Characteristics of ferrofluid flow over a stretching sheet with suction and injection

Case Studies in Thermal Engineering 14 (2019) 100470

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Characteristics of ferrofluid flow over a stretching sheet with suction and injection

T

Bahram Jalilia, Sina Sadighia, Payam Jalilia, Davood Domiri Ganjib,∗ a b

Department of Mechanical Engineering, Faculty of Engineering, Tehran North Branch, Islamic Azad University, Tehran, Iran Department of Mechanical Engineering, Babol Noshirvani University of Technology, P.O. Box 484, Babol, Iran

A R T IC LE I N F O

ABS TRA CT

Keywords: Ferrofluid Stretching sheet Suction and injection Homotopy perturbation method (HPM) and Akbari-Ganji's method (AGM)

Ferrofluid is a liquid that becomes magnetized in the presence of a magnetic field. Recently, ferrofluids have attracted the attention of researchers due to their paramagnetic properties. Many applications have conducted in electronic devices, mechanical engineering, materials science research and loudspeakers. In this paper, microstructure and inertial characteristics of a magnetite ferrofluid over a stretching sheet utilizing effective thermal conductivity model are studied through two semi-analytical methods which are homotopy perturbation method (HPM) and Akbari-Ganji's method (AGM). It is assumed that Fe3O4 as nanoparticle and water as a base fluid being together nanofluid model formularized by Tiwari-Das model. The effect of related parameters on stream function, velocity, micro-rotation velocity and temperature have obtained for positive and negative mass transfer flow. Results show that the velocity of the fluid in the vicinity of the sheet is maximized and decreases by moving away from the sheet. A significant variation in the behavior of stream function between the positive and negative mass transfer flow over a sheet are exhibited. However, such a behavior is attenuated after a certain distance from the sheet and then stream function is uniformed. Also, micro polar ferrofluid has the higher velocity than classical nanofluid. Furthermore, comparison of the results of this study with other researchers shows that the applied methods are efficient and useful for calculating the parameters of the present problem and have acceptable accuracy.

1. Introduction Ferrofluid is a type of fluid that contains suspended micro particles of iron, magnetite or cobalt in a solvent. Ferrofluids are made up of tiny magnetic fragments of iron suspended in oil often kerosene with a surfactant to prevent usually clumping oleic acid. Some applications of ferrofluids are in rotary seals in computer hard drives and other rotating shaft motors, and in loudspeakers to dampen vibrations. In medicine, ferrofluid is used as a contrast agent for magnetic resonance imaging. A growing number of researchers have begun to appreciate the potential for ferrofluid applications in the quickly emerging fields of microelectromechanical systems (MEMS) and nanoelectromechanical systems (NEMS), due to the 10 nm spherical diameter of the permanently magnetized particles. This size scale naturally makes ferrofluids a viable option for adaptation to MEMS and NEMS devices. It proved that when the nanoparticles added to pure water, the heat transfer coefficient increases. Nano-fluids change the temperature and velocity in the boundary layer. In fact, the amount of these changes depends on the type of nanoparticle and its amount. At first, Choi [1], Sakiadis [2] and Crane [3] investigated the nanofluid and stretching sheets. After this, many researchers interested and carried out extensive

∗

Corresponding author. E-mail address: [email protected] (D.D. Ganji).

https://doi.org/10.1016/j.csite.2019.100470 Received 6 April 2019; Received in revised form 23 May 2019; Accepted 26 May 2019 Available online 28 May 2019 2214-157X/ © 2019 Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/BY-NC-ND/4.0/).

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Nomenclature

I u φ, λ j v μf K V μnf k∗ T ϕ kf Tw δ ks T∞ κ knf F σf N F′

σs B0 F″ σnf Cp, nf G σ∗ g θ f p ρf s Pr ρs nf qr ρnf w R

Body couple per unit mass x-component of velocity Spin gradient viscosity coefficients Micro-inertia density y-component of velocity Base fluid dynamic viscosity Micro-rotation parameter Velocity vector Ferrofluid dynamic viscosity Mean absorption coefficient Temperature of the fluid Ferroparticles volume fraction Base fluid thermal conductivity Wall temperature Boundary parameter Ferroparticle thermal conductivity Ambient temperature Vortex viscosity Ferroparticle thermal conductivity Dimensionless stream function Base fluid electric conductivity Micro-rotation vector Dimensionless velocity

∞ S γnf

Ferroparticles electric conductivity Magnetic field intensity Coefficient of skin-friction Ferrofluid electric conductivity Ferrofluid heat capacity Dimensionless micro-rotation velocity Stefan-Boltzmann constant Acceleration due to gravity Dimensionless temperature Base fluid Pressure Base fluid density Ferroparticle Prandtl number Ferroparticles density Ferrofluid Radiative heat flux Ferrofluid density Condition at wall Radiative parameter α Stretching/shrinking parameter Condition at infinity Suction/injection parameter Spin-gradient viscosity

research in this field. On the other hand, some nonlinear equations are governed by these problems. Many of the mathematical problems in nature do not have a precise solution, and they must obtain approximate solutions with some relative method i.e. semi-analytic methods [4-10]. Ariel [11] studied about Magneto hydrodynamic viscoelastic fluid flow over a stretching sheet. He solved flow over a stretching sheet by utilizing HPM. Fathizadeh and Rashidi [12] analyzed the convective heat transfer by HPM. Ganji [13] compared the semianalytical methods for nonlinear equations in heat transfer. Fathizadeh et al [14] compared MHPM with HPM for MHD viscous flow past a stretching sheet and they show MHPM is more efficiency. Sheikholeslami and Ganji [15] worked out about the nanofluid and heat transfer between parallel plates with Brownian motion using DTM. Mabood et al. [16] depicted that the correlation that the magnetic parameter has a direct relation with Nusselt and Sherwood numbers. Lin et al. [17] reported about MHD pseudo-plastic nanofluid unsteady flow over a stretching sheet with inertial heat generation with four nanoparticles, namely Cu, CuO, Al2O3, and TiO2. Uddin et al. [18] solved numerically by RK45 and show that nanoparticle concentration decreases for stretching sheet and by contrast, flow is accelerated for stretching sheet. Abbas et al. [19] displayed that the concentration boundary layer thickness in one hand, has an adverse relation with the dimensionless velocity slip parameter for the first solution and the other hand, has a direct relation in the second solution. Mansur et al. [20] Observed there is an inverse relation between Brownian motion and thermophoresis parameter with the local Nusselt number. Ahmad Khan et al. [21], investigated the three-dimensional flow of nanofluid on the stretching sheet. Lin et al. [22] studied an unsteady flow over a stretching sheet with variable thermal conductivity. Animasaun et al. [23] have done an analytical work on Casson fluid flow on an exponentially stretching sheet with suction. It has been noted by Rashidi et al. [24] that the reduction of the critical suction point is due to the presence of the nanoparticles. Mahmood et al. [25] employed CSNIS for the computational fluid dynamic and found that the magnetic force can influence on the range increment of solution. Waqas et al. [26] claimed that the local Nusselt number has a direct relation with Prandtl and Biot numbers. Awaludin et al. [27] described that the first answer is physically reliable and stable. Turkyilmazoglu [28] scrutinized micropolar fluid past a stretching sheet and heat transfer. Hsiao [29] utilized finite difference method to show a direct relation between temperature effect and Prandtl and Eckert numbers. Eid et al. [30] focused on Carreau nanofluid past throughout a porous stretching sheet by numerical technique. Ganji et al. [31] solved a problem including squeeze number, radiation parameter and heat generation/absorption parameter by Differential Transform Method (DTM). Golafshan and Rahimi [32] used the HAM and the results show that decreasing in velocity profiles yields an increment of the magnetic field parameter. Hussanan et al. [33], applied Whittaker function and figures Table 1 Thermo physical properties of H2O and Fe3O4.

Fe3O4 H2O

ρ(kg/m3)

Cp(J/kg K)

k (W/m k)

σ(Ω m)-1

5180 997.1

670 4179

9.7 0.613

25000 0.05

2

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and then show that classical nanofluid have lower velocity than micro polar ferrofluid. In this paper the Ferrofluid problem is analyzed with numerical and semi-analytical methods for the first time. The semi-analytical methods, namely HPM and AGM [34-37] are applied on stretching sheet with suction/injection and a magnetic field perpendicular to the sheet (see Table 1). 2. Formulation of problem and basic equations Many of researchers have investigated the problem of micropolar [38] and classical nanofluid over a stretching sheet. In this paper it was assumed the steady state two-dimensional flow of a micropolar/classical ferrofluid on a stretching sheet in the y > 0 region. The surface in the x-direction gets stretch and the y-direction is normal at x = 0. The stretching surface velocity is uw(x) =ax, where a is an optional constant and positive for stretching sheet(see Fig.1). Also, there is a magnetic field normal to the surface. Under the above assumptions, here are governing equations of micro polar ferrofluid:

d (ρ ) = ∇ . (ρnf V) dt nf ρnf (

(1)

dV ) = −∇p + (2μnf + κ ) ∇ (∇ . V) − (μnf + κ ) ∇ × (∇ × V) + κ (∇ × N) + J × B + ρnf g dt

ρnf j (

dN ) = (φ + λ + γnf ) ∇ . (ρnf ⋅N) − γnf ∇ × (∇ × N) + κ (∇ × V) − 2κ N + ρnf I dt

(2) (3)

where effective density, effective dynamic viscosity, and spin gradient viscosity are presented by Brinkman [39], Bourantas and Loukopoulos [40], respectively:

ρnf = (1 − φ) ρf + ϕρs μnf =

(4)

μf (1 − φ)2.5

(5)

κ + )j 2

(6)

γnf = (μnf

Also, flow is incompressible with no external forces and equations will be:

ρnf (

dV + (V . ∇) V) = (μnf + κ ) ∇2 V + κ (∇ × N) + J × B dt

ρnf j (

(7)

dN + (N⋅∇) N) = γnf ∇2 N + κ (∇ × V) − 2κ N dt

(8)

and the body force proposed by Hussanan et al. [41] is:

J × B = −σnf B02 V

(9)

and the ferrofluid electric conductivity represented by Sheikholeslami [42]:

σnf = [1 +

3(σ − 1) ϕ ] σf (σ + 2) − (σ − 1) ϕ

(10)

And the governing equations go to:

ρnf (u

∂u ∂u ∂ 2u ∂N + v ) = (μnf + κ ) 2 + κ − σnf B02 u ∂x ∂y ∂y ∂y

(11)

Fig. 1. Stretching sheet (α > 0) [33]. 3

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Fig. 2. Comparison of temperature for suction

Fig. 3. Comparison of stream function for suction and stretching with K = 2, δ = 0.5, ϕ = 0.03.

ρnf j (u

∂N ∂N ∂ 2N ∂u +v ) = γnf 2 − κ (2N + ) ∂x ∂y ∂y ∂y

(12)

The specific heat capacity and the effective thermal conductivity reported by Khan et al. [43] and Mohyud-Din [44].

Cp, nf =

knf kf

=

Cp) s + (1 − ϕ)(ρCp) f ρnf

(13)

ks + 2kf − 2ϕ (kf − ks ) ks + 2kf + ϕ (kf − ks )

(14)

The energy equation can be written as

u

knf ∂2T ∂q ∂T ∂T +v = − r ∂x ∂y Cp) nf ∂y 2 ∂y

(15)

and conforming to Rosseland's approximation is expressed as 4

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Fig. 4. Comparison of velocity for suction and stretching K = 2, δ = 0.5, ϕ = 0.03.

Fig. 5. Comparison of velocity for injection and stretching K = 2, δ = 0.5, ϕ = 0.03.

u

3 16σ ∗T∞ 1 ∂T ∂T ∂ 2T ) 2 (knf + +v = ∗ 3k ∂x ∂y Cp) nf ∂y

(16)

The boundary conditions of the problem are

u = αu w (x ), v = vw , at y = 0, u → 0, y → ∞

N = −δ T = Tw

∂u ∂y

at y = 0, N → 0,

at y = 0, T → Tw,

(17)

y→∞

(18)

y→∞

(19)

For the acquaintance with these formulas, we assume that velocity components along the x and y directions consider as u and v, respectively. Surface mass transfer velocity is vw, the positive and negative value of vw are corresponded to the suction and injection, respectively. N is the angular velocity. μ is dynamic viscosity, δ is a constant in the 0≤ δ ≤ 1 region. When δ=0, specified that microelements near to the wall surface cannot rotate. If δ=1/2 illustrated the weakness of concentration of microelements. Moreover, δ=1 when δ requires to the turbulent boundary layer flow. Micro inertia per unit mass is j, γ is spin gradient viscosity and κ is the vortex viscosity. To solve comfortably these nonlinear differential equations, we utilized similarity variables such as

η=y

a a T − T∞ , u = axF ′ (η), ν = − aνf F (η), N = ax G (η), θ (η) = . νf νf Tw − T∞ 5

(20)

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Fig. 6. Comparison of micro rotation velocity for injection and stretching K = 2, δ = 0.5, ϕ = 0.03.

By substituting the variables (20) into the Eqs. (11), (12) and (16), we have

(

ρ ρ 3(σ − 1) ϕ 1 ) F ′ (η) + KG′ (η) = 0 + K ) F‴ (η) + (1 − ϕ + ϕ s ) F (η) F ″ (η) − (1 − ϕ + ϕ s ) F′2 (η) − M (1 + σ + 2 − (σ − 1) ϕ ρf ρf (1 − ϕ)2.5 (21)

(

1 (1 − ϕ)2.5

ρ ρ K + ) G″ (η) + (1 − ϕ + ϕ s ) F (η) G′ (η) − (1 − ϕ + ϕ s ) F ′ (η) G (η) − K (2G (η) + F ″ (η)) = 0 ρf ρf 2

Cp) s 1 ks + 2kf − 2ϕ (kf − ks ) ) F (η) θ′ (η) = 0 ( + R) θ″ (η) + (1 − ϕ + ϕ Pr ks + 2kf + ϕ (kf − ks ) Cp) f

(22)

(23)

and here the boundary conditions are

F (η) = S, F ′ (η) = α, G (η) = −δF ″ (η), θ (η) = 1 at η = 0

(24)

F ′ (η) → 0, G (η) → 0, θ (η) → 0 at η → ∞

(25)

and then the non-dimensional variables are introduced as

M2 =

σf B02 aρf

,K=

3 νf Cp) fνf 16σ ∗T∞ v κ ,S=− w , j = , Pr = ,R= 3k ∗kf aνf μf a kf

(26)

3. Basic idea of homotopy perturbation method (HPM) To illustrate the basic ideas of this method, we consider the following equation:

A (u) − f (r ) = 0

r∈Ω

(27)

With the boundary condition of:

B (u,

∂u )=0 ∂n

r∈Γ

(28)

Where A is a general differential operator, B a boundary operator, f(r) a known analytical function and Γ is the boundary of the domain Ω . A can be divided into two parts which are L and N, where L is linear and N is nonlinear. Eq. (29) can, therefore, be rewritten as follows:

L (u) + N (u) − f (r ) = 0

r∈Ω

(29)

Homotopy perturbation structure is shown as follows

H (v, p) = (1 − p)[L (v ) − L (u 0)] + p [A (v ) − f (r )] = 0

(30)

where: 6

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v (r , p): Ω × [0,1] → R

(31)

In Eq. (30), p ∈ [0,1] is an embedding parameter and u 0 is the first approximation that satisfies the boundary condition. We can assume that the solution of Eq. (30) can be written as a power series in p, as follows:

v = v0 + pv1 + p2 v2 + ...

(32)

and the best approximation for the solution is:

u = limp → 1v = v0 + v1 + v2 + ...

(33)

4. Application of homotopy perturbation method According to the defined method in the previous section, we build a homotopy to solve the equations.

H (F , p) = (1 − p)(F‴ (η) + F ″ (η) − F ′ (η) − F‴0 (η) − F ″0 (η) − F ′0 (η)) + p (F‴ (η) + F (η) F ″ (η) − (F ′ (η))2) = 0

(34)

H (G, p) = (1 − p)(G″ (η) + G′ (η) − G (η)) − G″0 (η) + G′0 (η) − G0 (η)) + p (G″ (η) + F (η) G′ (η) − F ′ (η) G (η)) = 0

(35)

So, F and G Polynomials will be like: n

F (η) = F0 (η) + F1 (η) + ...= ∑ Fi (η)

(36)

i=0

n

G (η) = G0 (η) + G1 (η) + ...= ∑ Gi (η)

(37)

i=0

After substituting F and G from Eqs. (36) and (37) into Eqs. (34) and (35), respectively, with some simplification and rearranging based on powers of p-terms, we have:

p0 :

d3 d2 d F0 (η) + F0 (η) − F0 (η) dη3 dη2 dη

(38)

d2 d G0 (η) + G0 (η) − G0 (η) dη2 dη

(39)

And the boundary conditions are:

F0 (0) = 1, F ′0 (0) = 1, F0′ (1000) = 0

(40)

G0 (0) = −0.5, G0 (1000) = 0

(41)

Solving the Eqs. (38) and (39) with those boundary conditions yield the answers which are:

F (η) = F0 (η) = − e−500

1 1 (e500 5 − 500 2 e500 5 − 500 − e−500 5 − 500

5 − 500 + 12 η 5 − 12 η

G (η) = G0 (η) = −

− e−500

1 2 e500

5 − 500

5 − 500

− e−500

1 − e−500

5 − 500

− e500

5 − 500 − 12 η − 12 η 5

5 − 500 + 12 η 5 − 12 η

(−e−500

5 +

5 + e500

5 e−500

5 − 500 + 1 η 5 − 1 η 2 2

+ e500

5 − 500 − 12 η − 12 η 5

+

5 e500

5 − 500

5 − 500 )

5 − 500 − 1 η − 1 η 5 2 2 )

(42) (43)

5. Basic idea of Akbari-Ganji's method (AGM) Generally, the differential equations are confirmed as this form:

pk : f (u, u′, u″, ..., u(n) ) = 0, u = u (x )

(44)

So, pk is a nonlinear differential equation and n is the order of magnitude of it and the form of boundary conditions is:

u (0) = u 0 , u′ (0) = u1, ..., u(m − 1) (0) = um − 1

(45)

u (L) = u L0 , u′ (L) = u L1, ..., u(m − 1) (L) = u Lm − 1

(46)

Next step of AGM is to apply polynomials with constant coefficients such as: n

f (η) =

∑ ak e−0.8kη = a0 + a1 e−0.8η + a2 e−1.6η + a3 e−2.4η + a4 e−3.2η + a5 e−4η + a6 e−4.8η + a7 e−5.6η k=0

(47)

n

g (η) =

∑ bk e−0.8kη = b0 + b1 e−0.8η + b2 e−1.6η + b3 e−2.4η + b4 e−3.2η + b5 e−4η + b6 e−4.8η k=0

7

(48)

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Therefore, when η = 0 :

u (0) = a0 = u 0 u′ (0) = a1 = u1 u″ (0) = a2 = u2

(49)

and when η = L :

u (L) = a0 + a1 L + a2 L2 +...+an Ln = u L0 u′ (L) = a1 + 2a2 L + 3a3 L2 +...+nan Ln − 1 = u L1 u″ (L) = 2a2 + 6a3 L + 12a4 L2 +...+n (n − 1) an Ln − 2 = u Lm − 1

(50)

After substituting the Eqs. (47) and (48) into the Eqs. (21)–(23) by considering conditions such as ϕ = 0, M = 0, S = 1 and α = 1 we have:

f (0) = 1 → a0 + a1 + a2 + a3 + a4 + a5 + a6 + a7 = 1

(51)

f ′ (0) = 1 → −0.8a1 − 1.6a2 − 2.4a3 − 3.2a4 − 4a5 − 4.8a6 − 56a7 = 1

(52)

f ′ (1000) = 0 → −2.934299667 ∗ 10−348a1 − 2.152528635 ∗ 10−695a2 − 1.184280761 ∗ 10−1042a3 − 5.791724403 ∗ 10−1390a4 − 2.655414842 ∗ 10−1737a5 − 1.168767433 ∗ 10−2084a6 − 5.001374423 ∗ 10−2432a7 = 0

(53)

g (0) = −0.5 → b0 + b1 + b2 + b3 + b4 + b5 + b6 = −0.5

(54)

g (1000) = 0 → b0 + 3.667874584 ∗ 10−348b1 + 4.934503169 ∗ 10−1043b3 + 1.809913876 ∗ 10−1390b4 + 6.638537105 ∗ 10−1738b5 + 2.434932152 ∗ 10−2085b6 = 0

(55)

It is needed 15 equations with 15 unknowns to solve those nonlinear differential equations. Thus, it can make a set of algebraic equations that should be solved. After solving such equations, the coefficients are calculated and expressed as

a0 = 1.618866809, a1 = −2.423602691 ∗ 10−44 , a2 = −0.5987852601, a3 = −0.03237184197, a4 = 0.01832942350, a5 = −0.007843682798, a6 = 0.002045098151, a7 = −0.0002405461017 , b0 = 1.589744335 ∗ 10−44 , b1 = −1.855003303 ∗ 10−11 , b2 = −8.432384211, b3 = 23.49332007 , b4 = −27.11167390, b5 = 14.60925386 , b6 = −3.058515830

(56)

By substituting the values of Eq. (56) into Eqs. (47) and (48), the semi-analytical answer of the nonlinear differential equations by AGM is:

f (η) = 1.618866809 − 2.423602691 ∗ 10−44e−0.8η − 0.5987852601e−1.6η − 0.03237184197e−2.4η + 0.01832942350e3.2η − 0.007843682798e 4η + 0.002045098151e−4.8η − 0.0002405461017e−5.6η

(57)

g (η) = 1.589744335 ∗ 10−44 − 1.855003303 ∗ 10−11e−0.8η − 8.432384211e1.6η + 23.49332007e−2.4η − 0.05865415520e−3.2η + 0.03137473119e−4η − 0.009816471125e−4.8η + 0.001347058170e−5.6η

(58)

6. Results and discussions In this paper, some of the physical properties of the ferrofluid flow, including velocity, micro-rotation velocity, temperature, and stream function have been studied and analyzed by semi-analytical methods. In order to show more precisely, stream function F (η) , velocity F ′ (η) , micro-rotation velocity G (η) and temperature θ (η) for stretching α > 0 sheet with suction S > 0 and injection S < 0 were investigated. For verification purposes, the results of this study have been compared with the results of other researchers such as Hussanan [33] and Turkyilmazoglu [28] as well as with the results of numerical solution. Looking at the graphs shown, it is clear that the results of this study are consistent with the results as those reported by other researchers. It is observed carefully in charts and graphs that the methods used are of great efficiency and accuracy. Fig. 2 illustrates the effect of the material parameter on the temperature field for suction and stretching. The stream function for suction and stretching compared with the numerical method in Fig. 3 and both of them have uniform behavior after η = 2 . As it can be seen in the Fig. 2 by increasing the amount of η, the temperature function has a downward trend. Also Fig. 3 shows that after a certain amount of η, the stream function approaches a constant value. The results obtained from the semi-analytical methods applied in this paper have a good accuracy and the convergence of the solutions is observable. Figs. 4 and 5 show the effect of the magnetic parameter on the velocity field for suction and injection on a stretching sheet. An increasing of thermal boundary layer thickness is exhibited due to the presence of the Magnetic parameter. Fig. 6 shows the micro8

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rotation velocity for injection over a stretching sheet. By keeping constant the micro-rotation and boundary parameters and reducing the magnetic parameter, the equations show their exclusive behaviors. Overall, as it is shown in all the figures, Akbari-Ganji's and Homotopy Perturbation semi-analytic methods are in good agreement with the method used in previous research, and these methods are efficient and capable and can be used in other engineering issues. 7. Conclusion In this study, microstructure and inertial characteristics of a magnetite ferrofluid over a stretching sheet using effective thermal conductivity model are examined. Governing formulae are solved by HPM and AGM. The effect of related parameters on stream function, velocity, micro-rotation velocity and temperature are demonstrated graphically. Results revealed that. 1. These semi-analytical methods applied for the first time to present the solutions. 2. On the stream function subject; there is a significant difference between suction and injection on a stretching sheet of which the volumetric flow rate of injection is much more than suction. 3. On the velocity subject, there is an inverse relation between magnetic and micro-rotation parameters. 4. On the micro-rotation velocity subject, in the lack of magnetic parameter, G (η) is on the lowest form. 5. On the temperature subject, the highest temperature is because of the absence of micro-rotation parameter. 6. It is noteworthy to mention that our results show classical nanofluid has higher velocity and micro-rotation speed than micropolar Ferro fluid in the absence of magnetic parameter. References [1] S.U. Choi, J.A. Eastman, Enhancing Thermal Conductivity of Fluids with Nanoparticles (No. ANL/MSD/CP-84938; CONF-951135–29), Argonne National Lab, IL (United States), 1995. [2] B.C. Sakiadis, Boundary-layer behavior on continuous solid surfaces: II. 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