Abstract
The problem of a non-Newtonian fluid flow past an upper surface of an object that is neither a perfect horizontal/vertical nor inclined/cone in which dissipation of energy is associated with temperature-dependent plastic dynamic viscosity is considered. An attempt has been made to focus on the case of two-dimensional Casson fluid flow over a horizontal melting surface embedded in a thermally stratified medium. Since the viscosity of the non-Newtonian fluid tends to take energy from the motion (kinetic energy) and transform it into internal energy, the viscous dissipation term is accommodated in the energy equation. Due to the existence of internal space-dependent heat source; plastic dynamic viscosity and thermal conductivity of the non-Newtonian fluid are assumed to vary linearly with temperature. Based on the boundary layer assumptions, suitable similarity variables are applied to nondimensionalized, parameterized and reduce the governing partial differential equations into a coupled ordinary differential equations. These equations along with the boundary conditions are solved numerically using the shooting method together with the Runge-Kutta technique. The effects of pertinent parameters are established. A significant increases in is guaranteed with when magnitude of is large. decreases with and .
1. Introduction
Within the last thirty years, the study of non-Newtonian fluid flow over a stretching surface has received significant attention due to its industrial applications. Such interest is fueled by its pertinent engineering applications in a number of fields as in spinning of filaments, continuous casting of metal, extrusion of polymers, crystal growing, glass fiber production, extrusion of plastic sheets, paper production, and process of condensation of metallic plates. Boundary layer analysis of fluid flow passing through a thick needle with variable diameter was investigated by Lee [1]. Historically, this can be referred to as the first report of flow adjacent to a surface with variable thickness where the effect of viscosity is highly significant. Thereafter, extensive studies were conducted on the boundary layer flows over a thin needle. Cebeci and Na [2] investigated the laminar free convection heat transfer from a needle. Ahmad et al. [3] examined the boundary layer flow over a moving thin needle with variable heat flux. The boundary layer flow over a stretching surface with variable thickness was analyzed by Fang et al. [4]. Recently, the flow of different fluids over an upper horizontal surface with variable thickness has been investigated extensively in [5–7]. This attracted Makinde and Animasaun [8, 9] to focus on the case of quartic autocatalysis kind of chemical reaction in the flow of an electrically conducting nanofluid containing gyrotactic-microorganism over an upper horizontal surface of a paraboloid of revolution in the presence and absence of thermophoresis and Brownian motion. In most cases, plastic dynamic viscosity of non-Newtonian Casson fluid tends to take energy away from the motion and transform it into internal energy. However, this area has been neglected.
In fluid mechanics, destruction of fluctuating velocity gradients due to viscous stresses is known as viscous dissipation. This partial irreversible processes is often referred to as transformation of kinetic energy into internal energy of the fluid (heating up the fluid due to viscosity since dissipation is high in the regions with large gradients). Pop [10] remarked that understanding the concept of energy dissipation and transport in nanoscale structures is of great importance for the design of energy-efficient circuits and energy-conversion systems. However, energy dissipation and transport of non-Newtonian fluid are also of importance to engineers and scientists. Motsumi and Makinde [11] examined the effects of viscous dissipation parameter (i.e., Eckert number), thermal diffusion, and thermal radiation on boundary layer flow of -water and -water nanofluids over a moving flat plate. In another theoretical study on combined effects of Newtonian heating and viscous dissipation parameter on boundary layer flow of copper and titania in water over stretchable wall, Makinde [12] reported an increase in the moving plate surface temperature and thermal boundary layer thickness. Depending on the admissible grouping of variables (parameterization), Eckert number and Brinkmann number () may be used to quantify viscous dissipation. In addition, unsteady mixed convection in the flow of air over a semi-infinite stretching sheet taking into account the effect of viscous dissipation was carried out by Abd El-Aziz [13]. Both at steady stage () and unsteady stage (), velocity and temperature of the flow increase with an increase in the magnitude of Eckert number. Recently, effects of viscous dissipation, Joule heating, and partial velocity slip on two-dimensional stagnation point flow were reported by Yasin et al. [14]. In another study conducted by Animasaun and Aluko [15], it is reported that when dynamic viscosity of air is assumed to vary linearly with temperature, normal negligible effect of Eckert number on velocity profiles will be noticed. Raju and Sandeep [16] focused on the motion of Casson fluid over a moving wedge with slip and observed a decrement in the temperature field with rising values of Eckert number. In the flow of non-Newtonian Casson fluid over an upper horizontal thermally stratified melting surface of a paraboloid of revolution, dissipation of smaller eddies due to molecular viscosity near the wall is significant. In the presence of constant magnetic field, electrically conducting Casson fluid flow over object that is neither a perfect horizontal/vertical nor inclined/cone is also an important issue.
The study of electrically conducting fluid flow is of considerable interest in modern metallurgical. This can be traced to the fact that most fluids in this sector are electrically conducting fluid. Historically, the first report on the motion of electrically conducting fluid in the presence of magnetic field was presented by Rossow [17]. Thereafter, Alfvén [18] reported that if a conducting liquid is placed in a constant magnetic field, every motion of the liquid generates a force called electromotive force (e.m.f.) which produces electric currents. One of the most significant importance of these contributions are its applications in engineering problems such as MHD generators, plasma studies, nuclear reactors, and geothermal energy extractions. Soundalgekar and Murty [19] investigated heat transfer in MHD flow with pressure gradient, suction, and injection. It was observed that an increase in the magnetic field parameter leads to an increase in fluid velocity, skin friction, rate of heat transfer, and a fall in temperature. Rajeswari et al. [20] observed that, due to the uniform magnetic field and suction at the wall of the surface, the concentration of the fluid decreases with the increase in chemical reaction parameter. Ghosh et al. [21] found that an increase in inclination of the applied magnetic field opposes primary flow and also reduces Grashof numbers. Das [22] concluded that increasing magnetic field and thermal radiation leads to deceleration of velocity but reverse is the effect for the melting parameter when the solid surface and the free stream move in the same direction. Motsa and Animasaun [23] presented the behavior of unsteady non-Darcian magnetohydrodynamic fluid flow past an impulsively using bivariate spectral local linearization analysis. Koriko et al. [24] illustrated the dynamics of two-dimensional magnetohydrodynamics (MHD) free convective flow of micropolar fluid along a vertical porous surface embedded in a thermally stratified medium. It was concluded that velocity profiles and microrotation profiles are strongly influenced by the magnetic field in the boundary layer, which decreases with an increase in the magnitude of magnetic parameter. Recently, theoretical investigation of MHD natural convection flow in vertical microchannel formed by two electrically nonconducting infinite vertical parallel plates and effects of MHD mixed convection on the flow through vertical pipe with time periodic boundary condition was presented explicitly by Jha and Aina [25, 26].
Several processes involving melting heating transfer in non-Newtonian fluids have promising applications in thermal engineering, such as melting permafrost, oil extraction, magma solidification, and thermal insulation. As such, a lot of experimental and theoretical work has been conducted in the kinetics of heat transfer accompanied with melting or solidification effect. The process of melting of ice placed in a hot stream of air at a steady state was first reported by Roberts [27]. Historically, this report can be referred to as the pioneering analysis of the melting phenomenon. Another novel report on melting phenomenon during forced convection heat transfer when an iceberg drifts in warm sea water was presented by Tien and Yen [28]. From their investigation, they observed that melting at the interface results in a decrease in the Nusselt number. Epstein and Cho [29] discussed the laminar film condensation on a vertical surface. Much later, melting heat transfer in a nanofluid boundary layer on a stretching circular cylinder was examined by Gorla et al. [30]. In the study of the effect of radiation on MHD mixed convection flow from a vertical plate embedded in a saturated porous media with melting, Adegbie et al. [31] reported that the Nusselt number decreases with increase in melting parameter. Adegbie et al. [31] stated that the temperature of UCM fluid flow over a melting surface is an increasing function of variable thermal conductivity parameter. Omowaye and Animasaun [32] investigated boundary layer analysis in the flow of upper convected Maxwell fluid flow. Due to the fact that temperature at the wall is zero, classical temperature-dependent viscosity and thermal conductivity linear models were modified to suit the case of both melting heat transfer and thermal stratification. In another study of micropolar fluid flow in the presence of temperature-dependent and space heat source, the analysis of the case where vortex viscosity is a constant function of temperature was reported in [33].
Within the past two decades, the effects of temperature-dependent viscosity on the fluid flow have become more important to engineers dealings with geothermal systems, crude oil extraction, and machinery lubrication. Due to friction and internal heat generated between two layers of fluid, viscosity and thermal conductivity of fluid substance may be affected by temperature; for more details see Batchelor [34], Lai and Kulacki [35], and Abd El-Aziz [36]. Proper consideration of this fact in the study on inherent irreversibility in a variable viscosity Couette flow by Makinde and Maserumule [37], numerical investigation of micropolar fluid flow over a nonlinear stretching sheet taking into account the effects of a temperature-dependent viscosity by Rahman et al. [38], effects of MHD on Casson fluid flow in the presence of Cattaneo-Christov heat flux by Malik et al. [39], fluid flow through a pipe with variation in viscosity by Makinde [40], Casson fluid flow within boundary layer over an exponentially stretching surface embedded in a thermally stratified medium by Animasaun [41], steady fully developed natural convection flow in a vertical annular microchannel having temperature-dependent viscosity in the presence of velocity slip and temperature jump at the annular microchannel surfaces by Jha et al. [42] have enhanced the body of knowledge on fluid flow, boundary layer analysis, and heat/mass transfer. In a recent experiment, Alam et al. [43] concluded that thermal boundary layer decreases with an increasing temperature-dependent viscosity. Hayat et al. [44] discussed the effect of variable thermal conductivity on the mixed convective flow over a porous medium stretching surface. In the article, the kinematics viscosity of Casson fluid was considered as a function depending on plastic dynamic viscosity, density, and Casson parameter and hence reported that increase in the magnitude of temperature-dependent viscosity parameter leads to an increase in fluid’s velocity. The above literature review shows that there exists no published article on the effects of viscous dissipation in the flow of non-Newtonian Casson fluid over a upper horizontal thermally stratified melting surface of a paraboloid of revolution.
2. Formulation of the Problem
Consider a steady, incompressible, laminar flow of an electrically conducting non-Newtonian (Casson) fluid over a melting surface on upper horizontal paraboloid of revolution in the presence of viscous dissipation and thermal stratification. The -axis is taken in the direction of motion and -axis is normal to the flow as shown in Figures 1(a) and 1(b). A uniform magnetic field of strength is applied normal to the flow. The induced magnetic field due to the motion of an electrically conducting Casson fluid is assumed to be so small; hence it is neglected. The stretching velocity is and the wall is assumed impermeable. It is further assumed that the immediate fluid layer adjacent to the surface is specified as , where . Using Boussinesq approximation, the difference in inertia is negligible but gravity is sufficiently strong to make the specific weight appreciably different between any two layers of Casson fluid on the surface; hence the rate flow on this kind of surface is referred to as free convection. The body force term suitable to induce the flow over a surface which is neither a perfect horizontal/vertical nor inclined/cone isFollowing the theory stated in Casson [45] and boundary layer assumptions, the rheological equation for an isotropic Casson fluid flow together with heat transfer is of the formSuitable boundary conditions governing the flow along upper horizontal surface of a paraboloid of revolution areThe formulation of the second term in boundary (5) states that the heat conducted to the melting surface on paraboloid of revolution is equal to the combination of heat of melting and the sensible heat required to raise the solid temperature to its stratified melting temperature (for details, see [29]). For lubricating fluids, heat generated by the internal friction and the corresponding increase in temperature affects the viscosity of the fluid; hence, it may not be realistic to constant function of plastic dynamic viscosity knowing fully well that space-dependent internal heat source is significant. In order to account for this variation, it is valid to consider the modified mathematical models of both the temperature-dependent viscosity and thermal conductivity models proposed in [46] and adopted in [31, 32] asMeanwhile, the models are still in good agreement with experimental data of Batchelor [34]. It is worth mentioning that the first and second terms in (7) are valid and reliable since in this study, whereas thermal stratification () at the melting wall and at the free stream is defined asUsing similarity variable for temperature in (7) to simplify we obtain suitable temperature difference in flow past thermally stratified horizontal melting surface of paraboloid of revolution asFrom thermal stratification models in (9), the following relations can be easily obtainedwhere is known as reference temperature. A significant difference between and can be easily obtained from (10). This can be traced to the fact that the linear stratification occurs at all points of “” on the wall and at all points of “” at the free stream as . In view of this, it is valid to define temperature-dependent viscous parameter by considering the second term in (10) since . Mathematically, the ratio of the first two terms in (10) can thus produce dimensionless thermal stratification parameter () asThe stream function and similarity variable are of the formIt is important to note that the stream function automatically satisfies continuity (2). The nonlinear partial differential equations (3) and (4) are reduced to the following nonlinear coupled ordinary differential equations
(a)
(b)
In (13) and (14), melting parameter , Prandtl number , magnetic parameter , Eckert number , temperature-dependent thermal conductivity parameter , space-dependent internal heat source parameter , skin friction coefficient , Nusselt number , and buoyancy parameter depending on volumetric-expansion coefficient due to temperature are defined aswhere is the shear stress (skin friction) between Casson fluid and upper surface of horizontal paraboloid of revolution and is the heat flux at all points on the surfaceIn order to nondimensionalize the boundary conditions (5) and (6), it is pertinent to note that the minimum value of is not the starting point of the slot. This implies that all the conditions in (5) are not imposed at . As shown in Figures 1(a) and 1(b), it is obvious that it may not be realistic to say that at all points on upper horizontal melting surface of a paraboloid of revolution. Hence, it is not valid to set in similarity variable . Upon using the minimum value of which accurately corresponds to minimum value of similarity variableThis implies that, at the surface , the boundary condition suitable to scale the boundary layer flow is . The boundary condition becomesMoreover, dimensionless governing equations (13) and (14) are depending on while the boundary conditions (18) and (19) are functions and/or derivatives depending on . In order to transform the domain from to it is valid to adopt and ; for more details, see Figure 1(b). Considering the fact that Prandtl number is strongly dependent on plastic dynamic viscosity and thermal conductivity, and it is assumed that both properties vary linearly with temperature; hence for more accurate analysis of boundary layer as suggested in [47, 48], the in (14) isEquation (18) reveals that Prandlt number at free stream is denoted as . The final dimensionless governing equation (coupled system of nonlinear ordinary differential equation) isDue to the fact that , the influence of temperature-dependent thermal conductivity on heat conduction during melting process diminishes. Dimensionless boundary condition reduces toUpon substituting the similarity variables (12) and models of physical quantities (i.e., and ) at the wall into (16) we obtain
3. Numerical Solution
Numerical solutions of the boundary valued problem (21)–(24) are obtained using classical Runge-Kutta method with shooting techniques and MATLAB package (bvp5c). The boundary value problem cannot be solved on an infinite interval and it would be impractical to solve it for even a very large finite interval; hence at infinity is . Using the method of superposition by Na [49], the boundary value problem of ODE has been reduced to a system of five simultaneous equations of first order (IVP) for five unknowns following the method of superposition. In order to integrate the corresponding IVP, the values of and are required. However, such values do not exist after the nondimensionalization of the boundary conditions (5) and (6). The suitable guess values for and are chosen and then integration is carried out. The calculated values of and at infinity () are compared with the given boundary conditions in (24) and the estimated values and are adjusted to give a better approximation for the solution. Series of values for and are considered and applied with fourth-order classical Runge-Kutta method using step size . The above procedure is repeated until asymptotically converged results are obtained within a tolerance level of . It is very important to remark that setting , all profiles are compatible with the boundary layer theory and asymptotically satisfies the conditions at free stream as suggested by Pantokratoras [50]. It is worth mentioning that there exist no related published articles that can be used to validate the accuracy of the numerical results. In view of this, (21)–(24) can easily be solved using ODE solvers such as MATLAB’s bvp5c as explained in Kierzenka and Shampine [51] and Gökhan [52].
3.1. Verification of the Results
In order to verify the accuracy of the present analysis, the results of classical Runge-Kutta together with shooting have been compared with that of bvp5c solution for the limiting case when , , , , , , , , , , , and at various values of within the range . As shown in Table 1, the comparison in the above case is found to be in good agreement. This good agreement is an encouragement for further study of the effects of other parameters.
4. Results and Discussion
The numerical computations have been carried out for various values of major parameters using the numerical scheme discussed in the previous section. This section presents the effect of different embedded physical parameters on the flow. Following Mustafa et al. [53], the ratio of momentum diffusivity to thermal diffusivity is considered to be unity (i.e., ) due to the fact that Casson fluid flow under consideration possesses substantial yield stress.
4.1. Influence of Velocity Index Parameter and Eckert Number
At metalimnion level of thermal stratification (), when , , , , , , , , , , and , it is worth noting that both vertical and horizontal velocities decrease with ; see Figures 2 and 3. Combination of these practical meanings of velocity index parameter justifies the decrease in both vertical and horizontal velocities profiles we obtained. Meanwhile, temperature distribution within the flow increases with from a few distance after wall till free stream; see Figure 4. With an increase in the magnitude of , Figure 5 reveals that temperature gradient profile increases near the wall and decreases thereafter as . It is very important to notice that velocity profile when perfectly satisfies free stream condition asymptotically. This can be traced to the fact that and as , . In addition, graphical illustration of the function which describes the immediate layer of Casson fluid next to an upper horizontal surface of a paraboloid of revolution against (i.e., and ) at various values of less than 1 and stretching velocity at the wall when stretching rate (i.e., are presented in Figure 6. It is noticed that as increases within , the thickness of the paraboloid of revolution decreases but corresponding influence of stretching on the flow is an increasing function. The variation in local skin friction coefficient and local Nusselt number which is proportional to local heat transfer rate as stated in (25) as a function of viscous dissipation term and velocity index parameter is shown in Figure 7 and Table 2. Figure 7 shows that, at a fixed value of , decreases with Eckert number . At constant value of , unequal decrease in with is also observed. In addition, decreases with at various values of . Table 2 shows that increases highly significant when magnitude of is large.
4.2. Influence of Non-Newtonian Casson Parameter , Thermal Stratification Parameter , and Eckert Number
Figures 8–12 illustrate the effect of increasing the magnitude of non-Newtonian Casson parameter on all the five profiles when , , , , , , , , , , , and . With an increase in the magnitude of , it is observed that vertical velocity decreases, horizontal velocity decreases, temperature distribution increases only within the fluid domain (), and the temperature gradient increases only near the wall. Figure 10 reveals that a distinct significant increase in is guarantee with an initial increase in from to . Physically, increase in the magnitude of non-Newtonian Casson parameter ( implies sharp transition in the flow behavior from non-Newtonian fluid flow to Newtonian fluid flow. In view of this, resistance in the fluid flow is produced. It is worth mentioning that an increase in implies a decrease in yield stress of the Casson fluid and increase in the magnitude of plastic dynamic viscosity .
It is observed that the present study complements related studies on Casson fluid flow with temperature-dependent plastic dynamic viscosity on nonmelting surface; see Figures and in [41], Figures and in [54], and Figure reported by Jasmine Benazir et al. [55]. The relationship between non-Newtonian Casson parameter and Eckert number is sought for and illustrated graphically in Figures 13 and 14. Within , there exists no significant difference in with . As shown in Figure 13, when the magnitude of , a distinct significant increase in is observed due to an increase in the magnitude of viscous dissipation parameter. At small magnitude of , local Nusselt number () which is proportional to local heat transfer rate is found to be decreasing with . At large value of , increases with ; see Figure 14. The simulation was further extended to unravels the relationship between non-Newtonian Casson parameter , thermal stratification parameter , and local skin friction coefficients when , , , , , , , , , , and . It is revealed in Figure 15 that increases with at epilimnion stage which is known as the highest and warmest layer (). In addition, a significant decrease in is observed with an increase in at hypolimnion stage which can be referred to as the coolest layer. Mathematically, when , this implies that and maximum wall temperature at the wall explains the increase in since increase in corresponds to a decrease in yield stress .
5. Conclusion
The boundary layer analysis of non-Newtonian Casson fluid flow over a horizontal melting surface embedded in a thermally stratified medium in the presence of viscous dissipation internal space heat source has been investigated numerically. The effects of the velocity power index, melting parameter, temperature-dependent viscous parameter, Eckert number, thermal conductivity, and magnetic interaction parameter were examined. Conclusions of the present analysis are as follows:(1)Increase in the magnitude of velocity index parameter leads to a decrease in velocity and increase in temperature due to combine practical influence of the parameter.(2) decreases with at various values of within the interval . decreases with and .(3)At various values of viscous dissipation within and due to the nature of the flow past melting surface, a valid non-Newtonian parameter falls within the interval . Equivalent Newtonian fluid flow is guaranteed for .(4)Local skin friction coefficient increases negligible with when magnitude of is small. A significant increases in is guaranteed with when magnitude of is large.(5)In the case of Casson fluid flow over an upper horizontal thermally stratified melting surface of a paraboloid of revolution, decrease in horizontal velocity is guaranteed with an increase and .(6)With an increase in the magnitude of , the influence of stretching velocity at the wall on horizontal and vertical velocities is stronger than that of which describes the immediate fluid’s layer next to upper horizontal surface of a paraboloid of revolution due to melting heat transfer.
An extension of the present study to the case of Williamson and Prandtl fluid flow over an upper horizontal thermally stratified melting surface of a paraboloid of revolution is hereby recommended. For suitable parametrization to achieve a comparative study, see [7].
Nomenclature
| : | Velocity component in direction |
| : | Velocity component in direction |
| : | Coefficient related to stretching sheet |
| : | Magnetic field strength |
| : | Skin friction coefficient |
| : | Specific heat at constant pressure |
| : | Distance along the surface |
| : | Distance normal to the surface |
| : | Parameter related to stretching sheet |
| : | Local Nusselt number |
| : | Prandtl number |
| : | Heat transfer |
| : | Dimensional fluid temperature |
| : | Stretching velocity at the wall |
| : | Thermal property of the fluid |
| : | Eckert number |
| : | Dimensionless vertical velocity |
| : | Thermal conductivity of the Casson fluid |
| : | Velocity power index |
| : | Melting temperature at wall |
| : | Ambient Temperature |
| : | Local Reynolds number |
| : | Non-Newtonian Casson parameter |
| : | Wall thickness parameter |
| : | Plastic dynamic viscosity |
| : | Latent heat of the fluid |
| : | Melting parameter |
| : | Dimensionless temperature |
| : | Kinematic viscosity |
| : | Temperature-dependent viscosity parameter |
| : | Temperature-dependent thermal conductivity parameter |
| : | Density of Casson fluid |
| : | Electrical conductivity of the fluid |
| : | Stream function. |
Competing Interests
The authors declare that they have no competing interests.