Meccanica 38: 325–334, 2003. © 2003 Kluwer Academic Publishers. Printed in the Netherlands.
Unsteady Flow due to Concentric Rotation of Eccentric Rotating Disks H. VOLKAN ERSOY Department of Mechanics, Faculty of Mechanical Engineering, Istanbul Technical University, Gümü¸ssuyu; ˙ 80191, Istanbul, Turkey (Received: 1 February 2001; accepted in revised form: 4 June 2002) Abstract. While two parallel disks are initially rotating with the same angular velocity about non-coincident axes, the axes are suddenly made coincident. The development of the flow is examined until the fluid rotates as a rigid body in the steady-state. The velocity field and the shear stress components on the disks are found exactly by a Fourier series solution. Furthermore, a series solution that converges rapidly at small times is obtained with the aid of the Laplace transform technique. Key words: Eccentric and concentric rotating disks, Unsteady flow, Newtonian fluid, Fluid mechanics.
1. Introduction The study of flow between eccentric rotating disks has received a great deal of attention for about 40 years. It is possible to determine the material moduli of non-Newtonian fluids if an instrument which consists of two parallel disks rotating with the same angular velocity about two different axes normal to the disks is used. Maxwell and Chartoff [1] were the first researchers to consider this fact and claimed that the two components of the tangential force on one of the disks should be measured. A detailed list of references about eccentric rotating disks can be found in [2, 3]. Time-dependent flows induced by eccentric rotating disks have been studied by several authors. Rao and Kasiviswanathan [4] investigated the unsteady flow in which the streamlines at any given instant are concentric circles in each plane parallel to a fixed plane and each point of the plane is performing non-torsional oscillations. Later, Kasiviswanathan and Rao [5] extended this flow to the unsteady heat transfer. They [6] also studied the unsteady MHD flow when the eccentric rotating disks are subjected to non-torsional oscillations. Erdoˇgan [7] found the velocity field for the symmetric unsteady flow induced by the rotation about noncoincident axes while both the disks are initially rotating with the same angular velocity about a common axis. He took the initial condition to be the rigid body rotation of the fluid. Later, Erdoˇgan [8, 9] studied the unsteady flows produced by both rotation about non-coincident axes and non-torsional oscillations. The initial condition is the axisymmetric flow caused by the rotation of the disks with the same angular velocity about a common axis. The disks suddenly start both to rotate about non-coincident axes and to execute non-torsional oscillations. In [8, 9] the lower disk and the two disks execute non-torsional oscillations, respectively. The flows are not symmetric since the oscillations of the disks are out of symmetric. Recently, Ersoy [10] has obtained an exact solution for the unsteady flow due to a sudden pull with
326 H. Volkan Ersoy constant velocities of the disks rotating eccentrically at the same speed. The initial condition is the solution given by Abbott and Walters [11]. Each of the disks is suddenly pulled with equal velocities in the opposite directions and as a consequence the flow remains symmetric throughout the motion of the disks. In this paper, we study the unsteady flow produced by a sudden coincidence of the axes while two disks are initially rotating with the same angular velocity about non-coincident axes. The initial condition is the solution obtained by Abbott and Walters [11]. Thus, our purpose is to study the unsteady flow induced until the fluid rotates as a rigid body in the steady-state. The velocity field and the shear stress components that are related to the force components in the x- and y-directions exerted by the fluid on the disks are found by a Fourier series solution. Moreover, a more appropriate series solution, which is valid at small times, is obtained using the Laplace transform technique.
2. Basic Equations Two infinite parallel disks are initially rotating in the same sense with the same angular velocity about two non-coincident axes normal to the disks. A Cartesian coordinate system Oxyz is taken with the z-axis along the bisection of the two axes of rotation. The upper and lower disks located at z = ∓h are rotating about the axes through the points Pu (O, , h) and Pl (O, − , −h), respectively. The disks suddenly start to rotate with their initial angular velocity about the z-axis. Therefore, the initial and boundary conditions are u = −y + fˆ(z),
u = −y,
v = x + g(z), ˆ
v = x,
w=0
at t = 0 for − h z h, (2.1)
w = 0 at z = ∓h for t > 0,
(2.2)
where u, v, w denote the velocity components along the x, y, z-directions, respectively. The functions fˆ(z) and g(z), ˆ obtained by Abbott and Walters [11], represent the eccentric symmetric rotation for a Newtonian fluid and are given by sinh kz , fˆ(z) + i g(z) ˆ =
sinh kh
(2.3)
√ √ where i = −1, k = /(2ν)(1 + i), and ν is the kinematic viscosity of the fluid. Because of the characteristic of the flow, a symmetric condition should also be written, that is, u = −y,
v = x,
w = 0 at z = 0 for t 0.
(2.4)
We assume that is much smaller than h. Thus, it is reasonable to try a solution of the form u = −y + f (z, t),
v = x + g(z, t),
w = 0.
(2.5)
Concentric Rotation of Eccentric Rotating Disks 327 Using (2.1), (2.2), (2.4) and (2.5), one obtains f (z, 0) = fˆ(z), f (∓h, t) = 0, f (0, t) = 0,
g(z, 0) = g(z) ˆ
for − h z h,
g(∓h, t) = 0 for t > 0, g(0, t) = 0 for t 0.
(2.6a) (2.6b) (2.6c)
Introducing F (z, t) = f (z, t)+ig(z, t) and substituting equation (2.5) into the Navier–Stokes equations, we get ν
∂F ∂ 2F − iF = C1 (t) + iC2 (t), − ∂z2 ∂t
p − p0 = 12 ρ2 (x 2 + y 2 ) + ρ(C1 (t)x + C2 (t)y),
(2.7) (2.8)
where p0 is a constant, ρ is the density of the fluid, C1 (t) and C2 (t) are the unknown functions. However, the Poiseuille type pressure gradient is removed since the flow is symmetric, thus we get that C1 (t) = C2 (t) = 0. The conditions for F (z, t) become F (z, 0) =
sinh kz sinh kh
(−h z h),
(2.9a)
F (∓h, t) = 0 (t > 0),
(2.9b)
F (0, t) = 0 (t 0).
(2.9c)
3. Solution to the Problem Using the conditions (2.9b)–(2.9c), the function F (z, t) can be written as F (z, t) =
∞ n=1
Vn (t) sin
nπ z . h
(3.1)
By means of the initial condition (2.9a), we obtain ∞ n(−1)n f 2 2 = (2π ) (R sin τ − n2 π 2 cos τ )e−n π τ/R sin nπ ζ, 4π 4 + R2
n n=1
(3.2a)
∞ n(−1)n g 2 2 = (2π ) (R cos τ + n2 π 2 sin τ )e−n π τ/R sin nπ ζ, 4 4 2
n π +R n=1
(3.2b)
where the Reynolds number R, the dimensionless time τ , and the dimensionless vertical distance ζ are given by R=
h2 , ν
τ = t,
ζ =
z . h
(3.3)
328 H. Volkan Ersoy The variations of f/ and g/ for various values of the parameters with ζ are shown in Figure 1. The dimensionless shear stress components on the disks are (S¯xz )ζ =∓1 = 2π 2
∞ n=1
(S¯yz )ζ =∓1 = 2π 2
∞ n=1
n2 2 2 (R sin τ − n2 π 2 cos τ )e−n π τ/R , 4 4 2 n π +R
(3.4a)
n2 2 2 (R cos τ + n2 π 2 sin τ )e−n π τ/R , 4 4 2 n π +R
(3.4b)
where S¯xz = Sxz /(µ / h), S¯yz = Syz /(µ / h), and µ is the dynamic viscosity of the fluid; (S¯ xz )ζ =∓1 and (S¯yz )ζ =∓1 are related to the x- and y-components of the force on the disks, respectively. The variations of (S¯xz )ζ =∓1 and (S¯yz )ζ =∓1 for different values of the Reynolds number R with the dimensionless time τ are displayed in Figure 2. 4. Solution at Small Times Since the solution obtained above converges slowly at small times, a different method that converges rapidly for small values of the time is used. By defining F (z, t) = H (z, t)e−it ,
(4.1)
equation (2.9) becomes ν
∂H ∂ 2H = 0, − ∂z2 ∂t
(4.2)
with the conditions as follows: H (z, 0) =
sinh kz sinh kh
(−h z h),
(4.3a)
H (∓h, t) = 0 (t > 0),
(4.3b)
H (0, t) = 0 (t 0).
(4.3c)
After taking the Laplace transform of equation (4.2) and the conditions (4.3b) and (4.3c), we find that −
s sinh kz, H¯ − H¯ = ν ν sinh kh
(4.4a)
¯ H(∓h, s) = 0,
(4.4b)
¯ H(0, s) = 0,
(4.4c)
Concentric Rotation of Eccentric Rotating Disks 329
Figure 1. Variations of f/ and g/ versus ζ at various values of τ and R = 10, 30.
330 H. Volkan Ersoy
Figure 2. Variations of (S¯xz )ζ =∓1 and (S¯yz )ζ =∓1 versus τ ; R = 10, 20, 30, 40.
where H¯ is the Laplace transform of H , s is the Laplace transform variable and a prime denotes differentiation with respect to z. We obtain the transformed solution as
[e−ϕ(h−z) − e−ϕ(h+z) ] 1
sinh kz 1 , (4.5) + H¯ = − s (1 − i/s) 1 − e−2ϕh s (1 − i/s) sinh kh √ n −1 for |χ| < 1. where ϕ = s/ν. It is well known that the series ∞ n=0 χ converges to (1−χ) Using this binomial series, it is possible to obtain the solution for small values of the time,
Concentric Rotation of Eccentric Rotating Disks 331 corresponding to large s [12]. Thus, we obtain ∞ ∞ 1 − ζ + 2m 1 + ζ + 2m H n n 2n 2n − i erfc √ + = − (4τ ) (i) i erfc √
2 τ/R 2 τ/R n=0 m=0 ∞
+
sinh kz τ p p i , sinh kh p=0 p!
(4.6)
where in erfc(.) denotes the repeated integrals of the complementary error function [13]. From equations (4.1) and (4.6), we have f = (− cos τ )[T0 − (4τ )2 T4 + (4τ )4 T8 − · · ·] +
+ (sin τ )[−(4τ )T2 + (4τ )3 T6 − (4τ )5 T10 + · · ·] + P (1)P (ζ ) + Q(1)Q(ζ ) , + [P (1)]2 + [Q(1)]2
(4.7a)
g = (− cos τ )[(4τ )T2 − (4τ )3 T6 + (4τ )5 T10 − · · ·] +
+ (sin τ )[T0 − (4τ )2 T4 + (4τ )4 T8 − · · ·] + +
P (1)Q(ζ ) − Q(1)P (ζ ) , [P (1)]2 + [Q(1)]2
(4.7b)
where
∞ 1 − ζ + 2m 1 + ζ + 2m r r − i erfc √ (r = 0, 2, 4, . . .), i erfc √ Tr = 2 τ/R 2 τ/R m=0 Q(ζ ) = cosh 12 Rζ sin 12 Rζ. P (ζ ) = sinh 12 Rζ cos 12 Rζ,
(4.8a–c)
Tr is an odd function of ζ . The functions Tr have maximum values at ζ = 1, and are zero at larger but takes the same value at ζ = ∓1. ζ = 0. When τ/R increases, Tr becomes √ Since ∂Tr /∂ζ = Yr−1 / 2 τ/R , the solutions in equations (3.4a) and (3.4b) become (S¯xz )ζ =∓1 =
(S¯yz )ζ =∓1 =
(− cos τ ) [Y−1 − (4τ )2 Y3 + (4τ )4 Y7 − · · ·]ζ =∓1 + √ 2 τ/R (sin τ ) [−(4τ )Y1 + (4τ )3 Y5 − (4τ )5 Y9 + · · ·]ζ =∓1 + + √ 2 τ/R P (1)P (1) + Q(1)Q (1) , + [P (1)]2 + [Q(1)]2 (− cos τ ) √ [(4τ )Y1 − (4τ )3 Y5 + (4τ )5 Y9 − · · ·]ζ =∓1 + 2 τ/R (sin τ ) [Y−1 − (4τ )2 Y3 + (4τ )4 Y7 − · · ·]ζ =∓1 + + √ 2 τ/R P (1)Q (1) − Q(1)P (1) , + [P (1)]2 + [Q(1)]2
(4.9a)
(4.9b)
332 H. Volkan Ersoy
Figure 3. Flow induced by the rotation about non-coincident axes while the disks are initially rotating about a common axis.
where Yr =
∞ m=0
1 + 2m − ζ 1 + 2m + ζ + ir erfc √ i erfc √ 2 τ/R 2 τ/R r
(r = −1, 1, 3, 5, . . .)
(4.10)
Concentric Rotation of Eccentric Rotating Disks 333 and a prime now denotes differentiation with respect to ζ . Yr is an even function of ζ . The functions Yr have maximum and minimum values at ζ = ∓1 and ζ = 0, respectively. When τ/R increases, Yr also increases for fixed ζ . The functions Yr have the relation Y−1 > Y1 > Y3 > Y5 > · · · for fixed τ/R and ζ . The exact solutions given by equations (3.2a), (3.2b), (3.4a) and (3.4b) converge rapidly at large times but slowly at small times. However, the series solutions shown by equations (4.7a), (4.7b), (4.9a) and (4.9b) converge rapidly for small values of the time, and it is clear that they are very convenient for τ 0.25. 5. Discussion The velocity field is determined with the help of the functions f (z, t) and g(z, t), which are the components of the time-dependent rigid body translation. These components oscillate with the periods shown in Figure 1 but decay rapidly, since the fluid tends to a rigid body rotation. The intervals on which these velocity components oscillate are independent of the Reynolds number R (Figure 1). Since the shear stress components Sxz and Syz do not depend on x and y, they are related to the x- and y-components of the force per unit area exerted by the fluid on the disks, respectively. Each of these forces is equal in magnitude and opposite in direction on the top and bottom disks. Figure 2 shows the positive direction of the force for the bottom disk. For small values of the time, the force components become negative and positive in the x- and y-directions, respectively; however, the x-component is larger than the y-component. Of course, the forces change their directions continuously and finally go to zero, because the fluid rotates as a rigid body in the steady-state. When the Reynolds number increases, the forces become larger in general and the time required for the steady-state becomes longer. Figure 3 summarizes the unsteady flow, studied by Erdoˇgan [7], between two disks rotating about non-coincident axes while they are initially rotating about a common axis. We acquire some remarkable results about the shear stress components that are related to the force components in the x- and y-directions on the disks. The x-component is always larger than the y-component for small values of the time and this fact is always valid for small Reynolds numbers. The values of these components depend on the Reynolds number in the ¯ ¯ steady-state, √ whereas they are zero in our present paper. Moreover, (Sxz )ζ =∓1 and (Syz )ζ =∓1 approach R/2 at high values of the Reynolds number and in the steady-state (Figure 3). Acknowledgements The author would like to express his gratitude to Prof. M.E. Erdoˇgan for his helpful comments. He is very grateful to the referees for their valuable suggestions. References 1. 2. 3. 4.
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