The stagnation point flow over a linearly stretching or shrinking sheet is considered in the present study. The transformed ordinary differential equations are solved numerically. Dual solutions are possible for the shrinking case, while the solution is unique for the stretching case. For the shrinking case, a linear temporal stability analysis is performed to determine which one of the solution is stable and thus physically reliable.

The problem of two-dimensional stagnation point flow towards a horizontal surface was first considered by Hiemenz1 who discovered that this problem can be analysed exactly by the Navier-Stokes equations. The analysis of the temperature distribution was done by Goldstein.2 The axisymmetric velocity distribution was analysed by Homann3 while that of the temperature distribution was reported by Sibulkin.4 

Crane5 is the first to study the incompressible viscous fluid over a linearly stretching sheet, and reported an exact solution for the velocity and the thermal fields. Grubka and Bobba6 considered the flow over a linearly stretching sheet with a power-law temperature distribution, and reported the exact solution for the thermal field in terms of Kummer’s function. They realized that the solution obtained by Crane5 for the boundary layer equations is also an exact solution to the Navier-Stokes equations. It is worth mentioning that the flow over a stretching sheet was studied recently by Ahmed et al.,7 Khan and Khan,8 Mustafa et al.9 and Khan and Rahman,10 among others.

The combination of both stagnation flow and stretching surface was considered by Mahapatra and Gupta,11,12 Nazar et al.13 and Ishak et al.14 Wang15 considered the stagnation flow over both stretching and shrinking sheets, and found that the solutions for the shrinking case are not unique. This problem was then extended by Ishak et al.16 to a micropolar fluid. The aim of the present paper is to study the stability of the dual solutions obtained by Wang.15 

The presentation begins with the mathematical formulation in §II followed by the linear temporal stability analysis in §III. The presentation of results are given in §IV and a summary and concluding remarks are given in §V.

Consider a steady stagnation point flow of a viscous fluid toward a linearly stretching/shrinking sheet of constant temperature as shown in Figure 1. The stretching/shrinking velocity uw(x) and the ambient fluid velocity ue(x) are assumed to vary linearly from the stagnation point, i.e. uw(x) = ax and ue(x) = bx, where a and b are constants with b > 0. We note that a > 0 and a < 0 correspond to stretching and shrinking sheets, respectively. Under these assumptions, the boundary layer equations are

(1)
(2)
(3)

where all symbols have their usual meaning. The boundary conditions are

(4)
FIG. 1.

Physical model and coordinate system.

FIG. 1.

Physical model and coordinate system.

Close modal

In order to solve Eqs. (1)(3) subject to the boundary conditions (4), we introduce the following similarity transformation:

(5)

where ψ(x, y) is the stream function defined as u = ∂ ψ/∂y and v = −  ∂ψ/∂x. Eq. (1) is identically satisfied, while Eqs. (2) and (3) become

(6)
(7)

where Pr = ν/α is the Prandtl number and prime denotes differentiation with respect to η. The transformed boundary conditions are

(8)

where λ = b/a is the stretching/shrinking parameter with λ > 0 for stretching and λ < 0 for shrinking.

The quantities of physical interest are the skin friction coefficient Cf, and the local Nusselt number Nux defined as

(9)

where τw is the surface shear stress along the plate and qw is the heat flux from the plate, defined as

(10)

Using (5) one finds

(11)

where Rex = uex/ν is the local Reynolds number.

In order to perform a stability analysis, we consider the unsteady problem. Equation (1) holds, while Eqs. (2) and (3) are replaced by

(12)
(13)

The following transformation is introduced:

(14)

Eqs. (12) and (13) now can be written as

(15)
(16)

and are subjected to the boundary conditions

(17)

To test the stability of the steady flow solution f(η) = f0(η) and θ(η) = θ0(η) satisfying the boundary-value problem (1)-(4), we write (Weidman et al.,17 Postelnicu and Pop,18 Roşca and Pop,19)

(18)

where γ is an unknown eigenvalue, and F(η, τ) and G(η, τ) are small relative to f0(η) and θ0(η). Eqs. (15)-(17) give an infinite set of eigenvalues γ1 < γ2 < ⋯ . There is an initial growth of disturbance if the smallest eigenvalue is negative, and thus the flow becomes unstable. On the other hand, if the smallest eigenvalue is positive, there is an initial decay and the flow is stable. Inserting (18) into (15) and (16), one finds the linearized problem

(19)
(20)

along with the boundary conditions

(21)

By setting τ = 0, the solutions f(η) = f0(η) and θ(η) = θ0(η) of the steady equations (6) and (7) are obtained. The functions F = F0(η) and G = G0(η) in (19) and (20) identify initial growth or decay of the solution (18). In this respect, we have to solve the linear eigenvalue problem

(22)
(23)

along with the boundary conditions

(24)

The variation of the skin friction coefficient is shown in Figure 2, while that of the local Nusselt number indicative of the rate of heat transfer is presented in Figure 3. Both figures show that for larger shrinking rates, dual solutions and non-existence of the similarity solutions exist, whereas all solutions are unique for the stretching case. In particular, there are unique solutions for λ ≥ − 1, dual solutions for λc < λ < − 1 and no solution for λ < λc.

FIG. 2.

Variation of the skin friction coefficient f″(0) with λ.

FIG. 2.

Variation of the skin friction coefficient f″(0) with λ.

Close modal
FIG. 3.

Variation of the local Nusselt number −θ′(0) with λ when Pr = 1.

FIG. 3.

Variation of the local Nusselt number −θ′(0) with λ when Pr = 1.

Close modal

Based on our computation, the turning point is λc = − 1.24658, which is in good agreement with that reported by Wang.15 At λ = λc, the upper (first solution) branch meets the lower (second solution) branch. The paper by Miklavčič and Wang20 is the first that reported the existence of dual solutions for the flow over a shrinking sheet. Fang and Zhang21 successfully obtained the closed form analytical solution for the MHD viscous flow over a shrinking sheet.

The validity of the numerical solutions presented in Figures 2 and 3 is supported by the velocity and temperature profiles presented in Figures 4 and 5. These figures show that the far field boundary conditions (8) are satisfied asymptotically.

FIG. 4.

Velocity profiles when λ = − 1.24 (shrinking case).

FIG. 4.

Velocity profiles when λ = − 1.24 (shrinking case).

Close modal
FIG. 5.

Temperature profiles when Pr = 1 and λ = − 1.24 (shrinking case).

FIG. 5.

Temperature profiles when Pr = 1 and λ = − 1.24 (shrinking case).

Close modal

For the shrinking case where dual solutions exist, we determine the stability of the solutions by finding the eigenvalues γ in (18). Negative smallest eigenvalue produces an initial growth of disturbance and thus the flow becomes unstable. In contrast, the positive smallest eigenvalue results in an initial decay of disturbance, thus the flow is stable. Table I shows the smallest eigenvalues γ for selected values of λ. It is seen from this table that γ is positive for the first solution, while negative for the second solution. Thus, the first solution is stable, and the second solution is not.

TABLE I.

Smallest eigenvalues γ at selected values of λ.

λ First solution Second solution
−1  1.3690  – 
−1.1  1.0463  −0.8437 
−1.2  0.5780  −0.5173 
−1.24  0.2121  −0.2036 
−1.245  0.1030  −0.1010 
−1.246  0.0622  −0.0614 
−1.24658  0.0008  −0.0229 
λ First solution Second solution
−1  1.3690  – 
−1.1  1.0463  −0.8437 
−1.2  0.5780  −0.5173 
−1.24  0.2121  −0.2036 
−1.245  0.1030  −0.1010 
−1.246  0.0622  −0.0614 
−1.24658  0.0008  −0.0229 

Numerical results showed that non-unique solutions are possible for a certain range of the shrinking strength. A linear temporal stability analysis showed that there is an initial decay of disturbance for the first solution, while the second solution showed an initial growth of disturbance. Hence, the first solution is stable and thus physically reliable, while the second solution is not.

The authors wish to express their thanks to the reviewers for their very good comments and suggestions. The financial supports received from the Ministry of Higher Education Malaysia (Project Code: FRGS/1/2015/SG04/UKM/01/1) and the Universiti Kebangsaan Malaysia (Project Code: DIP- 2015-010) are gratefully acknowledged.

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