Role of thermophoresis on unsteady/steady mixed convective flow in a vertical channel having convective boundary conditions

A numerical as well as analytical investigation is performed to study time dependent/steady state mixed convection flow in a vertical channel in presence of thermophoresis having convective boundary conditions. The main objective of this paper is to present exact solutions for steady state mathematical model under relevant boundary conditions and numerical solutions for time dependent situation under relevant initial and boundary conditions. The impact of various controlling parameters such as dimensionless time t\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t$$\end{document}, Schmidt number Sc\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Sc$$\end{document}, Biot numbers Bi1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Bi_{1}$$\end{document} and Bi2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Bi_{2}$$\end{document}, buoyancy ratio b\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b$$\end{document} and thermophoretic coefficient k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k$$\end{document} have on the flow patterns is discussed. Graphical results show that the effect of increasing two or more of the parameters simultaneously with high mixed convection parameter changes the flow from bi-directional to unidirectional. Also, the occurrence of a flow reversal depends on buoyancy and dimensional time. Furthermore, there exists a convergence of Sherwood number values at the cold plate when thermophoretic values are high. When buoyancy dominates the flow, a decrease of reverse flow at the cold plate is observed as thermophoresis approaches its peak. If Schmidt number increases simultaneously with time, a wider region across the channel where concentration of particles is zero is observed. When thermophoretic values are low, an increment in time reduces Sherwood numbers. Sherwood number values are observed to converge at the cold plate when thermophoretic values are high. These findings have implications in the developments of thermophoretic cells used in biochemical and medical researches because Sherwood numbers imply negative concentration fluxes at boundaries. When buoyancy dominates the flow, a decrease of reverse flow at the cold plate is observed as thermophoresis approaches its peak. If Schmidt number increases simultaneously with time, a wider region across the channel where concentration of particles is zero is observed. When thermophoretic values are low, an increment in time reduces Sherwood numbers. Sherwood number values are observed to converge at the cold plate when thermophoretic values are high. These findings have implications in the developments of thermophoretic cells used in biochemical and medical researches because Sherwood numbers imply negative concentration fluxes at boundaries.


Article Highlights
• When buoyancy dominates the flow, a decrease of reverse flow at the cold plate is observed as thermophoresis approaches its peak. • If Schmidt number increases simultaneously with time, a wider region across the channel where concentration of particles is zero is observed.
• When thermophoretic values are low, an increment in time reduces Sherwood numbers. Sherwood number values are observed to converge at the cold plate when thermophoretic values are high. These findings have implications in the developments of thermophoretic cells used in biochemical and medical researches because Sherwood numbers imply negative concentration fluxes at boundaries.

Keywords Mixed convection flow · Thermophoresis · Biot numbers
List of symbols a Effective thermophoretic coefficient Matrix of coefficients |A| max Maximum absolute value of A i.j b Buoyancy ratio Bi 1 ,Bi 2 Biot numbers at y = 0, 1 C ′ Dimensional fluid particle concentration C Cunningham correction factor

Introduction
An aerosol is a mixture of ultrafine-particle suspension such as the mixture of water vapor, ash, dust and soot in atmospheric air as a result of volcanic eruptions, transportation and urbanization. One of the most observable effects of a temperature gradient on aerosols is referred to as thermophoresis. We can observe this phenomenon in the exodus of particles from a hot room to settle on a cooler fan when the fan is turned off. A force referred to as the thermophoretic force is responsible for the push on the cooler slower particles by the faster kinetically energized particles to move forward towards a cold region. The Epstein model and Stokes model for thermophoretic force coefficient for the continuum regime; Brock model and Basset model for slip flow regime; Waldmann and Epstein thermophoretic force coefficient for the free molecule regime all contributed to the Talbot [1] thermophoretic force coefficient suitable for all regimes, by making adjustments to thermal slip coefficient C s , momentum exchange coefficient C m in Brock [2]. Young [3] presented a re-evaluation of thermophoresis in the transportation of spherical particles. Guha and Samanta [4] investigated the effect of natural convection on aerosol particle transportation and deposition in a flow considering two mathematical models of flows on vertical and horizontal plates. Kuznetsov and Nield [5] revised concentration boundary conditions as zero mass flux of particles in their study on free-convective boundary layer flow problems past a vertical plate. Flow reversal analyses in fully developed mixed convection flows were presented in Cheng et al. [6], Gau et al. [7] and Grosan and Pop [8] by employing various boundary conditions. Barletta and Zanchini [9] provided a general method for selecting fluid reference temperatures in mixed convection flows for the fully developed regime. The effect of natural convection in a flow bounded by two vertical parallel plates has been presented by Olsson [10]. Mathematical analysis and formulation of models in a mixed convection flow between two vertical parallel electrically conducting plates that are immersed in a liquid of high Prandtl number can be seen in Gradinger and Laneryd [11].
Double-diffusive flows are created by natural convection currents caused by a combined effect of temperature and concentration gradients. Some of these flows can be seen in Jha et al. [12] and Jha et al. [13] where they respectively studied the effects of free convective Couette flow in a vertical channel. Kaurangini and Jha [14] presented an analysis of unsteady flow with a pressure gradient and suction/injection in a parallel plated channel having an interface area in between a clean liquid and a porous medium. Kumar et al. [15] employed Runge-Kutta-45based shooting method and non-uniform thermal flux at boundaries in investigating water-Al 2 O 3 nanofluid transport in an annulus. Results show that heat transfer rate was better in a divergent channel as compared to a convergent one. In Parvin et al. [16], the Chon model and the Maxwell Garrett model are used to evaluate heat transfer characteristics in an annulus holding a water-alumina nanofluid. Tayebi et al. [17] studied natural convection of a CNT-water nanofluid in a space between two cylinders. Tayebi et al. [18] showed that the combined influences of internal thermal generation/absorption and hybrid nanoliquid significantly affected generation of entropy, heat transfer and hydrothermal properties.
Thermophoretic effect usually occurs with other mechanisms of heat/mass transfer, as can be seen in Jha and Sani [19]. They observed the effects of thermophoresis in both steady and unsteady states in a Couette flow. Makinde et al. [20] studied thermophoresis, heat transfer due to radiation and variable viscosity on an MHD flow over a heated rod embedded in a porous medium. Studies of thermophoretic transport and deposition of particles can be found in Partha [21]; Walker et al. [22], Jayaraj [23], Grosan et al. [24] and Magyari [25].
Finite difference Schemes are employed in heat and mass heat flow because they are efficient, stable, accurate and cost-effective. They sometimes provide alternatives to experimental investigation. They are also used in dynamical analysis of self-propelling soft bodied unicellular organisms like nematodes, rod-shaped bacteria and spermatozoa in non-Newtonian slime or Carreau fluids. Ali et al. [26] employed the finite difference method to study bacterial gliding on a film of non-Newtonian slime. These computations reveal that the velocity of glider declines as the rheology of fluid changes from pseudo-plastic to dilatant. Asghar et al. [27] described the dynamics of micro-organisms propelling through a cervical canal. A combined analysis of Implicit finite difference method and Newton-Raphson method reveal that micro-organisms could achieve a maximum speed when rheology of surrounding fluid was adjustable. A combination of Implicit Finite Difference method and Broyden method was used in Asghar et al. [28] to analysed creeping flow of active spermatozoa in an inclined channel and in Asghar et al. [29] to demonstrate propulsion speed and undulation amplitude in a swimmerbody.
If a fluid moves by natural convection, we say it moves by buoyancy of warmer fluid against a cooler fluid then the boundary condition we must impose is a homogenous Robin condition unless the temperature or heat flux are fixed. Robin boundary conditions in mixed convection can be found in Umavathi et al. [30] where they discussed the Soret effects using Robin boundary conditions on a mixed convective fluid flow. Patrulescu et al. [31] used boundary conditions of the third kind in a parallel-plated vertical channel holding a mixed convection flow. Kumar et al. [32] studied mixed convection in a vertical channel with Robin boundary conditions of a viscous and electrically conducting fluid. Kumar et al. [33] examined mixed convection in a vertical channel with boundary conditions of the third kind in a composite-porous medium. Umavathi et al. [34] presented the effect of chemical reaction, thermophoresis and convection in a vertical channel with Robin boundary conditions.
The purpose of this study is to consider the steady and unsteady states of a mixed convective thermophoretic flow and deposition of suspended particles in a fluid bounded by two parallel-walls. This study employs a numerical algorithm by Langtangen [35] uses equations from Talbot et al. [1], Grosan et al. [24], Magyari et al. [25], Guha [36] and Flagan and Seinfeld [37] in a special case that uses Robin boundary conditions at the two boundaries. An effective finite difference scheme was employed and verified with analytic results. Results obtained confirm the effectiveness of the thermophoresis phenomena on migration and mass transfer of particles like soot. The study reveals that flow reversal depends mostly on buoyancy and time. Sherwood numbers are shown to be parameters that can be used to monitor fouling and concentration fluxes in thermophoretic cells.

Mathematical analysis
Let us imagine a time-dependent fully developed regime of natural convective thermophoretic flow within an infinite vertical channel. Let us describe the situation in question in Fig. 1, where we identify x ′ -axis along both channel walls and the y ′ -axis taken normal to both walls. Also identify breadth of channel as L . If both channel walls are at rest, let time have the condition t ′ ≤ 0 , and temperature of walls and fluid assumed to be T 0 . At t ′ > 0 , If T ′ is thermodynamic temperature, then let T h denote the temperature within the wall at y � = 0 but define the temperature at y � = 0 be determined by the heat flux from the end of wall to the fluid within channel wall. And so also, at y � = L , T c denote the temperature of (within) wall and define the temperature at y � = L be determined by the heat flux from the within channel wall to the end of wall. Hence employing Fourier's constitutive equations at the boundaries, the boundary conditions we must impose on the two solid boundaries are ��� ⃗ Double-diffusive flows are created by natural convection currents caused by a combined effect of temperature and concentration gradients. We as heat expansion coefficient and the concentration expansion coefficient respectively. The present physical situation in dimensional form under investigation as in Grosan et al. [24]: , the initial and boundary conditions of interest in this situation are: The pressure gradient is obtained from the fact that: Thermophoretic deposition velocity in the y-direction v T is written as: where the coefficient of diffusion due to temperature gradient given by Guha [36] is, where k is the non-dimensional thermophoretic force coefficient defined by Talbot et al. [1] suitable for all regimes: C = 1 + Kn 2.514 + 0.8e −0.55∕Kn (Flagan and Seinfeld [37]), Kn = ∕D p with Guha [36] defining the relation = � √ p �� 2 . Using the following dimensionless quantities as defined in Grosan et al. [24], Magyari [25] and Barletta and Zanchini [9], https://doi.org/10.1007/s42452-022-04971-8 Research Article the nondimensionalized thermophoretic velocity is now defined as: Equation (5) nondimensionalized can be written in dimensionless form as: The dimensionless momentum, energy and concentration equations are respectively: With Eqs. (4), (7), N t > 1 2 and T � (y)∕ΔT > 1 , and the following functions are given as: The boundary conditions for concentration below correlates with Kuznetsov and Nield [5] suggesting zero mass flux of particles at surfaces. Therefore the initial and later boundary conditions in dimensionless form become:

Analytical solutions
To obtain closed form solutions of energy, concentration and momentum equations, the Eqs. (12)- (14) with (16) are presented in steady state: The boundary conditions are with a = kSc (See Magyari [25]) and z = d 1 y + d 2 + N t . Three cases of solutions are: Case I: when Bi 1 ≠ Bi 2 K 1 …K 4 are all constants given in the "Appendix".  While the Sherwood numbers on the two plates are given as:

Numerical solutions
The finite difference method is not only reliable but also the most efficient method for 1-dimensional flows. Our computational method is meant to be an alternative for experimental investigations, so can therefore be used in verification and validation. Taylor series expansion is used to approximate derivatives of t and y in Eqs. (12)- (14). We use centred space approximations for spatial derivative and the backwards difference approximations for time derivatives of the second order hence ensuring that Δt ≈ (Δy) 2 . We assume Sc ≠ 0 and Pr ≠ 0 , then in order to approximate the initial-boundary value problem, we introduce a rectangular mesh of points (iΔy, jΔt) in the channel [0, 1].
For our 1-dimensional case, the coefficients of (34)-(36) form a tridiagonal matrix and we call on the efficient Thomas algorithm to invert the tridiagonal matrices. The solution procedure is iterative and stops when a converged solution ∑ �Ai,j+1−Ai,j� M�A� max < 10 −5 is assumed.
Numerical integration is performed for every value of temperature, concentration and velocity, that is A i,j for internal non-boundary mesh points initiating at the most accessible mesh to the hot wall to the last mesh away from the cold wall. Tables 1 and 2 show numerical values obtained using Implicit finite difference method (IFDM) and those of analytical method (AM) at t = 1.0 , for particles with Sc = 0.6 (33)

Results and discussion
In this study, fog and dusty air were considered with Schmidt values of Sc = 0.6 and Sc = 600 respectively. The effect of thermophoretic coefficient k is considered and dimensionless time is varied in steady and unsteady states. The value k = 0 represents the absence of thermophoresis, the value k = 1.2 denotes value for water vapor or soot particles for particle diameter 0.1 < D p ≤ 1 μm . is taken as = 100 or = 1.5 which represents a buoyancy dominated flow or a typical mixed convective flow respectively. Through all the various profiles, N t = 8.0 is a value representing typical value for room temperature and N c = 2.0 represents usual concentration of particles [24]. Except where it is varied, buoyancy ratio b = 10 is assumed, suggesting a Robin boundary condition and the dominance of heat transfer. Bi is taken to represent Bi 1 = Bi 2 . The cases considered are: When Bi = 10 , Bi 1 ≫ Bi 2 and Bi 1 ≪ Bi 2 (an unequal conduction resistance of walls) and when Bi < 0.1 (when convection resistance of air to heat transfer dominates conduction resistance of both walls to heat transfer). Pressure gradient which is a resistance to fluid flow is taken as = 10. There is a decrease in reversal flow at the cold plate as thermophoresis approaches its peak, these observations are identical to results in Grosan et al. [24]. (1) absence of thermophoresis and (2) with k = 1.2 for t = 2, 4, 6 and steady state. For equal Biot numbers Bi = 10 with values of = 100 and Sc = 600 (Soot particles), it is observed that as Schmidt number increases and time increases also, the reverse flow disappears as time approaches steady state value. Observable in Fig. 2a, b, velocity is about zero at channel centerline. At the centre of channel the temperature differences about the borders cancel each other out at the centre. Comparisons of Fig. 2a-d suggest that an increase in either a combination, of one or two of Schmidt number, thermophoresis and buoyancy (high ) will change a flow from bidirectional to unidirectional. The results obtained are identical to observations in Magyari [25]. Figure 2e-h show the effect of Biot numbers on a thermophoretic velocity (k = 1.2) of particles with (Sc = 0.6) . Figure 2e shows a case Bi 1 = Bi 2 = 0.05 . The figures show that the more equal the Biot numbers are the more likely steady velocity would pass through the channel centreline. The smaller the Biot number at the hot plate, the smaller the velocity at the hot plate. The larger the Biot number at the hot plate, the larger the velocity at the hot plate. The larger the Biot number at cold plate, the bigger the velocity at the cold plate. The larger the Biot number at hot plate means the better the convection of fluid and more conduction resistance from inside the hot wall. Biot number increases flow reversal.

Velocity profiles
There is no appreciable difference between Fig. 2b, i showing Bi 1 = Bi 2 = 10 and Bi 1 = Bi 2 = 500 . There is also no appreciable difference between Fig. 2e, h showing Bi 1 = Bi 2 = 0.05 and Bi 1 = Bi 2 = 0 . In general, when Biot numbers are equal, all graphs pass through channel centreline. But when the Biot numbers are unequal, then the steady state graph doesn't pass through channel centerline i.e. velocity is not zero. Figure 2j-o show effects of buoyancy ratio (1) in the absence of thermophoresis k = 0 and (2) in the presence of thermophoresis k = 2.0 of air laden with particles of Sc = 600 with equal Biot numbers for different times. These figures together show the effects of thermophoresis, buoyancy ratio and time at a higher Schmidt number. The pairs, Fig. 2j, k; Fig. 2l, m depict the effect of a change in time from t = 0.02 to t = 0.2 on velocity profile in the absence of thermophoresis and in the presence of thermophoresis. We observe the disturbed velocity at t = 0.02 and a stable velocity at t = 0.2 . Figure 2j and l, k and m show: there is negligible effect of buoyancy on velocity of particles when thermophoretic parameter is significantly increased. Figure 2m depicts variations in buoyancy for velocity profile. It can be observed that the flow reversal increases with an increase in buoyancy. Figure 2m, n show velocity profiles when time is significantly increased from t = 0.2 to t = 2 we observe a disappearance of flow reversal as buoyancy ratio moves from positive to negative. This result suggests that occurrence of a flow reversal depends on buoyancy and most importantly time.
At small time, buoyancy has no effect on flow reversal. At large time, flows with a negative buoyancy ratio will observe no reverse flow. Negative buoyancy implies that particles are denser than the fluid and hence will control the flow in a negative direction as time passes. Hence Schmidt number, pressure gradient and buoyancy will work together. Figure 2o displays variation of buoyancy for velocity profile at a time very close to achieving a steady state velocity, showing that at steady state velocity or a time close to achieving steady state velocity, a small flow reversal will occur only when b = 0 and b = 0.5 or 0 ≤ b < 1. It is observed from Fig. 3a-e that time enhances concentration and thermophoresis has no appreciable change in (t, y) for Sc = 0.6 . Figure 3c, d show (t, y) profile for Sc = 600 . Figure 3a, c shows that when time and Schmidt number are increased simultaneously, a wider region across the channel where concentration is zero is created. Figure 3d reveals effect of a combined action of thermophoresis, Schmidt number and time: showing that when concentration is in steady state there is a sharp decline very close to the cold wall. Let 0 and 1 denote skin friction at hot and cold walls. Figure 4a, b show that both time and thermophoretic coefficient improve 0 . Figure 4a shows that the above statement is true except at steady state where 0 becomes almost constant after k = 0.7 . Figure 5a, b show that time decreases 1 . Figure 5a shows that the above statement is true except at steady state 1 . Figures show that time needs to be significantly increased to reach steady state skin friction. At steady state, thermophoresis increases skin friction at the cold wall until about k = 0.5.

Skin frictions and Sherwood numbers
Let Sh 0 and Sh 1 denote Shewood numbers at hot and cold walls. Figure 6a Sh 0 until about k = 0.5 . Figure 6b, b show a combined effect of time and thermophoresis on negative concentration gradient. Figure 7a, b show graphs for negative concentration gradient at the cold wall for two variations in time. We observe a decline in Sherwood number as time increases at low values of thermophoretic coefficient and a convergence of Sherwood number at the cold plate at higher end of the thermophoretic value. A small difference between values of Sh 1 is observed with increase in time. Generally, thermophoresis improves Sherwood number at the cooler boundary.

Conclusions
A numerical analysis of Grosan et al. [24] [25]. In general, Results from tables and graphs reveal: • Thermophoretic effect has negligible effect on water vapor particles (Sc = 0.6) relative to soot particles (Sc = 600) in relation to the velocity of the fluids and concentration of paticles. • When buoyancy dominates the flow, there is a decrease of reversal flow at the cold plate as thermophoresis approaches its peak. Hence as migration mass transfer of particles reaches its peak, the reverse flow in the opposite direction starts to slow down. • An increase in either a combination of one or two of the parameters Schmidt number, thermophoresis coefficient and buoyancy (high ) will undo a reserve flow. Hence the size of a particle, their ability to migrate and rise in the opposite direction of gravity will determine whether the flow will go upward, downward or bidirectional.
• The smaller the Biot number at the hot or cold plate, the smaller the velocity at that plate and vice-versa. The present analysis is limited to 1-dimensional transient laminar fully developed vertical flows. In the future the phenomena of electrophoresis and photophoresis will be investigated using various physical conditions. Future studies could look into new methods of solutions, investigating with various parameters or new applications for this study. Findings in the role of steady and unsteady state free convection flow in the presence of thermophoresis with Robin boundary conditions have implications in the developments of thermophoretic cells used in biochemical and medical researches. Funding No funds, grants or other support was received.

Availability of data and material Not applicable.
Code availability Codes used in the study are not provided but could be requested.