Mathematics in Applied Sciences and Engineering https://ojs.lib.uwo.ca/mase
Online First, pp.1-12 https://doi.org/10.5206/mase/14626

MACROSCOPIC ANALYSIS OF THE VISCOUS-DIFFUSIVE TRAFFIC FLOW

MODEL

GABRIEL OBED FOSU, ALBERT ADU-SACKEY, AND JOSEPH ACKORA-PRAH

Abstract. Second-order macroscopic traffic models are characterized by a continuity equation and

an acceleration equation. Convection, anticipation, relaxation, diffusion, and viscosity are the predom-

inant features of the different classes of the acceleration equation. As a unique approach, this paper

presents a new macro-model that accounts for all these dynamic speed quantities. This is done to

determine the collective role of these traffic quantities in macroscopic modeling. The proposed model

is solved numerically to explain some phenomena of a multilane traffic flow. It also includes a linear

stability analysis. Furthermore, the evolution of speed and density wave profiles are presented under

the perturbation of some parameters.

1. Introduction

The main classifications of macroscopic traffic models are the first-order and second-order models

[13, 21, 20, 36]. Second-order equations were introduced to overcome the shortcoming of the first-order

equation [6, 30]. The first-order category is also known as the continuity or LWR equation. The

second-order branch encompasses the first-order equation together with a dynamic velocity equation.

Oftentimes the dynamic velocity equation is also called the acceleration or momentum equation. Payne

[31] set forth the precedence with his classical dynamic velocity equation with little revision by Whitham

[37] some few years afterwards to form the Payne-Whitham (PW) model. The constitutive terms of the

acceleration equation are convection, anticipation, relaxation, diffusion, and viscosity. A review of these

dynamic terms as either accounted or unaccounted for within a given model formulation is presented

in Table 1.

From the reviewed models in Table 1, it was observed that convection, anticipation, and relaxation

are always present, but that of diffusion and viscosity are barely modeled. As a novel formulation, all

these terms are brought together to form a new momentum equation.

The proposed momentum equation is coupled with the continuity equation to form a system of partial

differential equation. Note that each of these terms under consideration describes some realistic traffic

phenomena, and hence the need to examine their contributive role in vehicular traffic analysis. The

definition and mathematical representation of these terms are detailed in the ensuing paragraphs.

In a traffic sense, the convection or transport term explains the movement of vehicles along with

their density/velocity profiles. (A density profile in the case of the continuity equation, and a velocity

profile for that of the acceleration equation). These are respectively defined as

q′(x,t)
∂k(x,t)

∂x
and v(x,t)

∂v(x,t)

∂x
(1.1)

Received by the editors 24 January 2022; accepted 5 July 2022; published online 9 July 2022.

2010 Mathematics Subject Classification. 35Q35, 65M06, 35L40.

Key words and phrases. Second-order model, viscosity, multilane flow, diffusion, speed-density profiles.

1

https://ojs.lib.uwo.ca/mase
https://dx.doi.org/https://doi.org/10.5206/mase/14626


2 G. O. FOSU, A. ADU-SACKEY, AND J. ACKORA-PRAH

Table 1. Constituents of the Acceleration Equation

Author(s) Convection Anticipation Relaxation Diffusion Viscosity

vvx (·)kx or (·)vx (·)(Ve −v) (·)vxx (·)vy
[31] X X X × ×
[23] X X X × ×
[25] X X X X ×
[28] X X X × ×
[16] X X X × X
[26] X X X × ×
[29] X X X X ×
[34] X X X × ×
[38] X X X X ×
[39] X X X × ×
[40] X X X × ×
[14] X X X × X

k(x,t),v(x,t) and q(x,t) denotes traffic density, speed, and flow rate in that order. The term v(x,t) ·
∂v(x,t)/∂x is to ensure that vehicles do not adjust their speeds in a haste manner. Rather, they are to

gradually change their speed to that of the downstream traffic and describes the coordination between

the speeds upstream and downstream.

The anticipation term describes the dispersion effect of the speed of heterogeneous traffic. That is,

how identical driver-vehicle react to flow conditions in a close neighborhood, more specifically frontal

effects. Drivers usually slow down when approaching heavy traffic downstream. The term is a diverging

point for most Navier-Stokes-like traffic models. In most Payne-like models, anticipation is modeled by

1

2τk

dVe(k)

dk

∂k

∂x
or −

1

k

dP(k)

dk

∂k

∂x
(1.2)

where P(k) = −Ve(k)/2τ is the pressure term which describes how drivers perceive ahead and adjust
to the density downstream. The anticipation term is the core component contributing to the backward

traveling wave of Payne-like models. Because of this critique by [9], Aw and Rascle postulated a new

pressure term of the form P(k) = kγ with γ > 0 [3, 32]. However, the AR model was found to be

ill-posed for a virtual road [27]. Besides, [23] in their bid to correct the deficiency of negative wave

speed replaced the density gradient term with a velocity gradient component, hence defined anticipation

as

c(k)
∂v(x,t)

∂x
(1.3)

The authors specifically used constant c > 0 in place of the functional c(k) to denotes the propagation

speed of some perturbations.

Another important characterization of traffic is through the relaxation term. This is sometimes called

the speed adaptation term. It details how vehicles adapt their speed to the steady-state speed. Given

the average speed v(x,t) and Ve(k) the steady-speed as a function of density, then the relaxation term

can be defined as
Ve(k(x,t)) −v(x,t)

τ
(1.4)

This explains how vehicles at average speed could adjust to the situational density-dependent speed.

The quantity τ > 0 is the relaxation time for the entire process of adapting to the localized speed. It



VISCOUS-DIFFUSIVE TRAFFIC FLOW MODEL 3

is a constant that does not depend on either speed or density. Under normal condition, it is estimated

to be between 20 to 30 seconds for highway traffic and something less for urban flow [36].

On certain occasions, the diffusive term

D
∂2v(x,t)

∂x2
(1.5)

is introduced as part of the macroscopic equations. It was earlier fused into the acceleration equation

[25, 29], and later as part of the LWR model [35]. The same expression (Equation 1.5) is used in

both the LWR and acceleration equations but may differ in the relevant application. The entire term

(1.5) explains how vehicles adjust their velocities to surrounding traffic conditions. It was introduced

to smooth shocks and abrupt transitions of traffic between different regimes. That is, smoothing the

density profiles in the event of a diffusive first-order model, and smoothing the velocity profile for a

diffusive acceleration equation. The diffusive term enhances the numerical properties of the model and

also eradicate shockwaves if there is the existence of numerical instabilities.

The last term to discuss is the lateral viscosity term. In the immediate recent past, [16] introduced

lateral viscosity to explain traffic resistance on a multilane highway. The term was deduced from the

no-slip condition of fluids and is given as

sd
µ2

k(x,t)

∂v(x,t)

∂y
(1.6)

where µ2 is the lateral viscous rate, sd is to model the sensitivity of safe vertical distance between

vehicles moving on neighboring lanes in the same direction. The velocity gradient ∂v(x,t)/∂y account

for the speed variations with respect to changes in usage of road lanes.

All these terms together with the local derivative ∂v(x,t)/∂t yields a generalized equation of the

form

∂v(x,t)

∂t
+ v(x,t)

∂v(x,t)

∂x
= f

(
c(k)

∂v(x,t)

∂x
,c2o(k)

∂k(x,t)

∂x
,
Ve(k) −v(x,t)

τ
,D

∂2v(x,t)

∂x2
,φ
∂v(x,t)

∂y

)
(1.7)

f is an arbitrary function, c2o(k) = −
1

2τk(x,t)

dVe(k)

dk
and φ = sd

µ2
k(x,t)

.

Equation (1.7) is decoupled as

∂v(x,t)

∂t
+ v(x,t)

∂v(x,t)

∂x
=
Ve(k) −v(x,t)

τ
− c2o(k)

∂k(x,t)

∂x
+ D

∂2v(x,t)

∂x2
−φ

∂v(x,t)

∂y
(1.8)

and

∂v(x,t)

∂t
+ v(x,t)

∂v(x,t)

∂x
= c

∂v(x,t)

∂x
+
Ve(k) −v(x,t)

τ
+ D

∂2v(x,t)

∂x2
−φ

∂v(x,t)

∂y
(1.9)

Equation (1.8) has a density-gradient anticipation term, while (1.9) has a velocity-gradient anticipation

term. Equation (1.8) together with the continuity equation is the isotropic viscous-diffusive model. It

connotes an extension of all Payne-like models. Equation (1.9) is the anisotropic correspondent.

As stated in Table 1, earlier formulations are devoid of either the diffusion or viscous term. As

such, most recent model presentations and analyses also fellow suite [2, 4, 7, 17, 27, 41] with few

considering the effect of stochasticity [4, 42]. Concerning traffic stability, the steady-state condition of a

viscous second-order macroscopic traffic flow model was performed by [1]. These authors obtained the

equilibrium points, the stability criterion, and the phase plane solution of the two velocity difference

model (TVDM) [18], which is considered as a simple extension the classical anisotropic macro model.

Similarly, [5] conducted a similar analysis to determine the global stability and the bifurcation of a



4 G. O. FOSU, A. ADU-SACKEY, AND J. ACKORA-PRAH

second-order continuum model. The authors observed the presence of subcritical Hopf bifurcation for

their derived macroscopic model. On the other hand, [8] established that the impact of viscosity under

a stable traffic condition is approximately zero. These authors used a series of proofs to confirm this

near-zero assertion.

This paper also presents a graphical simulation with a stability analysis to determine the effect of

both viscosity and diffusion for a multilane traffic flow. The next section begins with a derivation

of the instability criterion of a proposed anisotropic model. Specifically, a linearization analysis to

determine either stable or unstable traffic flow is presented. It is followed by a numerical simulation;

investigating the potency of the model to reproduce some relevant flow phenomena. These simulations

are presented within the domain of a multilane infrastructure. The final part of this paper is reserved

as the concluding section. However, the analysis concerning the proposed density-gradient model could

be considered as future research work.

2. Derivation of Instability Criterion

The stability criterion of the proposed model is determined using the linearization technique. Assum-

ing a homogeneous solution k(x,y,t) = ke and v(x,y,t) = Ve(k). Any deviation from these stationary

solutions are given as

δk = k(x,y,t) −ke, and δv = v(x,y,t) −Ve(k) (2.1)
Consequently, the viscous-diffusive anisotropic macroscopic traffic flow model is linearized as

∂(δk)

∂t
+ Ve

∂(δk)

∂x
+ ke

∂(δv)

∂x
= 0

∂(δv)

∂t
+ Ve

∂(δv)

∂x
− c

∂(δv)

∂x
=

1

τ

(
dV

dk
·δk − δv

)
+ D

∂2(δv)

∂x2
−φ

∂(δv)

∂y

(2.2)

By carefully examining vehicle trajectories, it is realized that traffic behaves wave-like. Due to this

proposition, the propagation of disturbance of flow can be inferred from the theory of waves. Therefore,

the underlying simple wave functions (2.3) is adopted to probe whether a disturbance will escalate or

decay over time.

δk = k̂ exp[is1x + is2y + (λ− iω)t] and δv = v̂ exp[is1x + is2y + (λ− iω)t] (2.3)

s1,2 are the spatial wave-numbers. These delimitate the wavelength along the longitudinal and lateral

axis respectively, ω is the wave frequency, λ is the wave dumping, k̂ and v̂ are the amplitudes at some

time t.

Further, equation (2.3) and its derivatives are substituted into (2.2). Note that higher order terms

are not considered in subsequent computations. The arrived simplification is

k̂(λ− iω)M̃ + iVesik̂M̃ + ikes1v̂M̃ = 0

v̂(λ− iω)M̃ + (Ve − c)is1v̂M̃−
M̃
τ

(
dVe
dk

k̂ − v̂
)

+ Ds21v̂M̃ + is2v̂M̃ = 0
(2.4)

where M̃ := exp[is1x + is2y + (λ− iω)t] 6= 0. Equation (2.4) is then represented in its vector form as
 λ̃ ikes1
−

1

τ

dVe
dk

λ̃ +
1

τ
− ics1 + iφs2 + Ds21


[k̂

v̂

]
=

[
0

0

]
(2.5)

This (2.5) is of the typical form Ax̂ = 0. The unknown vector x̂ to be determined are the amplitudes

[k̂ v̂]′. λ̃ = λ− iω and ω̃ = ω −Ves1 are abbreviations used to alleviate the computational process.
The solution to equation (2.5) is non-trivial if the determinant of the matrix A is zero. The deter-

minant of A produces the quadratic equation:



VISCOUS-DIFFUSIVE TRAFFIC FLOW MODEL 5

λ̃ + λ̃

(
1

τ̆
− iη̆

)
+
i

τ
ke
dVe
dk

s1 = 0 (2.6)

where 1/τ̆ = 1/τ + Ds21 > 0 and η̆ = cs1 −φs2 > 0. The solution to the characteristic polynomial (2.6)
is:

λ̃±(s) =
1

2

(
η̆ −

1

τ̆

)
±

√
1

4

(
1

τ̆2
− η̆2

)
+

(
−

1

τ

dVe
dk

kes1 −
η̆

2τ̆

)
(2.7)

As it can be seen, the square root term would yield a complex output, therefore the underlying equation

(2.8) is used to simplify this root term. The variable R is used to denote the real part, while the
imaginary part is denoted by I. From [19]

√
R± iI =

√
1

2

(√
R2 + I2 + R

)
± i
√

1

2

(√
R2 + I2 −R

)
(2.8)

The discriminant for stability dwells on the real part (Re) of the eigenvalues. Deductively, the real

part of the eigenvalues are

Re
(
λ̃±(s)

)
= −

1

2τ̆
±
√

1

2

(√
R2 + I2 + R

)
(2.9)

A choice is made between Re(λ̃−(s)) and Re(λ̃+(s)) depending on which is more non-negative. Observ-

ably, Re(λ̃−(s)) < Re(λ̃+(s)), which implies that any condition satisfying Re(λ̃+(s)) will automatically

satisfy Re(λ̃−(s)). Hence, the relevant eigenvalue to determine transitions from stationary traffic to

unstable flow is Re(λ̃+(s)). That is

−
1

2τ̆
+

√
1

2

(√
R2 + I2 + R

)
≥ 0

Simplified as

I2 ≥
1

4τ̆4
−

R
τ̆2

(2.10)

The result of substituting the values of R =
1

4

(
1

τ̆2
− η̆2

)
and ±|I| =

1

τ

∣∣∣∣dVedk
∣∣∣∣kes1 − η̆2τ̆ into (2.10) is(

1

τ

∣∣∣∣dVedk
∣∣∣∣kes1 − η̆2τ̆

)2
≥

1

4τ̆4
−

1

4

1

τ̆2

(
1

τ̆2
− η̆2

)
(2.11)

simplified as

1

τ

∣∣∣∣dVedk
∣∣∣∣kes1 ·

(
1

τ

∣∣∣∣dVedk
∣∣∣∣kes1 − η̆τ̆

)
≥ 0

Finally, the instability condition is obtained as

1

τ

∣∣∣∣dVedk
∣∣∣∣kes1 ≥ η̆τ̆ = (cs1 −φs2)

(
1

τ
+ Ds21

)
(2.12)

The instability condition is gratified if the change in velocity as a result of change in density is quite

larger. This scenario is evident within the synchronized regime (medium densities) of traffic flow. The

convergence of individual vehicular velocity to the steady-state velocity is realized during either the

free-flow regime or the congested regime. During these regimes, values for the velocity-density gradient

term are quite smaller. Hence, the instability criterion (2.12) will be violated. But in the absence of



6 G. O. FOSU, A. ADU-SACKEY, AND J. ACKORA-PRAH

the diffusion and viscosity rates, the threshold for equilibrium traffic corresponds to the value of the

sonic speed c. This is given by the expression in equation (2.13) below.∣∣∣∣dVedk
∣∣∣∣ke ≥ c (2.13)

3. Model Analysis

The discrete multilane version of the continuous viscous-diffusive model (1.9) is presented here by

the introduction of the lane index l, that is

∂kl(x,t)

∂t
+
∂ql(x,t)

∂x
= 0

∂vl(x,t)

∂t
+ vl(x,t)

∂vl(x,t)

∂x
= c

∂vl(x,t)

∂x
+
Ve(kl) −vl(x,t)

τ
+ D

∂2vl(x,t)

∂x2
−φ

∂vl(x,t)

∂y

(3.1)

ql(x,t),kl(x,t) and vl(x,t) are the flow rate, density, and speed for the lth lane respectively. This

equation (3.1) is presented to explain the inter-lane dependency of a multilane flow. Existing models

could only explain the association among a limited number of lanes, usually two lanes[12, 24]. Here, an

investigation for more than two lanes is presented.

The model is solved numerically because of the computational difficulties associated with the ana-

lytical approach to obtaining a solution to this system. Thus, the upwind finite difference scheme is

employed to numerically solve this macroscopic system (3.1).

The lane index continuity equation is discretized as

klm(n + 1) = k
l
m(n) + Φv

l
m(n)

[
klm−1(n) −k

l
m(n)

]
+ Φklm(n)

[
vlm(n) −v

l
m+1(n)

]
(3.2)

The discretized acceleration equation is given by either of the following depending on the intensity

of the traffic. When vlm(n) > c; being a lighter flow regime, then the acceleration equation is given as

vlm(n + 1) = v
l
m(n) + Φ

[
c−vlm(n)

][
vlm(n) −v

l
m−1(n)

]
− Θ

µ ·sd
klm(n)

(
vlm(n) −v

l+1
m (n)

)
+

∆t

τ

(
Ve −vlm(n)

)
+ DΩ

(
vlm−1(n) − 2v

l
m(n) + v

l
m+1(n)

)
(3.3)

In the case of heavy flow (vlm(n) < c), then the acceleration equation becomes

vlm(n + 1) = v
l
m(n) + Φ

[
c−vlm(n)

][
vlm+1(n) −v

l
m(n)

]
− Θ

µ2 ·sd
klm(n)

(
vlm(n) −v

l+1
m (n)

)
+

∆t

τ

(
Ve −vlm(n)

)
+ DΨ

(
vlm−1(n) − 2v

l
m(n) + v

l
m+1(n)

)
(3.4)

where ∆t/∆y = Θ, ∆t/∆x = Φ, and ∆t/∆x2 = Ψ. m,n,l are positive integers. xm = m∆x, yl =

l∆y, and tn = n∆t. k
l
m(n) ≈ k(xm,yl, tn) is the density of vehicles at region m of lane l at time

n. vlm(n) ≈ v(xm,yl, tn) is the velocity of vehicles at position m on lane l at time n. Moreover, the
steady-state velocity equation Ve is defined as [10, 11, 15]:

Ve = vf

{
1 − exp

[
1 − exp

(
kw
vf

(
kf

klm(n)
− 1
))]}

(3.5)

kw is the kinematic wave speed during a heavy traffic domain, vf and kf are respectively the maximum

speed and density.

For the initial condition, each lane has a specified density value with speed as derived.



VISCOUS-DIFFUSIVE TRAFFIC FLOW MODEL 7

Kl(x, 0) =



k1, if l = 1

k2, if l = 2
...

...
...

kL, if l = L

V l(x, 0) =



v(k1), if l = 1

v(k2), if l = 2
...

...
...

v(kL), if l = L

3.1. Simulation Result and Analysis. A graphical solution of this lane-index macro-model is pre-

sented in this section. In addition, the information flow of the wave profiles under the perturbation of

some key parameters are also presented.

Throughout the simulation, the total distance is taken as 3000m. A relatively shorter distance is

chosen so that one could simultaneously observe both lateral and longitudinal happenings on a given

multilane stretch. A simulation for five-lane infrastructure is presented with all vehicles moving along

the same direction, with inter-lane interval ∆y = 1.5m. ∆x = 300 and ∆t = 4s are chosen to satisfy

the Courant-Friedrichs-Lewy numerical stability condition (3.6).

max{vf − c,q′(k(x,t))} ·
∆t

∆x
≤ 1 (3.6)

The initial density profile for the five lanes is given as

Kl(x, 0) =




0.50000, if l = 1

0.38750, if l = 2

0.27500, if l = 3

0.16250, if l = 4

0.05000, if l = 5

The initial values for the first and last lanes are used as the boundary condition. The lanes are numbered

as l = 1, 2, · · · , 5. Lane one is the extreme outer lane, while five is the extreme inner lane. The initial
density for lane one is denser due to parking on that lane and on the shoulders of the road; this following

the no-slip condition of fluid [22, 33]. For the five lane analysis, lanes four and five are the high-speed

lanes, while the outer lanes are the low-speed lanes. The following are details of the other parameters

[16, 23, 38]:

µ2 = 0.0041 D = 10 sd = 0.37 kf = 1

c = 11m/s vf = 20m/s τ = 10s kw = 11m/s
Figure 1 describes some traffic dynamics of a five lane carriageway. It could be observed that the

density of the traffic becomes lighter moving from the outer lanes through to the inner lanes. Lane five

is less dense, and as such vehicles traversing on this lane have higher velocity compared to the other

lanes. Almost all vehicles on lane five could speed up to the maximum, but the situation is different

on the outer lanes. This scenario typifies a realistic multilane traffic state in developing countries of

which Ghana is not an exception. Here, passenger cars often transverse on the outer lanes, due to their

intermittent stopping to pick and drop passengers, making these lanes denser.

Nonetheless, drivers change lanes oftentimes to the left as the density of their driving lane becomes

compact. This is depicted through the simulation results with a speed drop and density rise in the higher

speed lanes. Deductively, a two-lane highway would easily get clumsy when there is an obstruction on

the outer lane. But the situation is different on roadways with arbitrary many lanes. The case of a

twenty-lane carriageway is shown in Figure 2. The density of the traffic reduces moving towards the

inner lanes because the model hinges on the no-slip condition. For this twenty-lane representation,

maximum flow is achieved somewhere around lane fifteen through to twenty.



8 G. O. FOSU, A. ADU-SACKEY, AND J. ACKORA-PRAH

Figure 1. Speed and density profiles for a five-lane carriageway

Figure 2. Multilane traffic profiles for a twenty-lane carriageway

In Figure 3, the sonic speed c is varied to determine it effect on flow. It was observed that predicting

vehicular traffic using this macroscopic model would fail when the value of the sonic speed far exceeds

vehicles speed. This is illustrated graphically with c = 150m/s. The absence of the anticipation term

(c = 0m/s) produced quite shorter wave profiles vis-a-vis the benchmark value of c = 11m/s in Figure

1.

Figure 3. Multilane speed profiles for c = 0, left; c = 50m/s, middle; and c = 150m/s, right

From the stability criterion (2.12), it was stated that the time taken for traffic to align with the

velocity-density dependent value is key in determining the stationarity of the flow. Thus, the adaptation

time τ is varied to determine its effect on these multilane wave profiles. From Figure 4, if drivers have

enough time to adapt, their speed profiles are finer. That is to say drivers have a longer time to react

without any repercussions. Comparing the plot right of Figure 4 to the other on the left, the speed

profiles become unstable as τ converges to zero.



VISCOUS-DIFFUSIVE TRAFFIC FLOW MODEL 9

Figure 4. Multilane speed profiles for τ = 1s, left; τ = 5s, middle; and τ = 120s, right

A similar perturbation of the lateral viscosity rate is represented in Figure 5. The viscous rate ranges

between zero and one; zero denotes the absence of any form of lateral resistance, and one denotes a

highly viscous flow. From the simulation plots 5, the effect of viscosity is apparent as µ2 assumes values

closer to the upper limit. It is seen that some amount of lateral obstruction has a consequential effect

on the speed of vehicles.

Figure 5. Multilane speed profiles under the perturbation of the viscous rate. Top

left: µ2 = 0, top right: µ2 = 0.041, bottom left: µ2 = 0.41, and bottom right: µ2 = 1.00

From Figure 6, the prediction of traffic using this multilane model was found to be feasible whether

diffusion was present or not. The diffusive term was barely substantiated in this simulation work.

Nonetheless, a highly enormous value showed inclinations of abnormal flow.

4. Conclusion

Macroscopic traffic models of second-order consist of the continuity equation and the dynamic velocity

equation. The dynamic velocity equation was introduced to rectify the shortcomings of the continuity

equation. The components of the dynamic velocity equation are convection, anticipation, relaxation,

diffusion, and viscosity. In this paper, a new macro-model that features all these dynamic speed

quantities is presented. Then after, the linear-stability condition of the new continuum second-order



10 G. O. FOSU, A. ADU-SACKEY, AND J. ACKORA-PRAH

Figure 6. Multilane speed profiles under the perturbation of the diffusion rate. Top

left: D = 0, top right: D = 50, bottom left: D = 500, and bottom right: D = 15000

model was determined through the analysis of wave profiles. The gradient of the velocity-density curve,

the average density, and the sonic speed were found to be the determining variables for either stable

or unstable flow. The proposed anisotropic model was again recast as a lane-indexed model and was

solved numerically using the upwind finite difference scheme. This reformulation was done to remove the

restriction on the proposed model as being single-piped. The viscosity rate, anticipation, diffusion, and

relaxation time were perturbed to examine its effect on flow. By the simulation plots, it was observed

that a smaller relaxation time, a larger anticipation rate, and a unit size lateral viscosity rate have an

adverse effect on the flow rate. The speed profiles were observed to be awkward for some rage of values.

However, the diffusive term did not have any substantial impact on the speed profiles, except for the

case of extremely large values.

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Corresponding author, Department of Mathematics, Kwame Nkrumah University of Science and Technol-

ogy, Kumasi, Ghana

Email address: gabriel.of@knust.edu.gh

Department of Applied Mathematics, Koforidua Technical University, Ghana

Email address: albert.adu-sackey@ktu.edu.gh

Department of Mathematics, Kwame Nkrumah University of Science and Technology, Kumasi, Ghana

Email address: jaackora-prah.cos@knust.edu.gh


	1. Introduction
	2. Derivation of Instability Criterion
	3. Model Analysis
	3.1. Simulation Result and Analysis

	4. Conclusion
	References