (44), gives

ut(r, t) — a(r, t) + a (r) cos (xt) + b (r) sin (xt)

+ i{è(r, t) + a (r) sin (xt) — b (r) cos (xt)}, (48)

a(r, t) — 2yB^

J0(k.r)

Î=Î Ji(k.)

Bik. k4 + x2 (ak. + i)2J

k.t "ak. + i

a(r) — 2cBi^

J0 (k„

Ji(k») k4 + x2(ak2 + i)

,, A , „ ^ J0(k„r)

b(r't)—2cBi g

"K + i)

bi(r) — 2cBi^

J0 (k.

k„ k4 + x2(ak2 + i)

Ji(k.) k.[k4 + x2(ak. + if]

where J1 is the Bessel function of first kind of order one and kn is the eigenvalue of the Bessel function of first kind of order zero.

The solution above is to the best of the present author's knowledge, the first known solution of the transient second grade problem when oscillatory pressure gradient is considered.

It should be noted that the solution (48) can be reduced to the permanent solution also called steady-state solution, for large time t ! 1, a(r, 1) ! 0, b(r, 1) ! 0.

Figure 2 Velocity profiles u(r, t) for different values of time parameter t with the cosine pressure gradient when C — 0.4, B0 — 0.7, B1 — 0.8, m — p and a — 0.53.

The permanent solution of (48) can be written in the simpler form as

upt(r, t) — a (r) cos (xt) + b (r) sin (xt)

+ i{ai (r) sin (xt) — b (r) cos (xt)}.

4. Graphical results

Figs. 2 and 3 display the effects of time on fluid velocity for both cosine and sine pressure gradient. It is noticed that fluid velocity u increases on increasing time t — 0.25,0.75,1.0 in the boundary layer region which implies that there is an enhancement in fluid velocity as time progresses for both cosine and sine pressure gradient. It is also noted that the solution corresponding for cosine oscillating pressure gradient contains the steady part.

The starting velocity u(r, t) is written as the sum of the steady solution us(r) given by Eq. (24), the transient solution ut(r, t) given by Eq. (48) and the permanent solution up(r, t)

Figure 3 Velocity profiles u(r, t) for different values of time parameter t with the sine pressure gradient when C — 0.4, B0 — 0.7, B1 — 0.8, m — p and a — 0.53.

. ■ ' "..... Starting velocity Permanent velocity i = 0.0 ♦♦♦ eee r = 0.5 . *** r = 0.8

"• - - r/'......■ a-

*** Steady-state

capillary tube has been established in explicit form. This solution presents as a sum of steady-state and transient solutions, the motion of the fluid for small and large times. However, for growing values of time t, the starting solution reduces to the well-known steady solution that is periodic in time and independent of the initial condition. This method appears to become helpful and may be used to obtain other analytical solutions for similar flow situations. It is wished this short review would bring out more research in this significant area.

Acknowledgment

Figure 4 Profiles of starting and permanent solutions for various values of radius parameter r with the cosine pressure gradient when c — 0.4,B0 — 0.7, Bi — 0.8,x — i.0 and a — 0.53.

The authors would like to acknowledge the financial support received from the Universiti Tun Hussein Onn Malaysia (UTHM) U112.

References

Starting velocity Permanent velocity

r = 0.0 ♦♦♦

r = 0.5 •••

*** r = 0S ***

X\ ------

o.i-1-1-

0 2 4 6

Figure 5 Profiles of starting and permanent solutions for various values of values of radius parameter r with the sine pressure gradient when c — 0.4, B0 — 0.7, Bi — 0.8, x — i.0 and a — 0.53.

given by Eq. (53). Since the limt!1ut(r, t) — 0, the transient solution can be neglected for large values of time t. In this case the flow is according to the steady and permanent solutions. These aspects are shown in Figs. 4 and 5 for cosine and sine pressure gradient, respectively. For three different values of the radial coordinate r — 0.0,0.5,0.8. It is noted that for small values of the time t, the difference between the starting solutions and the permanent solutions is significant. For large values of the time t, the curves corresponding to the starting solutions become identical with the curves corresponding to the permanent solutions for both cosine and sine pressure gradient.

5. Conclusion

In this paper, an exact analytical solution for the velocity field corresponding to the motion of a second grade fluid in a

[1] J.E. Dunn, K.R. Rajagopal, Fluids of differential type. Critical review and thermodynamic analysis, Int. J. Eng. Sci. 33 (i995) 689-729.

[2] M. Jamil, A. Rauf, C. Fetecau, NA. Khan, Helical flows of second grade fluid due to constantly accelerated shear stresses, Commun. Nonlinear Sci. Numer. Simul. i6 (20ii) i959-i969.

[3] M. Massoudi, TX. Phuoc, On the motion of a second grade fluid due to longitudinal and torsional oscillations of a cylinder: a numerical study, Appl. Math. Comput. 203 (2008) 47i-48i.

[4] D. Vieru, C. Fetecau, A. Sohail, Flow due to a plate that applies an accelerated shear to a second grade fluid between two parallel walls perpendicular to the plate, J. Appl. Math. Phys. (ZAMP) 62 (20ii) i6i-i72.

[5] P.A. Razafimandimby, M. Sango, Strong solution for a stochastic model of two-dimensional second grade fluids: Existence, uniqueness and asymptotic behavior, Non-linear Anal. Theory. Meth. Appl. 75 (20i2) 425i-4270.

[6] M.E. Erdogan, C.E. Imrak, Steady circulatory flow of a second grade fluid about a rotating porous cylinder, Int. J. Eng. Sci. 48 (20i0) i225-i232.

[7] M. Jamil, A.A. Zafar, A. Rauf, N.A. Khan, Some new exact analytical solutions for helical flows of second grade fluids, Commun. Nonlinear Sci. Numer. Simul. i7 (20i2) i4i-i53.

[8] M. Nazar, Corina Fetecau, A.U. Awanc, A note on the unsteady flow of a generalized second-grade fluid through a circular cylinder subject to a time dependent shear stress, Nonlinear Anal.: Real World Appl. ii (20i0) 2207-22i4.

[9] S. Nadeem, Abdul Rehman, Changhoon Lee, Jinho Lee, Boundary layer flow of second grade fluid in a cylinder with heat transfer, Math. Prob. Eng. 20i2, 640289, http://dx.doi.org/ i0.ii55/20i2/640289 (i3 pages).

[10] R.A. Shah, S. Islam, A.M. Siddiqui, Exact solution of a differential equation arising in the wire coating analysis of an unsteady second grade fluid, Math. Comput. Model. 57 (20i3) i284-i288.

[11] R. A Shah, S. Islam, A.M. Siddiqui, T. Haroon, Optimal homotopy asymptotic method solution of unsteady second grade fluid in wire coating analysis, J. Ksiam i5 (20ii) 20i-222.