1 Introduction

Stagnation point flows occur in many practical situations when there is a flow impinging on an object. Yang^[1] was the first who investigated the unsteady stagnation point flow over a flat plate using a similarity transformation technique. He analyzed both the momentum and thermal boundary layers, and his solution is an exact solution of the whole unsteady Navier-Stokes (NS) equations. Yang's work was further generalized by Williams Ⅲ^[2] to the stagnation point flow in an axisymmetric setting and by Jankowski and Gersting^[3] to a general three-dimensional configuration. Teipel^[4] studied the heat transfer problem of the three-dimensional unsteady stagnation point flow and analyzed the heat transfer characteristics under different unsteadiness parameters. Wang^[5] further extended the problem to the unsteady stagnation point flow considering the impinging directions. Rajappa^[6] investigated the mass transpiration effects for Yang's probem. Burde^[7] studied the unsteady NS equations and proposed some new solutions to the unsteady stagnation point flow. Ludlow et al.^[8] further analyzed the unsteady boundary layer equations and obtained new solutions. Ma and Hui^[9] performed an analysis on the two-dimensional boundary layer equations and proposed a new type of unsteady separated stagnation point flow.

Until recent years, there are still many active research activities in the literature in studying the unsteady stagnation point problem. Takhar et al.^[10] investigated the unsteady axisymmetric stagnation point flow over a circular cylinder. Unsteady mixed convection near the three-dimensional stagnation point region was studied by Eswara and Nath^[11], and the effects of large mass injection were analyzed. The flow and heat transfer problems of a magnetohydrodynamic (MHD) fluid in the stagnation region of a three-dimensional body were investigated by Kumari and Nath^[12]. Unsteady three-dimensional viscoelastic fluid flow near the stagnation point was analyzed by Seshadri et al.^[13]. Xu et al.^[14] employed an analytical method to obtain series solutions to the unsteady stagnation point flow of non-Newtonian flows. Unsteady stagnation point flow over a stretching sheet was investigated by Nazar et al.^[15], and this problem was extended to a second grade fluid by Baris and Dokuz^[16]. The mass transfer effects on Yang's problem were studied in detail by Fang et al.^[17], and multiple solution branches were found for both momentum and thermal boundary layers. The unsteady stagnation point flow over a plate moving along the flow impingement direction was studied by Zhong and Fang^[18] for both two-dimensional and axisymmetric cases. It should be noted that in all these works, the solutions to the similarity equations were not given in a closed form. Magyari and Weidman^[19] provided a closed form solution for a special condition of Yang's problem. Fang and Zhong^[20] derived a closed form solution to Yang's problem including mass transpiration and wall movement.

Moreover, the unsteady stagnation point flow of MHD fluids also received much attention in the literature due to its important engineering applications. A few recent examples are discussed here. Soid et al.^[21] investigated the MHD stagnation point flow over a stretching/shrinking sheet using numerical techniques. Multiple solutions were found in the work depending on the wall movement parameter. Chen et al.^[22] extended the unsteady MHD stagnation point flow over a shrinking sheet by including thermal radiation and slip effects. Turkyilmazoglu et al.^[23] generalized the unsteady MHD flow to the rear stagnation point configuration over off-centred deformable surfaces. Zaib et al.^[24] studied the heat and mass transfer of an unsteady MHD stagnation point flow of a nanofluid by considering the effects of thermophoresis. More related works on flows over moving boundaries of different types of fluids and the MHD flows for different flow configurations can be found in Refs.[25]-[30]. It should be pointed out that for the forward unsteady MHD stagnation point flow, there is no solution in a closed form reported in the literature. Therefore, in this paper we will further extend the work in Ref.[20] to include the hydromagnetic effects on the general unsteady stagnation point flow passing a moving surface. The energy equation will also be investigated for a constant wall temperature case and a uniform wall heat flux case.

2 Theoretical derivation 2.1 Momentum and energy equations

We consider a two-dimensional laminar viscous incompressible stagnation point flow of MHD fluids over a flat plate with an unsteady free stream velocity U_∞=U₀x(1-γt)^-1, where U₀ and γ are constants. The x-axis points to the free stream direction. The y-axis is perpendicular to the x-axis. The wall moves in the x-direction with a velocity of U_w=λU₀x(1-γt)^-1 and . A mass transfer velocity in the y-direction exists at the wall as V_w(x, t), which is to be determined. Assume the fluid is electrically conductive, and a magnetic field B(x, t) is applied normal to the wall in the y-direction. It is noticed that for this given flow configuration, the magnetic field does not depend on x as with B₀ being a positive constant. The induced magnetic field is negligible as compared with the external magnetic field. The boundary condition of the energy equation at the wall is either maintained at a constant temperature, T_w, or a time-dependent uniform heat flux, q_w, and the fluid temperature at the free stream is kept constant as T_∞. The two-dimensional NS equations and the boundary layer energy equation are given as follows^[12]:

(1)

(2)

(3)

(4)

with the velocity boundary conditions (BCs)

(5)

where u and v are the velocities in the x- and y-directions, T is the temperature of the fluid, ν is the kinetic viscosity, p is the fluid pressure, ρ is the fluid density, σ is the fluid electrical conductivity, and α is the thermal diffusivity of the fluid. The momentum equations are full NS equations, and the energy equation is based on the boundary layer assumption by ignoring . Similarity equations are obtained by defining the stream function and the similarity variable in the following forms,

(6)

With these definitions, the velocity components are given as

and

respectively. The wall mass transfer velocity becomes

based on the second term of Eq. (5). The boundary layer energy equation can be nondimensionlized by defining a dimensionless temperature as , where T_Ref is a reference temperature to be determined later. The transformed similarity equations are

(7)

(8)

with the BCs

(9)

where is called the unsteadiness parameter, is the mass transfer parameter, is the wall moving parameter, and is the magnetic parameter. Positive s means a mass suction at the wall, while a negative value is for mass injection. Positive λ shows that the wall is moving in the same direction as the free stream (stretching wall), and a negative value means that the wall is moving opposite to the free stream (shrinking wall). Based on Eq. (3), it is found that is a function of t and y. That is, and integrating once yields , where G(t, x) is the constant of the integration. Therefore, is independent of y. Applying the x-momentum at the free stream yields

(10)

The y-direction pressure gradient can be derived from Eq. (3) as

(11)

The pressure can be obtained by integrating Eqs. (10) and (11) as follows:

(12)

where p₀ is a constant from the integration.

2.2 Analytical solution for momentum problem

In this study, we consider a special case with β=2. For β=2, the momentum similarity equation becomes

(13)

with the same BCs as Eq. (9). The analytical solutions of Eq. (13) can be found by a function transformation as F(η)=f(η)-η. It is obtained

(14)

with the transformed BCs as

(15)

Following the logic from previous publications^[31-33], the solution to Eq. (14) has the following form,

(16)

Substituting Eq. (16) into Eq. (14) and Eq. (15) yields

(17)

(18)

and

(18)

Then, Eq. (16) can be rewritten as

(20)

The value of δ can be found by

(21)

For a physically feasible solution, δ must be a positive real number. Equation (19) must satisfy s²+4(λ+M+3)≥0 to have real solutions. Under certain given values of s, M, and λ, two solutions may exist for Eq. (19), as shown in Fig. 1. Only positive solutions to Eq. (19) are physically practical. For positive s, namely, mass suction, there exists a dual solution region all the time. However, for mass injection, there is only one solution for λ+M>-3. For λ+M=-3, there is only one solution for mass suction as δ=s. Then, the solution to the general problem (see Eq. 13) is given by

(22a)

and/or

(22b)

The velocity becomes

(23a)

and/or

(23b)

For M=0, the solution reduces to the one found by Fang and Zhong^[20]. For s=λ=M=0, it becomes the solution derived by Magyari and Weidman^[19].

2.3 Energy equation solution

Two heat transfer cases are solved analytically and presented in this section. The first case is for a constant wall temperature one. The second is for uniform time-varying wall heat flux.

2.3.1 Constant wall temperature case

For the case of constant wall temperature, the energy equation BCs are T=T_w at y=0, and T=T_∞ at y→∞. By defining with T_Ref=T_w-T_∞, the energy equation becomes

(24)

with the BCs

(25)

Equation (19) gives that . Substituting it into Eq. (24) yields

(26)

A new variable as can be defined, and in terms of Eq. (26) becomes

(27)

with the transformed BCs

(28)

The general solution to Eq. (27) reads

(29)

With η as the variable, the solution is given as

(30)

Applying the BCs, it is obtained as

(31)

where Γ(a, x) is the incomplete Gamma function^[34]. The heat flux at the wall is given by

(32)

Based on the property of the incomplete Gamma function, there is no finite value of Γ(a, 0) for a negative value of a. This adds another constraint for the similarity energy equation, say M+4 < δ², which is equivalent to . From the solution of δ, we have

which implies for the upper solution branch,

(33a)

and for the lower solution branch,

(33b)

This constraint can also be obtained from the energy equation (8). In order to have finite solutions, for β=2, it must have f(∞)-η>0. This gives a requirement , which is equivalent to the above-derived condition.

2.3.2 Uniform wall heat flux case with time-varying rate

For the time-dependent uniform wall heat flux case, the BCs become

(34)

and

(35)

where k is the fluid thermal conductivity, and q_w is the wall heat flux constant. Defining with , the energy equation becomes

(36)

with the BCs

(37)

Substituting the momentum solution yields

(38)

Applying the BCs, the solution reads

(39)

Again, for the uniform wall flux case, the parameters should also satisfy the same constraints, namely, .

2.4 Stability analysis for the momentum solution

Since there are two possible solutions for the momentum equation, it is useful to conduct a stability analysis to find out if the flow is stable or not. Following the commonly used techniques for the boundary layer flow stability analysis^[35-36], we can use the following new dimensionless variables and functions,

(40)

where τ is a dimensionless time. Substituting Eq. (40) into Eq. (2) yields

(41)

In order to analyze the stability of the solution, we introduce a small disturbance with a growth rate ω into the solution as

(42)

where ω is an unknown eigenvalue, and G(η) is a function much smaller than f₀(η)=f(η). Plugging Eq. (42) into (41) and neglecting all the small terms yield the linearized eigenvalue problem,

(43)

with the BCs as

(44)

Without loss of generality, we study the flow stability at τ=0 for G₀(η) of β=2, and Eq. (43) becomes

(45)

Because Eq. (45) is a linear differential equation of G₀(η), it can be normalized by G₀^"(0) as without loss of generality. This leads to an initial value problem as

(46)

with the BCs as

(47)

Boundary condition (47) provides the constraint for the eigenvalues of ω. Substituting the solution of f(η), it is obtained

(48)

Although Eq. (48) is a linear differential equation, it is too complicated to find an analytical solution to it. Examples are calculated for a set of parameters, namely, M=1.0, λ=-6, and s=5. From the results in Fig. 1, it is found that there are two solutions δ_upper=4.561 55 and δ_lower=0.438 447. By numerically solving Eq. (48), we can find the smallest eigenvalue for the upper solution branch is ω₁=8.029 and for the lower solution branch ω₁=-5.303. This clearly shows that the lower solution branch is not stable and could not be physically achieved in reality.

3 Results and discussion

Results of different parameters are shown in this section to demonstrate the flow and heat transfer characteristics. The general solution for both cases of the thermal boundary condition involves the computation of the incomplete Gamma function. This is calculated using the embedded functions in MATLAB software, which is also used for generating all the plots.

3.1 Solutions of the flow problem

The results of dimensionless wall drag, namely, f"(0)=(λ-1)δ, and the velocity profiles, say f'(η), are presented for different values of the three controlling parameters including M, λ, and s. The solution domain of δ for the momentum equation is shown in Fig. 1 for different values of M, λ, and s. As comparing M=0.0 with M=1.0, the solution domain shifts towards the negative direction of λ for the given values of the mass transfer parameters. As mentioned in Subsection 2.1, for s≤0, there is one solution for a given value of λ with λ+M>-3. For s>0, there are two solutions to the flow problem as shown in Fig. 1. For completeness, the solution boundary for the energy equation is also shown in Fig. 1 for the two values of the magnetic parameters as discussed in Subsection 2.3.1.

Effects of the three parameters on the velocity distribution in the boundary layers are shown in Figs. 2-4. The effects of mass transfer parameters are depicted in Fig. 2. Increasing mass suction results in a thinner boundary layer for both fixed (λ=0) and stretching (λ>0) wall problems. For a shrinking wall problem (λ < 0), it is possible that there are two solutions as shown in Fig. 3. Again, for a shrinking wall problem, a higher mass suction helps reduce the boundary layer thickness for the upper solution branch. However, a higher mass suction leads to thicker boundary layers for the lower solution branch as shown in the bottom plot in Fig. 3. The effects of the magnetic parameters are shown in Fig. 4. Larger magnetic parameters result in a thinner boundary layer with given values of s and λ for the upper solution branch, which is the only solution branch for the given combination of parameters.

3.2 Heat transfer solutions 3.2.1 Constant wall temperature case

The effects of M, λ, s, and Pr are shown in Figs. 5-8, respectively. As shown in Fig. 5, the effects of the magnetic parameter M on the temperature profiles depend on the combinations of the three parameters. It is found from numerical experiments that the values of λ dominate the velocity profile trends. It should be noted that different mass transfer parameters are used in Fig. 5 for different wall stretching parameters in order to show a clear trend of the profiles as well as considering the existence of solutions for different wall moving parameters. For a stretching wall (i.e., λ>1), the thermal boundary layer becomes thicker with the increase in the magnetic parameter. However, for λ < 1, the boundary layer thickness decreases with the increase in M. For λ=1, there is a simple analytical solution for the momentum equations as f(η)=η+s, and the energy equation solution is given by θ(η)=e^-Prsη. It is seen that for λ=1, the magnetic parameter has no impact on the solutions.

As illustrated in Fig. 6, an increase in the wall moving parameter leads to a thinner boundary layer under given conditions of s, M, and Pr. Accordingly, the wall heat flux increases with the increase in the wall moving parameter λ. Similar trends are found for the effects of the mass transfer parameter s as shown in Fig. 7. The wall heat flux becomes higher for larger mass suction parameters. The effect of the Prandtl number is quite straightforward, and an increase in the Prandtl number results in a thinner thermal boundary layer and a higher wall heat flux as depicted in Fig. 8.

3.2.2 Time-dependent uniform wall heat flux case

For the cases with a uniform time-varying heat flux, temperature distributions are shown in Figs. 9-12. The observed trends of the effects of four parameters are similar to the constant wall temperature case. The variation of the thermal boundary layer thickness is similar to the results with a constant wall temperature. Because the wall heat flux is uniform, a thinner thermal boundary layer generally indicates less heat penetration length into the fluid and a lower wall temperature. Generally speaking, the wall temperature decreases when s, λ, and Pr increase. The effects of the magnetic parameter do depend on the wall moving parameter. Again different mass transfer parameters are used in Fig. 9 for different wall stretching parameters in order to show a clear trend of the profiles as well as considering the existence of solutions for different wall moving parameters. For λ >1, the wall temperature increases with the increase in M, while for λ < 1, the wall temperature decreases with the increase in M. For λ=1, the solution to energy equation solution is , which does not depend on M.

4 Conclusions

The flow and heat transfer of an unsteady MHD fluid flow near the stagnation point region over a moving flat surface with mass transpiration are investigated. The solutions are given in a closed form for both momentum and thermal similarity equations. The fluid flow solution is an exact solution to the NS equations. Under certain conditions with a shrinking wall, there might exist two solution branches for both flow and heat transfer problems. Stability analysis indicates that the lower solution branch is not stable. Results show important influences of the governing parameters on the fluid flow and heat transfer. The closed-form solution provides a rare case of the MHD unsteady solution for the NS equations and can be used for validating the numerical code for computational fluid dynamics.