Abstract

A novel mathematical computing analysis for steady magnetohydrodynamic convective flows of radiative Casson fluids moving over a nonlinearly elongating elastic sheet with a nonuniform thickness is established successfully in this numerical exploration. Also, the significance of an externally applied magnetic field with space-dependent strength on the development of MHD convective flows of Casson viscoplastic fluids is evaluated thoroughly by including the momentous influence of linear thermal radiation along with the temperature-dependent viscosity and thermal conductivity effects. By combining the assumption of the low-inducing magnetic field with the boundary layer approximations, the governing partial differential equations monitoring the current flow model are transmuted accordingly into a set of nonlinear coupled ordinary differential equations by invoking appropriate similarity transformations. Moreover, these derived differential equations are resolved numerically by utilizing a new innovative GDQLLM algorithm integrating the local linearization technique with the generalized differential quadrature method. On the other hand, the behaviours of velocity and temperature fields are deliberated properly through various graphical illustrations and different sets of flow parameters. However, the accurate datasets generated for the skin friction coefficient and local Nusselt number are presented separately in tabular displays, whose physical insights are discussed comprehensively via the slope linear regression method (SLRM). As main results, it is demonstrated that the higher values of the Casson viscoplastic parameter reduce significantly the fluid velocity within the boundary layer region, while a partial reverse tendency is observed near the stretching sheet as long as the wall thickness parameter is increased. Besides the previously mentioned hydrodynamical features, it is also depicted that the thermal field throughout the medium is enhanced considerably with the elevating values of these parameters.

1. Introduction

Linear and nonlinear rheological features of complex fluidic mediums have currently attracted widespread interests and considerable physical insights in the scientific debates owing to their valuable applications and widespread benefits over the whole usual fluids. Additionally, the non-Newtonian fluids have been extensively investigated during the past few decades by many scientists to explore their distinguishing rheological properties and establish new realistic constitutive laws permitting detailed explanations for some anomalous experimental observations. Indeed, the dynamic of most fluids together with their diverse characteristics cannot be properly described by the classical Navier–Stokes equations because the most common fluids used in industries are of a non-Newtonian kind, whose Cauchy stress and strain tensors can be expressed mathematically by quasilinear or purely nonlinear relationships depending greatly on their rheological behaviours, such as shear-thinning, shear-thickening, viscoelasticity, viscoplasticity, material memory, Weissenberg effect, yield stress, die-swelling, and relaxation and retardation times, which are not defined in the Newtonian model. For this reason, the non-Newtonian fluids are widely used in various branches of science and bioengineering, for instance in the modelling of biological structures, drilling muds, cement slurries, petroleum, coated sheets, polymer manufacturing, foodstuffs, lubrication fields, and biomedical flows. Generally, the rheological attitudes of non-Newtonian fluids are dissimilar from those of viscous fluids. Besides, the transport equations governing the dynamical evolution of non-Newtonian fluids are fully nonlinear and highly complicated from the handling point of view compared with the Newtonian fluids. Furthermore, an analysis of the literature survey shows that there is no single constitutive equation that reveals all properties of such fluids. So, several constitutive non-Newtonian models have been suggested by the scientific communities. These models are principally classified into the differential, integral, and rate-type categories. In the rate-type rheological fluid models (e.g., Maxwell, Oldroyd-B, Jeffery, and Burgers models), the shear stress is an implicit relationship of the velocity gradient and its higher temporal derivatives. However, the differential rheological model (e.g., Leonov model) has been established mathematically to explain the existence of a set of discrete relaxation times. This phenomenological model is based on the presence of an internal thermodynamic equilibrium state, where the equilibrium elastic deformations have been memorized and any aberration kind from this state can cause a new nonequilibrium state. In contrast to the aforesaid models, the extra stress tensor for the integral constitutive model (e.g., K-BKZ and Wagner models) is expressed mathematically as the temporal integral of the memory function and the elastic potential distributions as well as the Cauchy–Green tensor and its inverse (i.e., Finger tensor). Moreover, this model shows an excellent capability of giving comparable upshots against the experimental rheological results for long dies towards the extrudate swell occurrence witnessed in some viscoelastic materials, such as acrylonitrile-butadiene-styrene nanocomposites and other polymer melts. This rheological phenomenon is due mainly to the elastic recovery of deformation, which reflects the relaxation effect on the material after leaving from the die state.

Numerous constitutive models have been proposed by pioneering researchers to describe complex convective boundary layer flows. For instance, Bilal et al. [1] adopted the tangent hyperbolic rheological model to examine the characteristics of an unsteady MHD convective flow of a non-Newtonian fluid moving over a nonlinear stretching porous sheet under the combined influence of convective heating and time-dependent magnetic field in the presence of magnetized dusty particles, heat generation/absorption, and variable thermal conductivity. Similarly, Bibi et al. [2] accomplished comprehensive computational estimations with the aid of the shooting technique along with the built-in MATLAB bvp4c package to handle approximatively an unsteady MHD convective flow of a tangent hyperbolic fluid conveying tiny magnetized particles over a linear stretching permeable sheet characterized by a uniform suction/injection velocity and affected by the significant effects of variable thermal conductivity, internal heat source/sink, convective heating, and time-dependent magnetic field. Otherwise, Mohebbi et al. [3] utilized the power-law model to evaluate the shear-thinning and shear-thickening features of a non-Newtonian fluid by conducting a numerical investigation on a fully developed forced heat transfer convection for a laminar non-Newtonian flow confined between two parallel plates containing partially porous media in the presence of a uniform internal arrangement of circular obstacles. In the same way, Liu et al. [4] developed an approximate approach to scrutinize the non-Newtonian rheological behaviour of fresh concrete based on the Herschel–Bulkley model [5] and various available experimental data sources. Sequel to the aforesaid background, Zhu et al. [6] investigated numerically the mutual influence of fluid yield stress and finite-size particles on a turbulent flow of a Bingham fluid [7] flowing inside a vertical channel under the effect of no-slip velocity boundary condition and the periodicity physical constraints in the streamwise and spanwise Cartesian directions. Based on the Carreau–Yasuda rheological model and Buongiorno’s nonhomogeneous nanofluid model, Waqas et al. [8] proposed a realistic physical model to analyze the impacts of second-order velocity slip, zero nanoparticles mass flux condition, thermal radiation, and chemical reaction on the features of MHD convective flows over a convectively heated stretching surface in a porous medium by considering the coexistence of gyrotactic motile microorganisms and tiny solid particles and taken into account the thermophoresis and Brownian motion aspects. Keeping in mind the nonsimilarity characteristic of some flow problems causing by the non-Newtonian behaviour of the flowing fluids, Amanulla et al. [9] carried out an accurate numerical implementation to find nonsimilar solutions for the momentum and thermal boundary layers developed in a non-Darcy porous medium during the steady two-dimensional MHD free convection flow of a Prandtl–Eyring fluid past over an isothermal sphere under the effective influences of buoyancy forces and various slip conditions. Earlier, Muhammad et al. [10] exploited Buongiorno’s mathematical formulation along with the generalized Fourier–Fick's law and Rosseland's approximation to establish a theoretical nanofluid model allowing to evaluate more efficiently the Cattaneo–Christov heat transfer for a three-dimensional MHD-EMHD convective flow of an Eyring–Powell nanofluid flowing horizontally over an elongated Riga plate by considering the effects of nonlinear thermal radiation, activation energy, velocity slip condition, and convective heating. To explore the impact of shear-thinning and shear-thickening rheological properties on different kinds of transportation phenomena in non-Newtonian nanofluids under the combined effect of Brownian motion and thermophoresis diffusion of solid nanomaterials, Hayat et al. [11] extended the Sisko rheological fluid model [12] along with Buongiorno’s mathematical formulation [13] and the vanishing mass flux condition for nanoparticles [14, 15] to simulate a three-dimensional MHD forced convective nanofluid flow over a nonisothermal surface, which is taken to be materially impermeable, heated convectively, and stretched bidirectionally via linear velocities. In the objective of inspecting the surface drag forces in nanofluidic mediums, as well as the feasibility of enhancing their heat and mass transportations, Hayat et al. [16] addressed an imperative unsteady three-dimensional MHD squeezing viscoelastic flow of a second-grade nanofluid between two isothermal parallel disks, whose chemical species are controlled actively via Buongiorno’s two-phase model under the assumption of small magnetic Reynolds number theory along with the blowing and suction effects. To foresee the impacts of shear-thinning and shear-thickening rheological properties, Sajid et al. [17] performed an approximative analytical investigation on the two-dimensional coating process for incompressible third-grade fluid flows in a thin slit located between a moving substrate and flexible/stiff coater by supposing that the coating layer thickness is greatly less than the blade length. Likewise, Arifuzzaman et al. [18] utilized the explicit finite difference technique to analyze the heat and mass transfer features of unsteady MHD radiative-convective flows of fourth-grade fluids moving perpendicularly from an isothermal permeable surface in a porous medium under the effects of magnetic field, suction/injection velocity, nonlinear order chemical reaction, heat generation, viscous dissipation, and buoyancy forces.

Due to the progress in the rheology of homogeneous and nonhomogeneous viscoelastic mixtures employed in widespread industrial and engineering areas, various constitutive rheological relationships have been developed for biological and polymeric rate-type liquids to reveal the memory and elasticity features of these liquids by introducing the relaxation and retardation times via the Oldroyd-B rheological model [19]. In the range of low shear rate model, the dynamic viscosity of Oldroyd-B fluids found to be increasingly varied, while this trend is reversed for higher shear rates indicating that these materials possess both solid and liquid characteristics. In particular, these properties can be constrained rheologically in some cases to deduce the Maxwell model [20] by disappearing the retardation time characterized by the solid tendency of viscoelastic fluids, in which the fluidic medium cannot respond instantaneously to any change in shear stress. In this direction, it is worth noting here that the Oldroyd-B model is incapable to explain the shear-thinning and shear-thickening rheological phenomena observed in some physiological and biological fluids (e.g., blood, serum, urine, saliva, and tears). Further, this model exhibits a quasilinear relationship between the shear stress and the shear strain rate, while this rheological dependency becomes a linear relationship for the Maxwell model. Among the most common rate-type model that having a time derivative in its linear rheological constitutive law and offering the possibility to investigate easily the viscoelastic behaviour of non-Newtonian fluids, we find the constitutive rheological Jeffrey’s fluid model [21]. Bearing in mind the importance of this rheological model and its simplicity in simulating many realistic viscoelastic flows encountered widely in the bioscience sector and technology, this linear rheological law has lately obtained exceptional interest by researchers. In this regard, Hayat et al. [22] exploited the constitutive rheological equation of the Jeffrey model to examine the probable effects of an applied magnetic field, viscous dissipation, and internal heat source on the stagnation point flow of a non-Newtonian viscoelastic fluid towards a nonlinear stretching surface, which is characterized by a variable surface thickness and heated isothermally via an appropriate melting process. In another relevant investigation, Hayat et al. [23] inspected a three-dimensional MHD convective flow of an Oldroyd-B nanofluid moving horizontally over a bidirectionally stretching surface under the effects of zero nanoparticles mass flux condition, convective heating, Brownian motion, and thermophoresis. Because of the substantial interest given to these non-Newtonian models, Koteswara Reddy et al. [24] proposed an adjustable rheological model to analyze the features of MHD mixed convection viscoelastic flows of Maxwell/Jeffery/Oldroyd-B nanofluids moving vertically over a nonisothermal conical geometry by considering the existence of an internal heat source/sink along with the impact of thermophoresis and Brownian motion of solid nanoparticles. In a particular case, Awias et al. [25] carried out a numerical implementation via the bvp4c technique to study three-dimensional mixed convection flows of Maxwellian nanofluids driven by a vertically rotating and stretching isothermal cylinder, under the influences of various interactions including thermal/mass buoyancy, magnetic, thermophoretic, and Brownian forces. On the other hand, the existence of multiple relaxation times in the response of some materials (e.g., Earth’s mantle and asphalt mastics) in creep and stress relaxation examinations can be adequately explained by another specified rate-type fluid model of higher order. For this purpose, many researchers prefer modelling numerous complex flow problems with the help of the Burgers rheological model [26] due to its capability of describing the coexistence of two different relaxation times. In the framework of this advanced rate-type fluid category, Hayat et al. [27] assumed the generalized Fourier–Fick’s law along with Buongiorno’s nanofluid model to analyze the heat and mass transportation phenomena occurred during the steady two-dimensional convective flows of Burgers nanofluids over a linearly stretching isothermal sheet.

Later on, various experimental investigations indicated that the Casson rheological fluid model [28] is still the best rheological way to understand the non-Newtonian behaviour of many viscoplastic fluids, like chocolate, blood, tomato sauce, paint, shampoo syrup, some polymer solutions, and other colloidal suspensions. This kind of fluids exhibits a shear-thinning rheological behaviour (i.e., pseudoplastic feature), whose viscosity decreases with the increasing shearing rate as long as the stress upsurges above a critical value recognized as the yield stress. More precisely, the presence of mineral ingredients in a liquid increases significantly the dynamic viscosity of mediums and modifies its rheological characteristic to a non-Newtonian behaviour with a rheological tendency to an increase in the fluidity, when the resulting mixture is sheared. However, the fluidic mediums show the capability to gel under certain situations, when it is not sheared. In this context, Kelessidis and Maglione [29] realized an exhaustive statistical analysis on the Casson and Robertson–Stiff rheological models [28, 30] via various existing experimental viscometric data generated especially for water-Greek bentonite suspensions to choose the excellent rheological category permitting a more realistic description of the studied aqueous medium. After successful attempts, they concluded that both suggested models can describe well the addressed sample data, in which the true shear rates for the Casson and Robertson–Stiff models are found to be higher compared with the Newtonian shear rates, whose differences become lesser at higher shear rates. These findings indicate an assessment in the Newtonian behaviour at greater shear rates. In the framework of the practical use of non-Newtonian Casson fluids, Qasim and Noreen [31] exploited the stability analysis theory to illustrate more evidently the possibility of generating multiple solutions for dissipative boundary layer flows of Casson fluids moving over a porous shrinking surface. More recently, Bhatti et al. [32] studied semianalytically MHD radiative Casson fluid flows from a nonisothermal linear stretching surface by including the Hall and ion-slip effects in the momentum equation via the generalized form of Ohm’s law and imposing local convective heating as a feasible thermal condition at the horizontal boundary for the energy conservation equation.

Building upon the aforementioned literature review, the authors tackled their flow problems by utilizing various numerical and semianalytical procedures, such as Runge–Kutta method (RKM), spectral local linearization method (SLLM), spectral quasilinearization method (SQLM), spectral relaxation method (SRM), Keller box method (KBM), homotopy perturbation method (HPM), and homotopy analysis method (HAM), which are extensively employed in other related investigations. Besides, the analysis of these previous research works reveals that there is a greater lack of innovative research studies on the numerical analysis topic dealing realistically with the heat transportation phenomenon via non-Newtonian flows over complex geometries (e.g., irregular sheet, needle, cone, sphere, and cylinder). Motivating by these facts, a new numerical method is established in this scrutiny to explore the rheological fluid flow and heat transfer characteristics of radiative Casson viscoplastic fluids having temperature-dependent thermophysical properties (i.e., dynamic viscosity and thermal conductivity), surrounding by a nonuniform applied magnetic field and moving horizontally over an isothermal stretching sheet with variable thickness. Moreover, the graphical and tabular highlights produced for every physical quantity showing up in the present numerical assessment (i.e., velocity, temperature, skin friction coefficient, and heat transfer rate) are discussed physically through plausible explanations. Furthermore, the flow patterns are further displayed for velocity boundary layers and streamlines to inspect the consequence of applying a nonuniform magnetic field on the fluid flow properties.

2. Problem Statement

Let us consider a steady two-dimensional MHD convective flow of a Casson viscoplastic fluid exhibiting a temperature dependency for the dynamic viscosity and thermal conductivity and moving over a nonlinearly slender sheet of variable thickness , which is heated uniformly via an imposed temperature and stretched horizontally in the positive sense of the -axis with a spatially variable velocity under the impacts of thermal radiation and variable magnetic field , where , , and . Here, the dimensional constants are utilized in the expressions of and to characterize the slender sheet geometry and its dynamical behaviour, while the exponent designates the velocity power parameter. Also, the constant is selected properly in this physical problem, in such a way that the sheet is sufficiently thin compared to its width . As schematically portrayed in Figure 1, the mathematical characteristics of the studied flow geometry are presented in a Cartesian coordinate system , whose unit vectors are . In this configuration, the origin is located at the leading edge of the transverse section centre of the sheet, in which the -axis is positioned horizontally along the symmetrical axis of the elastic sheet, while the -axis is taken perpendicularly to the flow direction. As assumptions, the radiative fluid flow is supposed laminar, incompressible, and having a weaker electrical conductivity and can be developed with a low Reynolds magnetic number, in which the induced magnetic field can be neglected reasonably.

Figure 1 

Schematic diagram of the present flow problem.

By adopting the linearized form of the Rosseland radiative heat flux model [33] along with the boundary layer approximations, the various transport equations (i.e., mass, momentum, and energy conservation equations) governing the present flow problem are stated as follows:

As emphasized above, and represent the velocity components in the and directions, respectively, means the fluid density, corresponds to the specific heat of the Casson fluid, indicates the Casson fluid parameter, refers to the plastic dynamic viscosity of the medium, symbolizes the fluid temperature, specifies the thermal conductivity of the fluid, signifies the Stefan–Boltzmann constant, designates the mean absorption coefficient, defines the electrical conductivity of the medium, and shows the strength of the variable magnetic field.

By assuming a linear temperature dependency for the thermophysical quantities , we get [34]

Moreover, the thermophysical properties shown above designate the effective values of the variable quantities at the stream temperature , whereas dimensionless quantities and represent the variable viscosity and thermal conductivity parameters, correspondingly.

For the flow geometry under consideration, the prescribed boundary conditions regulating equations (1)–(3) are written as follows:

Accordingly, equations (1)–(3) can be nondimensionalized by invoking the following authentic similarity transformations:

In this state, the resulting nonlinear ordinary differential equations (ODEs) with their proper boundary conditions are given by

As underlined above, the prime superscripts designate the derivation with respect to . Also, the physical parameters , , , and involved in equations (8)–(10) are defined formally as follows:

For further simplifications, we set the following feasible changes:

Therefore, the nonlinear differential system described by equations (8)–(11) is altered to

Here, the prime notations are utilized to refer to the derivation for the variable .

3. Engineering Quantities of Interest

In this physical problem, the significance of the principal engineering quantities of major importance can be scrutinized appropriately by estimating locally the magnitude of dragging forces and heat transfer rate at the stretching surface in terms of the skin friction coefficient and Nusselt number . These physical quantities are defined as follows:in which

By making use of equations (5), (7), and (13), we get the following expressions:

As mentioned above, the physical quantities and represent the reduced forms of the local skin friction coefficient and Nusselt number , which are given by

Here, denotes the local Reynolds number, where .

4. Numerical Modeling Strategy

A new efficient numerical procedure is presented exclusively in this investigation to solve the nonlinear differential system described in the fluid motion and heat transfer features of a non-Newtonian viscoplastic fluid possessing temperature-dependent viscosity and thermal conductivity and flowing over an isothermal stretching sheet with irregular and axisymmetric geometry under the influence of a nonuniform applied magnetic field with the presence of thermal radiation. The proposed numerical algorithm is built on a purely theoretical framework. In this regard, the resulting nonlinear differential system of equations (14)–(17) can be linearized and decoupled accordingly to reduce them into a sequence of dependent subsystems of linear ODEs. After several successful rearrangements, the constructed differential subsystems are discretized spatially to be tackled numerically by utilizing the generalized differential quadrature local linearization method (GDQLLM).

Based on the pioneering mathematical findings reported by Bellman and Kalaba [35], the present flow model can be handled iteratively by applying the local linearization technique (LLT) to the following generated system:

As highlighted in equation (22), the value corresponds to the estimated dimensionless boundary layer thickness assuring the asymptotical achievement of the stream boundary conditions with higher precision. Consequently, the linearized form of the nonlinear differential system is written as follows:

Moreover, the solutions of the linear differential system can be adjusted iteratively through the functions , which are expressed as follows:

Here, the iterative subscripts and are utilized in equations (23)–(26) to indicate the previous and current estimations, respectively.

Furthermore, to boost the numerical procedure approved during the resolution steps, it is preferable to initialize the iterative process by testing solutions verifying all the boundary conditions imposed on the differential system . These initial guessed solutions take the following expressions:

By applying the generalized differential quadrature method (GDQM) along with the modified Gauss–Lobatto collocation points [36–41], the resulting linearized system can be discretized spatially to give three decoupled linear algebraic subsystems , , and , whose unknowns are the approximate discrete values of the functions , , and in the computational domain . These subsystems can be written in the following matrix forms:

Additionally, the square matrices and vector columns are expressed as follows:

As shown above, and symbolize the first- and second-order differentiation matrices, respectively, denotes the unit matrix, the set highlights the modified Gauss–Lobatto collocation points, and represents the number of collocation nodal points .

According to Shu [42], the components of the modified differentiation matrices and can be computed from the following general system:

Furthermore, it bears noting that, after discretizing the differential system , the derivative terms can be handled numerically in each collocation point as follows:where is the order of derivation concerning the variable .

During the progression in the iterative procedure, the reduced quantities and are computed as follows: