MHD viscoelastic flow and heat transfer over a vertical stretching sheet with Cattaneo-Christov heat flux effects
Graphical abstract
The physical model of a vertical stretching sheet.
Introduction
The study of the flow and heat transfer over a stretching surface has attracted much attention [1], [2], [3] for its applications in many fields of science and technology. Such as extrusion processes, hot rolling, paper production, artificial fibers, wire drawing, ion exchange columns, distillation towers and subterranean chemical waste migration, the study is important for controlling the quality of the final products [4], [5] and many researchers have studied the effects of various parameters on the velocity profiles and the temperature distribution [6], [7]. Moreover, the slip boundary [8], [9] and chemical reaction [10] also play an important role in the temperature and velocity. Aman et al. [11] found that the heat transfer rate at the surface increases while the skin friction coefficient decreases with slip. Recently, the flow and heat transfer over a vertical surface also attract many scholars' attention, such as Ishak et al. [12], [13], Saleh et al. [14] and Rashidi et al. [15], some author considered different heat transfer fluids such as engine oil, ethyleneglycol and water over the vertical stretching sheet [16], [17]. The magnetic field effects were also widely used in physics and engineering [18]. Pavlov [19] observed that increased stretching decreases the surface shear stress and the surface heat transfer rate while increases the fluid velocity and temperature. Ishak et al. [20] studied the steady MHD mixed convection flow adjacent to a vertical surface with prescribed heat flux. Freidoonimehr et al. [21] studied the transient MHD laminar free convection flow of nano-fluid over a vertical surface. They found that the transverse magnetic field opposes the transport phenomena and the temperature distribution increases slightly with the increase in the magnetic parameter.
Heat transfer is a widespread phenomenon in the nature which exists due to temperature difference between objects or within the same body. Fourier's law of heat conduction law [22] is successfully for understanding the phenomena of heat transfer. However, the main shortcoming of the Fourier's laws contradicts the principle of causality. The propagation speed of heat disturbance is finite by introducing the thermal relaxation time in the Fourier's model [23]. Moreover, Christov [24] modified the time derivative in the Maxwell-Cattaneo model with the Oldroyd upper-convected derivative which successful preserves the material-invariant formulation. For the incompressible fluid with the Cattaneo-Christov heat conduction model, Tibullo and Zampoli [25] proved the uniqueness of the solution. Han et al. [26] studied coupled flow and heat transfer in viscoelastic fluid with Cattaneo-Christov heat flux model. He successfully studied the effects of the relevant parameters on the temperature and velocity. Mustafa [27] explored the rotating flow of Maxwell fluid over a linear stretching sheet with the Cattaneo-Christov heat flux. Hayat et al. [28], [29] studied the flow over a stretching sheet with variable thickness and MHD flow of Oldroyed-B fluid with homogeneous-heterogeneous reactions with Cattaneo-Christov heat flux, respectively. Recently, the interest has been extended to the problem of the flow and heat transfer over sheet by utilizing Cattaneo-Christov heat flux model [4], [32], [33]. However, the model has seldom been used to study coupled flow and heat transfer of boundary layer in viscoelastic fluid.
The purpose of this article is to study the MHD flow and heat transfer over a vertical stretching sheet with Cattaneo-Christov heat flux effects. The governing partial differential equations are reduced into a class of ordinary differential equations by local similarity transformation. The numerical solutions are obtained and the effects of physical flow parameters such as the magnetic number, local Reynolds number, Grashof number, Prandtl number and the relaxation time are studied. The graphs are plotted and discussed for the skin friction coefficient and heat transfer rate with velocity slip parameter.
Section snippets
Problem formulation
This paper considers the MHD incompressible flow over a vertical stretching sheet under steady state conditions. The physical model is shown in Fig. 1, it is assumed that the sheet moving with a velocity of U = U0xm(U0 is a positive constant). There is a magnetic field B = B0x(m − 1)/2 in the direction of the y-axis and there exists a velocity slip on boundary. The governing equations of mass, momentum and energy are expressed as follows
Results and discussion
The local similarity solution ordinary differential Eqs. (10), (11) subject to the boundary condition (Eq. (12)) are solved for different values of the magnetic number M, local Reynolds number Rex, Grashof number Grx, Prandtl number Pr and mixed convection parameter λ versus x. The results listed in Table 1 shows that the parameter R influences the Rex1/2Cf = f″(0) and NuxRex− 1/2 = − θ′(0). In order to analyze the general trend of Rex1/2Cf and NuxRex− 1/2, we choose a limited number of points. Fig. 2
Conclusions
In this paper, we investigate the steady boundary layer MHD flow past a vertical stretching sheet with Cattaneo-Christov heat flux model effects. We consider the steady boundary layer flow with a partial slip velocity and the relaxation time parameters. The function bvp4c technique in Matlab is used to obtain the numerical solutions. The influences of velocity slip number, magnetic number, Prandtl number, mixed convection parameter and relaxation time parameters on dimensionless velocity and
Acknowledgement
The work is supported by the National Natural Science Foundations of China (Nos. 51276014, 51476191).
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