**1. Introduction**

The most common fundamental type of flow through a channel is known as "Poiseuille flow." The Poiseuille flow has gained notable attention of various researchers due to its significant industrial applications. As an example, Siddiqui et al. [1] studied the plane Poiseuille flow with heat transfer. Alamri et al. [2] examined the plane Poiseuille flow with simultaneous effects of heat, magnetohydrodynamics (MHD) and second-order slip. They reported that the Stefan blowing prominently influenced on velocity and temperature distributions. Babic [3] has numerically investigated unsteady granular flows, namely transient Couette flow and cyclic Couette flow. In the first case velocity of the wall varies from one constant to another while in the second case the velocity is a harmonic function of time.

Moreover, non-Newtonian fluids have tremendous applications in the textile industry. Among the different models of non-Newtonian fluids, couple stress fluid has only lubricant viscosity. Consequently, in the absence of microstructure, couple stress in fluid arises which creates rotation without translation. Devakar et al. [4] investigated the couple stress fluid for three different cases. Ilani et al. [5] presented the unsteady nature of couple stress fluid between two parallel plates. Srinivasacharya et al. [6] discussed the laminar flow of couple stress fluid by means of quasi-linearization technique. Murthy and Nagaraju [7] conducted a study of couple stresses on the surface of a cylinder. The rotation of the

container generates the flow employing super adherence condition. All reported studies on couple stress eventually concluded that velocity always slows down in the fluid due to couple stresses by all means. Significant contributions on couple stress fluid can be seen in [8–10].

Furthermore, it is a well-known fact that most of the chemical and mechanical processes are of multiphase types. Consequently, several investigations have been performed up until now. In near past, different physical aspects of bi-phase flows were investigated by numerous researchers, such as Wu et al. [11], who discussed granular flow between opposite inclined plates for second-grade fluid containing spherical particles. Bognar et al. [12] offered flow analysis of non-Newtonian fluid on an oblique plane with material properties. Latz and Schmidt [13] presented numerical solutions for fast-moving and very slow-moving granular flows. Latz and Schmidt [14] provided numerical solutions for fast-moving and very slow-moving granular flows. The constitutive relations at small and intermediate densities were equivalent to those derived from the kinetic theory of granular flow which nevertheless recovers many aspects of dense granular flow. Two-phase fluids are inspected by Armanini [15]. The article provides full detail as to how granular fluid mechanics work. Interaction of solid–fluid for particulate flow with heat transfer is analyzed by Dan et al. [15]. Distributed Lagrange multipliers are used to obtain the expressions for velocity and temperature fields. The Boussinesq approximation is used for temperature and flow fields. The positioning of the particle is tracked by using the discrete element method.

In addition, the performance of lubricated coatings with magnetic, nanoparticles, heat transfer, and slip is very much ubiquitous in daily life. For instance, dish washing, replacement of lubricated cardiac valves, and industrial dye, as well as blood pressure control of a patient, are some common examples of slip and magnetization combination. Wang et al. [16] reported the effects of CeO2 nanoparticles on laser cladding of Ti-based ceramic coatings. Wang et al. [17] studied nanostructure with heating treatment on thin carbon films. Ellahi et al. [18] conducted a comparative study on shiny film coating on multi-fluid flows suspended with nano-sized particles. Khan et al. [19] have used double-layer optical fiber using Phan-Thien-Tanner fluid as a coating material. Lu et al. [20] inspected nonlinear thermal radiation and entropy optimization coatings with hybrid nanoliquid flow. Riaz et al. [21] proposed a model on mass transport peristaltic flow coated with Synovial fluid. Khan [22] has analyzed the effects of slip on MHD flow of a nanofluid in a vertical channel. Bhatti et al. [23] have investigated nanofluid influenced by externally applied magnetic fields. A new slip model is proposed by Zhu and Ye [24]. They used modeling approaches for submicrometer gas-phase heat conduction over a broad pressure range. Zhang et al. [25] rectified the classical second-order boundary condition for the fundamental flows. A list of core investigations on coatings [26,27], MHD [28,29], and nanoparticles [30–37] related to proposed is given for readers to get detail understating.

Unlike all the cited literature, this article addresses a biphasic flow that has yet not been reported. Much has been done with couple stress fluid; suspension of nanoparticles, slip, and magnetic fields, but the current innovative idea, which reflects the mechanism of industrial and geophysical multiphase flows is missed. Theory of couple stress fluid which is based upon the polarity of fluid entices to incorporate metallic particles that display high magnetic susceptibility. Therefore, no choice is left other than the suspension of Hafnium particles that fit the best. An additional contribution of lubrication and heating wall distinguish the present work by changing the morpho-hydrodynamics of bi-phase flow, which is, so far, a new and different prospect in the relevant field.

#### **2. Mathematical Analysis**

The particulate couple stress fluid containing Hafnium particles of spherical shape between two flat plates apart from each other at distance *h* is considered. Flow is generated by the constant pressure gradient, as shown in Figure 1.

**Figure 1.** Particulate flow through slippery plates.

### *2.1. Governing Equations*

The governing equations, such as continuity, momentum, and energy describing the particulate flow of couple stress, are given as:

• Conservation of mass

$$
\overrightarrow{\nabla}\overrightarrow{V} = 0\tag{1}
$$

• Conservation of momentum

$$
\rho \frac{d\vec{V}}{dt} = \nabla T + \frac{\mathbb{C}\mathbf{S}}{(1-\mathbb{C})} \left(\vec{V}\_p - \vec{V}\_f\right) + \frac{\rho \vec{f}}{(1-\mathbb{C})} \tag{2}
$$

where *T* denotes the Cauchy stress tensor and is defined by

$$T = -pI + s \tag{3}$$

where *I* is a unit tensor and *s* is an extra stress tensor. This can be obtained by the product of Rivlin–Ericksen tensor and coefficient of dynamic viscosity as follows:

$$s = \mu\_{\mathbb{S}} A\_1 \tag{4}$$

$$A\_1 = L + L^t \tag{5}$$

• Conservation of energy

$$
\rho\_f \text{(C}\_p\text{)} \frac{d\Theta}{dt} = k \,\nabla^2 V + TL \tag{6}
$$

The steady and laminar velocities flows in each phase is given by:

$$
\overrightarrow{\dot{V}}\_f = \begin{bmatrix} u\_f(\mathbf{x}, \mathbf{y}) & \mathbf{0} \ \mathbf{0} \end{bmatrix} \tag{7}
$$

$$
\overrightarrow{\dot{V}}\_{\mathcal{P}} = \begin{bmatrix} \mu\_{\mathcal{P}}(\mathbf{x}, \mathbf{y}) & \mathbf{0} & \mathbf{0} \end{bmatrix} \tag{8}
$$

The flow is under the simultaneous influences of transversely applied magnetic fields. Moreover, the plates transmit the heat into the system being thermally charged by an external source; consequently, temperature factor can be written as:

$$
\Theta = \begin{bmatrix} \Theta(y) \ 0 \ 0 \end{bmatrix} \tag{9}
$$

In view of Equations (7)–(9), the above governing equations (Equations (1)–(6)) take the following components forms:

$$0 = -\frac{\partial p}{\partial \mathbf{x}} + \mu\_s \left(\frac{\partial^2 u\_f}{\partial y^2}\right) - \eta\_1 \left(\frac{\partial^4 u\_f}{\partial y^4}\right) + \frac{CS}{(1-C)} \left(u\_p - u\_f\right) - \frac{\sigma B\_0^2}{(1-C)} u\_f \tag{10}$$

$$
\mu\_p = \mu\_f - \frac{1}{S} \left(\frac{\partial p}{\partial \mathbf{x}}\right) \tag{11}
$$

$$0 = \frac{\partial^2 \Theta}{\partial y^2} + \frac{\mu\_s}{k} \left(\frac{\partial u\_f}{\partial y}\right)^2 - \frac{\eta\_1}{k} \left(\frac{\partial u\_f}{\partial y}\right) \left(\frac{\partial^3 u\_f}{\partial y^3}\right) \tag{12}$$
