1. Introduction
Recent technological advancements have renewed interest in non-Newtonian fluid flows. These fluids play an important role in many situations. Engineers, mathematicians, modelers, physicists, and computer scientists have been interested in the flows of these fluids in recent years due to their applications in industry and technology. In addition to the Navier–Stokes equations, the governing equations of non-Newtonian fluids are of a higher order and highly nonlinear, which are presented in [
1,
2,
3]. Non-Newtonian fluid flows have been the subject of even more research. Micropolar fluids differ from non-Newtonian fluids in that they exhibit microscopic characteristics such as micro-rotational and rotational inertia. The theory of micropolar fluids was initially suggested by [
4]. Micropolar fluids are essential for the movement of colloidal suspensions, liquid crystals, additive-containing fluids, suspension solutions, animal blood, and other fluids. Ariman T [
5] discusses the application of micro-continuum fluid mechanics. Ezzat [
6] and Helmy [
7] presented some uses of the magnetohydrodynamic (MHD) boundary layer flow. Raees [
8] and Jena [
9] incorporate the applications of the Free Convection Flow of a Micropolar Fluid from a Vertical Flat Plate. The application of stagnation point flow of micropolar fluid flow was investigated by Guram [
10] and Nazar [
11]. Similarly, Ahmadi [
12] and Takhar [
13] have investigated the flow of micropolar fluids in various configurations to accomplish such applications. Recently, Ramesh et al. [
14] have investigated the time-dependent Casson-micropolar nanofluid squeezing flow with suction/blowing and slip effects. By contrast, Kumbinarasaiah et al. [
15] used the Hermite wavelet approach to solve coupled nonlinear differential equations occurring in a rotating micropolar nanofluid flow system. Later on, Shamshuddin et al. [
16] demonstrated the influences of different aspects of micropolar nanofluids such as a chemically reactive Casson fluid treated to a rotatory system with electrical and Hall currents between parallel plates. Hussain et al. [
17] investigated multiple slip effects in a biothermal MHD convection flow of a micropolar nanofluid between two parallel discs, while Sastry et al. [
18] incorporated the application to cardiovascular problems of unsteady 3D micropolar nanofluid flow through a squeezing channel.
Squeezing flows are also common in hydrodynamic machines, polymer processing, compression and injection molding models, lubrication processes, and polymer processing procedures, among other things. The time-dependent squeezed flow is frequently addressed in industrial situations to characterize fluid movement along a prescribed length of contracting area. A preliminary sluggish liquid may be placed between a pair of rectangular parallel discs and planes to show the associated mathematical models upon such planes of coordinates, and this flow can be set up. Usha [
19] found an exact solution to a similar problem in a couple of elliptic plates using a multifold time-dependent series. The time-dependent viscous squeezing flow between two parallel plates with a constant temperature was investigated by [
20]. Grimm [
21] looked at the squeezing flows of Newtonian liquid plates.
Furthermore, in the last two decades engineers have turned their attention to a new form of fluid known as “nanofluid.” Nanofluids are fluids that include nanometer-sized particles in a suspension, also known as nanoparticles. Nanomaterials are important because of their high thermal and mechanical properties. The properties of the nanoliquids created by each nanomaterial are considerably altered by these materials. Among nanomaterials, there is a substance known as aluminum alloys, in which aluminum plays a significant role. Aluminum alloys are divided into two categories: heat-treatable and non-heat-treatable. Some of the most commonly used nanofluids are water
, ethyl glycol
and oil. Choi [
22] was the first to demonstrate that putting these nanoparticles into a base fluid enhanced its thermal conductivity. Buongiorno [
23] proposed a mathematical model to emphasize the significant impacts of the Brownian motion and thermophoretic diffusion of nanoparticles.
Soundings into the flow dynamics of
nanofluids have revealed their importance in cooling operations by Popa [
24], T.M.O. Sow [
25], and H. Beiki [
26]. To cool an engine, Moghaieb [
27] employed an
nanofluid. Ganesh [
28] and Rashidi [
29] demonstrated that similarity solutions improved the flow properties of
nanofluids over stretchy surfaces. They used experimentally-based thermophysical properties in their studies to ensure the physical pertinence of gamma alumina nanoparticles.
A careful review of the literature finds that no one has studied the effects of an experimentally based Prandtl number model on the squeezing flow of micropolar fluids flow of (gamma alumina and water) and (gamma alumina and ethyl glycol) across parallel plates, to the author’s knowledge. In light of this, we investigate the impact of an effective Prandtl number model on the squeezing flow of nanoparticle micropolar fluid between parallel planes. As a result, the squeezing flow of (gamma alumina and water) and (gamma alumina and ethyl glycol) between two parallel planes is explored in this study under the impact of the effective Prandtl number. The analytical solutions generated are being utilized to investigate the influence of various factors on the squeezing flow between two parallel planes.
2. Mathematical Analysis
Consider the two-dimensional axisymmetric flow of incompressible nanofluid between two parallel planes with vertical magnetic fields proportional to
. As for low Reynold numbers, the magnetic field is neglected. It contains nanoparticles such as aluminum oxide
and gamma aluminium oxide
in base fluids such as water
and ethylene glycol
. The lower plane is at Z = 0, and the upper plane is separated by
As shown in the
Figure 1, the upper plane moves toward or away from the lower plane at a velocity of
.
The following are the governing equations for two-dimensional axisymmetric and unsteady fluid flow, as well as heat transfer in viscous fluids:
Subject to auxiliary conditions [
30]
The velocity components in the
and
directions are represented by
and
, respectively, while the azimuthal component of micro-rotation is represented by
. The concentration of nanoparticles in the fluid is denoted by
.
and
represent nanoparticle concentrations, while
and
represent constant temperature at the lower and upper planes, respectively.
is the effective density,
is efficient thermal diffusion,
is dynamic viscosity of nanofluids,
is vortex viscosity,
is kinematic viscosity,
indicates pressure, and
represents specific heat capacitance. Furthermore,
is the mass diffusion coefficient,
is the Brownian motion coefficient, and
is the thermophoretic diffusion coefficient.
is temperature, and
is the mean fluid temperature. Furthermore,
represents a non-dimensional parameter, which is the ratio of the nanoparticle’s effective heat capacity to the fluid’s heat capacity.
is assumed to be constant and represents wall injection and suction velocity.
is assumed to be constant and
, where
represents the refrence length.
is the effective electrical conductivity of nano liquids. Moreover, the effective density
and dynamic viscosity
of the nanofluid [
24] are as follows:
Nanofluid’s effective Prandtl number [
30,
31] is demarcated as
Nanofluid’s effective electrical conductivity is given as
The thermophysical characreristics of nanoparticles are painted in
Table 1 [
31].
Under the following similarity transformation [
32] to reduce the constitutive Equations (2)–(6) as:
2.1. Model of
Using similarity transformations (15), we get transform equations for
where,
2.2. Model of
Using similarity transformations (15), we get transform equations for
where,
The non-dimensionalized boundary conditions are
The following non-dimensionized parameters are used in the flow model.
where
is Micropolar parameter,
is Magnetic parameter (Hartmann number),
is Squeezed Reynolds number,
is Lewis number,
is Thermophoresis parameter,
is Brownian motion parameter
is Prandtl number, and
is suction or injuction parameter.
The skin friction coefficient
, local nusselt number
, and sherwood number
at both upper and lower disks are demarcated [
30] as
where,
The transformed form of
,
for the model
is
and for the model
is
where
is local squeezed Reynolds number.
5. Numerical Results
To validate our present study, the impact of different emerging parameters such as Hartmann number , microrotation parameter and local squeezed Reynold number on shear stresses, rate of heat transfer, and mass transfer are portrayed in tabulated form.
Table 2 and
Table 3 illustrate how different emerging parameters of the model
affect the skin friction coefficient in upper and lower planes with strong
and weak
. interactions. When
, increasing Hartmann number M in the upper plane causes the skin friction coefficient to rise, but increasing Hartmann number M in the top plane causes the skin friction coefficient to fall. However, for both strong and weak connections, opposing behaviors are observed at the lower and higher planes for
. As the local compressed Reynolds number
Rr increases, the skin friction coefficient in the upper and lower planes decreases, resulting in both weak and strong interactions for
and
. The influence of the micropolar parameter
K on the skin friction coefficient in both the upper and lower planes is equivalent to the Hartmann number
M. As a result, for both strong and weak contacts, the skin friction coefficient at the top plane increases with increasing values of
and
for S > 0, but the opposite phenomenon occurs for rising values of
and
for
. Similarly,
Table 4 and
Table 5 report on gamma alumina with ethyl glycol
which we have observed for gamma alumina with water
The influence of the local Nusselt number
on the model of Gamma Alumina with Water
under the influence of physical parameters with strong and weak contacts at the upper and lower disks is explored in
Table 6 and
Table 7. For rising values of local squeezed Reynold number, in the presence of strong and weak contacts, the rate of heat transfer increases at the upper disc but decreases at the lower disc for both suction and injection. When the Brownian motion parameter is present, the rate of heat transfer decreases as the Brownian motion
at the upper disc grows for both strong and weak contacts at a lower disc. However, for both
and
, the local Nusselt number
is proportional to the thermophoresis parameter
in the upper disc. In both circumstances of suction and injection, the inverse proportionality is observed at the lower discs. Similarly, in
Table 8 and
Table 9, the same phenomenon occurred for
.
Table 10 and
Table 11 indicate the effect of critical flow parameters on the local Sherwood number
at the upper and lower discs for the model
in the presence of weak and strong interactions. For
, the rate of mass transfer at the upper plane rises as
increases, whereas for both weak and strong contacts, the rate of mass transfer at the bottom plane decreases. However, the opposite behavior is expected for
. The mass transfer rate is increased when there is an increase in Brownian motion parameter
only at the upper plane, while at the lower plane the rate of mass transfer decreases for
. Also, it decreases at both planes for
. Similarly, when increases occur in the thermophoresis parameter
, decreasing mass transfer occurs at the upper plane at both suction and injection, while the opposite behavior occurs at the lower plane at both suction and injection for both weak and strong interaction, respectively. However, for
, the local Sherwood number
is directly proportional to the Lewis number Le, but for
, it is inversely proportional.
Table 12 and
Table 13 indicate the effect of critical flow parameters on the local Sherwood number
at the upper and lower planes for the model
at both strong and weak contacts. For both strong and weak contacts, this model is the same as the above model for
in both upper and lower planes. By contrast, for
only the squeezed Reynolds number and Brownian motion parameters differ from the above model, indicating that as
increases in the upper plane, it decreases in the lower plane. When
increases, the Sherwood number also increases in the upper plane while it decreases in the lower plane.