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

Abstract

Background

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.

Results

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.

Conclusions

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 $H_s$. The concentration of the fluid is inversely proportional to the thermal diffusion parameter Td and the heat source or heat sink parameter $H_s$.

1 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 two-dimensional 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 visco-elastic 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.

2 Method

2.1 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

Fig. 1

Geometry of the flow

• 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:

$$\begin{aligned} \frac{\partial u}{\partial t}= & {} -\frac{1}{\rho } \frac{\partial P}{\partial x}+g \beta \left( T-T_0\right) +g \beta \left( C-C_0\right) +\gamma \left( 1-\lambda \frac{\partial }{\partial t}\right) \frac{\partial ^2 u}{\partial y^2}+\frac{k N}{\gamma }(v-u)-\frac{\gamma }{k} u \end{aligned}$$

(1)

$$\begin{aligned} \frac{\partial v}{\partial t}= & {} \frac{k}{m}(u-v) \end{aligned}$$

(2)

$$\begin{aligned} \frac{\partial T}{\partial t}= & {} \frac{K_T}{\rho C_p} \frac{\partial ^2 T}{\partial y^2}-\frac{Q_1}{\rho C_p}\left( T-T_0\right) \end{aligned}$$

(3)

$$\begin{aligned} \frac{\partial C}{\partial t}= & {} D \frac{\partial ^2 C}{\partial y^2}-\alpha \left( C-C_0\right) +D_T \frac{\partial ^2 T}{\partial y^2} \end{aligned}$$

(4)

The initial and boundary condition are

$$\begin{aligned}&t=0, u=v=0, T=T_0,\textrm{C}=\textrm{C}_{\textrm{o}}, \quad -\textrm{h}<y<\textrm{h} \\&t>0, u=v=0,=T \textrm{o}+\left( T_{\textrm{w}}-T_{\textrm{o}}\right) \left( 1-e^{-a t}\right) , and \\&C=\textrm{Co}+(\textrm{Cw}-\textrm{Co})\left( 1-e^{-a t}\right) , \quad y=\pm h \end{aligned}$$

(5)

where u—fluid velocity (m/s), t—time (s), $\rho$—fluid density $(\mathrm{kg/m}^3)$, P—pressure of the fluid $(\mathrm{N/m}^2)$, x-axis of co-ordinates oriented along the flow direction. (m), g—gravity acceleration $(\mathrm{m/s}^2)$, $\beta$—coefficient of volumetric thermal expansion $(1/^{\circ } C)$, T—fluid temperature $(^{\circ } C)$, $T_{o}$—initial temperature $(^{\circ } C)$, C fluid concentration $(\mathrm{kg/m}^3)$, $C_0$—initial uniform concentration at $T_0$, $\gamma$—kinematic viscosity $(\mathrm{N s/m}^2)$, $\lambda$—the diffusing particles’ mean free path, y—normal to the plate coordinate axis (m), k—non-dimensional chemical reaction coefficient, N—the particle density number of the dust, v—dust particles velocity (m/s), m—mass of the dust particles (kg), $K_{\textrm{T}}-$ the fluid’s thermal conductivity coefficient (W m$^{-1}$ K$^{-1})$ $C_{\textrm{P}-}$ specific heat at constant pressure (J/kg K), $Q_1$—volumetric heat generation or absorption rate (W/m$^2)$, D—rate of mass diffusion (m$^2$/s), $\alpha$—chemical reaction coefficient, $D_{\textrm{T}}$—thermal diffusion coefficient (m$^2$/s). $C_w$—concentration at the wall, and $T_w$—temperature at the wall

Considering the following dimensionless quantities

$$\begin{aligned} P^*=\frac{P}{\gamma \rho }, \quad y^*=\frac{y}{h}, x^*=\frac{x}{h} \\ u^*=\frac{u}{h}, \quad v^*=\frac{v}{h}, t^*=\frac{\gamma t}{h^2} \\ a^*=\frac{a h^2}{\gamma }, \quad T^*=\frac{T_0-T}{T_0-T_w}, C^*=\frac{C_0-C}{C_0-C_w} \end{aligned}$$

(6)

The governing equations are reduced to

$$\begin{aligned}{} & {} \frac{\partial u}{\partial t}=-\frac{\partial P}{\partial x}+G_r T+G_m C+\gamma \left( 1-V_e \frac{\partial }{\partial t}\right) \frac{\partial ^2 u}{\partial y^2}+\frac{1}{\tau }(v-u)-\frac{u}{M} \end{aligned}$$

(7)

$$\begin{aligned}{} & {} \quad \frac{\partial v}{\partial t}=\frac{1}{r}(u-v) \end{aligned}$$

(8)

$$\begin{aligned}{} & {} \quad \frac{\partial ^2 T}{\partial y^2}-P_r \frac{\partial T}{\partial t}-H_s T=0 \end{aligned}$$

(9)

$$\begin{aligned}{} & {} \quad \frac{\partial ^2 C}{\partial y^2}-S_c \frac{\partial C}{\partial t}-C_r S_c C+T_d \frac{\partial ^2 T}{\partial y^2}=0 \end{aligned}$$

(10)

The initial and boundary conditions are given by

$$\begin{aligned} &t=0, u=v=T=C=0,-\textrm{1}<y<\textrm{1} \\&t>0, u=v=0, T=\left( 1-e^{-a t}\right) \text{ and } C=\left( 1-e^{-a t}\right) , y=\pm 1 \end{aligned}$$

(11)

where Grashof number $(1/^{\circ }$K) $\quad G_r=g \beta h \frac{T_w-T_0}{\gamma }$, modified Grashof number $(1/^{\circ }$K) $\quad G_m=g \beta h \frac{C_w-C_0}{\gamma }$, visco-elastic parameter $\quad V_e=\frac{\gamma \lambda }{h^2}$, mass concentration of dust particle g/m$^3$ $\quad l=\frac{\gamma m}{k h^2}$, Prandtl number $\quad P_r=\mu \frac{C_P}{K_T}$, source or sink parameter for heat $\quad H_s=\frac{Q_1}{K_T} h^2$, Schmidt number $\quad S_c=\frac{\gamma }{D}$, dimensionless parameter of a chemical reaction $\quad C_r=\frac{\alpha h^2}{\gamma }$, parameter of thermal diffusion $\quad T_d=\frac{D_T}{D}\left( \frac{T_W-T_0}{C_W-C_0}\right)$, the fluid viscosity $\mu =\gamma \rho$, and chemical reaction parameter $\quad M=\frac{K}{h^2}$

2.2 Analytical solution

Starting with the homogenous differential equation for temperature

$$\begin{aligned} \frac{\partial ^2 T(y, t)}{\partial y^2}-P_r \frac{\partial T(y, t)}{\partial t}-H_s T(y, t)=0 \end{aligned}$$

(12)

it is subjected to non-homogenous boundary conditions

$$\begin{aligned} &t=0, T=0, y \epsilon (-1,1) \\&t>0, T=\left( 1-e^{-a t}\right) , y=\pm 1 \end{aligned}$$

(13)

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 functions, each of which is only dependent on a single independent variable  [23]. We first transform the non-homogenous boundary conditions into homogeneous ones by assuming that

$$\begin{aligned} T(y, t)=\left( 1-e^{-a t}\right) \frac{\cosh \left( m_1 y\right) }{\cosh \left( m_1\right) }+T^*(y, t) \end{aligned}$$

(14)

Substituting Eq. 14 in Eq. 12 we get

$$\begin{aligned} \begin{array}{rcc} \frac{\partial ^2 T^*(y, t)}{\partial y^2}-P_r \frac{\partial T^*(y, t)}{\partial t}-H_s &{} T^*(y, t)=F(y, t) \\ \text{ at } t=0 &{} T^*=0 &{} y \in (-1,1) \\ \text{ at } t>0 &{} T^*=0 &{} y=\pm 1 \end{array} \end{aligned}$$

(15)

where

$$\begin{aligned} F(y, t)=\left( \mathrm {\gamma }_3+\mathrm {\gamma }_2 e^{-a t}\right) \frac{\cosh \left( m_1 y\right) }{\cosh \left( m_1\right) }, \mathrm {\gamma }_3=\left( H_s-m_1^2\right) , \mathrm {\gamma }_2=\left( P_r a-\mathrm {\gamma }_3\right) \end{aligned}$$

(16)

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

$$\begin{aligned} \frac{\partial ^2 T^*(y, t)}{\partial y^2}-P_r \frac{\partial T^*(y, t)}{\partial t}-H_s T^*(y, t)=0 \end{aligned}$$

(17)

assume

$$\begin{aligned} T^*(y, t)=T^*(t) T^*(y) \end{aligned}$$

(18)

by substituting Eq. 18 in Eq. 17 we get

$$\begin{aligned}{} & {} T^*(t) T^{* \prime \prime }(y)-P_r T^{*^{\prime }}(t) T^*(y)-H_s T^*(t) \quad T^*(y)=0 \end{aligned}$$

(19)

$$\begin{aligned}{} & {} \quad \therefore \frac{T^{* \prime \prime }(y)}{T^*(y)}-P_r \frac{T^{* \prime }(t)}{T^*(t)}-H_s=0 \end{aligned}$$

(20)

$$\begin{aligned}{} & {} \quad \therefore \frac{T^{* \prime \prime }(y)}{T^*(y)}=P_r \frac{T^{* \prime }(t)}{T^*(t)}+H_s=-\lambda ^2 \end{aligned}$$

(21)

$$\begin{aligned}{} & {} \therefore T^{* \prime \prime }(y)+\lambda ^2 T^*(y)=0 \end{aligned}$$

(22)

and

$$\begin{aligned}{} & {} P_r \frac{T^{* \prime }(t)}{T^*(t)}+H_s=-\lambda ^2 \end{aligned}$$

(23)

$$\begin{aligned}{} & {} \quad \therefore P_r T^{* \prime }(y)+\left( H_s+\lambda ^2\right) T^*(t)=0 \end{aligned}$$

(24)

The solution of Eq. 22 is

$$\begin{aligned}{} & {} T^*(y)=l_1 \cos (\lambda y)+l_2 \sin (\lambda y) \nonumber \\{} & &\text{ at } y=1 \quad T^*=0 \rightarrow 0=l_1 \cos \lambda +l_2 \sin \lambda \\&\text{ at } y=-1 \quad T^*=0 \rightarrow 0=l_1 \cos \lambda -l_2 \sin \lambda \\&\text{ Then } \text{ by } \text{ addition, } \text{ we } \text{ get } \\&2 l_1 \cos \lambda =0 \\&l_1 \ne 0\quad \therefore \cos \lambda =0\rightarrow \quad \lambda =\frac{2 n-1}{2} \pi , \quad n=1,2,3, \ldots \\&0=l_1 \cos \lambda -l_2 \sin \lambda \\&0=C \cos \left( \frac{2 n-1}{2} \pi \right) +l_2 \sin \left( \frac{2 n-1}{2} \pi \right) \\&\therefore l_2=0 \end{aligned}$$

(25)

$$\begin{aligned}{} & {} \quad \therefore T^*(y)=l_1 \cos \left( \frac{2 n-1}{2} \pi \right) y \end{aligned}$$

(26)

This function is the eigen function for solving the non-homogeneous partial differential equation

$$\begin{aligned}{} & {} T^*(y, t)=\sum _{n=1}^{\infty } T_n^*(t) \cdot \cos \left( \frac{2 n-1}{2} \pi \right) y \end{aligned}$$

(27)

$$\begin{aligned}{} & {} \quad \frac{\partial ^2 T^*(y, t)}{\partial y^2}-P_r \frac{\partial T^*(y, t)}{\partial t}-H_s T^*(y, t)=F(y, t) \end{aligned}$$

(28)

$$\begin{aligned}{} & {} \quad F(y, t)=\sum _{n=1}^{\infty } F_n(t) \cdot \cos \left( \frac{2 n-1}{2} \pi \right) y \end{aligned}$$

(29)

$$\begin{gathered} \frac{{\partial ^{2} T^{*} (y,t)}}{{\partial y^{2} }} - P_{r} \frac{{\partial T^{*} (y,t)}}{{\partial t}} - H_{s} T^{*} (y,t) = \sum\limits_{{n = 1}}^{\infty } {F_{n} } (t) \cdot \cos \left( {\frac{{2n - 1}}{2}\pi } \right)y \hfill \\ T^{*} = 0\quad {\text{ at }}\quad t = 0 \hfill \\ T^{*} = 0\quad {\text{ at }}\quad y = \pm 1 \hfill \\ \end{gathered}$$

(30)

Then the solution is

$$\begin{aligned} T^*(y, t)=\sum _{n=1}^{\infty } T_n^*(t) \cdot \cos \left( \frac{2 n-1}{2} \pi \right) y \end{aligned}$$

(31)

Substitute Eq. 31 in Eq. 30 we can get

$$\begin{aligned} &\sum _{n=1}^{\infty }-\left( \frac{2 n-1}{2} \pi \right) ^2 T_n^*(t) \cos \left( \frac{2 n-1}{2} \pi \right) -\sum _{n=1}^{\infty } P_r \frac{d T_n^*(t)}{d t} \cos \left( \frac{2 n-1}{2} \pi \right) y \\&-H_s \sum _{n=1}^{\infty } T_n^*(t) \cos \left( \frac{2 n-1}{2} \pi \right) y=\sum _{n=1}^{\infty } F_n(t) \cdot \cos \left( \frac{2 n-1}{2} \pi \right) y \end{aligned}$$

(32)

where

$$\begin{aligned} F(y, t)=\sum _{n=1}^{\infty } F_n(t) \cos \left( \frac{2 n-1}{2} \pi \right) y \end{aligned}$$

(33)

By orthogonality

$$\int_{{ - 1}}^{1} F (y,t)\cos \left( {\frac{{2m - 1}}{2}\pi } \right)ydy = \sum\limits_{{n = 1}} {\int_{{ - 1}}^{1} {F_{n} (t)\left\{ {\cos \left( {\frac{{2n - 1}}{2}\pi } \right)y} \right\}\left\{ {\cos \left( {\frac{{2m - 1}}{2}\pi } \right)y} \right\}dy} }$$

(34)

For

$$\begin{aligned} m=n \quad \int _{-1}^1 \cos \left( \frac{2 n-1}{2} \pi \right) y \cos \left( \frac{2 m-1}{2} \pi \right) y d y=1 \\ m \ne n \quad \int _{-1}^1 \cos \left( \frac{2 n-1}{2} \pi \right) y \cos \left( \frac{2 n-1}{2} \pi \right) y d y=0 \\ \therefore F_n(t)=\int _{-1}^1 F(y, t) \cos \left( \frac{2 n-1}{2} \pi \right) y d y \\ F_n(t)=\int _{-1}^1\left[ \left( \textrm{X}_3+\gamma _2 e^{-a t}\right) \frac{\cosh m_1 y}{\cosh m_1}\right] \cos \left( \frac{2 n-1}{2} \pi \right) y d y \\ =\frac{2\left[ \gamma _3+\gamma _2 e^{-a t}\right] }{\cosh m_1} \int _0^1 \cosh \left( m_1 y\right) \cos \left( \frac{2 n-1}{2} \pi \right) y d y \end{aligned}$$

(35)

$$\begin{aligned}{} & {} F_n(t)=\gamma _1 \gamma _3+\gamma _2 \gamma _1 e^{-a t} \nonumber \\{} & {} Where\quad \gamma _1=\frac{(-1)^{n-1}(2 n-1) \pi }{m_1^2+\left( \frac{2 n-1}{2} \pi \right) ^2} \end{aligned}$$

(36)

From Eq. 34

$$\begin{aligned} \sum _{n=1}^{\infty }\left[ P_r \frac{d}{d t} T_n^*(t)+\left( H_s+\left( \frac{2 n-1}{2} \pi \right) ^2\right) T_n^*(t)\right] \cos \left( \frac{2 n-1}{2} \pi \right) =-\sum _{n=1}^{\infty } F_n(t) \cos \left( \frac{2 n-1}{2} \pi \right) y \end{aligned}$$

(37)

By orthogonality, we can get

$$\begin{aligned}{} & {} \frac{d T_n^*(t)}{d t}+\frac{\left( H_s+\left( \frac{2 n-1}{2} \pi \right) ^2\right) }{P_r} T_n^*(t)=-\frac{F_n(t)}{P_r} \end{aligned}$$

(38)

$$\begin{aligned}{} & {} \quad \frac{d T_n^*(t)}{d t}+\propto T_n^*(t)=\frac{-F_n(t)}{P_r} \end{aligned}$$

(39)

where $\propto =\frac{H_s+\left( \frac{2 n-1}{2} \pi \right) ^2}{P_r}$, Then

$$\begin{aligned} T_n^*(t)=e^{-\propto t} \int \frac{-F_n(t)}{P_r} e^{\alpha t} d t+l_1 e^{-\propto t} \end{aligned}$$

(40)

at $t=0 \quad T_n^*(t)=0$ then    $l_1=\frac{\gamma _1}{P_r}\left[ \frac{\gamma _3}{\alpha }+\frac{\gamma _2}{(\propto -a)}\right]$

$$\begin{aligned}{} & {} T_n(t)=\frac{\gamma _1}{P_r}\left[ -\frac{\gamma _3}{\alpha }\left( 1-e^{-\propto -t}\right) +\frac{\gamma _2}{(\propto -a)}\left( e^{-\propto t}-e^{-a t}\right) \right] \end{aligned}$$

(41)

$$\begin{aligned} \therefore T(t, y)=&\left[ \left( 1-e^{-a t}\right) \frac{\cosh m_1 y}{\cosh m_1}\right] +\sum _{n=1}^{\infty } \frac{\gamma _1}{P_r}[ -\frac{\gamma _3}{\alpha }\left( 1-e^{-\alpha t}\right) \\&\left. +\frac{\gamma _2}{(\propto -a)}\left( e^{-\alpha t}-e^{-a t}\right) \right] \cos \left( \frac{2 n-1}{2} \pi \right) y \end{aligned}$$

(42)

Substituting Eq. 42 in 10 to get

$$\begin{aligned}&\frac{\partial ^2 C(y, t)}{\partial y^2}-S_c \frac{\partial C(y, t)}{\partial t}-C_r S_c C(y, t)=-T_d \frac{\partial ^2 T(y, t)}{\partial y^2} \\&\text{ at } \quad t=0 \quad C=0 \quad \text{ For } y \in (-1,1) \\&\text{ at } \quad t>0 \quad C=\left( 1-e^{-a t}\right) \text{ For } y=\pm 1 \end{aligned}$$

(43)

To solve Eq. 43 by separation of variables method we first transform the non-homogenous boundary conditions to homogenous one. Let $C(y, t)=\left( 1-e^{-a t}\right) \frac{\cosh \left( m_2 y\right) }{\cosh \left( m_1\right) }+C^*(y, t)$ For $t=0\quad C=0 \quad \therefore C^*=0,t>0 \quad C=\left( 1-e^{-a t}\right) \text{ at } y=\pm 1 \therefore C^*=0$

$$\begin{aligned}{} & {} \therefore \frac{\partial ^2 C^*(y, t)}{\partial y^2}-S_c \frac{\partial C^*(y, t)}{\partial t}-C_r S_c C^*(y, t)=F_c(y, t) \end{aligned}$$

(44)

$$\begin{aligned} F_c(y, t)=\left( \gamma _4+\gamma _5 e^{-a t}\right) \frac{\cosh \left( m_2 y\right) }{\cosh \left( m_2\right) }-T_d\left( m_1\right) ^2\left( 1-e^{-a t}\right) \frac{\cosh \left( m_1 y\right) }{\cosh \left( m_1\right) } \\ +T_d \sum _{n=1}^{\infty }\left( \frac{2 n-1}{2} \pi \right) ^2 T_n(t) \cdot \cos \left( \frac{2 n-1}{2} \pi \right) y \end{aligned}$$

(45)

which can be written as

$$ \begin{gathered} F_{c} (y,t) = \sum\limits_{{n = 1}}^{\infty } {F_{{nc}} } (t) \cdot \cos \left( {\frac{{2{\text{n}} - 1}}{2}\pi } \right){\text{y}} \hfill \\ {\text{By}}\;{\text{orthogonality}} \hfill \\ \therefore F_{{nc}} (t) = \int_{{ - 1}}^{1} F (y,t)\cos \left( {\frac{{2n - 1}}{2}\pi } \right)ydy \hfill \\ F_{{nc}} (t) = \left( {\gamma _{4} + \gamma _{5} e^{{ - at}} } \right)\gamma _{6} - \gamma _{7} \gamma _{8} \left( {1 - e^{{ - at}} } \right) + \gamma _{9} T_{{nT}} (t) \hfill \\ \gamma _{4} = \left( {C_{r} S_{c} - m_{2}^{2} } \right),\quad \gamma _{5} = \left[ {m_{2}^{2} + S_{c} a^{2} - C_{r} S_{c} } \right] \hfill \\ \gamma _{6} = \frac{{( - 1)^{{n - 1}} (2n - 1)\pi }}{{m_{2}^{2} + \left( {\frac{{2n - 1}}{2}\pi } \right)^{2} }}\quad ,\gamma _{7} = \gamma _{1} = \frac{{( - 1)^{{n - 1}} (2n - 1)\pi }}{{m_{1}^{2} + \left( {\frac{{2n - 1}}{2}\pi } \right)^{2} }} \hfill \\ \gamma _{8} = \left( { - T_{d} m_{1}^{2} } \right),\gamma _{9} = \left[ {\left( {\frac{{2n - 1}}{2}\pi } \right)^{2} T_{d} } \right] \hfill \\ \therefore \frac{{\partial ^{2} C^{*} (y,t)}}{{\partial y^{2} }} - S_{c} \frac{{\partial C^{*} (y,t)}}{{\partial t}} - C_{r} S_{c} \hfill \\ C^{*} (y,t) = \sum\limits_{{n = 1}}^{\infty } {F_{{nc}} } (t) \cdot \cos \left( {\frac{{2n - 1}}{2}\pi } \right)y \hfill \\ \end{gathered} $$

(46)

$$\begin{aligned}{} & {} \quad C^*(y, t)=\sum _{n=1}^{\infty } T_{n c}(t) \cdot \cos \left( \frac{2 n-1}{2} \pi \right) y \end{aligned}$$

(47)

By orthogonality, we can get

$$\begin{aligned}{} & {} \frac{d T_{nc}(t)}{dt}+\frac{\left( C_r S_c+\left( \frac{2 n-1}{2} \pi \right) ^2\right) }{S_c} T_{nc}(t)=-\frac{F_{n c}(t)}{S_c} \end{aligned}$$

(48)

$$\begin{aligned}{} & {} \quad T_{n c}(t)=\frac{-1}{S_c} \frac{\beta _4}{\alpha _1}\left( 1-e^{-\alpha _1 t}\right) -\frac{1}{S_c} \frac{\beta _5}{\left( \alpha _1-a\right) }\left( e^{-a t}-e^{-\alpha _1 t}\right) -\frac{\beta _6}{S_c\left( \alpha _1-\alpha \right) }\left( e^{-\alpha t}-e^{-\alpha _1 t}\right) \end{aligned}$$

(49)

where $\beta _1=\left[ \frac{\gamma _1}{P_r}\right] , \quad \beta _2=\left[ \frac{\gamma _3}{\alpha }\right] , \quad \beta _3=\left[ \frac{\gamma _2}{\alpha -a}\right] ,\beta _4=\gamma _4 \gamma _6-\gamma _7 \gamma _8+\beta _1 \beta _2 \gamma _9, \beta _5=\gamma _5 \gamma _6+\gamma _7 \gamma _8-\beta _1 \beta _3 \beta _9,\beta _6=\beta _1 \beta _3 \gamma _9-\beta _1 \beta _2 \gamma _9,\alpha _1=\frac{C_r S_c+\left( \frac{2 n-1}{2} \pi \right) ^2}{S_c}$

$$\begin{aligned} \therefore C(y, t)=\left[ \left( 1-e^{-a t}\right) \frac{\cosh m_2 y}{\cosh m_2}\right] +\sum _{n=1}^{\infty } T_{n c}(t) \cdot \cos \left( \frac{2 n-1}{2} \pi \right) y \end{aligned}$$

(50)

Solve Eq. 8 for u in to get

$$\begin{aligned} u=\tau \frac{\partial v}{\partial t}+v \end{aligned}$$

(51)

Substitute Eq. 51 into Eq. 8 to get

$$\begin{aligned}&\tau \frac{\partial ^2 v}{\partial t^2}+\frac{\partial v}{\partial t}=\frac{d p}{d x}+G_r T+G_m C\\&+\left( 1-V_e \frac{\partial }{\partial t}\right) \left( \tau \frac{\partial ^3 v}{\partial t \partial y^2}+\frac{\partial ^2 v}{\partial y^2}\right) -L \frac{\partial v}{\partial t}-\frac{1}{M}\left( \tau \frac{\partial v}{\partial t}+v\right) \end{aligned}$$

(52)

$$\begin{aligned}&\tau V_e \frac{\partial ^4 v}{\partial t^2 \partial y^2}+V_e \frac{\partial ^3 v}{\partial t \partial y^2}-\tau \frac{\partial ^3 v}{\partial t \partial y^2}-\frac{\partial ^2 v}{\partial y^2}+\left( 1+\frac{\tau }{M}+L\right) \frac{\partial v}{\partial t}+\frac{v}{M}+\tau \frac{\partial ^2 v}{\partial t^2}\\&=G_r T+G_m T-\frac{d p}{d x} \end{aligned}$$

(53)

by substituting $v(y, t)=\sum _{n=1}^{\infty } T_{n v}(t) \cos \left( \frac{2 n-1}{2} \pi \right) y$

$$\begin{aligned} \sum _{n=1}^{\infty }\left\{ a_2 T_{n v}^{\prime \prime }(t)+b_2 T_{n v}^{\prime }(t)+l_2 T_{n v}(t)\right\}&\cdot \cos \left( \frac{2 n-1}{2} \pi \right) y\\&=\left( 1-e^{-a t}\right) \left( G_r \frac{\cosh m_1 y}{\cosh m_1}+G_m \frac{\cosh m_2 y}{\cosh m_2}\right) \\&+\sum _{n=1}^{\infty }\left( G_r T_{n T}(t)+G_m T_{n c}(t)\right) \cos \left( \frac{2 n-1}{2} \pi \right) y-\frac{d P}{d x} \nonumber \\ Where\quad{} & {} &a_2=\tau \left( 1-V_e\left( \frac{2 n-1}{2} \pi \right) ^2\right) , \frac{d P}{d x}=P(t), b_2=\left( 1+L+\frac{\tau }{M}\right) +\left( \tau -V_e\right) \cdot \left( \frac{2 n-1}{2} \pi \right) ^2 \\&l_2=\frac{1}{M}+\left( \frac{2 n-1}{2} \pi \right) ^2,T_{n T}(t)=\beta _1 \beta _2+\left( \beta _1 \beta _3-\beta _1 \beta _2\right) e^{-\propto t}-\beta _1 \beta _3 e^{-a t} \\&T_{nc}(t)=-\frac{\beta _4}{\propto S_c}-\frac{\beta _6}{\left( \alpha _1-\propto \right) S_c} e^{-\propto t}-\frac{\beta _5}{\left( \alpha _1-\propto \right) S_c} e^{-a t}+\left( \frac{\beta _5}{S_c\left( \alpha _1-a\right) }+\frac{\beta _6}{S_c\left( \alpha _1-\propto \right) }\right) e^{-\alpha _1 t} \end{aligned}$$

(54)

From Eq. 54 by using orthogonality

$$ \begin{aligned} & \int_{{ - 1}}^{1} {\cos } \left( {\frac{{2n - 1}}{2}\pi } \right)y\cos \left( {\frac{{2m - 1}}{2}\pi } \right)ydy = \left\{ {\begin{array}{*{20}l} 0 \hfill & {m \ne n} \hfill \\ 1 \hfill & {m = n} \hfill \\ \end{array} } \right. \\ & \int_{{ - 1}}^{1} {\cos } \left( {\frac{{2m - 1}}{2}\pi } \right)y\frac{{\cosh m_{1} y}}{{\cosh m_{1} }}dy = \frac{{( - 1)^{n} (2n - 1)\pi }}{{m_{1}^{2} + \left( {\frac{{2n - 1}}{2}\pi } \right)^{2} }} = \gamma _{7} \\ & \int_{{ - 1}}^{1} {\cos } \left( {\frac{{2m - 1}}{2}\pi } \right)y\frac{{\cosh m_{2} y}}{{\cosh m_{2} }}ydy = \frac{{( - 1)^{n} (2n - 1)\pi }}{{m_{2}^{2} + \left( {\frac{{2n - 1}}{2}\pi } \right)^{2} }} = \gamma _{8} \\ & \int_{{ - 1}}^{1} P (t)\cos \left( {\frac{{2m - 1}}{2}\pi } \right)ydy = 2P(t)\frac{{( - 1)^{{n - 1}} }}{{\left( {\frac{{2n - 1}}{2}\pi } \right)}} = P(t).\gamma _{{15}} ,\quad \gamma _{{15}} = \frac{{2( - 1)^{{n - 1}} }}{{\left( {\frac{{2n - 1}}{2}\pi } \right)}} \\ & a_{2} T_{{nv}}^{{\prime \prime }} (t) + b_{2} T_{{nv}}^{\prime } (t) + l_{2} T_{{nv}} (t) = \gamma _{{11}} + \gamma _{{12}} e^{{ - at}} + \gamma _{{13}} e^{{ - \alpha t}} + \gamma _{{14}} e^{{ - \alpha _{1} t}} - P(t)\gamma _{{15}} \\ & {\text{Where}} \\ & \gamma _{{11}} = \left[ {\left( {\gamma _{7} + \beta _{1} \beta _{2} } \right)G_{r} + \left( {\gamma _{8} - \frac{{\beta _{4} }}{{\alpha S_{c} }}} \right)G_{m} } \right] \\ & \gamma _{{12}} - \left[ {\left( {\gamma _{7} - \beta _{1} \beta _{3} } \right)G_{r} + G_{m} \left( {\gamma _{8} - \frac{{\beta _{5} }}{{\left( {\alpha _{1} - a} \right)S_{c} }}} \right)} \right]e^{{ - at}} \\ & \gamma _{{13}} = \left[ {G_{r} \left( {\beta _{1} \beta _{3} - \beta _{1} \beta _{2} } \right) - G_{m} \left( {\frac{{\beta _{6} }}{{\left( {\alpha _{1} - \alpha } \right)S_{c} }}} \right)} \right]e^{{ - \alpha t}} \\ & \gamma _{{14}} = \left[ {\frac{{\beta _{5} }}{{S_{c} \left( {\alpha _{1} - \alpha } \right)}} + \frac{{\beta _{6} }}{{S_{c} \left( {\alpha _{1} - \alpha } \right)}}} \right]G_{m} e^{{ - \alpha _{1} t}} \\ \end{aligned} $$

(55)

Consider the homogenous one

$$\begin{aligned} \textrm{a}_2 \mathrm {~T}_{\textrm{nv}}^{\prime \prime }(\textrm{t})+\textrm{b}_2 \mathrm {~T}_{\textrm{nv}}^{\prime }(\textrm{t})+\textrm{C}_2 \mathrm {~T}_{\textrm{nv}}(\textrm{t})=0 \end{aligned}$$

(56)

The auxiliary equation and its roots are

$$\begin{aligned}{} & {} a_2 m^2+b_2 m+l_2=0, \quad m_1=\frac{-b_2+\sqrt{b_2^2-4 a_2 l_2}}{2 a_2},\quad m_2=\frac{-b_2-\sqrt{b_2^2-4 a_2 l_2}}{2 a_2} \end{aligned}$$

(57)

$$\begin{aligned}{} & {} \therefore T_{nv_h}(t)=l_1 e^{m_1 t}+l_2 e^{m_2 t} \end{aligned}$$

(58)

For non-homogenous one, the particular solution is

$$\begin{aligned} T_{nv_p}(t)=\frac{1}{a_2 D^2+b_2 D+l_2}\left[ \gamma _{11}+\gamma _{12} e^{-a t}+\gamma _{13} e^{-\alpha t}+\gamma _{14} e^{-\alpha _1 t}-\gamma _{15} P(t)\right] \end{aligned}$$

(59)

Case 1: $P(t)=a_0 e^{-a_1 t}$ $a_0,a_1$ are constants. The exponential 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

$$\begin{aligned} T_{n v}(t)=\gamma _{16}+\gamma _{17} e^{-a t}+\gamma _{18} e^{-\propto t}+\gamma _{19} e^{-\alpha _1 t}-\gamma _{20} e^{-a_1 t} \end{aligned}$$

(60)

The general solution is

$$\begin{aligned} \textrm{T}_{\textrm{nv}}=\textrm{C}_1 \textrm{e}^{\textrm{m}_1 \textrm{t}}+\textrm{C}_2 \textrm{e}^{\textrm{m}_2 \textrm{t}}+\gamma _{16}+\gamma _{17} \textrm{e}^{-\textrm{at}}+\gamma _{18} \textrm{e}^{-\alpha \textrm{t}}+\gamma _{19} e^{-\alpha _1 t}-\gamma _{20} e^{-a_1 t} \end{aligned}$$

(61)

where

$$ \begin{aligned} & \gamma _{{16}} = \frac{{\gamma _{{11}} }}{{l_{2} }},\quad \gamma _{{17}} = \frac{{\gamma _{{12}} }}{{\left( {a_{2} a^{2} - b_{2} a + l_{2} } \right)}},\gamma _{{18}} = \frac{{\gamma _{{13}} }}{{\left( {a_{2} \alpha ^{2} - b_{2} \alpha + l_{2} } \right)}}, \\ & \gamma _{{19}} = \frac{{\gamma _{{14}} }}{{\left( {a_{2} \alpha _{1}^{2} - b_{2} \alpha _{1}^{2} + l_{2} } \right)}},\gamma _{{20}} = \frac{{\gamma _{{15}} a_{0} }}{{\left( {a_{2} a_{1}^{2} - b_{2} a_{1} + l_{2} } \right)}} \\ & \because {\text{ at }}t = 0\quad T_{{nv}} = 0,\therefore 0 = l_{1} + l_{2} + \gamma _{{16}} + \gamma _{{17}} + \gamma _{{18}} + \gamma _{{19}} - \gamma _{{20}} \\ & \because \frac{{aI_{{nv}} }}{{dt}} = 0\therefore 0 = m_{1} l_{1} + m_{2} l_{2} - a\gamma _{{17}} - \propto \gamma _{{18}} - \alpha _{1} \gamma _{{19}} + a_{1} \gamma _{{20}} \\ & \therefore l_{1} = \frac{1}{{\left( {m_{2} - m_{1} } \right)}}\left( { - m_{2} \gamma _{{16}} - \left( {m_{2} + a} \right)\gamma _{{17}} - \left( {m_{2} + \propto } \right)\gamma _{{18}} - \left( {m_{2} + \propto _{1} } \right)\gamma _{{19}} + \left( {m_{2} - a_{1} } \right)\gamma _{{20}} } \right) \\ & \therefore l_{2} = \frac{1}{{\left( {m_{2} - m_{1} } \right)}}\left( {m_{1} \gamma _{{16}} + \left( {m_{1} + a} \right)\gamma _{{17}} + \left( {m_{1} + \propto } \right)\gamma _{{18}} + \left( {m_{1} + \alpha _{1} } \right)\gamma _{{19}} - \left( {m_{1} - a_{1} } \right)\gamma _{{20}} } \right) \\ & v(y.t) = \sum\limits_{{n = 1}}^{\infty } {T_{{nv}} } (t)\cos \left( {\frac{{2n - 1}}{2}\pi } \right)y \\ \end{aligned} $$

(62)

$$\begin{aligned} T_{n u}(t)=\tau \frac{d T_{n v}}{d t}+&T_{n v}=l_1\left( 1+\tau m_1\right) e^{m_1 t}+l_2\left( 1+\tau m_2\right) e^{m_2 t}+\gamma _{16}+(1-\tau a) \gamma _{17} e^{-a t} \\&+(1-\tau \propto ) \gamma _{18} e^{-\propto t}+\left( 1-\tau \propto _1\right) \gamma _{19} e^{-\alpha _1 t}-\left( 1-a_1 \tau \right) \gamma _{20} e^{-a_1 t} \end{aligned}$$

(63)

$$\begin{aligned}{} & {} u(y, t)=\sum _{n=1}^{\infty } T_{n u}(t) \cos \left( \frac{2 n-1}{2} \pi \right) y \end{aligned}$$

(64)

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.

$$\begin{aligned}&\frac{1}{\left( a_2 D^2+b_2 D+l_2\right) } \gamma _{15} P(t)=\frac{1}{a_2\left( D-m_1\right) \left( D-m_2\right) } \gamma _{15} P(t) \\&=\frac{1}{a_2\left( m_1-m_2\right) }\left( \frac{1}{D-m_1}-\frac{1}{D-m_2}\right) \gamma _{15} P(t) \\&=\frac{1}{a_2\left( m_1-m_2\right) }\left[ \frac{-1}{m_1}\left( 1-\frac{D}{m_1}\right) ^{-1}+\right. \\&\left. \frac{1}{m_2}\left( 1-D / m_2\right) ^{-1}\right] P(t) \gamma _{15} \end{aligned}$$

(65)

$$\begin{aligned}{} & {} \frac{1}{\left( a_2 D^2+b_2 D+l_2\right) } \gamma _{15} P(t)=\frac{1}{a_2}\left[ \left( \frac{a_3}{m_1 m_2}+\frac{a_4\left( m_1+m_2\right) }{\left( m_1 m_2\right) ^2}\right) +\frac{a_4}{m_1 m_2} t\right] \gamma _{15}=\left[ a_5+a_6 t\right] \gamma _{15} \end{aligned}$$

(66)

where

$$\begin{aligned} a_5=\left[ \frac{a_3}{m_1 m_2 a_2}+\frac{a_4\left( m_1+m_2\right) }{\left( m_1 m_2\right) 2 a_2}\right\rfloor * \frac{1}{a_2}, a_6=\frac{a_4}{m_1 m_2 a_2} * \frac{1}{a_2} \nonumber \\{} & {} T_{n v}(t)=l_1 e^{m_1 t}+l_2 e^{m_2 t}+\gamma _{16}+\gamma _{17} e^{-a t}+\gamma _{18} e^{-\propto t}+\gamma _{19} e^{-\alpha _1 t}+\left( a_5+a_6 t\right) \gamma _{15} \end{aligned}$$

(67)

By applying the boundary conditions $at\quad t=0 \quad T=0, \quad \frac{d T}{d t}=0, we get$

$$\begin{aligned}&l_1=\frac{1}{\left( m_2-m_1\right) }\left[ -m_2 \gamma _{16}-\left( m_2+a\right) \gamma _{17}-\left( m_2+\propto \right) \gamma _{18}-\left( m_2+\propto _1\right) \gamma _{19}-a_5 m_2 \gamma _{15}+a_6 \gamma _{15}\right] \\&l_2=\frac{1}{\left( m_2-m_1\right) }\left[ +m_1 \gamma _{16}+\left( m_1+a\right) \gamma _{17}+\left( m_1+\propto \right) \gamma _{18}+\left( m_1+\alpha _1\right) \gamma _{19}+a_5 m_2 \gamma _{15}-a_6 \gamma _{15}\right] \nonumber \\{} & {} T_{n v}(t)=l_1 e^{m_1 t}+l_2 e^{m_2 t}+\gamma _{16}+\gamma _{17} e^{-a t}+\gamma _{18} e^{-\alpha t}+\gamma _{19} e^{-\alpha _1 t}+\left( a_5+a_6 t\right) \gamma _{15} \end{aligned}$$

(68)

$$\begin{aligned}{} & {} v(y, t)=\sum _{n=1}^{\infty } T_{n v}(t) \cos \left( \frac{2 n-1}{2} \pi \right) y \end{aligned}$$

(69)

$$\begin{aligned} T_{n u}(t)=\tau \frac{d T_{n v}}{d t}+&T_{n v}=l_1\left( \tau m_1+1\right) e^{m_1 t} \\&+l_2\left( \tau m_2+1\right) e^{m_2 t}+\gamma _{16}+(1-\tau a) \gamma _{17} e^{-a t}+(1-\tau \propto ) \gamma _{18} e^{-\alpha t}\\ {}&+\left( 1-\tau \propto _1\right) \gamma _{19} e^{-\alpha _1 t} +\left( a_5+a_6 t+\tau a_6\right) \gamma _{15} \end{aligned}$$

(70)

$$\begin{aligned}{} & {} u(y, t)=\sum _{n=1}^{\infty } T_{nu}(t) \cos \left( \frac{2 n-1}{2} \pi \right) y \end{aligned}$$

(71)

3 Results

The analytical solutions for partial differential equations have been carried out, and results are displayed for temperature, concentration, and velocity profiles for two different cases of pressure distribution. The 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.

Fig. 2

Variation of temperature with $H_s$

Fig. 3

Variation of temperature with $P_r$

Fig. 4

Variation of concentration with $S_c$

Fig. 5

Variation of concentration with $T_d$

Fig. 6

Variation of concentration with $H_s$

Fig. 7

Variation of concentration with $P_r$

Fig. 8

Variation of fluid velocity with $T_d$-case 1

Fig. 9

Variation of dust velocity with $T_d$-case 1

Fig. 10

Variation of fluid velocity with $C_r$-case 1

Fig. 11

Variation of dust velocity with $C_r$-case 1

Fig. 12

Variation of fluid velocity with $T_d$-case 2

Fig. 13

Variation of dust velocity with $T_d$-case 2

Fig. 14

Variation of fluid velocity with $C_r$-case 2

Fig. 15

Variation of dust velocity with $C_r$-case 2

Fig. 16

Variation of temperature with time

Fig. 17

Variation of concentration with time

The physical quantities for engineering applications, local Nusselt number $Nu=- (\delta T/\delta y)_{y=0}$, and local Sherwood number $Sh=- (\delta C/\delta y)_{y=0}$ 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.

Table 1 Comparison of Nusselt number and local Sherwood number

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

Table 2 Comparisons of temperature and concentration

Table 3 Comparisons of fluid and dust velocities

4 Discussions

Figures 2 and 3 show the influence of Prandtl number $P_r$ and the heat source or heat sink coefficient $H_s$ on temperature. It has been noted that the more temperature increase, the more Pr and $H_s$ decrease. Figures 4 and 5 confirm that the concentration of the fluid increases when the thermal diffusion parameter Td increases and the Schmidt number Sc decreases, respectively. Figure 6 confirms that the concentration of the fluid increases when the heat source or heat sink coefficient $H_s$ decreases. Figure 7 confirms 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, respectively. The temperature and concentration increase as time increases. The graphs are based on the following values:

$$\begin{aligned}&c_{\textrm{r}}=0.4, a=0.2, H_{\textrm{s}}=0.2, t=1, P_{\textrm{r}}=0.4, S_{\textrm{c}}=0.6, M=10, l=0.1, D=1, V_{\textrm{e}}=0.3, \tau =0.9, G_{\textrm{m}}=5, G_{\textrm{r}}=10 \\&n=2, a_0=2.5, a_1=0.3, a_3=2.5, a_4=0.2 \text{ and } m_2=1.5 \end{aligned}$$

5 Conclusion

The exact expressions for temperature, concentration, and velocity profiles for fluid and dusty particles are derived analytically. Results are displayed graphically for the various values of parameters. The key discovery may be summed up as follows:

• The temperature is inversely proportional to both the Prandtl number Pr and the heat source and sink parameter $H_s$.

• The concentration of the fluid is inversely proportional to the thermal diffusion parameter Td and the heat source or heat sink parameter $H_s$.

• The fluid concentration is proportional to Schmidt’s number Sc and Prandtl’s number Pr

• Fluid and dust particle velocities increase as Td increases.

• Fluid and dust particle velocities increase as Cr decreases.

• If the dust particles are very fine, i.e., the mass of the dust particles is very small, and as $\tau -> 0$ the velocities of the fluid and dust particles will be the same.

• The temperature and concentration increase with time.

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.