Investigation of generalized Fick's and Fourier's laws in the second-grade fluid flow
1 Introduction The fluid flow due to rotating disks have numerous
applications in many fields, e.g., computer storage devices, rotating machinery, gas or marine turbines, electronic devices, rotating disk reactors for biofuels production, rotating heat
exchangers, and medical equipment [1-14].
Karman [1] investigated the hydrodynamic flow by a rotating
infinite disk, and developed an appropriate transformation to
transform the governing PDEs into ODEs. Reddy et al. [2] studied
the heat and mass transport in the MHD flow of copper and silver
water due to rotating porous disks, and implemented the finite
element technique. Sheikholeslami et al. [3] explored the
nanofluid flow with rotating inclined disks. Turkyilmazoglu [4]
evaluated the flow due to rotating disks with nanomaterials. Hayat
et al. [5] investigated the viscoelastic nanomaterial convective
flow between two stretchable rotating disks. Imtiaz et al. [6]
presented the radiative flow of carbon-water nanomaterials between
two convectively heated rotating disks. Hayat et al. [7]
analyzed the probable error and statistical declaration for the
radiative two-phase flow submerged in Ag-H2O and Cu-H2O
nanoparticals. Takhar et al. [8] presented the steady
axisymetric flow between two porous disks.
The stratification in heat and mass transfer arises through the
variation of temperature, concentration differences, or the presence
of different densities. Double stratification involved various
practical applications, e.g., the thermal stratification of
reservoirs, oceans, rivers, and ground-water reservoirs, heterogenous mixtures, manufacturing processing, and the density
stratification of atmosphere. Srinivasacharya and Surender [15]
discussed the significance of doubly stratified laminar flow of a
viscous fluid with nanoparticles. Hayat et al. [16] explored the
double stratified nanomaterial flow with thixotropic materials in
presence of heat source/sink. Babu and Sandeep [17] examined the
doubly stratified melting flow in the rate type nanoliquid with heat
generation. Nadeem and Muhammad [18] addressed the non-Fourier
heat flux and double stratification in the rate type nanoliquid flow
over a stretched sheet. Waqas et al. [19] explored the
stratified flow of an Oldroyd-B nanofluid in presence of heat
generation.
The perception of heat transport is affective in technological and
industrial processes. These applications include energy production, atomic reactors cooling, power generation, etc. Due to this fact, many scientists and engineers are interested in analyzing the
mechanism of heat transport. Fourier [20] proposed the
well-known heat conduction law. Cattaneo [21] modified the
Fourier law by adding the time relaxation term. Christov [22]
replaced the time derivative by the upper convective derivative in
the theory proposed by Cattaneo [23], and names the new theory
as the Cattaneo-Christov heat flux theory (generalized Fourier's
law). Hayat et al. [24] examined the Cattaneo-Christov relation
for heat flux to explore the flow of rate type liquid over a
stretchable sheet. Shehzad et al. [25] examined the effects of
the non-Fourier heat flux and autocatalysis cubic chemical reaction on viscoelastic fluid flow. Some recent relative
contributions to the Cattaneo-Christov theory can be referred to
Refs. [26]-[28].
On the basis of the above discussion, in this paper, we aim to
discuss the flow of second-grade liquid due to a rotating porous
stretchable disk. Second-grade fluid is the principal subclass of
differential type fluids. The model proposed by Rivlin and
Ericksen [29] can predict the effects of normal stress. It
concentrates on the effects of the non-zero and unequal normal
stress differences in the shearing flows flagging way for their
applicability in various mechanisms such as die swell, rod-climbing, cooling of electronic devices, and solar energy collectors. The
Darcy relation is employed for a porous medium. The heat and mass
transfer are explored subject to Fourier's and Fick's laws. The
thermal and solutal stratifications at the surface are considered.
Appropriate transformations are used to obtain the nonlinear
systems. The homotopy analysis method (HAM) is utilized to obtained
the series solutions of the nonlinear systems [30-39].
The effects of involved variables on the temperature, velocities, concentration, skin frictions, and heat and mass transfer rates are
examined.
2 Formulation Here, the three-dimensional steady incompressible flow of the
second-grade fluid due to a rotating porous stretchable disk is
discussed. The flow is due to the linear stretching velocity. The
energy equation is discussed in the absence of thermal radiation and
viscous dissipation. The heat and mass transfer are described by
Fourier's and Fick's laws. The disk rotates at z=0 with the
angular speed ω and the stretching rate a1 (in the axial
direction, see Fig. 1). Solutal and thermal stratifications are accounted. The relevant expressions are as follows [10, 26, 40]:
|
(1) |
|
(2) |
|
(3) |
|
(4) |
|
(5) |
with
|
(6) |
where u, v, and w indicate the velocity components, ρ
indicates the fluid density, ν indicates the kinematic
viscosity, α1 indicates the normal stresses modulus, k*
indicates the porosity permeability, cp indicates the specific
heat, T indicates the temperature, C indicates the
concentration, q°; indicates the heat flux, and J°;
indicates the mass flux. With the Cattaneo-Christov theory, we can
express the heat and mass fluxes as follows:
|
(7) |
|
(8) |
where λE and λC represent the relaxation time of
heat and mass, respectively, k represents the thermal
conductivity, and DB represents the mass diffusion coefficient. For steady and incompressible flow, Eqs. (7) and (8) become
|
(9) |
|
(10) |
Eliminating q°; and J°; from Eqs. (4) and (5) and
Eqs. (9) and (10), we have the temperature and concentration
equations as follows:
|
(11) |
|
(12) |
We consider [14]
|
(13) |
Then, Eq. (1) is trivially verified, and we have
|
(14) |
|
(15) |
|
(16) |
|
(17) |
|
(18) |
where
|
(19) |
In the above equations, Re represents the Reynolds number, ϖ represents the material parameter, β represents the
porosity parameter, Pr represents the Prandtl number, Sc
represents the Schmidt number, A1 represents the disk stretching
parameter, γ1 represents the thermal relaxation, γ2
represents the concentration relaxation, S represents the thermal
stratification, and Z represents the solutal stratification.
Mathematically, the skin friction coefficients Cfr and
Cfϑ, heat transfer rate Nur, and mass transfer rate
Shr are defined as follows:
|
(20) |
where
|
(21) |
Substituting Eq. (21) into Eq. (20), we have
|
(22) |
where τzr and τϑ r indicate the shear
stresses, qw indicates the heat flux, Jw indicates the mass
flux, and 
indicates the local Reynolds
number.
3 Series solution The initial approximations for nonlinear expression and auxiliary
linear operators are
|
(23) |
with
|
(24) |
where bi (i=0, 1, ..., 8) denote the arbitrary constants.
4 Convergence The auxiliary variables ℏf, ℏg, ℏθ, and
ℏϕ have significant roles in controlling and adjusting the
convergence region. For the acceptable estimation of such variables, we have ℏ-curves at the 40th-order of approximations (see
Fig. 2 and Table 1). Figure 2 declares that, the convergence
intervals are as follows:
Table 1 Different orders of approximations when
Re=0.3, ϖ=0.1, Pr=1.5, A1=γ1=Z=0.4, β=0.2,
γ2=S=0.5, and Sc=1.0
Table 1 presents
that, the 25th-, 28th-, 22th-, and 30th-orders of deformations are
suitable for the convergence to the series solutions for f''(0),
g'(0), θ'(0), and ϕ'(0), respectively.
5 Discussion In this section, we intend to provide the approximate (series)
solutions of the resulting systems with the HAM. Here, the effects
of different variables on the velocity, concentration, temperature, coefficient of skin friction, temperature gradient, and concentration
gradient are physically focused on. Attention is particularly given
to the consequences of the Reynolds number Re, porosity parameter
β, material parameter ϖ, Prandtl number Pr, thermal
relaxation γ1, solutal relaxation γ2, stretching
parameter A1, thermal stratification S, and solutal
stratification Z.
5.1 Radial, axial, and tangential velocities Figures 3-13 describe the effects of different physical variables
on the radial velocity f'(η), axial velocity f(η), and
tangential velocity g(η).
Figures 3-5 show that the radial f'(η), axial f(η), and
tangential g(η) velocities decay for the Reynolds number Re.
From the figures, we can see that Re is an increasing function of
the angular speed ω, which is responsible for the decay in
the fluid velocities.
The effects of the material parameterϖ on the
radial f'(η), axial f(η), and tangential g(η)
velocities are portrayed in Figs. 6-8. Clearly, the velocities
(radial, axial, and tangential) increase with the material parameter
ϖ.
Figures 9-11 sketch the behaviors of the radial f'(η), axial
f(η) and tangential g(η) velocities via the porosity
parameter β. It is noticed that the velocities
f'(η), f(η), and g(η) decay with β.
Figures 12 and 13 are portrayed to show the effects of the parameter
A1 on the axial and radial velocities. For larger A1, the
axial and radial velocities are enhanced. This is due to the
increase in the stretching rate.
5.2 Temperature profile Figures 14-17 are displayed to interpret the effects of different
parameters on the temperature field θ(η).
Figure 14 is plotted to discuss θ(η) for Pr. From the figure, we can see that the temperature is
a decreasing function of Pr. When Pr increases, the thermal
conductivity weakens, and therefore θ(η) decays.
Figure 15 is sketched to show the effects of the Reynolds number
Re on θ(η). As can be seen in the figure, when the
Reynolds number increases, less temperature field can be noticed.
Figure 16 shows the effects of γ1 on the temperature field.
From the figure, we can see that an augmentation in γ1
indicates a decrease in θ(η). Physically, the material particles need much time to transfer heat to its contiguous
particles. Thus, θ(η) is larger for γ1=0 in case
of Fourier's law when compared with non-Fourier's heat flux model.
Figure 17 depicts the behavior of the temperature field via the
thermal stratification. From the figure, we can see that temperature
is a decreasing function of the thermal stratification parameter.
Physically, the difference between the surface temperature and the
ambient temperature decreases gradually. As a result, the
temperature decreases.
5.3 Concentration profile Figures 18-21 are sketched to analyze the effects of the Reynolds
number, Schmidt number, concentration relaxation, and solutal
stratification on the concentration field.
The effects of the
Reynolds number Re on the concentration field ϕ(η) is
shown in Fig. 18. From the figure, we can see that the
concentration decays for the Reynolds number.
Figure 19 is sketched for the concentration field via the Schmidt
number. It shows a decreasing behavior of concentration for the
Schmidt number. Since the Schmidt number is the ratio of momentum
to mass diffusivity and the mass diffusivity decays for larger
Schmidt numbers, the concentration field decays when the Schmidt
number increases.
Figures 20 and 21 show the effects of the concentration relaxation
and solutal stratifications on ϕ(η). From the figures, we
can see that when the concentration relaxation variable
γ2 increases, the concentration field decays. A similar
behavior is observed for the solutal stratification variable (see
Fig. 21). This is because that the concentration difference
diminishes between the surface and the ambient fluid.
5.4 Velocity, temperature, and concentration gradients Figures 22-31 are portrayed to reveal the effects of different
involved variables, e.g., the Reynolds number, material parameter, porosity parameter, Prandtl number, thermal relaxation, thermal
stratification, Schmidt number, concentration relaxation, and
solutal stratification on the skin friction coefficients and heat and mass transfer
rates. From the figure, we can see that, the surface drag force is
an increasing function of the Reynolds number, porosity parameter, and material parameter. Similarly, the heat transfer rate enhances
for larger Reynolds numbers, Prandtl numbers, and thermal
relaxation variables while decreases for fixed values of the thermal
stratification variable. Moreover, the mass transfer rate increases
for larger Schmidt numbers and solutal stratification and
concentration relaxation variables.
6 Concluding remarks The second-grade fluid flow due to a rotating stretchable disk in
a porous medium is investigated. The main concluding remarks are
listed as follows:
(ⅰ) The radial velocity increases with ϖ and A1 while
decays with Re and β.
(ⅱ) The axial velocity boosts via Re, ϖ, and A1
whereas decays with β.
(ⅲ) The tangential velocity enhances for larger ϖ while
reduces for larger Re and β.
(ⅳ) The elevation in Pr, Re, γ1, and S correspond to
smaller temperature.
(ⅴ) Concentration decays with Re, Sc, γ2, and Z.
(ⅵ) The temperature gradient enhances for larger ϖ and
β.
(ⅶ) The velocity gradient is higher when Re, Pr, and
γ1 are larger, while is smaller when S is larger.
(ⅷ) The Sherwood number boosts via Re, Sc, γ2, and
Z.
2018, Vol. 39
