New insight into the nano-fluid flow in a channel with tempered fractional operators

While studying time fractional fluid flow problems it is typical to consider the Caputo derivative, however, these models have limitations including a singular kernel and an infinite waiting time from a random walk perspective. To help remedy this problem, this paper considers a tempered Caputo derivative, giving the system a finite waiting time. Initially, a fast approximation to a generalised tempered diffusion problem is developed using a sum of exponential approximation. The scheme is then proven to be unconditionally stable and convergent. The convergence properties are also tested on a sample solution. The fast scheme is then applied to a system of coupled tempered equations which describes the concentration, temperature and velocity of a nanofluid under the Boussinesq approximation. The most notable finding is that increasing both the fractional and tempering parameters reduces the heat transfer ability of the nanofluid system.

1. Introduction The ability to remove heat from a system is of vital importance in many areas, such as cooling and solar energy, among others [1].Most commonly used fluids, however, have poor heat conductivity, especially compared to solid materials such as metals and carbon nanotubes.Fortunately combining these materials into nanofluids has been shown to greatly improve these properties, notably thermal conductivity [2].This has inspired a number of applications such as desalination [3], oil recovery [4], cooling of electronic equipment [5], solar energy [6,7], and other applications [8].These factors have caused nanofluids to become a topic of intense research in recent years.Nanofluids are created by mixing one or more nanometre-scale particles with desirable properties into a base fluid.The most common base fluids considered in nanofluids are water, ethylene glycol, or some type of oil [9].And the most common classes of nanoparticles used are metals, metal oxides and carbon nanotubes (both single and multiwalled) [10].Depending on the chosen nanoparticle and the volume fraction of it introduced to the base fluid, the physical properties of the nanofluid can be radically changed.These property changes depending not only on the amount of nanoparticle in the solution but also on the shape, chemical reactions, distribution, and other effects of the nanoparticle.Exactly how they behave has been the subject of intense research, and many models and approximations have been proposed for thermal conductivity [11], electrical conductivity [12], density [13], heat capacitance [14], viscosity [15], and other thermophysical properties.
Fractional derivatives have been found used in many study areas [16,17], including finance [18], medical imaging [19], as well as many physical systems viscoelasticity [20,21], quantum mechanics [22], and anomalous diffusion [23].Especially focused on problems with complex domains, in which the dynamics are more conducive to memory and non-local effects.The intersection between these two areas, fractional nanofluid flow, has been intensely studied recently and is the primary interest of this paper.All of these models consider fluid velocity, however, it is common to incorporate temperature and occasionally nanofluid concentration equations and coupling terms [24,25].These allow a more complete physical model of the system, especially when an application is being considered, with the price of the ease of finding solutions.Most of these models replace the time or space derivatives of the integer order model with a single fractional equivalent to incorporate non-local or memory effects into the models.These typically include the Riemann-Liouville [26], Riesz [27], or the fractional Laplacian [28] for the spatial component, and the Riemann-Liouville [29], Caputo [30], Caputo-Fabrizi [31], Atangana-Baleanu [32] for the time derivatives.Also, more complex multi-term [33] or distributed order [34] fractional diffusion problems and nanofluid models have been studied, which incorporate fractional derivatives into the derivation, such as fractional Oldroyd-B nanofluids [35].
Of particular interest are the tempered derivatives, a generalisation of the regular fractional derivative.The tempered derivative introduces a new parameter which, from a random walk perspective, reduces the particle waiting time [36].This is important, especially for fluid flow contexts, where the infinite wait times of the regular Caputo fractional derivatives [37] have awkward physical interpretations.It has been shown that this allows the tempered Caputo derivative to effectively describe experimental data [38,39].
A variety of solution methods for fractional nanofluid flow have been developed.Analytic solutions for fractional nanofluids in one dimension have been found by using the Laplace transform [30], however, these solutions are unwieldy and highly problem specific.A more tractable approach is to by solving the Laplace transformed ordinary differential eqution before numerically inverting to create semi-analytic solutions [40].The most common solution type for fractional nanofluid models is to derive a numerical scheme, and many methods have been used to approximate a solution.Most often finite difference methods for operators such as the L1 [41], L2 1−σ [42], Grunwald-Letnikov [26,43] have been used.Finite element methods have been used [25], especially in problems with higher numbers of spatial dimensions on complex domains.
Tempered fractional diffusion problems, in both time and space, have been studied in the literature.And many fractional nanofluid models also have been reported.However, to the best of our knowledge, no work has been done on tempered fractional nanofluid flow.Therefore, we aim to develop a system of tempered fractional PDEs and solve them using a newly developed fast approximation of the tempered derivative [44,45] to effectively deal with their highly coupled nature.
The main novel contributions of this paper are: 1.A new tempered time-fractional Nano-fluid flow accounting for the memory effects in a one-dimensional channel is proposed, which circumvents the infinite characteristic waiting times for the general time-fractional derivatives from a continuous random walk perspective.2. The finite difference method together with a fast algorithm is applied to solve the coupled Nanofluid flow system, which can reduce the computation time significantly.The rigorous theoretical analysis of the numerical scheme is established.3. The numerical method is validated by a numerical example and the influence of the important physical parameters on the concentration, velocity and temperature fields is investigated.The main findings are that increasing both the fractional and tempering parameters reduces the heat transfer ability of the nanofluid system, larger volume fractions of nanoparticles increase the ability of the nanofluid to transfer heat, and increasing the mean fluid velocity increases the heat transfer capabilities of the system.This paper will have the following structure.In section 2, a tempered fractional model of nanofluid flow is conducted.Section 3 proposes a fast numerical scheme for the tempered fractional diffusion problem.The convergence of the numerical scheme is verified in section 4, then applied to the nanofluid model with a wide variety of examined parameters.In section 5 the overall conclusions of the paper are presented.The stability and convergence of the scheme are established in section appendix.

Nanofluid model
The coupled system of PDEs under consideration describes the nanofluid flow in a one-dimensional channel by using the Boussinesq approximation of the Navier-Stokes equation, similar to [30] and [46].Consider the nano-fluid flow in a thin channel with left wall x l and right wall x r , where x is the normal direction in the Cartesian coordinate system and the y axis lies along the centre of the channel [46].A uniform magnetic field of intensity B is imposed along either the x axis or z axis, inducing a Lorentz force.The concentration (C) is driven by diffusion while temperature (T) is affected by diffusion and radiative heat flux.The fluid velocity (U) itself diffuses in a porous material with a perpendicular magnetic field.External acceleration is due to an applied pressure gradient (P(t)) and coupling with concentration and temperature.The governing equations of the tempered fractional fluid flow are then [30] ( ) ( ) 3 with constant initial conditions and Dirichlet boundary conditions imposed at the left (x = x l ) and right (x = x r ) walls of the channel The viscosity (μ nf ), density (ρ nf ), electrical conductivity (σ nf ), thermal expansion coefficient (β Tnf ), thermal conductivity (k nf ), and heat capacity (c nf ) are defined similarly to [47].
Although in principle any model depending only on the nanoparticle volume fraction (f) can be used.Here the subscripts represent the base fluid ( f ), nanoparticle (n), and nanofluid (nf ) respectively.We choose [47] ( ) To simplify the governing equations somewhat the following nondimensionalisation was used where L = x r − x l , alongside any other specific nondimensionalisations which are required for a chosen pressure gradient.In this paper, we investigate the case of the nanofluid flow with memory effects, which means the stress on a fluid element depends on the history of the deformation imposed on that element.The integer order time derivatives are replaced with the tempered Caputo derivative in each equation by introducing a fractional parameter α and a tempered parameter p. Rearranging then gives the fractional PDEs of interest with the nondimensional boundary and initial conditions where the resulting nondimensional coefficients are given by ¯¯( ) The tempered Caputo derivative is a generalisation of the regular Caputo fractional derivative, which has two parameters: the fractional order α (characterising the memory) and the tempering parameter p.It is important to note the limiting cases: as p → 0 the normal Caputo fractional derivative is recovered, and in the case that p → 0 and α → 1 the regular derivative returns.It can be defined by its relationship to the traditional Caputo derivative as in [44] ( ) Notably the tempered derivative of a constant is not zero, but can be found easily by using the Caputo derivative of the exponential [48].This is where the new source terms in the concentration (2.13) and temperature (2.12) equations originate where E a,b (t) is the two-parameter Mittag-Leffler function Each equation has a corresponding number which is commonly used to study transport phenomena between the boundaries and fluid.In this particular system of PDEs, there are two boundaries and two associated numbers, in the nondimensional form: the Sherwood number (Sh) for concentration [30] ( ) the Nusselt number (Nu) for temperature [30] ( ) and the skin friction coefficient (Sk) for velocity [30] ( ) ( ) ( ) It is important to note that these depend primarily on the internal dynamics of the channel at the boundary and not directly on the boundary conditions.

Numerical method
The numerical method proposed in this section is designed to solve tempered Caputo problems with greatly improved speed when compared to traditional finite difference methods for fractional operators, with a minimal loss of accuracy.To solve the nano-fluid flow equations, observing that each equation individually has the same general form Therefore, the finite difference scheme which solves this general equation can be applied repeatedly to solve the coupled system (2.12)-(2.14).It shoud be noted that dividing this by e − pt and making appropriate substitutions will convert this to a regular Caputo derivative, of which a large variety of solution methods are known.But a direct approximation of the tempered Caputo derivative is chosen because it is more applicable to other, more general, problems.
Firstly, a uniform discretisation of time and space is chosen, which also allows the numerical solution and error to be defined ( ) The second order space derivative is discretised using the standard central difference method The general form (3.1) can then be placed into a semi-discrete matrix-vector form where ( ) To discretise the tempered Caputo derivative, a fast finite difference method is chosen due to the high coupling of the nano-fluid flow model (2.12).This approximation is developed for the regular Caputo derivative in [45] and extended to the tempered Caputo derivative by [44].The fast finite difference method operates by splitting the tempered Caputo derivative into a local part on [t k , t k+1 ] and a historical part on [0, t k ].The local part is approximated similarly to the L1 method for the Caputo derivative.And the historical part uses a sum of exponentials approximation to remove the singular integral from the tempered Caputo derivative.The sum of exponentials approximation can be calculated using the code from [45], which allows the approximation to be arbitrarily precise at the cost of additional summation terms and weights (w j , s j ) By using this approximation, and integration by parts, the singular integral in the tempered Caputo derivative can be replaced with a recursion relationship, reducing the number of timesteps required to calculate the next timestep from all previous timesteps to merely the previous two timesteps and the initial condition, which means that the fast approximation has vast memory and speed improvements over a direct approximation.The approximation is split into two cases.The very first timestep, k = 0, is And for k 1, the historical component (H) has to be updated at each timestep before solving the for the next timestep.

H H F
F H e e q q e q e q 0 1 1 , , 3 .9 where , and q j = p + s j .Thus applying this fast finite difference approximation to the semidiscrete equation (3.5) and rearranging gives two systems of linear equations to solve for the next timestep.At k = 0 And for every subsequent timestep, k 1

Validation of the numerical method
Theorem 6 shows the fast scheme to be convergent, with an expected convergence accuracy of O(h 2 + τ 2− α + ò).To validate this, a convergence test in time was undertaken.The convergence order was calculated with the space step (h = 1/ 2000) and the sum of exponentials parameter (ò = 10 −24 ), both chosen to be far smaller than the timestep.Therefore, in order to test the convergence order of the fast tempered algorithm the following analytic solution and corresponding source term were chosen where A = D = 5 was chosen.Six tests were run for various α and p values (see tables 1 and 2).It can be seen that the temporal errors of the numerical scheme converge with the theoretical order 2 − α, which verifies the correctness and effectiveness of the numerical method.

Nanofluid behaviour
Next, we apply the proposed method to a system of coupled tempered equations which describes the concentration, temperature and velocity of a nanofluid under the Boussinesq approximation.A copper-water nanofluid was used for these simulations, and the following fluid and particle properties (table 3) were applied to the nanofluid property models (2.11)-(2.13).These were assumed not to vary with velocity, temperature, or concentration so that the effect of the tempered Caputo derivative and the volume fraction could be focused on.This flaw is found in most nanofluid research and more accurate nanofluid models which vary with velocity, By using these values with the nanofluid models (2.8) all of the nanofluid properties used in this paper can be calculated (table 4).
In addition, except when being examined, the free parameters were chosen to be as in table 5.
All results were calculated using the fast numerical method with timestep τ = 0.0005, spacestep h = 0.0005, and the sum of exponential approximation tolerance ò = 10 −12 .Unless otherwise noted.It is important to note that whenever p = 0 has been chosen those solutions are of the same class of solutions to regular Caputo derivative problems as studied in [30], however, the default parameters were chosen differently from this source to more clearly visualise the effect that p has on the velocity, temperature, and concentration profile in this paper.
4.2.1.Concentration.In general, the behaviour of the concentration profile is characterised by a rapid approach to a linear profile between the two boundary conditions.The fractional parameter α has a pronounced effect, increasing the concentration profile with a lower value (figure 1).While the tempering parameter p has a less noticeable effect, decreasing the profile as it increases (figure 2).The Schmidt number and mean velocity both decrease the concentration profile as they increase, as expected (figures 3 and 5).The effect of f is relatively small, an increase corresponds to a decrease in the height of the concentration profile (figure 4).Even though the volume fraction has a significant effect on the values of density and viscosity individually.
The Sherwood number gives insight into the behaviour of the nanofluid at the boundary because it is directly analogous to the Nusselt number (temperature), and skin friction coefficient (velocity).Additionally, because the concentration governing equation is quite simple the Sherwood number at both boundaries correlates well with the internal behaviour of the concentration profile.The Sherwood number is affected by the fractional parameter, with a decrease of α resulting in a significantly flatter profile which is higher at the left boundary and lower at the right (figure 6).At the left boundary, the Sherwood number increases with a decreasing tempering parameter, while the opposite is true at the right boundary (figure 7).Increasing f increases the Sherwood number at both boundaries (figure 8).In all cases, as time increases the behaviour of the Sherwood number asymptotes to a constant value as the behaviour at the boundaries stabilises.The effect of the tempered fractional derivative on the temperature profile is very similar to the concentration profile.Decreasing α increases the temperature profile (figure 9).While increasing the tempering parameter reduces the profile (figure 10).As mentioned previously higher mean fluid velocity carries heat away from the channel, reducing the height of the temperature profile (figure 11).Increasing the absorption coefficient (λ) gives a proportional increase to the temperature profile as can be seen in the governing equation (2.12), even to above the temperature at the right boundary (figure 12).This could potentially have ramifications for cooling applications as heat can flow back out of the channel and into the hot wall (x = 1).The volume fraction has a mild effect, counterintuitively reducing the height of the temperature profile (figure 13).For a different base fluid/nanoparticle combination or different nanofluid models, it is conceivable that f may have a more noticeable effect on the temperature profile.
However, the volume fraction has a significant effect on heat transfer at the boundaries.The effect of the tempered Caputo derivative on the behaviour of the Nusselt numbers is quite similar to the Sherwood numbers.As α approaches one 0.5918 401 [50] (figure 14) it increases the magnitude of the Nusselt number at the left wall but increases it at the right wall.An increasing tempering parameter decreases the Nussult number at the left wall and increases it at the right wall (figure 15).As can be seen in (figure 16) f has a significant impact on the Nusselt number.An extreme one at the left boundary and doubling the magnitude between f = 0 and f = 0.2.Thus having a higher volume fraction in a copper-water nanofluid is a critical factor in transferring heat from the channel walls.So increasing the tempering and fractional parameters will improve heat transfer at the right wall.At the right boundary, the Nusselt number asymptotes to a constant as time increases, so eventually a regular amount of heat flux occurs into the fluid from the boundary.

Velocity.
In both the governing equation (2.11) and numerical scheme (3.10), (3.11) the pressure gradient can be an arbitrary function of time.So for this section, an oscillatory pressure gradient ( ( ) ( ) = P t t sin ) was chosen.The general shape of the velocity profile, for mild pressure gradients, is a parabola between the boundary conditions with a slight lean towards x = 1 (figure 18), caused by the coupling components with temperature and concentration in the governing equation (2.11).As the fractional parameter α increases the height of the velocity profile Table 4.The properties of the nanofluid for specific values of f.Note the f = 0.2 pushes the limit of many of these nanofluid models.decreases (figure 17).Similarly, increasing the tempering parameter significantly decreases the height of the profile (figure 18).While it may seem counterintuitive the shape of the velocity profile is highly dependent upon the mean velocity (figure 19), with a mean higher velocity resulting in a lower, flatter profile.This is due to a combination of a lower mean velocity increasing the temperature (figure 5) and concentration (figure 11) profiles, thus increasing the coupling terms And a lower velocity decreases the Reynolds number, which improves diffusivity.Introducing nanoparticles to the base fluid increases the viscosity of the nanofluid; so as expected increasing f reduces the height of the velocity profile and increases the prominence of the coupling effects (figure 20).The porosity of the channel only has a mild effect upon the velocity profile, with values of κ approaching very close to zero having a significant difference from unhindered flow (figure 21).A similar effect is seen with the applied magnetic field strength (B), as unrealistically strong values are required to have a visually distinct effect (figure 22).This does not mean that these effects are always insignificant, just that specifically for a water-copper nanofluid using these specific nanofluid models.It is entirely conceivable that a more viscous or conductive base fluid/nanoparticle combination could vastly increase the contribution of these two effects.The solutal expansion coefficient (β C ) has a positive correlation with the height of the velocity profile (figure 23), because it directly controls the coupling between concentration and velocity.It is important to note that the thermal expansion coefficient is not a free variable, as it has its own nanofluid model (2.8) and its contribution is implicitly linked to that of f.Because the pressure gradient can be arbitrarily chosen it is important to examine a variety of potential behaviours, specifically how the long-term behaviour is caused by the tempering parameter.Two gradients were chosen for their ubiquity.P(t) = 0, otherwise known as natural convection, in which the only external forces acting on the velocity profile is the coupling with temperature and concentration.As can be seen in figure (24) as time continues the velocity profile rapidly approaches a steady state with a characteristic lean caused by the coupling with temperature and concentration.Interestingly the untempered version (figure 25) exhibits long-term growth caused by the coupling which allows it to reach a far larger magnitude.
The pressure gradient is not restricted to be positive and can even be strongly negative enough to reverse the flow of the nanofluid.An oscillatory pressure gradient, ( ) ( ) = P t t sin , was picked to investigate how a sometimes negative gradient interacts with the tempering parameter.As seen in (figure 26) the sinusoidal pressure gradient evolves the velocity profile in step, even driving the velocity profile negative.Notably, the magnitude of the peaks remains roughly the same.This behaviour is quite different from the untempered problem (figure 27) which approximates untempered natural convection at the peaks, but with troughs corresponding to the negative parts of the pressure gradient.
In general, regardless of the pressure gradient chosen, having a nonzero tempering parameter has two effects on the velocity profile.Firstly tempering the system suppresses the magnitude of velocity profile, which is in accordance with the test varying the tempering parameter at the same timestep (figure 18).The second effect of tempering the system is that it homogenises the long-term behaviour of the system.As it appears that unless the pressure gradient has some form of    long-term growth to counter the tempering parameter the velocity profile approaches something akin to the long-term behaviour of the untempered natural convection problem.So it can be said that tempering a system significantly reduces the velocity of the fluid.
As the envisioned applications of this type of fractional PDE involve moving heat away from the boundary, the behaviour of the velocity at the boundary is very important.As a larger velocity in the immediate neighbourhood of the boundary indicates the fluid can move colder fluid into contact with the surface faster, thus more efficiently transferring heat away from the surface.For this reason, the skin friction coefficient for was examined at both walls, especially its interaction with the tempering parameter.
For this velocity profile both boundary conditions are zero, therefore the pressure gradient and temperature and concentration coupling cause the velocity to take a vague parabolic shape.This causes the space derivatives at the left wall to have a positive sign and a negative sign at the left wall.So the skin friction coefficients at each wall behave similarly but have the opposite sign, the difference between the two is caused by concentration and temperature coupling.
The fractional parameter α consistently decreases the magnitude of the skin friction coefficient as it increases at x = 0, and increases the magnitude at x = 1.This effect holds for most times (figure 28), and is most likely a holdover from the behaviour of the Sherwoods and Nusselt numbers due to coupling.So the closer to the integer order problem the system is the more effective it will be at transferring heat further down the channel.
The tempering parameter has the same effect on the skin friction coefficient at both walls.Reducing the magnitude as p    increases (figure 29).The nanofluid volume fraction.Similarly increasing the nanofluid volume fraction f will increase the magnitude of the skin friction coefficient at both boundaries (figure 30).

Conclusions
In this paper solutions for a coupled tempered Caputo fractional nano-fluid flow system, derived using the Boussinesq approximation, were studied.Concentration, temperature and velocity profiles were considered, with the temperature and concentration equations coupled to the velocity.Additional effects considered were radiative heat flux, flow in a porous medium, the Lorentz force with a perpendicular magnetic field, and a variety of pressure gradients with varied behaviours.The properties of the nanofluid used simple models which depend only on the volume fraction, and are valid for low-volume fractions.
The solution method proposed used a fast finite difference approximation of the tempered Caputo derivative and the standard central difference method for the second derivative.This numerical scheme has a truncation error of O(h 2 + τ 2− α + ò).The scheme was proven to be unconditionally stable and convergence tests agreed with the theoretical convergence rate.Additionally, solutions using this scheme were used to approximate the Sherwood number, Nusselt number, and skin friction coefficient.
The results were calculated using a copper-water nanofluid.Extensive graphical analysis for the behaviour of the concentration, temperature, and velocity profiles was conducted, with a focus on the effects of the pressure gradient and tempering parameter.As the problem was aimed towards applications such as cooling and solar energy the primary conclusions of this paper are:          • Reducing the fractional parameter α increases the ability of the system to transfer heat.• Increasing the tempering parameter p reduces the ability of the system to transfer heat.• Larger volume fractions of nanoparticle f increase the ability of the nanofluid to transfer heat.• Increasing the mean fluid velocity Ū increases the heat transfer capabilities of the system.• Extreme conditions are required for porosity and MHD effects to have a significant impact on a copper-water nanofluid.
In the future, this fast numerical technique could be expanded in a variety of ways.For example, higher spatial dimensions, nonlinear coefficients and nano-fluid models, more advanced boundary conditions, and considering effects that introduce even more extreme coupling.Additionally,     because this study considered the time fractional tempered version of this problem it is natural to consider the space fractional tempered version of the problem, and perhaps eventually the space-time fractional tempered problem.
, h , we define the following discrete inner products and induced norms: where which is equivalent to . Multiplying hu i k on both sides of (A9) and summing up for i from 1 to - N 1 leads to Multiplying τ on both sides of (A11) and summing up for k from 1 to n derives Utilising lemma appendix A.   Noting that approximating the historical part of the tempered Caputo derivative has a higher order error.Thus, the local part is the only error term (with respect to the timestep), which is not negligible., Lemma appendix A.5.Using (3.6) it is easy to show that the truncation error from the sum of exponentials approximation is first order with respect to the tolerance.where c is a positive constant independent of h and τ.

ORCID iDs
Libo Feng https:/ /orcid.org/0000-0002-1320-7946 Nomenclature B magnetic field intensity [kg/s 2 /A] c nf heat capacity of the nanofluid [J/K] C concentration of the fluid [kg/m 3 ] C l concentration at the left wall [kg/m 3 ] C r concentration at the right wall [kg/m 3 ] D nf mass diffusivity [m 2 /s] g gravitational acceleration [m/s 2 ] k nf thermal conductivity of the nanofluid [W/ m/K] Nu Nusselt number [-] p tempered parameter [-] P(t) pressure gradient [N/m 3 ] Sh Sherwood number [-] Sk skin friction coefficient [-] T temperature of the fluid [K] T l temperature at the left wall [K] T r temperature at the right wall [K] U fluid velocity in the x direction [m/s] Greek Symbols α fractional order [-] β C the volumetric coefficient of mass expansion [m 3 /kg] β T the volumetric coefficient of thermal expansion [1/K] ò error precision [-] κ constant parameter [-] Nanotechnology Nanotechnology 35 (2024) 085403 (17pp) https://doi.org/10.1088/1361-6528/ad0d24* Author to whom any correspondence should be addressed.Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.

4. 2 . 2 .
Temperature.Due to the quite low thermal conductivity and large heat capacity of water the ability for heat transfer away from the boundary before being removed by the perpendicular fluid flow is low.This results in the shape of the temperature profile changing slowly (relative to the concentration profile) as time progresses.And for most reasonable parameters the majority of the extra heat in the channel is trapped in the boundary layer near (x = 1).

Figure 2 .
Figure 2. Effect of the tempering parameter (p) on the concentration profile (t = 0.1).

Figure 3 .
Figure 3.Effect of the Schmidt number on the concentration profile (t = 1).

Figure 4 .
Figure 4. Effect of the volume fraction of the nanoparticle (f) on the concentration profile (t = 0.1).

Figure 7 .
Figure 7.The effect of the tempering parameter (p) on the Sherwood number at both boundaries.

Figure 6 .
Figure 6.The effect of the fractional parameter (α) on the Sherwood number at both boundaries.

Figure 10 .
Figure 10.Effect of the tempering parameter (p) on the temperature profile (t = 0.75).

Figure 11 .
Figure 11.Effect of the mean fluid velocity ( Ū ) on the temperature profile (t = 1).

Figure 8 .
Figure 8.Effect of the volume fraction of the nanoparticle (f) on the Sherwood number at each wall.

Figure 13 .
Figure 13.Effect of the volume fraction of the nanoparticle (f) on the temperature profile (t = 1).

Figure 14 .
Figure 14.The effect of the fractional parameter (α) on the Nusselt number at both boundaries.

Figure 15 .
Figure 15.The effect of the tempering parameter (p) on the Nusselt number at both boundaries.

Figure 16 .
Figure 16.Effect of the volume fraction of the nanoparticle (f) on the Nusselt number at each wall.

Figure 18 .
Figure 18.The effect of the tempering parameter (p) upon the velocity profile (t = 1).

Figure 19 .
Figure 19.Dependance of the shape of the velocity profile upon the mean velocity (t = 1).

Figure 21 .
Figure 21.The effect of the porosity of the channel (κ) upon the velocity profile (t = 1).

Figure 22 .
Figure 22.The effect of the magnetic field (B) upon the velocity profile (t = 1).

Figure 20 .
Figure 20.The effect of the volume fraction of nanoparticle (f) on the velocity profile (t = 1).Figure 23.Dependance of the shape of the velocity profile upon the solutal expansion coefficient (β) (t = 1).

Figure 23 .
Figure 20.The effect of the volume fraction of nanoparticle (f) on the velocity profile (t = 1).Figure 23.Dependance of the shape of the velocity profile upon the solutal expansion coefficient (β) (t = 1).

Figure 29 .
Figure 29.Effect of the tempering parameter (p) on the skin friction coefficient.( ) ( ) = P t t sin .

Figure 30 .
Figure 30.Effect of the volume fraction (f) on the skin friction coefficient.( ) ( ) = P t t sin .

Table 1 .
The error and convergence order of the untempered (p = 0) example problem for a variety of α.

Table 2 .
The error and convergence order of the tempered (p = 1) example problem for a variety of α.

Table 3 .
Particle and fluid properties used.

Table 5 .
Free parameters used to calculate results.
Thus the fast difference scheme is unconditionally stable with respect to the boundary condition, initial condition, and source term.,A.2.ConvergenceLemma appendix A.3.The error bound of the centered difference approximation is well known and is easy to find when using the Taylor series It is straightforward to derive Proof.
Similar to the derivation in theorem appendix A.2, we derive