The numerous technological applications of non-Newtonian fluids have stimulated research within this field. Non-Newtonian fluids are technologically applied in manufacturing of plastic sheets, pharmaceutical, physiology, fiber technology, crystal growth, etc. Many models of non-Newtonian fluids have been proposed since the characteristics of non-Newtonian fluids cannot be described by a single constitutive relationship. Forced convection alone is not enough to dissipate all heat in very high-power output devices. In such cases, mixed convection will often give desired results. Mixed convection phenomena mainly occur in industrial and technological applications, such as cooling of nuclear reactors during emergency shutdowns, heat exchangers in low velocity environments, solar collectors, and electronic devices cooled by fans.
Srinivasacharya et al.  and Ishak et al.  studied mixed convention boundary layers in stagnation point flow. Ahmad et al.  analysed film flow of a micropolar fluid. Recently, Khan et al.  investigated mixed convection of a Sisko fluid in the presence of convective boundary conditions.
A Jeffrey fluid is one of the models for non-Newtonian fluids that describes the influence of the ratio of the relaxation to retardation times. Kothandapani and Srinivas  applied the model for MHD peristaltic flow in an asymmetric channel. Hayat et al.  examined thermal radiation effects on a Jeffrey fluid. Nadeem et al.  presented an analysis of the boundary layer flow of a Jeffrey fluid. Hayat et al.  discussed the three dimensional flow of a Jeffrey fluid when the thermal conductivity varies with temperature. Turkyilmazoglu and Pop  investigated the flow and heat transfer of a Jeffrey fluid near the stagnation point on a stretching/shrinking sheet.
Kavita et al.  analytically studied the effect of heat transfer of an MHD oscillatory flow of Jeffrey fluid. Khan et al.  investigated the problem of MHD boundary layer flow. Bilal et al.  discussed flow of a Jeffrey nanofluid over a radially stretching sheet. Abbasi et al.  applied the homotopy analysis method to solve a Jef-frey nanofluid flow with thermal radiation and double stratification. More recently, Ali et al.  analytically and exactly studied an oscillating and incompressible MHD secondgrade fluid in porous medium. Xiao-Jun et al.  proposed a new numerical approach, embedding the differential transform and Laplace transform to solve the diffusion equation fractional heat transfer derivative. Xiao-Jun  proposed a new integral transform operator.
Ellahi et al.  discussed effects of nanoparticles to analyze Jeffrey fluid along a catheter. Hamad et al.  investigated the dynamics of a Jeffrey fluid near the stagnation point on a stretching sheet taking into account the thermal jump condition at the surface.
Recently, Das et al.  presented a numerical analysis of an electrically conducting Jeffrey fluid over a stretching sheet with partial surface slip.
Heat transfer process with radiation effects is very significant in electrical power generation, space vehicles, missiles, nuclear plants, astrophysical flows and many other applications. Little work has been done on mixed convection flow of MHD Jeffrey fluid over an exponentially stretching sheet in the presence of thermal radiation and chemical reaction. The governing equations of a Jeffrey fluid will be simplified by using a suitable similarity variable and then solved numerically by the Spectral Quasi-Linearization method (SQLM). Graphs and tables are presented to illustrate and discuss important hydrodynamic, thermal and solutal characteristics of the the flow.
2 Mathematical formulation
We consider a steady two-dimensional incompressible Jeffrey fluid over an exponentially stretching sheet. Thermal radiation and a uniform chemical reaction are significant in this study. The x-coordinate is along the stretching sheet and the y-coordinate is taken to be perpendicular to the stretching sheet. The sheet is stretched vertically with velocity where u0 is a constant and L is the reference length. We also assume that the sheet has surface temperature and surface concentration where T0 and C0 are constants. T∞ and C∞ are, respectively, the free stream temperature and concentration. The governing boundary layer equations for the model under consideration are:
where u and v are the velocity components in the x and y directions, respectively. λ1 is the ratio of relaxation time to retardation time, λ2 is the retardation time, T is fluid temperature, C is fluid concentration, D is mass diffusivity, kr is the chemical reaction rate constant, ν is kinematic viscosity, qr is the radiative heat flux, ρ is fluid density. Also, g, βT, βc, α are gravitational acceleration, thermal expansion coefficient, solutal expansion coefficient, and thermal diffusivity, respectively. B is the magnetic field which is assumed to be B = B0 where, B0 is a constant magnetic field.
The corresponding boundary conditions for the flow model are:
2.1 Similarity transformations
The stream function ψ is introduced such that, , , and we define the non-dimensional variables
The corresponding boundary conditions are
Here, prime denotes differentiation with respect to η, β = u0λ2/L exp(x/L) is the Deborah number, λ3 = gβT exp(−2x/L)ν L(Tw − T∞) is the thermal buoyancy parameter, λ4 = gβT exp(−2x/L)ν L(Cw − C∞) is the solutal buoyancy parameter, is the thermal radiation parameter, Pr = ν/α is the Prandtl number, Sc = ν/D is the Schmidt number, and γ = kr exp(−x/L) is the chemical reaction parameter.
3 Methods of solution
3.1 An overview of the Spectral Quasi-Linearization Method (SQLM)
We illustrate the main idea of SQLM using a single n-th order nonlinear differential equation
where a ≤ x ≤ b and ′,″, …,(n) denote differentiation with respect to x one time, two times, …, n times. SQLM comprises two basic steps where Quasi-linearization precedes Chebyshev differentiation.
Let u, ur and v denote (u, u′, u″, …, u(n)), (ur, u′r, u″r, …, ur(n)) and (v, v′, v″, …, v(n)) respectively, where r = 0, 1, 2, …. Taylor series expansion of (13) about v gives
upon neglecting higher order terms and re-arranging. Writing (14) in more details gives
which is an n-th order differential equation in u provided v is known. We replace v and u with ur and ur+1 respectively we get
where for each p = 1, 2, …, n and
3.1.2 Chebyshev differentiation
Before we apply Chebyshev differentiation on (16), we do the following:
If the physical domain is infinite or semi-infinite, we truncate it to a finite one [a, b] but such that |b − a| is large enough for the boundary conditions at the end points to still apply.
Use the change of variable x(ξ) = [a + b + (b − a)ξ]/2 to transform the differential equation (16) on the physical domain [a, b] to its counterpart
on the computational domain [−1, 1], where α = 2/(b − a).
On [−1, 1] define grid points ξ0, ξ1, …, ξN by ξj = cos(πj/N) for each j = 0, 1, …, N.
Chebyshev differentiation evaluates the p-th derivative of u with respect to ξ at ξ = ξj using the formula
where D̂ is the (N + 1) × (N + 1) Chebyshev differentiation matrix . If we evaluate Equation (18) at ξ = ξj for each j = 0, 1, …, N and then evaluate each derivative using Formula (19), we end up with a linear system
where A is a function of D. Upon solving (20) we get u at each grid point and
for any ξ ∈ [−1, 1], where Lj is the Lagrange polynomial of degree N associated with node ξj. Since (21) is the solution of (18), if we write ξ in terms of x then we get a solution u(x) of the original differential equation (13).
3.2 Application of SQLM to current problem
where u = (f, f′, f″, f″′, θ, ϕ). Upon applying Quasi-linearization on (22) we get
Similarly, Quasi-linearization transforms (9) to
Proceeding in a similar manner also transforms (10) to
Φr = [Φr(ξ0),Φr(ξ1) … Φr(ξN)]T, I is the (N + 1) × (N + 1) identity matrix and
is an (N + 1) × (N + 1) diagonal matrix. The linear system (29) is subject to boundary conditions
Solution of the linear system (29) is preceded by the following:
We choose initial approximations as
Since U0 is now known, if we solve Equation (29) for each r = 0, 1, ⋯ we get subsequent approximations U1, U2, ⋯.
4 Results and discussion
A parametric study is conducted and the results are presented in both graphical and tabular forms, as well as being discussed. The numerical results are iteratively generated by the Spectral Quasi-Linearization Method (SQLM) for the parameters that are significant in the current work. It is remarked that the SQLM results presented in this work were obtained using N = 50 allocation points with the infinity value n∞ taken as 30. The tolerance level was set to be ϵ = 10−8. The following are the default values for pertinent parameters considered in this study:
β = 0.5, λ1 = 0.3, λ3 = 0.5, λ4 = 0.5, M = 1, R = 1, Pr = 0.71, Sc = 0.22. The number of collocation points N was fixed as the smallest values of N which gives a consistent solution with the stipulated tolerance level ε. Figure 1 shows the convergence and stability of the solutions generated by the SQLM. We can observe from this figure that convergence is realised at as low as 5 iterations which shows that this method as equally good as other numerical and analytical methods such as the variational iteration method or the homotopy perturbation method.
Table 1 presents a comparison between SQLM approximate results and bvp4c results for default values of magnetic parameter M. We observe that there is an excellent agreement between the results from the two methods.
The influences of the Deborah number, the ratio of the relaxation to retardation times, thermal buoyancy, magnetic and thermal radiation parameters on the skin friction coefficient, and Nusselt and Sherwood numbers are displayed in Tables 2, 3 and 4, respectively. We observe in these respective tables that the Deborah number has very significant influence on skin friction coefficient and Nusselt and Sherwood numbers. We observe that the skin friction coefficient is reduced when the Deborah number β increases. Increasing the value of β implies that fluid elasticity is enhanced at the same time reducing viscosity, thus reducing the frictional force on the surface. However, the Nusselt number and Sherwood number are enhanced by increasing values of the Deborah number. Increasing the values of the ratio of relaxation time to retardation time significantly increases the skin friction coefficient but at the same time reduces the Nusselt and Sherwood numbers. The fluid particles need much more time to come back from a perturbed system to an equilibrium system, thus increasing frictional force on the surface at the same time reducing the rates of heat and mass transfer on the wall surface. Increasing the buoyancy forces, as expected, has significant effects on the skin friction coefficient, Nusselt number and Sherwood numbers. The skin friction coefficient is reduced with enhanced values of buoyancy forces, whereas the opposite effect is encountered on the rates of heat and mass transfer on the surface. We also observe in these tables that the skin friction coefficient increases with increasing values of the magnetic parameter, but the rates of heat and mass transfer are reduced. Lastly, we see in Table 2 that increasing the value of thermal radiation enhances the skin friction coefficient as well as the Nusselt number but has little effect on the Sherwood number.
Figure 2 depicts the influence of the Deborah number, β, on velocity profiles. As β depends on the retardation time λ2, increasing the value of β corresponds to increase in the retardation time. Physically, elasticity is enhanced by an increase in the retardation time. It is noted that elasticity and viscosity effects are inversely proportional to each other so the fluid velocity is enhanced by a decrease in viscosity. This explains why the velocity increases as the value of the Deborah number increases as can be clearly observed in Figure 2.
The influence of the ratio of relaxation time to retardation time, λ1, on the velocity is illustrated in Figure 3. Physically, an increase in λ1 corresponds to increase in the relaxation time and/or a decrease in the retardation time. This then means that fluid particles need much more time to come back from a disturbed system to an equilibrium system. Thus the velocity decreases with increasing values of λ1 as can been seen in Figure 3.
Figure 4 displays effects of the thermal buoyancy parameter λ3 on velocity profiles. As expected, it is clearly observed that the velocity profiles are enhanced by increasing the value of λ3. Increasing λ3 makes the buoyancy force stronger and stronger and this in turn increases the velocity profiles. The effect of the magnetic parameter on the velocity distribution is depicted in Figure 5. An increase in the strength of the magnetic field produces a drag force which reduces the fluid velocity. The application of a controlled transverse magnetic field can be used as a stabilizing mechanism to boundary layer flow. This can also be used to delay the transition from laminar to turbulent flow.
Figure 6 shows variation of the Prandtl number, thermal radiation number and Schmidt number on velocity profiles. A small value of the Prandtl number, Pr << 1, means that thermal diffusivity dominates while a large value, Pr >> 1, means that the momentum diffusivity dominates the flow behavior. This explains why the fluid velocity is greatly reduced when the values of the Prandtl number are increased as can be clearly seen in Figure 6(a). Figure 6 (b) clearly indicates that the fluid velocity profiles and boundary layer thickness increase with an increase in thermal radiation parameter (R). However, the velocity profiles and boundary layer thickness are greatly reduced by increasing values of the Schmidt number. Physically, Schmidt number is a dimensionless parameter which represents the ratio of momentum diffusion to mass diffusion in fluid. When the Schmidt number is large, it means that the momentum is transported by molecular means across a liquid much more effectively than by species. Thus the fluid becomes heavier thereby reducing the fluid velocity as can clearly be observed in Figure 6 (c).
The effect of the Deborah number on the temperature is depicted in Figure 7. It is observed that increasing the value of the Deborah number yields a reduction in both temperature and thermal boundary layer thickness. Figure 8 depicts the influences of the thermal buoyancy parameter, magnetic parameter and thermal radiation R on the temperature profiles. We observe that temperature decreases with increasing values of the thermal buoyancy force. Large values of λ3 produce large buoyancy force which in turn produces large kinetic energy. Thus the temperature and thermal boundary layer are reduced as λ3 increases. Increasing the strength of the magnetic field enhances the fluid temperature as can be seen in Figure 8(b). Increasing the value of the thermal radiation R has a tendency to increase conduction effects. Thus higher values of R imply higher surface flux, therefore reducing the temperature at each point away from the surface. This can be seen in Figure 7(c). It is observed that the thermal boundary layer thickness decreases as the value of R increases.
Figure 9 displays the influence of chemical reaction, the magnetic parameter and the Schmidt number on concentration profiles. As expected, the concentration profiles are reduced with increasing values of chemical reaction parameter. The velocity of the fluid is reduced as the magnetic field parameter increases this in turn enhances the species concentration. This explains why the concentration profile and solutal boundary layer increase as the magnetic parameter increases. Lastly, we observe that the concentration and the solutal boundary layer are significantly reduced as the value of the Schmidt number increases. This is because when the Schmidt number is large, momentum is transported by molecular means across liquid much more effectively than by species.
The current study investigated the problem of mixed convection flow of an MHD Jeffery fluid in the presence of thermal radiation and chemical reaction. The following are the main observations of this investigation:
Increases in the values of the Deborah number β, lead to increasing values of the velocity profiles as well as momentum boundary layer thickness. However, the fluid temperature and species concentration are decreasing functions of the Deborah number.
The fluid velocity is found to be an increasing function of the buoyancy parameters (λ3, λ4), thermal radiation and retardation time. Meanwhile, fluid velocity is found to decrease with increasing values of the magnetic parameter, Prandtl and Schmidt numbers.
Fluid temperature as well as the thermal boundary layer thickness increase with increasing values of the magnetic parameter and retardation time, but decreases with increasing values of the Deborah number, Prandtl number, thermal radiation and buoyancy parameter.
The species concentration and the solutal boundary layer thickness increase with increasing values of the magnetic parameter and retardation time. However, the opposite influences are observed when values of the Deborah number, buoyancy and chemical reaction parameters are increased.
We also observe that the SQLM is a very fast reliable method that can converge after as few as five iteration.
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About the article
Published Online: 2018-05-24
Citation Information: Open Physics, Volume 16, Issue 1, Pages 249–259, ISSN (Online) 2391-5471, DOI: https://doi.org/10.1515/phys-2018-0036.
© 2018 S. Shateyi and G. T. Marewo, published by De Gruyter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0