Analytical solution of thermal effect on unsteady visco-elastic dusty fluid between two parallel plates in the presence of different pressure gradients

Thermal diffusion of dusty fluids has valuable interference in various fields, including waste-water treatment, oil transportation, and power plant pipes. Dusty fluids are used in lots of industrial fields as a result of their improved heat transfer and heat management capabilities. These industries range from renewable energy systems to aerobic plastic sheet extrusion, manufacturing, and rolling and reaching metal sheet cooling. The work embodied in this paper presents the analytical solution performed to predict the effects of thermal diffusion on dusty, viscous, incompressible fluid flows between two porous, parallel vertical plates with a heat source or a heat sink. The mathematical equations are solved by the separation of variables and Laplace transform techniques. The influence of temperature is investigated for various values of Prandtl number and heat source or heat sink parameters. Also, the influences of various coefficients like the thermal diffusion coefficient, Schmidt number, Prandtl number, and heat source or heat sink coefficient on the concentration are observed. The fluid velocity distribution is graphically obtained. The solutions are discussed and exhibited graphically. The influences of the thermal diffusion parameter and chemical reaction parameter on fluid and dust particles’ velocities are examined. A parametric study on the effect of time on temperature and concentration is made. The exact expressions for temperature, concentration, and velocity variation for fluid and dusty particles are obtained analytically. The temperature is inversely proportional to both the Prandtl number Pr and the heat source or heat sink parameter Hs\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_s$$\end{document}. The concentration of the fluid is inversely proportional to the thermal diffusion parameter Td and the heat source or heat sink parameter Hs\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_s$$\end{document}.


Background
Thermal diffusion of dusty fluids has always been a point of interest to researchers as well as practitioners owing to its valuable interference in various fields, including wastewater treatment, oil transportation, and power plant pipes.Dusty fluids have paved their way through lots of industrial fields as a result of their improved heat transfer and heat management capabilities; these industries range from renewable energy systems to aerobic plastic sheet extrusion, manufacturing, and rolling metal sheet cooling.Nidhi et al. [1] have studied the unsteady magnetohydrodynamics (MHD) Walter's-B viscoelastic dust effect of vibration temperature on fluids passing through inclined porous plates with thermal radiation.
Farhad et al. [2] investigated the effects of a magnetic field paired with heat transfer on conductive, viscoelastic, incompressible, and dusty fluids moving between two non-conductive inflexible plates.Radhika [3] and [4] have studied the effect of heat transfer on dusty liquids with hybrid nanoparticles floating on the molten surface.Bilal et al. [5] discussed the Couette flow of dusty viscoelastic fluid in a rotating frame with heat transfer.Mallikarjuna et al. [6] investigated the effect of radiation and thermal diffusion on the MHD heat transfer flow of a dusty, viscous, incompressible conductive fluid between two parallel plates.Govindarajan et al. [7] studied the combined impacts of mass and heat transmission on a radiant MHD oscillating thin dusty fluid in a fully saturated permeable channel.
The study of chemical reactions and thermal diffusion has many applications, including liquid metal cooling of nuclear processes, sustained plasma confinement for controlled thermonuclear fusion, and electromagnetic metal casting.K. Suneeth et al. [8] have studied the effect of thermal radiation on the flow of MHD viscoelastic fluid through a porous movable plate with a primary chemical reaction.Debasish Dey [9] made a numerical model to study and analyze the dusty fluid flowing through a vertical surface with heat-generating and endothermic-type chemical reactions.Prasanthi et al. [10] have investigated the upshots of chemical reactions and radiations on MHD dusty flow over a sloping porous sheet immersed in a porous medium.
Furthermore, it is mentioned that other researchers studied the influence of pressure gradients on dusty fluid flow.Kanaka Lata Ojha et al. [11] have studied the influence of sinusoidal gradients of pressure upon twodimensional unsteady viscoelastic hydromagnetic currents through channels lined by porous materials.Dash et al. [12] have studied the effect of a sinusoidal pressure gradient on viscoelastic hydromagnetic flow between two porous parallel plates.MHD's dusty fluid flow has witnessed heavy research and has grabbed researchers' attention for quite some time.Jadav Konch [13] has developed computer codes to study outcomes at the unsteady glide of a viscous dusty fluid beyond an exponentially increased vertical plate with viscous dissipation within the existence of a warmth supply and magnetic field.Bilal [14] has numerically discussed the influence of Hall currents on the unsteady rotating current of carbon nanotubes in a permeable Darcy-Forchheimer media with dust grains and nonlinear heat radiation.Dawar [15] has investigated the Hall effect in dual-phase radiative dusty nanofluid flow across a stretched sheet.Neetu Singh et al. [16]] have investigated the effect of inertia on dusty fluid in a permeable medium.Chitra et al. [17] have studied the flow of an unsteady dusty fluid across an impermeable media in a circular pipe under the influence of a magnetic field and a time-varying pressure gradient with slip conditions.Sudhir Kumar [18] has studied, using linearized theory, the influences of dust particles, rotation, couple stress, and magnetization on a thin film of couple stress magnetized fluid's thermal stability.Sasikala [19] focused on adding dust particles between two parallel plates passing through a permeable media in the existence of a magnetic field.He observed that the upper plate has a uniform suction force, while the bottom plate has a constant injection.Farhad Ali et al. [20] have discussed the fluctuating natural convection of heat-absorbing viscoelastic dust liquid in horizontal channels with the MHD infusion.
Analytical methods are often the preferred approach for solving partial differential equations (PDEs) [21] when it is possible to do so.This is because analytical solutions are often more accurate than numerical solutions, and they can be more easily generalized to other problems [4,22].Some of the most common analytical methods for solving PDEs include separation of variables [23], Fourier transforms [24], Laplace transforms, and Green's functions.Marwan Al-Raeei [24] derived the bulk modulus relationship for the Morse interaction, by using the Fourier transformation and the mean-spherical approximation.
The purpose of this investigation is to derive an analytical solution by using the separation of variables method to discuss the effect of heat on the unsteady, dusty viscoelastic fluid in the presence of different pressure gradients.The problem is formulated and analytically solved, and the relevant results are discussed in depth graphically to explore the influence of various fluid parameters.

Governing equations
The problem considered is a flow of an incompressible viscous dusty fluid between two infinite plates separated by a distance 2h without body force as illustrated in Fig. 1 based on the aforementioned assumptions • The flow is unsteady, laminar, and subjected to pressure gradient varying over time the plate movement • The flow passes through two parallel plates that are vertically heated and have a heat source or sink.• In the beginning, the fluid and dust particle clouds will both be expected to be stationary.• The dust particles are considered to be homogeneous in size with a spherical shape.• The dust particle number density is assumed to be uniform.• The flow in a porous medium.
The governing equations [25] are: (1) The initial and boundary condition are where u-fluid velocity (m/s), t-time (s), ρ-fluid den- sity (kg/m 3 ) , P-pressure of the fluid (N/m 2 ) , x-axis of co-ordinates oriented along the flow direction.(m), ggravity acceleration , -the diffusing particles' mean free path, y-normal to the plate coordinate axis (m), k-nondimensional chemical reaction coefficient, N-the particle density number of the dust, v-dust particles velocity (m/s), m-mass of the dust particles (kg), C w -concentra- tion at the wall, and T w -temperature at the wall Considering the following dimensionless quantities (5) and The governing equations are reduced to The initial and boundary conditions are given by where Grashof number

Starting with the homogenous differential equation for temperature it is subjected to non-homogenous boundary conditions
To solve this differential equation by using the method of separation of variables, which is one of the most popular methods for solving partial differential equations, and assuming that the solution is separable, that is, the final solution can be formulated as a product of different (7) functions, each of which is only dependent on a single independent variable [23].We first transform the nonhomogenous boundary conditions into homogeneous ones by assuming that Substituting Eq. 14 in Eq. 12 we get where Equation 15 is a homogenous partial differential equation with homogenous boundary conditions that will be analyzed via the method of separation of variables.We can firstly solve the homogenous one assume by substituting Eq. 18 in Eq. 17 we get and ( 14) The solution of Eq. 22 is This function is the eigen function for solving the nonhomogeneous partial differential equation Then the solution is Substitute Eq. 31 in Eq. 30 we can get (24) Then by addition, we get 2l 1 cos = 0 where By orthogonality For (32) Where Page 6 of 14 Substituting Eq. 42 in 10 to get To solve Eq. 43 by separation of variables method we first transform the non-homogenous boundary conditions to homogenous one.Let C(y, t) = 1 − e −at cosh (m2y) which can be written as By orthogonality, we can get (45) where Solve Eq. 8 for u in to get Substitute Eq. 51 into Eq.8 to get by substituting v(y, t) = ∞ n=1 T nv (t) cos 2n−1 2 π y From Eq. 54 by using orthogonality (50) Consider the homogenous one The auxiliary equation and its roots are For non-homogenous one, the particular solution is (55 Where (59) γ 11 + γ 12 e −at + γ 13 e −αt +γ 14 e −α 1 t − γ 15 P(t) Case 1: P(t) = a 0 e −a 1 t a 0 , a 1 are constants.The expo- nential pressure variation arises because of the no-slip boundary condition, which states that the fluid sticks to the plates and has zero velocity at the plates' surface.As a result, fluid particles near the plates experience a higher drag force than those far away from the plates.This difference in drag force creates a pressure gradient that drives the fluid flow.The particular solution is The general solution is where (60) T nv (t) = γ 16 + γ 17 e −at + γ 18 e −∝t + γ 19 e −α 1 t − γ 20 e −a 1 t (61) T nv =C 1 e m 1 t + C 2 e m 2 t + γ 16 + γ 17 e −at + γ 18 e −αt + γ 19 e −α 1 t − γ 20 e −a 1 t (62) Case 2: P(t) = a 3 + a 4 t , a 3 , a 4 are constants.This case describes a uniform pressure gradient along the flow direction.The linear pressure variation arises when the fluid flow is in the fully developed region, i.e., when the velocity distribution has evolved completely and is maintained constant along the flow direction.In this case, the pressure drop proportionates to the distance between the plates, and the pressure variation is linear. (63) By applying the boundary conditions at t = 0 T = 0, calculation of flow rate for a constant pressure gradient is easier than for an exponential pressure gradient.As opposed to that, the exponential pressure gradient is more realistic than the constant pressure gradient.The exponential pressure gradient can lead to more stable flow conditions than the constant pressure gradient.The physical quantities for engineering applications, local Nusselt number Nu = −(δT /δy) y=0 , and local Sherwood number Sh = −(δC/δy) y=0 are defined as [26].The current results are compared with the previously published available results for different cases, which obtained numerically by Abass et al. [26], in order to investigate the validity of the current results and procedures.The values are shown in Table 1.Nusselt and Sherwood numbers have been compared for various parameters.The current results showed good agreement.
The analytical results for temperature, concentration, fluid velocity, and dust velocity showed good agreement with Madhura and Kalpana [25] as shown in Tables 2 and 3      The proposed analytical method will be used to study more complex problems that include chemical, ion slip, and magnetohydrodynamic effects on dusty fluid flow with mass and heat transfer in a porous media.

Fig. 1
Fig. 1 Geometry of the flow the fluid viscosity µ = γρ , and chemical reaction parameter M = K h 2

Figures 2 and 3
Figures2 and 3show the influence of Prandtl number P r and the heat source or heat sink coefficient H s on tem- perature.It has been noted that the more temperature increase, the more Pr and H s decrease.Figures4 and 5confirm that the concentration of the fluid increases when the thermal diffusion parameter Td increases and the Schmidt number Sc decreases, respectively.Figure6confirms that the concentration of the fluid increases when the heat source or heat sink coefficient H s decreases.Figure7confirms that the concentration of the fluid increases when the Prandtl number Pr increases.Figures 8, 9, 10, and 11 represent the fluid and dust velocities profiles.These figures confirm that fluid and dust velocities increase as Td increases.Figures 12, 13, 14, and 15 show that the fluid and dust velocities increase when Cr decreases.Figures 16 and 17 represent the influence of time on the temperature and the concentration,

Table 1
Comparison of Nusselt number and local Sherwood number

Table 2
Comparisons of temperature and concentration

Table 3
Comparisons of fluid and dust velocities