Abstract
In this analysis, steady, laminar, and two-dimensional boundary layer flow of nonlinear convection micropolar nanofluid due to a rotating disk is considered. The mathematical formulation for the flow problem has been made. By means of appropriate similarity transformation and dimensionless variables, the governing nonlinear boundary value problems were reduced into coupled high-order nonlinear ordinary differential equations with numerically solved. The equations were calculated using method bvp4c from matlab software for various quantities of main parameters. The influences of different parameters on skin friction coefficients and , wall duo stress coefficients , -, and -, the Nusselt number -, and Sherwood number , as well as the velocities, temperature, and concentration are analysed and discussed through tables and plotted graphs. The findings indicate that an increase in the values of thermal and solutal nonlinear convection parameters allow to increase the value of velocities and near surface of the disk and reduce at far away from the disk as well as thermal and solutal Grashof numbers tolerate to increase in the value of radial velocity near surface of the disk.
1. Introduction
Eringen [1] was the first researcher who presents the theory of micropolar fluid for which the classical Navier–Stokes theory is has a limitation for its full description. Consequently, Hamzeh [2] evaluated the behaviour of micropolar Casson fluid on natural convective flow past a solid sphere. It was found that material parameter diminishes the values of the narrow skin friction coefficient, but grows both values of the local Nusselt number and angular velocity profiles as its value rises. Wubshet and Chaluma [3] also reported the variation of density on micropolar nanofluid flow about an isothermal sphere. Moreover, wall shear stress and angular velocity gradient at the wall enhance with an enlargement in the material parameter as illustrated by Mandal and Mukhopadhyay [4]. Furthermore, the flow of MHD and heat transfer due to stretching rotating disk were examined by Mustafa [5] and Akhter et al. [6].
Mostafa and Shimaa [7] reported the influence of magnetic field on flow and heat transfer of a micropolar fluid about a stretching surface in the presence of heat generation (absorption). Also, the effects of microrotation parameter on flow between a rotating and stationary disk was examined by Anwar and Guram [8]. Moreover, Rashidi and Freideonimehr [9] presented the effects of velocity and temperature slip on the entropy generation past rotating disk. Also, the effects of diffusion-thermo and thermodiffusion on radially stretching disk have been evaluated by Khan et al. [10].
The impact of Prandtl number on radiative flow due to stretchable rotating disk with variable thickness was computed by Tasawar et al. [11]. Shamshuddin et al. [12] have evaluated the influence of this parameter on numerical study of heat transfer and viscous flow in a dual rotating extendable disk system employing a non-Fourier heat flux model. They indicated that, with an improvement in the Prandtl number, there is a strong decrease in temperature of fluid as well as radial skin friction. Moreover, the Stefan blowing effect on bioconvective flow and heat transfer of nanofluid over a rotating stretchable disk was reported by Lafiff et al. [13] and Yin et al. [14].
Muhammad and Naeem [15] and Noor et al. [16] have examined the influences of velocity slip with magnetic field in micropolar nanofluid flow along rotating disk and in mixed convection lower flow of a micropolar nanofluid along vertically elongating surface accordingly. Also, the flow and heat transfer computations for nanofluid fluid flow and the impact of heat generation/consumption and thermal radiation over a rotating disk were analysed by Narfifah et al. [17], Mushtaq and Mustafa [18], and Anwer et al. [19]. Moreover, Tasawar et al. [20] had discussed the impact of thermal slip condition on MHD flow of Cu-water nanofluid due to a rotating disk. They found that thermal boundary thickness increased for lower thermal slip parameter values, but increasing the values of it reduced heat transfer from the disk to the adjacent fluid. Moreover, the numerical study of nanofluid flow and heat transfer over a rotating disk was examined by Ahmad et al. [21]. MHD mixed convection movement of a nanofluid over nonlinear enlarging sheet including variable Brownian and thermophoretic diffusion coefficient have been evaluated by Sumalatha and Shanker [22], and the results illustrated that the influence of nonlinear stretching parameter drops both the flow of the fluid as well as temperature distribution.
All the above research articles have been on the flow over a plane, over a stretching surface, on MHD boundary flow, or in many other areas. In this paper, we evaluate numerically the nonlinear convection flow of micropolar nanofluid due to a rotating disk in the presence of multiple slip conditions, using bvp4c from matlab. The outcomes of physical parameters on fluid velocity, temperature, and concentration were discussed and indicated in graphs and tables as well.
2. Mathematical Formulation
Let us choose the cylindrical system in the direction component of the flow velocity and the angular velocity components correspondingly. The study considers incompressible, laminar, and nonlinear convection flow of micropolar nanofluid over a circular disk at . The disk rotates with uniform microrotation about the z-axis. As the result of revolving symmetry the end products in the azimuthal direction may be considered. The wall of the gyrating disk has constant temperature and concentration despite the fact the ambient temperature and concentration are represented by and correspondingly, as shown in Figure 1. By means of Mandal and Mukhopadhyay [4], Anwer and Guram [8], Tasawar et al. [11], Noor et al. [16], and Sajjad et al. [23] the governing differential equations of the flow are given as follows:with boundary conditionswhere are angular velocity in the r, , and z axes, correspondingly, are material constants (viscosity coefficients), L stands for momentum slip factor, n stands for angular slip factor, h represents thermal factor, m is solutal jump factor, stands for the coefficient of fluid viscosity, is the density, stands for the specific heat, represents the gravity, is the microinertia per unit mass, is the vortex viscosity, T stands for temperature, K is the thermal conductivity of the fluid, and represent the variable temperature and concentration at the surface, where and being constants give the rate of growth of temperature and concentration alongside the surface and and stand for the uniform temperature, concentration of the free stream, and and are constants, and are the constant coefficients of thermal and volumetric expansion, respectively. This relation will be nonlinear density, temperature, and concentration (NDTC) variation. is the ratio between the effective heat capability of the nanoparticle material and the heat capability of the fluid and and stand for the Brownian and the thermophoretic diffusion coefficient, respectively. Let be the radial distance from the symmetrical axis to the surface.
By using the nondimensional variables such as , , , , , , , , ,The continuity equation can be integrated so that it satisfies equation (1). Hence, equations (2)–(9) can be written in the following nonlinear system of ODEs:with the boundary conditions.
Atwhere prime represents differential with respect to , is the velocity slip parameter, denotes microrotation slip parameter, stands for thermal parameter, withstands for solutal parameter, (, , , , , and are angular velocity parameters, and denote thermal and solutal mixed convection parameters (or Archimedes numbers), represents thermal Grashof number, represents solutal Grashof number, is Reynolds number, and and are thermal and solutal nonlinear convection parameters, correspondingly. We observe that, for , the flow of equations (14) and (15) become mixed convective micropolar nanofluid. Also, the relation of buoyancy forces ( and ) seen in the these equations are the dimensionless parameters, indicating the ratios between the buoyancy forces caused by temperature, concentration change, and the ratio inertial to viscous forces owing to kinematic viscosity change. Furthermore, Gr and Gm are zero for nonbuoyancy influence because of thermal and mass diffusions, as , i.e., are for nonbuoyancy effect owing to momentum diffusion and when are for thermal, mass buoyancy forces, and inertial to viscous forces of the same strength. is Prandtl number, is Schmidt number, stands for Brownian motion parameter, and represents thermophoresis parameter.
It can appear that, due to the rotational symmetry, the derivative in the radial direction may be neglected and equations (14)–(21) are reduced to the next nonlinear system of ordinary differential equations:with the boundary conditions.
At
The physical measures of awareness in this problem are the narrow skin friction coefficients , surface couple stresses , the Nusselt number Nu, and the Sherwood Sh number, and they can be written as follows:where , , , , , , and . Using the dimensionless variables (12) and the boundary conditions (20), the narrow skin friction coefficients surface couple stresses, the Nusselt number and the Sherwood number are obtained:
3. Numerical Solution
Pairs of seven harmonized high order ordinary differential equations (14)–(20), subjected to the boundary conditions, equation (21), are answered numerically using the function bvp4c from matlab software for various values of physical parameters and numbers.
Statistical results are found using Matlab BVP solver bvp4c from matlab which is a finite difference code that realize the three-stage Lobatto IIIa formulation. To apply bvp4c from matlab, first, equations (14)–(20) are converted into a system of first-order equations.
Second, assemble a boundary value problem (bvp) and using the bvp solver in matlab to numerically solve this system, including the above boundary condition and income on a suitable finite value for the far field boundary condition, that is, , say and the step size is taken as , the numerical result is obtained; it is exact to the fifth decimal place as the measure of convergence. In solving the BVP by means of matlab, bvp4c has only two point of views: a function ODEs for calculation of the residual in the boundary conditions and a building solint that provides a guess for a mesh. The ODEs are handled exactly as in the Matlab IVP solvers. Further clarification on the procedure of bvp4c is found in the book by Shampine et al. [24].
Let , , , , , , , , , , , , , , , , , , , , , , , and y = [f, , , G, , , , , , , , , , , , , , , , , , , ]T give
4. Results and Discussion
In this section, the results of different governing physical parameters on nondimensional velocity, temperature, concentration, skin friction and wall couple stress coefficients and confined Nusselt and Sherwood numbers have been discussed. In this study, the comparison with the literature value is not compared, since there is no related article.
4.1. Velocity Profiles
The dimensionless velocity profile graphs of for different values of angular velocity parameter , thermal and solutal nonlinear convection parameters (, s), and thermal and solutal Grashof numbers (Gr, Gm) are shown in Figures 2–11. Figures 2–9 reveal that an increase in values of Gr, Gm, , and s reduces the kinematic viscosity of the fluid flow which falls opposite to the flow of the fluid that results in an increase of the magnitude of radial and azimuthal velocity profiles near the surface of the disk, but they decrease with an enhancement of Gm, , and s far away from the surface. Moreover, boost in the values of angular velocity parameter raises the vortex viscosity of the fluid flow which adds opposition to flow of the fluid that results in declination of the fluid velocity profile , but the reverse effects are observed on the fluid velocity profile as seen in Figures 10 and 11.
4.2. Temperature and Concentration Profiles
The influences of thermal and solutal jump parameters s2 and s3, nonlinear convection parameters (, s), and thermal and solutal Grashof numbers (Gr, Gm) on temperature and concentration sketches are prearranged in Figures 12–19. These figures indicate that the values of temperature and concentration disseminations and their boundary layer thickness decrease with increase of thermal and solutal jump parameters s2 and s3, nonlinear convection parameters (, s), and thermal and solutal Grashof numbers (Gr, Gm). These effects happen due to the decline in the kinematic viscosity of the fluid which reduces thermal diffusion, that leads to a decrease in thermal and solutal boundary layer thicknesses.
4.3. Skin Frictions and Wall Couple Stresses
The impacts of thermal and solutal nonlinear convection parameters (, s), thermal and solutal Grashof numbers (Gr, Gm), angular velocity parameters (B1–B5) on skin friction coefficients (, ), and wall duos coefficients () sketches are prearranged in Figures 20–35. Table 1 and Figures 20 and 29 illustrate that upsurge in values of B1 reduces the kinematic viscosity of the fluid flow which falls opposite to the flow of the fluid that results in decline of the skin friction coefficients and wall duos coefficients -; Table 1 demonstrates that increase in velocity slip parameter s0 reduces skin friction coefficients , whereas the opposite effects are observed on the skin friction coefficients and wall duos coefficients . Moreover, Figures 22–27 show that boost in values of , s, Gr, and Gm raise density of the fluid flow which increases resistance to the flow of the result upsurge skin friction coefficients . Similarly, Figures 30 and 31 indicate that an increase in values of B2 improves vortex viscosity of the fluid flow which enlarges resistance to rotate the fluid that results in upsurge wall duos coefficients and -. Furthermore, increase in values of B3, improve wall couple coefficients - but bring down wall couple coefficients as indicated in Figures 32 and 33. Next, wall couple coefficients - reduces with raise values of B4 and B5 in the boundary layer, as shown in Figures 34 and 35.
4.4. Nusselt and Sherwood Numbers
The impacts of Prandtl Pr, Lewis Le numbers, thermal, solutal (s2, s3), and Brownian motion Nb parameters on Nusselt , and Sherwood numbers are presented in Figures 36–41. Figure 36 illustrates that boost in values of Pr bring down thermal diffusion of the temperature that cause upsurge Nusselt number near the surface of the disk. Figure 37 demonstrates that a rise in the values of Le reduces the density of the fluid , as a result the Sherwood number increased, whereas an increment in the values s2 and s3 reduces the Nusselt number and Sherwood number as demonstrated in Figures 38-39. Figures 40 and 41 show the impact of Brownian motion parameter Nb on Nusselt number and Sherwood number . From the figures, it is possible to see that Nb favors the Nusselt number and opposite effect on the Sherwood number.
5. Conclusions
This report considers the effects of thermal and solutal nonlinear convection as well as manifold slip conditions on dimensional boundary layer flow of a micropolar nanofluid due to a rotating disk were analysed. The boundary layer equations of the flow problem are reduced into a pairs of high-order nonlinear ordinary differential equations by using the similarity transformation. The obtained differential equations are solved numerically using bvp4c from matlab software. Numerical outcomes are acquired for different main parameters of the flow problem. The effects of governing parameters are presented by means of figures and tables. The main findings are(1)An increasing in the values of thermal and solutal nonlinear convection parameters allow to increase in the value of velocities and near surface of the disk and reduce at far from the disk as well as thermal and solutal Grashof numbers tolerate to increase the value of radial velocity near surface of the disk.(2)Velocity profile is positive throughout its boundary, indicating in that way there is outward flow from the disk which reaches the particular point and it drops to zero when far from the surface of the disk.(3)The existence of the thermal and solutal jump, nonlinear convection parameters, and Grashof numbers allows to decline the temperature and concentration distributions near the disk surface.(4)Radial skin friction coefficients exhibit increasing behavior for increasing thermal and solutal nonlinear convection parameters and Grashof numbers; skin friction coefficients exhibit increasing behavior for increasing thermal and solutal nonlinear convection parameters.(5)An increase in quantities of Pr and Le causes enhancement in the Nusselt and Sherwood numbers while they fall as magnitudes of s2 and s3 increase, accordingly.(6)The value of wall couple coefficients () increase with B2. However, they rotate in opposite directions.(7)An improvement in the values of angular velocity parameters (B1, B3) allows to rotate the wall couple stresses ( and ) in opposite directions.
Nomenclature
| a, b, c: | Material constants (viscosity) |
| : | Skin friction coefficients |
| : | Specific heat |
| : | Microrotation parameters |
| C: | Concentration |
| : | Ambient concentration |
| : | Wall concentration |
| d: | Radius of disk (m) |
| : | Thermophoretic diffusion |
| (f, G): | Dimensionless functions |
| : | Thermal Grashof number |
| : | Solutal Grashof number |
| : | Nondimension angular velocity |
| j: | Microinertia density |
| h: | Thermal jump factor () |
| K: | Thermal conductivity |
| : | Wall couple stresses |
| L: | Momentum slip factor () |
| m: | Solutal jump factor |
| Le: | Lewis number |
| : | Solutal jump factor |
| n: | Angular slip factor |
| : | Angular components |
| : | Brownian motion parameter |
| : | Thermophoresis parameter |
| : | Local Nusselt number |
| Pr: | Prandtl number |
| : | Heat and mass fluxes |
| : | Polar coordinates |
| Re: | Reynolds number |
| Sh: | Sherwood number |
| : | Solutal jump parameter |
| s: | Solutal nonlinear convection parameter |
| : | Thermal jump parameter |
| : | Velocity slip parameters |
| : | Microrotation slip parameters |
| : | Wall temperature (K) |
| : | Ambient temperature (K) |
| : | Velocity components |
| T: | Temperature of the fluid (K) |
| : | Thermal nonlinear convection parameter |
| : | Thermal and volumetric expansion |
| , : | Constants |
| : | Dimensionless similarity variable |
| : | Dimensionless temperature |
| : | Dimensionless concentration |
| : | Dimensionless stream wise coordinate |
| : | Fluid viscosity (Pa·s) |
| : | Vortex viscosity coefficient (Pa·s) |
| : | Kinematic viscosity coeff. |
| : | Fluid density |
| : | Wall shear stress (Pa) |
| : | Conditions at the free stream |
| : | Condition at the surface. |
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.