In this study, we propose a boundary value problem that contains two arbitrary parameters in the differential equation and show that the results of a number of existing stretching problems (linear, power law, and exponential stretching) are the special cases of the proposed boundary value problem. A two-term analytic asymptotic solution of this problem is developed by introducing a small parameter in the differential equation. Interest lies in the finding of rare exact analytical solutions for the zeroth and first order systems. Surprisingly, only a two-term closed form of analytical solution shows an excellent match with the existing literature. The solution for second-order system is found numerically to improve the accuracy of the approximate solution. The generalised analytic solution is tested over a number of stretching problems for the velocity field and skin friction coefficient showing an excellent match. In conclusion, various stretching problems discussed in literature are special cases of this study.
Flow due to stretching surface plays a vital role in different fields of engineering and industry. Many real processes involve various type of stretching velocities such as linear, power-law, and exponential. Such flows are generally generated in fibres spinning, extrusion of polymers, hot rolling, glass blowing, continuous casting, manufacturing of plastic, and rubber sheet. Engineers and scientists have modelled a number of such physical problems in fluid mechanics and presented analytical and numerical results for better understanding of the fluid behaviour and adequate explanation of the empirical results. Its mathematical interest in attempting nonlinear systems cannot be ignored.
Sakiadis  first studied the boundary layer flow over a surface moving with a constant velocity in a viscous fluid. Crane  in his highly cited paper found a closed-form exact analytical solution for steady linearly stretching surface in a viscous fluid. Banks  and later on Liao  extended this work to nonlinear stretching of the surface for power-law stretching velocity. The flow of viscous fluid over exponential stretching surface was considered by Magyari and Keller . Similarity solutions for the nonlinear stretching velocity of the form x1/3 are found by Cortell . A great deal of stretching phenomena has been discussed by introducing various physical parameters for different geometries in the discussion of flow field in viscous fluid [7–10]. All of these problems are modelled in two-dimensions because the stretching has been taken as unidirectional, that is, along x-axis and the viscous diffusion takes place in the y-direction. In the case of stretching of the plate in two directions, the ensuing three-dimensional problems have been discussed by Wang .
Most of these stretching problems for viscous fluid are solved either numerically or using approximate analytical techniques except Crane and Angew . The approximate solutions are generally presented in series form which requires a lot of calculations and computer time. The aim of this study is to suggest a general approximate analytical solution of a proposed equation with stretching boundary conditions that represent the solution of all the stretching problems in viscous fluids. Perturbation method is used to find the solution by introducing a temporary small parameter (ε) and solving the resulting system of equations successively. A two-term perturbation result gives an excellent match with the known numerical results. The exact solution of the leading order nonlinear problem is obtained, and a two-term perturbation result is presented. The solution of the original problem can, in the end, be recovered by letting ε= 1. We mention that this is an old and standard way of perturbation technique  to solve a difficult problem by introducing a small parameter temporarily, when no such parameter already exists. This simple method is used elegantly to abridge a number of stretching problems into one.
Important physical quantities, such as the velocity field and the skin friction coefficient, are calculated and compared with the known, mostly numerical solutions, results to establish the assertion. An excellent agreement is found between the results of the proposed method and the existing literature for skin friction coefficient and the velocity profile near the surface. We accept that the velocity profiles are not in very good agreement far away from the surface; nevertheless, it is not crucial for the present study in that the velocity near the surface and the skin friction coefficient are important, particularly from engineering point of view.
Finally, where we candidly admit to have visited the known work only, we quickly add that the methodology adopted in this study works well, without any exception, for all the stretching velocities in the viscous fluid in two or three-dimensions. We understand that the analytical solutions have an advantage over the numerical solution in the sense that it gives a clearer picture of the physics of the problem when compared with the numerical solutions. The numerical solution is not sufficient in itself, and it always helps to substantiate numerical analysis with analytical or experimental results. This paper addresses this particular requirement for the stretching problems in viscous fluid and saves a lot of effort to visit every stretching problem separately.
2 Problem Formulation
Let us consider a two-dimensional laminar flow of a steady incompressible of viscous fluid over a semi-infinite surface. The surface is stretching with velocity uw(x) in x-direction, and y-axis is considered normal to the surface. Using Boussinesq approximations, the appropriate governing equations of continuity and momentum are as follows:
where u and v are the components of velocity in x and y directions and v is the kinematic viscosity of the fluid.
Boundary conditions of the problem are as follows:
The skin friction coefficient is defined by the following equation:
Analytical and numerical results of (1)–(4) have already been presented for linear, nonlinear, and exponential stretching velocities in viscous fluid. In this study, a closed form of generalized solution is discussed, which covers all these stretching velocities as special cases.
Table 1 gives a glossary of the stretching papers in the viscous fluid, which are discussed in the literature. In these papers, the governing boundary layer equations and the boundary conditions are written in a self-similar form, in terms of similarity function f(η), using the appropriate similarity transformations. The function f(η) is a dimensionless stream function and its derivative f′(η) is the dimensionless velocity, whereas η is a similarity variable.
|Stretching velocities||Self-similar equations||References|
|Linear stretching uw(x)= α x||f″′ + ff″ − f′2=0,||Crane |
|Exponential stretching uw(x)=U0ex/L||f″′ + f″ − 2f′2=0,||Magyari and Keller |
|Specific power-law stretching||Cortell |
|Power-law stretching uw(x)=U0xn||Banks |
|Generalized power-law stretching uw(x)= a(b + x)λ||Liao |
|Linear stretching in three-dimension||f″′ + mff″ − f′2=0,||Wang |
In this study, the similarity solution of the stretching problems was also discussed. This method entails that a similarity solution can be represented into one universal graph when plotted in terms of the similarity variable. The similarity solution keeps its dimensionless form because the similarity transformation converts the original dimensional variables into non-dimensional form. This provides a way to perform experiments by scaling the results from model to prototype. These experiments are conducted at smaller scale in order to save both time and money. The similarity transformation (whenever it exists) helps in reducing the governing partial differential equation into ordinary differential equation. The exact or analytical solution of these equations is possible in many model problems and quite helpful for better understanding of the flow behaviour. For example, Fang and Zhang  obtained an exact analytical solution of the Falkner–Skan equation with linearly stretching wall and Shahzad et al.  presented an exact analytical solution for axisymmetric flow over a radially stretching sheet having velocity of the form u= ar3.
We note that self-similar forms of the governing equations vary with the choice of the stretching velocity; however, the boundary conditions are the same.
The self-similar equations given in the Table 1 can be put in the following generalized form:
where particular choices of m and n construct the self-similar equations for various stretching velocities given in the Table 1. However, in principle, the other stretching velocities can end up into differential equations with other values of m and n. Interestingly, the exact solution of (5) and (6) is given by Crane  for m= n= 1. For other stretching velocities, no exact solutions have been obtained. However, approximate analytical solutions have been provided by a number of authors. These solutions are often in series form and require a lot of iterative computations to reach a good accuracy. Indeed, there is always a need to express the solutions in closed form for better understanding of the physical processes and a meaningful explanation for empirical results. In this backdrop, we introduce a new approach to give an almost exact analytical solution (when compared with the numerical solution) for the skin friction of the given stretching problems in viscous fluid with a considerable ease. In this study, we have considered two-dimensional flow with stretching in x-direction only; however; the three-dimensional problem with stretching in two directions (say x and y) can also be solved using the proposed method. An example for this case is also included in Tables 1 and 2.
|Stretching types||Values of m and n||Present||Referred paper||% Error|
|Linear||m= 1, n= 1||0.979796||1.0, Crane ||2.02|
|Exponential||m= 1, n= 2||1.27017||1.28181, Magyari and Keller ||0.91|
|Specific power-law||m= 2/3, n= 1/3||0.659966||0.677647, Cortell ||2.61|
|Power-law||m= 1, n= 2k/(k + 1),|
|k= 0.2||0.754247||0.766758, Cortell ||1.63|
|k= 0.5||0.866667||0.889477, −||2.56|
|k= 0.75||0.932312||0.953786, −||2.25|
|k= 1.5||1.04350||1.061587, −||1.70|
|k= 3.0||1.13333||1.148588, −||1.33|
|k= 7.0||1.20357||1.216847, −||1.09|
|k= 10.0||1.22207||1.234875, −||1.04|
|k= 20.0||1.24519||1.257418, −||0.97|
|k= 100.0||1.26502||1.276768, −||0.92|
|Generalized power- law||m= 1/2, n= β, β= 0.1||0.516398||0.504471, Liao ||−2.4|
|β= 0.2||0.547723||0.560408, −||2.26|
|β= 0.3||0.596285||0.612421, −||2.63|
|β= 0.4||0.645497||0.661155, −||2.37|
|β= 0.5||0.692820||0.707107, −||2.02|
|β= 0.6||0.737865||0.750665, −||1.71|
|β= 0.7||0.780720||0.792141, −||1.44|
|β= 0.8||0.821584||0.831785, −||1.23|
|β= 0.9||0.860663||0.869806, −||1.05|
|β= 1||0.898146||0.906376, −||0.91|
|β= 2||1.212440||1.216019, −||0.29|
|β= 5||1.862260||1.863220, −||0.05|
|β= 10||2.607810||2.608148, −||0.01|
|β= 20||3.669740||3.669861, −||0|
|Linear (3-Dim)||m= 2, n= 1||1.1431||1.1737, Wang ||2.61|
First, a relatively small term is identified in the governing equation and its contribution is relegated to the higher order perturbation term by introducing a small temporary parameter (ε) with it. After introducing the small parameter, the resulting equation is solved and the two terms regular perturbation solution is presented by taking ε= 1. This suggests the following equations:
Substituting (8) into (5) and boundary conditions (6), the zeroth- and the first-order systems are:
Zeroth-order system is given below:
First-order system is as follows:
The two systems are both nonlinear in nature and appear difficult to solve. However, the closed-form exact analytical solutions of the above systems are obtained (n ≠ 0) and are expressed as follows:
The velocity profile and the skin friction coefficient of the generalised problem are obtained by substituting (13) and (14) in (8), taking derivatives and finally putting ε= 1. It gives the following:
The above two-term perturbation solution is a general analytical solution for viscous fluid flow over a surface stretching with various velocities that are mentioned in Table 1.
3 Results and Discussion
Since the exact solution is available only for the case of linear stretching, the velocity profiles of other cases (Tab. 1) are first obtained numerically to verify the validity of our closed-form solutions. This should be done since the values of velocity profiles for other cases have not been tabulated. However, the skin friction is shown in tabular form, and we have provided a comparison of our results directly with the relevant papers. Numerical solutions are obtained using shooting method.
Figure 1a–d show the comparison of velocity profiles obtained in (15) with the exact (or numerical) solutions. A good comparison is found between the solutions for all types of stretching velocities near the surface. However, far away from the surface, velocity profiles are not in good agreement. The most important physical quantity is skin friction, and this compromise is affordable.
The focus of our attention is the skin friction coefficient (16) and its comparison with the existing literature given in Table 2, which shows an error of <2.7%. This is comfortably reached through exact solutions of the leading order and first-order perturbation results. Further accuracy can be achieved through the numerical solution of the next higher-order (third-order) term. Figure 2 shows the comparison of three terms perturbation results with the existing literature.
Since it is possible to overview all the stretching velocities, that is, linear and nonlinear, it can be readily concluded that the value of skin friction coefficient increases with an increasing ‘n’ (Tab. 1 and Fig. 2). This conclusion has been possible by looking on all the stretching velocities together, that is, the nonlinearity in the stretching surface velocity increases the skin friction. It is due to the fact that if every part of the surface is stretched unequally, the skin friction will increase accordingly. This observation may add distinctive values to this study besides presenting a new generalised solution of the stretching problems.
A new general approximate analytical solution is obtained for the flow of a viscous fluid for various stretching problems. Two-term perturbation solution provides a good approximation that matches with earlier solutions found numerically. This is an old and standard way to solve a difficult nonlinear problem by temporarily introducing a small parameter when no such parameter already exists. The advantage is a neat, less expensive, and less cumbersome solution in comparison with other series solutions given for the same set of problems by other authors. The way of finding the solution may help to draw our attention to some other difficult nonlinear problems showing the same success. In the backdrop of the observations made, it is concluded that the nonlinearity in the stretching velocity increases the skin friction on the surface.
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