This paper reports multiple slip effects on MHD unsteady flow heat and mass transfer over a stretching sheet with Soret effect; suction/injection and thermal radiation are numerically analyzed. We consider a time-dependent applied magnetic field and stretching sheet which moves with nonuniform velocity. Suitable similarity variables are used to transform governing partial differential equations into a system of coupled nonlinear ordinary differential equations. The transformed equations are then solved numerically by applying an implicit finite difference method with quasi-linearization technique. The influences of the various parameters on the velocity temperature and concentration profiles as well as on the skin friction coefficient and Sherwood and Nusselt numbers are discussed by the aid of graphs and tables.
1. Introduction
The Navier–Stokes theory is centered on the central idea of no-slip condition. Many authors have obtained both numerical and analytical solutions by applying no-slip boundary conditions to study velocity and temperature profiles. The importance of slip conditions in microchannel or nanochannels has stimulated much interest on the study of vibrating values [1]. It is now known that a slip can occur if the working fluids contain concentrated suspensions. Soltani and Yilmazor [2] performed using a parallel disk rheometer with emphasis on a wall slip phenomenon on the rheological characterization of highly filled suspensions consisting of a Newtonian matrix, mixed with two different sizes of aluminum power and two different sizes of glass beads. When the fluid is particulates such as suspensions, emulsions foams, and polymer solutions, a partial velocity slip may occur on the stretching boundary. Slip effects can arise in various industrial processes at boundaries of pipes, walls, and/or curved surfaces. A Navier velocity slip condition is a usual approach in studying slip phenomena. Mahanthesh et al. [3] examined three-dimensional flow of nanofluid for the effects of partial slip and chemical reaction towards an exponential stretching sheet. Hayat et al. [4] examined simultaneous effects of slip and heat transfer on peristaltic flow. Motsa and Shateyi [5] studied the problem of a nonlinear boundary value problem arising in rotating disk flow under the effects of a partial slip, thermal diffusion, and diffusion thermo. Shateyi and Mabood [6] have illustrated the impact of slip and viscous dissipation on MHD mixed convection stagnation-point flow over a nonlinear stretching sheet. Khan et al. [7] carried out a study on a two-dimensional flow of an incompressible Williamson fluid of Cattaneo–Christov heat flux type over a linearly stretched surface with the influence of magnetic field, thermal radiation-diffusion, heat generation, and viscous dissipation.
Due to the many applications in engineering and industries, the magnetohydrodynamic fluid flows on a stretching sheet have achieved much importance nowadays [8]. Such applications include the liquid coating on photographic films, the boundary layer through the liquid film in the concentration process, and aerodynamic extrusion of plastic sheets. In addition, a wide range of applications on MHD flow can be found in numerous fields such as electronic cooling, boilers, heat insulation and metal extrusion, liquid metal fluid oil reservoirs, geothermal systems, nuclear process, micro-MHD pumps, high temperature plasmas, groundwater systems, energy storage units, biological transportation, and thermal energy storage devices . Mabood et al. [9] examined the effects of heat source and chemical reaction on MHD rotating fluid towards a vertical plate influenced by a porous medium. Kumar et al. [10] investigated the impact of frictional heating on MHD ferrofluid with radiation.
Heat transfer, determined by thermal radiation, has vast applications in different technological processes, including missiles, nuclear power plants, satellites and space vehicles, gas turbines, and the numerous propulsion devices for aircraft. Linear radiation is not analytically valid for immense temperature difference. Abbas et al. [11] discussed the effects of radiation in the presence of uniform magnetic field for a nanofluid on a curved stretching surface by incorporating slip effect. Recently, Makinde et al. [12] developed a numerical study of radiation effects on chemically reacting MHD nanofluid influenced by heat source/sink and combined heat and mass transfer analysis for mixed convection flow over vertical surface with radiation and chemical reaction illustrated by Ibrahim et al. [13]. Prasannakumara et al. [14] studied the effects of velocity slip, temperature jump, solutal slip and thermal radiation on a steady flow, heat and mass transfer of an incompressible Jeffrey nanofluid over a horizontal stretching surface. Imtiaz et al. [15] examined unsteady MHD flow of curved stretching surface. Some interesting investigations relevant to flow and heat transfer can be viewed in [16–23].
To the author’s knowledge, no studies have this far been communicated with regard to the multiple slips on hydromagnetic unsteady flow and heat and mass transfer influenced by radiation in permeable frame of reference. Numerical solutions are provided for some special cases, while the physical interpretation for the various parameters is discussed with the help of graphs.
2. Governing Equations
A two-dimensional MHD flow of an incompressible electrically conducting fluid over a permeable stretching surface in the presence of thermal radiation is considered. A coordinate system is chosen in such a way that x-axis is measured along the sheet, and y-axis is normal to it as shown in Figure 1. The sheet is moving with nonuniform velocity Ux,t=ax/1−λt along x-axis where a is the stretching rate and λ is the positive constant with the property λt<1. A transverse magnetic field that is assumed to be the function of distance from origin is defined as Bx=B0x−1/2 with B0≠0, where x is the coordinate along the surface and B0 is the magnetic field strength. The induced magnetic field is negligible as compared to the applied magnetic field. Let T∞ and C∞ are the free stream temperature and free mass concentration. The governing equations for the continuity, momentum, energy, and concentration can be written:(1)∂u∂x+∂v∂y=0,(2)∂u∂t+u∂u∂x+v∂u∂y=ν∂2u∂y2−σB2xuρ+gβTT−T∞+gβCC−C∞,(3)∂T∂t+u∂T∂x+v∂T∂y=α1+16T∞3σ∗3k∗κ∂2T∂y2,(4)∂C∂t+u∂C∂x+v∂C∂y=DM∂2C∂y2+DT∂2T∂y2,where x and y are the coordinates along and normal to the sheet; u and v are the components of the velocity in the x and y directions, respectively; ρ is the density of the fluid; ν is the kinematic viscosity of the fluid; σ is the electrical conductivity; g is the acceleration due to gravity; βT is the thermal expansion coefficient; βC is the concentration expansion coefficient; α is the thermal diffusivity; T is the temperature; C is the concentration; DM is the molecular diffusivity; DT is the thermal diffusivity; σ∗ is the Stefan–Boltzmann constant; and k∗ is the mean absorption coefficient.
Sketch of the physical model.
The boundary conditions for the above described model are:(5)y=0:u=Ux,t+Uslip,v=vw,T=Twx,t+Tslip,C=Cwx,t+Cslip,y⟶∞:u⟶0,T⟶T∞,C⟶C∞,where vw=v0/x is the suction/injection velocity.
The temperature of the sheet Twx,t and the concentration Cwx,t at the surface are assumed as of the following form:(6)Twx,t=T∞+T0ax2ν1−λt−2,Cwx,t=C∞+C0ax2ν1−λt−2,where T0, and C0 are the reference temperature and reference concentration, respectively, such that 0≤T0≤Tw and 0≤C0≤Cw. The above expressions are valid if 1−λt>0.
As per usual, the stream function ψ is defined as u=∂ψ/∂y and v=−∂ψ/∂x so that equation (1) is satisfied. Let us introduce the following dimensionless functions f,θ,ϕ, and similarity variable η as follows:(7)η=aν1−λty,ψ=aν1−λtxfη,T=T∞+T0ax1−λt−22νθη,C=C∞+C0ax1−λt−22νϕη.
Now substituting equation (7) into equations (2)–(4), we get the following system of nonlinear ordinary differential equations:(8)f‴+ff″−f′2−δη2f″+f′−Mf′+λ1θ+λ2ϕ=0,(9)1Pr1+Rθ″+fθ′−f′θ−δη2θ′+2θ=0,(10)ϕ″+Scfϕ′−f′ϕ−Scδη2ϕ′+2ϕ+ScSrθ″=0.
The transformed boundary conditions of the problem are(11)f0=fw,f′0=1+Sff″0,f′∞=0,θ0=1+Sθθ′0,θ∞=0,ϕ0=1+Sϕϕ′0,ϕ∞=0,where δ=λ/a is the unsteadiness parameter; λ1=gβTT0/aν and λ2=gβCC0/aν are the buoyancy parameters; Pr=ν/α is the Prandtl number; R=16T∞3σ∗/3k∗κ is the thermal radiation parameter; M=σ/ρaB0 is the magnetic field parameter; Sc=ν/DM is the Schmidt number; Sr=DTT0/νC0 is the Soret number; and fw=−vw1−γt/νa is the suction/injection parameter. Here, fw=0 represents the impermeable surface, fw>0 represents the suction, and fw<0 represents the injection of the fluid through permeable surface.
The physical quantities of interest are local skin friction coefficient Cf, local Nusselt number Nu, and local Sherwood number Sh are defined as(12)Cf=μρUw2∂u∂yy=0,(13)Nu=xκTw−T∞κ∂T∂yy=0−4σ∗3k∗∂T4∂yy=0,(14)Sh=−xCw−C∞∂C∂yy=0.
Substitute equation (7) into equations (12)–(14) to obtain the final dimensionless form:(15)Cfr=RexCf=f″0,Nur=NuRex=−1+Rθ′0,Shr=ShRex=−ϕ′0,where Rex=Uwx/ν is the local Reynolds number, Cfr is the reduced skin friction, Nur is the reduced Nusselt number, and Shr is the reduced Sherwood number.
3. Numerical Method
The reduced ordinary differential equations (9)–(11) subject to the boundary conditions (12) were solved numerically using an implicit finite difference method with quasi-linearization technique. The details of the proposed method can be found in the studies of Mabood et al. [21], Inoyue and Tate [22], and Bellman and Kalaba [23]. The effects of the various parameters on the velocity, temperature, concentration, and heat and mass transfer rates are investigated. The step size and convergence criteria were taken as Δη=0.001 and 10^{−6}, respectively. The asymptotic boundary conditions in equation (12) were approximated by using a value of 10 for the similarity variable ηmax as follows:(16)ηmax=10,f′10=θ10=ϕ10=0.
The choice of ηmax=10 ensures that all numerical solutions approached the asymptotic values correctly.
4. Results and Discussion
In order to validate our current results and to determine the accuracy of the present analysis, comparisons with available results of the skin friction coefficient −f″0 for the unsteady flow of viscous incompressible Newtonian fluid are done. In Table 1, we compare our results of the skin friction coefficient generated by Chamkha et al. [16]. In this table, we observe that there is an excellent agreement between our results, and those observe in this table that the skin friction coefficient increases with the increasing values of the stretching parameter. In Table 2, we observe that there is an excellent agreement between our present results with those previously obtained by Mabood and Das [8]. We also observe in this table that, as the magnetic parameter M increases, the skin friction significantly increases due to the Lorentz drag force caused by electromagnetism increases. Lastly, Table 3 depicts the comparison of our present results to those obtained by Ali [17] when investigating the effect of the Prandtl number on the heat transfer rate. We observe that the heat transfer rate on the surface is greatly affected by the Prandtl number. Increasing the Prandtl number increases the rate of heat transfer on the stretching surface.
Comparison of −f″0 for various values of δ when fw=M=λ1=λ2=Sf=0.
δ
Chamkha et al. [16]
Present
0.8
1.261512
1.261042
1.2
1.378052
1.377724
Comparison of −f″0 for various values of M when fw=δ=Sf=0.
M
Mabood and Das [8]
Present
0
−1.000008
−1.0000084
1
1.4142135
1.41421356
5
2.4494897
2.44948974
10
3.3166247
3.31662479
50
7.1414284
7.14142843
100
10.049875
10.0498756
500
22.383029
22.3830293
1000
31.638584
31.6385840
Comparison of heat transfer rate −θ′0 when M=fw=Sf=Sθ=δ=λ1=λ2=Rd=0.
Pr
Ali [17]
Present
0.72
0.8058
0.8088
1
0.9691
1.0000
3
1.9144
1.9237
10
3.7006
3.7207
Figure 2 depicts the effect of increasing the magnetic field parameter M on the velocity profiles with (Figure 2(b)) and without (Figure 2(a)) hydrodynamic slip. We observe in both cases that the velocity profiles are reduced with increasing values of the magnetic parameter. Physically, the presence of the transverse magnetic field in the fluid flow produces a drag-like force called the Lorentz which in turn decelerates the fluid motion. However, the existence of the hydrodynamic slip as can be clearly seen in Figure 2 increases the velocity boundary layer. We also observe in Figure 2 that suction reduces the velocity boundary layer. Thus, suction can be used as a stabilizing mechanism to delay the transition from the laminar boundary layer flow to turbulent flow.
Effects of M and fw on dimensionless velocity. (a) No hydrodynamic slip. (b) With hydrodynamic slip.
The effect of varying the thermal buoyancy parameter on the velocity distribution is depicted in Figure 3 and that of solutal buoyancy parameter is shown in Figure 4. Increasing values of the buoyancy parameters leads to the increase in the temperature and solutal gradients. Physically, this explains why the velocity profiles increase as more forces are added with these increases in buoyancy parameters δ. We also observe in Figure 3 that the velocity profiles decrease as the unsteadiness parameter increases. Also increasing the values of thermal radiation parameter R causes the profiles of the velocity to increase (Figure 4).
Effects of λ1 and δ on dimensionless velocity. (a) No hydrodynamic slip. (b) With hydrodynamic slip.
Effects of λ2 and R on dimensionless velocity. (a) No hydrodynamic slip. (b) With hydrodynamic slip.
The influence of the magnetic parameter M and thermal radiation R on the temperature profiles is shown on Figure 5. Reduction of the flow velocity due to the increase of the magnetic field strength causes the temperature profiles (Figure 5) and concentration profiles (Figure 6) to increase. Physically, applying the magnetic field heats up the fluid and thus reduces the heat and mass transfer rates from the wall causing increases in fluid temperature and concentration distributions. We also observe in Figure 5 that the fluid temperature increases with increasing values of the thermal radiation parameter R. The effect of radiation in the thermal boundary layer (equation (3)) is equivalent with an increased thermal diffusivity.
Effects of M and R on dimensionless temperature. (a) No thermal slip. (b) With thermal slip.
Effects of M and Sc on dimensionless concentration. (a) No thermal slip. (b) With thermal slip.
Figure 7 depicts the influence of thermal buoyancy parameter on the dimensionless temperature profiles. It is clearly observed that an increase in the thermal Grashof number causes a decrease in the thermal boundary layer thickness and consequently the fluid temperature decreases due to the buoyancy effect. We also observe in Figure 7 that the temperature profiles are lower for suction fw>0. Figure 8 shows the effect of the stretching parameter δ and the Prandtl number on the temperature distribution. We observe in this figure that the temperature profiles are reduced with increasing values of the Prandtl number. It is also observed in this figure that the temperature profiles are reduced with increasing values of the stretching parameter. Figure 6 displays the effect of varying the Schmidt number and the magnetic field parameter on the concentration profiles. The Schmidt number represents the relative ease of the molecular momentum and mass transfer, and it is very important in multiphase flows. The effect of increasing values of the Schmidt number is to reduce the momentum boundary layer, and this leads to the thinning of the diffusion layer.
Effects of λ1 and fw on dimensionless temperature. (a) No thermal slip. (b) With thermal slip.
Effects of δ and Pr on dimensionless temperature. (a) No thermal slip. (b) With thermal slip.
Figure 9 depicts the effects of magnetic, unsteadiness, slip velocity, and suction/injection parameters on the skin friction coefficient. We observe in this figure that f″0 decreases with increasing values of the slip velocity Sf and buoyancy parameters. But it increases with increasing values of the magnetic and suction parameters. Figure 10 exhibits the nature of the heat transfer coefficient with the thermal radiation, thermal slips, buoyancy, magnetic, and suction and injection parameters. We observe in this figure that increasing the magnetic field strength M strongly suppresses the Nusselt numbers. Physically, an extra work done by dragging the fluid against the action of transverse magnetic field is then dissipated on heat energy in the boundary layer. This leads to the heating up of the boundary layer resulting in more heat transferred to the fluid flow. Thus heat transfer to the wall is reduced. The rate of heat transfer on the stretching surface is significantly enhanced by the increasing values of the thermal radiation parameter, suction parameter, and solutal buoyancy parameter. But opposite responses are observed with increasing values of thermal buoyancy and injection parameters.
Effects of M,δ,Sf,λ1,λ2, and fw on the skin friction coefficient.
Effects of M,R,Sθ,λ1,λ2, and fw on the reduced Nusselt number.
Finally, Figures 11(a) and 11(b) present the Sherwood distributions with variation of the Schmidt number, magnetic parameter, solutal slip parameter, Soret number, sectional injection, and unsteadiness parameter. We observe in these figures that the mass transfer rate decreases with magnetic and solutal slip parameters. We can clearly see in these figures that the rate of mass transfer increases with the Schmidt number and reduces the solutal boundary layer thickness. Finally, we observe that the Sherwood number is significantly elevated with increasing values of the unsteadiness parameter.
Effects of Sc,Sr,Sϕ,δ,M, and fw on the reduced Sherwood number.
5. Conclusions
In this article, a mathematical model has been developed to simulate an unsteady two-dimensional magnetohydrodynamic flow of an incompressible electrically conducting fluid over a permeable stretching surface in the presence of multiple slip effects, Soret effect, and thermal radiation. Suitable similarity variables have been used to transform the boundary layer conservation equations into a nondimensional, coupled, and nonlinear system of ordinary differential equations. The resultant system of ordinary differential equations was then numerically solved using an implicit finite difference method with quasi-linearization technique. We also used special cases of the present model to verify our results with previous studies. A parametric study was performed to explore the effects of various governing parameters on the flow and heat and mass transfer characteristics. The following conclusions can be drawn from the present study:
Increasing the values of the magnetic field parameter, suction parameter, slip parameters, and unsteady parameter leads to the deceleration of the fluid velocity near the boundary layer region.
The velocity profiles are increased with increasing values of the injection parameter, buoyancy parameters, and thermal radiation.
The velocity profiles are found to be reduced with increasing values of the magnetic parameter, suction parameter, and slip velocity parameter.
Increments in thermal radiation, magnetic field parameter, and slip parameters yield enhancement in the fluid temperature. This in turn leads to rapid reductions in the heat transfer rate. However, the opposite effect is experienced with increasing values of thermal buoyancy parameter, Prandtl number, and unsteadiness parameter.
The skin friction coefficient increases with the increase of the magnetic field, Prandtl and Schmidt numbers, unsteadiness parameter, and suction parameter, but the effect is opposite for increasing values of slip parameters and buoyancy parameters.
The concentration profiles are enhanced by increasing values of the magnetic field but are reduced by increasing values of the Schmidt number and buoyancy and slip parameters.
Nomenclaturea:
Stretching rate
Bo:
Magnetic field of constant strength
Cf:
Local skin friction coefficient
cp:
Specific heat
C:
Concentration of the fluid
Cw:
Stretching sheet concentration
C0:
Reference concentration
C∞:
Ambient concentration
Cfr:
Reduced skin friction
DM:
Molecular diffusivity
DT:
Thermal diffusivity
fw:
Suction/injection parameter
k∗:
Mean absorption coefficient
M:
Magnetic parameter
Nur:
Reduced Nusselt number
Pr:
Generalized Prandtl number
Rex:
Local Reynolds number
Shr:
Reduced Sherwood number
Sc:
Schmidt number
Sr:
Soret number
Sf:
Velocity slip
Sθ:
Thermal slip
Sϕ:
Solutal slip
T:
Temperature of the fluid
Tw:
Stretching sheet temperature
T0:
Reference temperature
T∞:
Ambient temperature
u:
Velocity along the x direction
v:
Velocity along the y direction
x,y:
Coordinates
θη,ϕη:
Nondimensional temperature and concentration parameter
η:
Similarity variable
υ:
Kinematic viscosity
α:
Thermal diffusivity
βT:
Thermal expansion coefficient
βC:
Concentration expansion coefficient
τ:
Stress tensor
δ:
Unsteadiness parameter
λ:
Constant
ρ:
Density of the fluid
λ1:
Thermal buoyancy parameter
λ2:
Solutal buoyancy parameter
σ∗:
Stefan–Boltzmann constant.
Data Availability
No data used in this manuscript.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
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