The issue of the boundary layer flow through a moving plate is increasingly attracting the attention of scholars and researchers because of its importance for many practical applications. Hence, many studied topics have been disclosed in recent decades.^[1–8] One of the topics is concerned with the numerical solutions. For example, Yang and Zhang et al.^[9] used the finite difference method to detect bifurcations and chaos of a nonlinear forced vibration of the system. Anuar and Yacob et al.^[10] studied thermal field of flow through a moving plate with the effect of thermal radiation. Numerical solutions are obtained by Runge–Kutta–Fehlberg method with shooting technique. In the study of Ali and Nazar et al.,^[11] the shooting method is applied to obtain the problem of flow through a movable plate with a magnetic field near the stagnation point. Sharma and Bhargava et al.^[12] considered the issue of free convection in a micro-polar fluid with viscous dissipation. Moreover, its numerical solution can be obtained by the meshless-Galerkin method. In order to study axially moving two-dimensional materials, Hatami and Azhari et al. proposed^[13] that a p-version finite element can be used to obtain the numerical results. However, as mentioned above, it can be seen that those numerical solutions do not have the characteristics of differentiation and integration because the numerical data is discrete. Hence, in order to resolve the issue, Puhov in 1976^[14] proposed the method of modified differential transform method (DTM) with shooting method to study this kind of problem. Because DTM is an extension of the Taylor series method, a numerical method can be used to obtain analytic solutions for ordinary differential equations (ODEs) problems,^[15–20] partial differential equations (PDEs) problems,^[21–24] differential-algebraic equations (DAEs) problems,^[25–28] differential difference equations (DDEs) problems,^[29–32] linear integro-differential equation problems^[33–39] and eigenvalue problems.^[40–44] Hence, in view of the above, a modified differential transform method (MDTM) will be used to obtain the numerical results for entropy generation in Flow through a movable plate with variable temperature in the x axis. Another important issue studied in this paper is irreversibility analysis, which is one of the energy conservation methods,^[45–50] because irreversibility often arises in energy transfer processes.^[51–52] Consequently, in order to study the irreversibility of a thermal system, many scholars have studied irreversibility due to heat convection. Such as, Bejan^[53–54] proposed the expression for the entropy generation rate, whose goal is to study entropy generation in a heat Transfer process. Since then, many researchers have studied the effect of various conditions on Irreversibility of flow field. Such as Rashidi and Mohammadi et al. presented the irreversibility analysis for a permeable surface with a variable heat source.^[55] Guillermo Ibanez^[56] presented entropy generation analysis for a channel with permeable plates and analyzed irreversibility due to the effect of hydrodynamic slip and convective boundary. Adnan Saeed Butt and Asif Ali^[57] examined entropy generation due to the effect of thermal radiation in a moving plate. Hedayati and Malvandi et al.^[58] presented the entropy generation minimization (EGM) for heat transfer on a moving wedge. Yazdi and Hashimc et al.^[59] proposed the second law analysis for Power-Law Fluid on Micro-patterned movable surface. For other interesting studies in solving heat transfer problem, Yang proposed integral transform so as to solve the heat transfer problems. The results indicated that the method is powerful for finding analytical solution for heat transfer equation, such as steady heat transfer equation^[60–61] and heat diffusion equation.^[62] However, unlike the above mentioned topics, this present work will investigate the effect of variable wall temperature in a moving plate on entropy generation.

Consider that the plate is movable with velocity. Moreover, the temperature of the plate along the axis direction is variable. Hence, the governing equations for boundary layer flow are in the following

The dimensionless parameters are proposed as the same method^[5] in the following

Because the boundary conditions are infinite space. Let , then . So we have the boundary value

For Eq. (10) are not a standard initial value ODE, it is difficult to get the numerical solution. We delete the boundary value and add the initial value . Then we can find the value p that the solution g(t) go through by shooting method. After g(t) is solved, then we use the same method to get m(t). For three terms of the analytic solution is

The issue of irreversibility is important because energy conservation cannot be ignored in many practical applications. Hence, in order to study irreversibility of a thermal system, the local entropy generation rate for a flow through a moving plate with variable temperature is defined as

Following Bejan.^[28,38–39] who proposed that the ratio of the local entropy generation rate to the characteristic entropy generation rate is equal to the entropy generation number. Therefore, from Eq. (6), the entropy generation number can be proposed as Eq. (12), where the Eckert number . Dimensionless temperature difference . In view of above, it can be concluded that the value of is constant. Moreover, the characteristic entropy generation is , the heat transfer irreversibility is , the fluid friction irreversibility is . Thus, Bejan number Be can be written^[40–41] as Eq. (13).

Now, consider a brief description of standard DTM. Let v(t) be an analytic function in a domain D. Here t = a represents any point in D. Then, Taylor series expansion for the function of v(t) is defined as

It is worth mentioning that the function of v(t) can be a Maclaurin series when a = 0

The differential transformed function v(t) is expressed as

In practical applications, because the differential transformation method is derived from the Taylor series expansion, it is found that the number of arguments required to restore the unknown function precisely can be reduced by specifying an appropriate value of the constant H. In other words, the function v(t) can be obtained in terms of a finite series as follows:

Next, we state some important properties of the Taylor differential transformation derived using the above expressions, which are needed in the sequel. The operation properties of differential transformation. If the transformed functions for both U(k) and V(k) are derived from u(x) and v(x), respectively, then the fundamental mathematical operations of differential transformation are listed as follows. (The other proofs can be seen in references.)

(i) If , then .

(ii) If , then .

(iii) If , then .

(iv) If , then .

(v) If , then .

We only give the proof of (v), the proof of the others can be seen in references. For simplicity, Considering H = 1,

Now, back to our model (10). Consider the differential equations (3) and (5)

So we have

After solving G(k), then we can have M(k) after substituting it into the second equation of Eq. (22).

The algorithms are coded in the computer algebra package-Mathematica. The environment variable Digits controlling the number of significant digits is set to 32 in all the calculations done in this paper. Case 1. Consider the case, as shown in Tables 1–3. Case 2. Consider the case, as shown in Tables 4–5.

Table 1

Table 1

Table 1 The present method f′, f′, f″ for case 1, iterations m = 6. . η Rk4 Present method f 0.0 0.0 0.0 1 0.786 0.785 2 1.218 1.211 3 1.432 1.433 5 1.577 1.544 9 1.609 1.544 10 1.609 1.544 f′ 0 0.001 0.001 8 0.000 0.000 9 0.000 0.000 f″ 0 0.000 0.000 8 0.000 0.000 9 0.000 0.000

Table 1 The present method f′, f′, f″ for case 1, iterations m = 6. .

Table 2

Table 2

Table 2 The present method θ, θ′, θ″ for case 1, iterations m = 6. . θ Rk4 Present method θ 0.0 0.0000 0.000 00 1 0.1375 0.1327 2 0.0151 0.0131 3 0.0004 0.0005 5 0.0000 0.0000 9 0.0000 0.0000 10 0.0000 0.0000 θ′ 0.0 0.0000 0.0000 8 0.0000 0.0000 9 0.0000 0.0000 θ″ 0.0 0.0000 0.0000 8 0.0001 0.0000 9 0.0000 0.0000

Table 2 The present method θ, θ′, θ″ for case 1, iterations m = 6. .

Table 3

Table 3

Table 3 The absolutely maximum error for Table 2 between the present and Rk4 methods as the iteration m = 8, 9, 10; . Interation m Absolutely Maximum Error m = 8 0.0056 m = 9 0.0053 m = 10 0.0051

Table 3 The absolutely maximum error for Table 2 between the present and Rk4 methods as the iteration m = 8, 9, 10; .

Table 4

Table 4

Table 4 The present method θ, θ′, θ″ for case 2, iterations m=6. . θ Rk4 Present method θ 0.0 0.0000 0.000 00 1 0.076 25 0.079 18 2 0.005 66 0.079 18 3 0.000 34 0.002 89 5 0.000 00 0.000 40 9 0.000 00 0.000 00 10 0.000 00 0.000 00 θ′ 0.0 0.000 00 0.000 00 0.008 0.000 00 0.000 00 9 0.000 00 0.000 00 θ″ 0.0 0.000 00 0.000 00 8 0.000 01 0.000 00 9 0.000 00 0.000 00

Table 4 The present method θ, θ′, θ″ for case 2, iterations m=6. .

Table 5

Table 5

Table 5 The absolutely maximum error for table 4 between the present and Rk4 methods as the iteration . . Iteration m Absolutely Maximum Error m = 4 0.0098 m = 5 0.0077 m = 6 0.0034

Table 5 The absolutely maximum error for table 4 between the present and Rk4 methods as the iteration . .

The irreversibility analysis is presented for variable temperature in a moving plate. Similarity solutions, such as velocity-gradient, temperature-gradient, heat transfer irreversibility , fluid friction irreversibility , entropy generation number and Bejan number Be, can be obtained by the modified differential transform method (DTM). As shown in Fig. 1, profiles will increase with increasing the value of Prandtl number. This shows that the temperature-gradient profiles will increase while the value of Prandtl number increases. In other words, Prandtl number will affect temperature-gradient in boundary layer flow. So, the profiles will increase with increasing the profile of temperature-gradient (as Fig. 7). Also, the three profiles in Fig. 1 have a maximum value near a point η = 0. Also, the slope of in Fig. 1 will drop down at the edge of the boundary layer. The results can be seen in other figures of profiles too.

Fig. 1

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	Fig. 1 Profiles with various Prandtl number while n = 1.

In Fig. 2, it is seen that profiles will increase with increasing the value of Prandtl number, but velocity-gradient profile is constant for various Prandtl number (Pr). This means that Prandtl number will not affect the velocity-gradient in boundary layer flow at all, as shown in Fig. 8.

Fig. 2

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	Fig. 2 Profiles with various Prandtl number while n = 1.

So, from Eq. (12), it is found that the Pr is a suitable factor for evaluating irreversibility for fluid friction . Moreover, the value of is constant for various n values. Now, paying attention to the entropy generation number in Fig. 3, it can be seen the results mentioned above match well with the profile of entropy generation number which increases with increasing the value of Pr. This is because the entropy generation number is equal to sum of irreversibility for heat transfer and the irreversibility for fluid friction. Then, in Fig. 4, it is found that a decrease in Bejan number Be results in an increase in the value of Prandtl number. Also, the three profiles in Fig. 4 have a maximum value near a point η = 0, where the slope will drop down at the edge of the boundary layer. This means that boundary layer flow irreversibility is dominated by heat convection near moving plate surfaces.

Fig. 3

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	Fig. 3 profiles with various Prandtl number while n = 1.

Fig. 4

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	Fig. 4 Be profiles with various Prandtl number while n = 1.

However, their irreversibility is dominated by friction when η reaches infinity. Finally, it is concluded that there is a boundary layer flow through a movable plate with variable temperature in x axis, its temperature profile and stream function is presented in Figs. 5 and 6, respectively. Then, various properties of fluid (such as Pr = 0.7, 2, 3) is applied to analyze the similarity profiles of entropy generation when plate temperature varies as , where n = 1. Moreover, it is noticed that the heat transfer process of a practical thermal system is irreversible. So, while various properties of boundary flow through a movable plate and the distribution of temperature along x axis is linear, the similarity profiles of entropy generation will be concerned because entropy generation plays a key role in evaluating the performance of the heat transfer process. It is worth mentioning that the temperature profile will decrease as a result of increasing Prandtl number. Meanwhile, the profile will increase. Consider that the heat transfer performance of gas is less than water. So, the profile will increase because of the increasing of Prandtl number. Hence, it is concluded that the results from the recent study clearly show that entropy generation will play a key role in evaluating the irreversibility of a practical thermal system.

Fig. 5

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	Fig. 5 Temperature profiles with various Prandtl number n = 1.

Fig. 6

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	Fig. 6 The stream function profile with various Prandtl number n = 1.

Fig. 7

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	Fig. 7 Temperature-Gradient profiles with various Prandtl number n = 1.

Fig. 8

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	Fig. 8 Velocity–Gradient profile with various Prandtl number n = 1.

A practical thermal system is irreversible, hence the issues of energy conservation have attracted much attention in recent years. In this paper, therefore, the irreversibility for boundary layer flow through a moving plate with variable temperature will be analyzed because of their importance in many practical applications. In order to obtain the similarity solution of boundary layer flow, modified ADM with shooting method will be used to obtain similarity solution for boundary layer. For example, velocity-gradient, temperature-gradient, heat transfer irreversibility , fluid friction irreversibility , entropy generation number and Bejan number (Be). Moreover, it is worth mentioning that ADM is advantageous because its numerical solution is differentiation and integration. Therefore, its analysis result is reliable and highly accurate. The recent study presented the following results about entropy generation analysis:

(i) The study of entropy generation is important because energy conservation can not be ignored in many practical applications.

(ii) In order to study Irreversibility of a moving plate with various temperature, suitable similarity variables are used to transform the local entropy generation rate to entropy generation number.

(iii) A modified differential transform method (DTM) with shooting method is used to obtain the similarity solution of the entropy generation, the proposed numerical method, their accuracy of analytic solution is very high.

(iv) The similarity solution profiles , and , will increase with increasing of the Prandtl number value. Moreover, profiles have a maximum value near a point η = 0. The slope of similarity solution profiles will drop down at the edge of the boundary layer.

(v) It is worth mentioning that the temperature profile will decrease as a result of the increasing of Prandtl number. Meanwhile, the profile will increase.