T.H. Alarabi et al.

2.1. Mathematical modeling relaxation-time of the dust elements, the thermal equilibrium time, the
solutal equilibrium time, respectively.
The model of non-Newtonian Casson nanomaterial and dusty liquid The simplified formula of the prevailing partial differential equa-
model of this problem is specified as26,47 : tions can be obtained by the following similarity variables48,49 :
For the fluid phase: √
−𝑑
𝑢 = 𝑏𝑠𝑓 ′ (𝜂), 𝑢𝑝 = 𝑢 = 𝑏𝑠𝐹 ′ (𝜂), 𝑣 = 𝑏𝜈𝑓 𝑓 (𝜂),
𝜕{(𝑟 + 𝑑)𝑣} 𝜕𝑢 𝑟+𝑑
+𝑑 =0 (1) √
𝜕𝑟 𝜕𝑠 √
−𝑑 𝑏
𝑣𝑝 = 𝑏𝜈𝑓 𝐹 (𝜂), 𝜂 = 𝑟,
1 2 1 𝜕𝑝 𝑟+𝑑 𝜈𝑓
𝑢 = (2)
𝑟+𝑑 𝜌𝑓 𝜕𝑟 𝑇 − 𝑇∞
𝑝 = 𝜌𝑓 𝑏2 𝑠2 𝑃 (𝜂), 𝑝𝑝 = 𝜌𝑝 𝑏2 𝑠2 𝑃𝑝 (𝜂), 𝜃(𝜂) = ,
( ) 𝑇𝑓 − 𝑇∞
𝜕𝑢 𝑑 𝜕𝑢 𝑢𝑣 1 𝑑 𝜕𝑝 1
𝑣 + 𝑢 + =− + 𝜈𝑓 1 + 𝑇𝑝 − 𝑇∞ 𝐶 − 𝐶∞
𝜕𝑟 𝑟 + 𝑑 𝜕𝑠 𝑟 + 𝑑 𝜌𝑓 𝑟 + 𝑑 𝜕𝑠 𝛾 𝜃𝑝 (𝜂) = , 𝜙(𝜂) = ,
[ 2 ] 𝑇𝑓 − 𝑇∞ 𝐶𝑓 − 𝐶∞
𝜕 𝑢 1 𝜕𝑢 1 (3)
× + − 𝑢 𝐶𝑝 − 𝐶∞
𝜕𝑟2 𝑟 + 𝑑 𝜕𝑟 (𝑟 + 𝑑)2 𝜙𝑝 (𝜂) = (12)
𝜎 2 𝐾𝑁 𝐶𝑓 − 𝐶∞
− 𝐵 𝑢+ (𝑢 − 𝑢)
𝜌𝑓 0 𝜌𝑓 𝑝
2.2. Dimensionless equations
( ) [( ) 𝐷 ( )2 ]
𝜕𝑇 𝑑 𝜕𝑇 𝜕2 𝑇 1 𝜕𝑇 𝜕𝐶 𝜕𝑇 𝜕𝑇
𝑣 + 𝑢 =𝛼 + + 𝜏 𝐷𝐵 + 𝑇 When including the dimensionless variables defined in Eq. (12), the
𝜕𝑟 𝑟 + 𝑑 𝜕𝑠 𝜕𝑟2 𝑟 + 𝑑 𝜕𝑟 𝜕𝑟 𝜕𝑟 𝑇∞ 𝜕𝑟
Eqs. (1)–(10) taking the form
𝑄0 𝜌𝑝 𝑐𝑠 𝜌𝑝
+ (𝑇 − 𝑇∞ ) + (𝑇 − 𝑇 ) + (𝑢 − 𝑢)2 For the fluid phase:
(𝜌𝑐𝑝 )𝑓 𝜏𝑇 (𝜌𝑐𝑝 )𝑓 𝑝 𝜏𝑚 (𝜌𝑐𝑝 )𝑓 𝑝
𝑓 ′2
(4) 𝑃′ = (13)
𝜂+𝐵
( )[ ]
2𝐵 1 ′′′ 1 1
( ) ( 2 ) 𝑃 = 1+ 𝑓 + 𝑓 ′′ − 𝑓′
𝜕𝐶 𝑑 𝜕𝐶 𝜕2 𝐶 1 𝜕𝐶 𝐷 𝜕 𝑇 1 𝜕𝑇 𝜂+𝐵 𝛾 𝜂+𝐵 (𝜂 + 𝐵)2
𝑣 + 𝑢 =𝐷𝐵 + + 𝑇 + 𝐵 𝐵 𝐵 (14)
𝜕𝑟 𝑟 + 𝑑 𝜕𝑠 𝜕𝑟2 𝑟 + 𝑑 𝜕𝑟 𝑇∞ 𝜕𝑟2 𝑟 + 𝑑 𝜕𝑟 + 𝑓 𝑓 ′′ − 𝑓 ′2 + 𝑓𝑓′
𝜌𝑝 𝜂+𝐵 𝜂+𝐵 (𝜂 + 𝐵)2
− 𝑅𝑐 (𝐶 − 𝐶∞ ) + (𝐶 − 𝐶) ( )
𝜌𝑓 𝜏 𝑐 𝑝 − 𝑀𝑓 ′ + 𝐷𝜌 𝛼𝑑 𝐹 ′ − 𝑓 ′

(5)
1 𝐵
𝜃 ′′ + 𝜃′ + 𝑃 𝑟𝑓 𝜃 ′ + 𝑃 𝑟𝑁𝑏𝜃 ′ 𝜙′ + 𝑃 𝑟𝑁𝑡𝜃 ′2 + 𝑃 𝑟𝑄𝜃 + 𝑃 𝑟𝐷𝜌 𝛤 𝛼𝑡 (𝜃𝑝 − 𝜃)
For the dust phase: 𝜂+𝐵 𝜂+𝐵
( )2
𝜕{(𝑟 + 𝑑)𝑣𝑝 } 𝜕𝑢𝑝 + 𝑃 𝑟𝐷𝜌 𝛼𝑑 𝐸𝑐 𝐹 ′ − 𝑓 ′ = 0
+𝑑 =0 (6)
𝜕𝑟 𝜕𝑠 (15)
1 2 1 𝜕𝑝𝑝
𝑢 = (7)
𝑟 + 𝑑 𝑝 𝜌𝑝 𝜕𝑟 ( )
1 𝐵 𝑁𝑡 1
𝜕𝑢𝑝 𝜕𝑢𝑝 𝑢 𝑝 𝑣𝑝 𝜙′′ + 𝜙′ + 𝑆𝑐𝑓 𝜙′ + 𝜃 ′′ + 𝜃 ′ − 𝑆𝑐𝛿𝜙 + 𝑆𝑐𝐷𝜌 𝛼𝑐 (𝜙𝑝 − 𝜙) = 0
𝑑 1 𝑑 𝜕𝑝𝑝 𝐾𝑁 𝜂+𝐵 𝜂+𝐵 𝑁𝑏 𝜂+𝐵
𝑣𝑝 + 𝑢 + =− − (𝑢 − 𝑢) (8)
𝜕𝑟 𝑟 + 𝑑 𝑝 𝜕𝑠 𝑟+𝑑 𝜌𝑝 𝑟 + 𝑑 𝜕𝑠 𝜌𝑝 𝑝 (16)
( )
𝜕𝑇𝑝 𝑑 𝜕𝑇𝑝 𝜌𝑝 𝑐 𝑠
𝜌𝑝 𝑐𝑠 𝑣𝑝 + 𝑢𝑝 =− (𝑇𝑝 − 𝑇 ) (9) For the dust phase:
𝜕𝑟 𝑟+𝑑 𝜕𝑠 𝜏𝑇
𝐹 ′2
𝜕𝐶𝑝 𝑑 𝜕𝐶𝑝 1 𝑃𝑝′ = (17)
𝑣𝑝 + 𝑢 = − (𝐶𝑝 − 𝐶) (10) 𝜂+𝐵
𝜕𝑟 𝑟 + 𝑑 𝑝 𝜕𝑠 𝜏𝑐
2𝐵 𝐵 𝐵 𝐵 ( )
𝑃 = 𝐹 𝐹 ′′ − 𝐹 ′2 + 𝐹 𝐹 ′ + 𝛼𝑑 𝑓 ′ − 𝐹 ′ (18)
For this model, the corresponding boundary conditions are47,48 𝜂+𝐵 𝑝 𝜂+𝐵 𝜂+𝐵 (𝜂 + 𝐵)2
𝜕𝑇 𝐵
𝑢 = 𝑢𝑤 (𝑠) = 𝑏𝑠, 𝑣 = 0, −𝑘 = ℎ𝑓 (𝑇𝑓 − 𝑇 ), 𝐹 𝜃 ′ + 𝛼𝑡 (𝜃 − 𝜃𝑝 ) = 0 (19)
𝜕𝑟 𝜂+𝐵 𝑝
𝜕𝐶
− 𝐷𝐵 = ℎ𝑐 (𝐶𝑓 − 𝐶), 𝑟=0 𝐵
𝜕𝑟 𝐹 𝜙′𝑝 + 𝛼𝑐 (𝜙 − 𝜙𝑝 ) = 0 (20)
𝜕𝑢 𝜕𝑢𝑝 𝜂+𝐵
𝑢, 𝑢𝑝 → 0, , → 0, 𝑣𝑝 → 𝑣, 𝑇 , 𝑇 𝑝 → 𝑇∞ , 𝐶, 𝐶𝑝 → 𝐶∞
𝜕𝑟 𝜕𝑟 The dimensionless reduced boundary conditions are
(11)
𝑓 ′ = 1, 𝑓 = 0, 𝜃 ′ = −𝐵𝑖(1 − 𝜃), 𝜙′ = −𝐵𝑗(1 − 𝜙), 𝜂 = 0,
Here 𝑟 and 𝑠 indicate the directions of the component’s velocity, (𝑢, 𝑣)− ′ ′
𝑓 , 𝐹 ⟶ 0, ′′
𝑓 , 𝐹 ⟶ 0,′′
𝐹 ⟶ 𝑓, 𝜃, 𝜃𝑝 ⟶ 0, 𝜙, 𝜙𝑝 ⟶ 0, 𝜂⟶∞
the components of the liquid velocity, (𝑢𝑝 , 𝑣𝑝 )− the components of
(21)
particle-phase velocity, (𝑝, 𝑝𝑝 )− the pressure of the liquid and dust
particles, respectively, (𝜌𝑓 , 𝜌𝑝 )− the density of liquid and dust particles, where (𝑓 , 𝐹 )− the non-dimensional stream function for liquid and
respectively, 𝛾− the Casson parameter, 𝜈𝑓 − the kinematic viscosity, 𝜎− particle speed, respectively, (𝜃, 𝜃𝑝 )− the dimensionless temperature of
the fluid electrical conductivity, 𝑁− the number density of the dust the liquid and dust particles, respectively, (𝜙, 𝜙𝑝 )− the dimensionless
particles, 𝐾 = 𝑚∕𝜏𝑚 − the resistance of stokes, 𝜏 = (𝜌𝑐𝑝 )𝑛𝑝 ∕(𝜌𝑐𝑝 )𝑓 − the concentration of the liquid and dust phases, respectively, (𝑃 , 𝑃𝑝 )− the
nanofluid heat capacity ratio, 𝛼 = 𝑘∕(𝜌𝑐𝑝 )𝑓 − the thermal diffusivity of dimensionless pressure of the liquid and dust elements, respectively,
the base fluid, 𝐷𝑇 − the thermophoretic diffusion coefficient, 𝑄0 − the 𝐵 = 𝑑(𝑏∕𝜈𝑓 )1∕2 − the curvature factor, 𝑀 = (𝜎𝐵02 ∕𝑏𝜌𝑓 )− the magnetic
volumetric of temperature generation or absorption, 𝐷𝐵 − the Brownian factor, 𝐷𝜌 = (𝑚𝑁∕𝜌𝑓 )− the mass concentration of the dust elements,
diffusion variable, 𝑅𝑐 − the reaction rate, (𝑐𝑠 , 𝑐𝑝 )− the dust elemets 𝛼𝑑 = (1∕𝑏𝜏𝑚 )− the liquid-particle interaction coefficient for speed,
specific heat and liquid specific heat, respectively, (𝜏𝑚 , 𝜏𝑇 , 𝜏𝑐 )− the 𝛼𝑡 = (1∕𝑏𝜏𝑇 )− the fluid–particle interaction factor for temperature, 𝛼𝑐 =