Unsteady Chemically Reacting Casson Fluid Flow in an Irregular Channel with Convective Boundary

1 Introduction

Fluids have always been a major area of research due to their applicability and diversity. Among these, there are two important branches, characterised due to a variety of properties, Newtonian and non-Newtonian fluids. Newtonian fluids have been studied [1–6] for their application in daily life, such as paper production, glass blowing, geothermal energy extraction, and nuclear reactors. The unique relation of non-linearity between shear stress and the shear rate gave birth to the field of non-Newtonian fluids. Non-Newtonian fluids are an active area of research due to its extensive and major applications for fluids like shampoos, blood, oils, paints, ice cream, etc. The non-Newtonian fluid when subjected to constant pressure and magnetic field made velocity distribution uniform and viscosity effects clear near the boundary [7]. Experiments have also been carried on Couette flow for non-Newtonian fluids, and the results matched the theoretical one [8]. The non-linearity of shear rate with viscosity and time dependence and elongation effects make a new type of non-Newtonian fluids, which are called viscoelastic fluids [9]. Viscoelastic fluids have been under observation since long time and are characterised as non-Newtonian fluids. These have been studied with Newtonian heating too [10]. These fluids between parallel plates have a significant role in many engineering fields such as petroleum production, solar power collectors, and chemical catalytic reactors.

Heat and mass transfer of a time-dependent magnetohydrodynamics flow of an electrically conducting viscoelastic fluid in non-uniform channel was discussed [11]. It was found that the Hall effect plays a vital role in controlling the velocity of the fluid. The boundary layer flows of these fluids with stretching flows make this field important and useful for plastic films, paper production, polymer extrusion, etc. Convective flow is also an important property, which is discussed [12, 13]. It was noticed that mixed convection parameter in momentum and thermal boundary layers have opposite role [12]. Recently, dusty viscoelastic fluid flow with convective boundary was discussed [14, 15], and it was shown that the velocity profiles of dusty fluid are higher than the dust particles. This dusty fluid showed velocity, temperature, and concentration profiles correspondingly for increasing time with varying mass diffusion. Adomian decomposition method was used to study the effect of couple stresses on the entropy generation rate of the viscous incompressible fluid flow with convective heating at the walls [16]. Homotopy analysis method was used to study the Dufour and Soret effects on study free convection flow of a couple stress fluid flowing through a vertical channel with heat and mass transfer [17]. It was observed that both velocity and temperature of the fluid decrease with the decrease in the Dufour number. Heat transfer chacteristics were studied for oscillatory hydromagnetic non-Newtonian couple stress fluid flow with non-uniform wall temperature between parallel porous plates [18]. The combined effect of heat source/sink and radiative heat transfer was studied for viscoelastic fluid with saturated porous medium through a channel [19].

Motivated from previous studies, the present article deals with unsteady, chemically reacting Casson fluid in an irregular vertical channel with convective boundary conditions. The non-dimensionlised equations for momentum, temperature, and mass concentration are solved by using perturbation method technique. Exact solutions are calculated for obtained ordinary differential equations. Results of various parameters such as thermal radiation parameter, chemical reaction parameter, thermal Grashof number, solutal Grashof number, Soret number, and Biot number are discussed by graphs. We also concluded the results of the present study at the end.

2 Mathematical Formulation

Consider an unsteady, incompressible two-dimensional Casson fluid flow in a long vertical channel where one wall is wavy while another is lying flat as shown in Figure 1. The x^*-axis is taken along the wavy wall, while the y^* -axis is perpendicular to it. The wavy wall (y^*=0) is at temperature T₁ and mass concentration C₁, while the flat wall (y^*=d) is at temperature T₂ and mass concentration C₂. The fluid is assumed to be electrically conducting. Furthermore, it is considered that the convective heat exchange from the walls of the channel follow Newton’s law of cooling. The fluid flow is exposed to the influence of heat absorption, radiation absorption, first-order chemically reactive species, and uniform magnetic field of strength B₀. Under these assumptions, model equations for momentum, temperature, and mass concentration are given as follows [15]:

Figure 1:

Schematic flow configuration.

(1)∂u*∂t* = ν(1 + 1β)∂2u*∂y*2 − νku* − σ B02ρu*+ gβ T (T* − T1) + gβc(C* − C1), (1)

(2)∂T*∂t* = kρcp∂2T*∂y*2 − Qtρcp(T* − T1) − 1ρcp∂qr∂y* + νcp(1 + 1β)(∂u*∂y*)2 + σ B02ρcpu*2 + Qcρcp(C* − C1), (2)

(3)∂C*∂t* = D ∂2C*∂y*2 − kR(C* − C1). (3)

The corresponding boundary conditions are given by [14]

(4)t* = 0:     u* = 0,    T* = T1 ,     C* = C1 for  y* ∈(ϵ cos(k2x*), d), (4)

(5)t* > 0 :       u* = 0,     −k ∂T*∂y* =  hf (T1 − T* + (T2 − T1)) ϵ e−in*t*, (5)

(6)C* = C1 + (C2 − C1)ϵ e−in*t* at  y* = ϵ cos(k2x*) C* = C1 + (C2 − C1)ϵ e−in*t* at  y* = d. (6)

Using the relation in [14], radiative heat flux q_r gas near equilibrium is given by

(7)∂qr∂y* = 4(T1 − T*) I′,   I′ = ∫​kλ1w       ∂ebλ1 ∂T dλ1. (7)

Here, q_r is the radiative heat flux,kλ1wis the radiation absorption coefficient at the wall, and ebλ1 is Plank’s constant.

Introducing the following non-dimensional quantities

x = x*d ,     y =   y*d ,     u = u*U0,     θ = T* − T1T2−T1 ,      φ = C* − C1C2−C1,

M2 = σeB02d2μ ,       1k = d2k*,       Gr = gβT(T2 − T1)d2υu0 , GC = gβc(C2 − C1)d2υu0 ,

Pr = μCpk,      αT = QTd2k ,    F = 4I′d2k,      ϵ =  ϵ*d  ,     h = ϵcos(λx),

αc = Qc(C2 − C1)d2k(T2 − T1) ,      Sc = νD ,     Bi = hf dk ,      Kr = KRd2 υ ,    λ = K2d, 

t = υ t*d2 ,  n = n* d2 υ.

Applying these dimensionless quantities, (1)–(3) take the form

(8)∂u∂t = (1 + 1β)∂2u∂y2 − (M2 + 1k)u + Gr θ +  GC φ, (8)

(9)∂2θ∂y2 − Pr∂θ∂t + (F − αT)θ +  Pr Ec (1 + 1β)(∂u∂y)2 + Pr Ec M2 u2 + αcφ = 0, (9)

(10)∂2φ∂y2 − Sc∂φ∂t − Sc Kr φ = 0. (10)

Boundary conditions (4)–(6) become

(11)t = 0:            u = 0,     θ = 0,        φ = 0 for y ∈(h, 1), (11)

(12)t > 0 :       u = 0,    θ′ = Bi( θ− ϵ e−int),  φ = ϵ e−int at y = h, (12)

u = 0,     θ′ = Bi( θ−1− ϵ e−int),         φ = 1 + ϵ e−int at y = 1.

Here, prime (′) denotes the derivative with respect to y.

3 Solution of the Problem

Equations (8)–(10) represent the set of coupled partial differential equations that cannot be solved in closed form. However, these partial differential equations can be reduced to a set of ordinary differential equations by using the perturbation technique in terms of ε as: taking Ec=εe^–int and Ec<<1 since the flow due to joule dissipation is supper imposed on the main flow:

(13)u(y, t) = u0(y) +  ε e−int u1(y) + o( ε2), (13)

(14)θ(y, t) = θ0(y) +  ε e−int θ1(y) + o( ε2), (14)

(15)φ(y, t) = φ0(y) +  ε e−int φ1(y) + o( ε2), (15)

Now by substituting (13)–(15) into (8)–(10), then equating harmonic and non-harmonic terms, and neglecting square and higher powers of ε, we have

(16)u″0−(M2 + 1k)(ββ + 1)u0 +  β Grβ + 1θ0 + β GCβ + 1φ0 = 0, (16)

(17)θ″0 + (F−αT)θ0 + αcφ0 = 0, (17)

(18)φ″0−Sc Kr φ0 = 0, (18)

(19)u″1−(M2 + 1k − n)(ββ + 1)u1 +  β Grβ + 1θ1 + β GCβ + 1φ1 = 0, (19)

(20)θ″1 + (F − αT + n Pr)θ1 + αcφ1 = 0, (20)

(21)φ″1 + n Sc φ1  − Sc Kr φ1 = 0. (21)

The corresponding boundary conditions (11) and (12) become

(22)u0 = 0,     θ′0 = Bi θ0,         φ0 = 0,   u1 = 0, (22)

(23)θ′1 = Bi(θ1−1),         φ1 = 1 at y = h (23)

u0 = 0,     θ′0 = Bi( θ0−1),          φ0 = 1,      u1 = 0,

(24)θ′1 = Bi(θ1−1),        φ1 = 1 at  y = 1. (24)

The exact solutions of (16)–(21) with boundary conditions (22)–(24) are given in the following:

(25)u(y, t) = e−y(2(A1 + A3) + A5)(C2ey((A1 + 2A3) + A5)A6A7 + C1ey(3A1 + 2A3 + A5)A6A7−(A32 − A12)(k(1 +  β)(A52 − A12)(C6ey(2A1 + A3 + A5)Grkβ + C5ey(2A1 + 3A3 + A5)Grkβ) + A7e2y(A1 + A3 + A5)C9 + e2y(A1 + A3)C10)/(A32 − A12)(k(1 + β)(A52 − A12)A7 − εe−nt(e−y(4(A1 − A2) + 2A4 + A8)(−C4ey(3(A1 − A2) + 2A4 + A8)kA9A10)−C3ey(5(A1 − A2) + 2A4 + A8)kA9A10 − (A12 − A22 − A42)(k(1 + β)(A52 − A12) + (1 + β)A22−nβ)))(C6ey(4(A1 − A2) + A4 + A8))Grkβ + C5ey(4(A1 − A2) + 3A4 + A8)Grkβ +e2y(2(A1 − A2) + A4)(1βA10)(e2yA8C11 + C12))))/(A12 − A22 − A42)1βA10(k(1 + β)(A52 − A12) + (1+β)A22 − nβ)))), (25)

(26)θ(y, t) = 1A12 − A32e−y(A1 + A3)−(C2eyA3αc − C1ey(2A1 + A3)αc + eA1y(A12 − A32)(e2yA3C5 + C6))−εe−nt(e−y(A1 − A2 + A4)(−C4eyA4αc − C3ey(2(A1 − A2) + A4)αc+e(A1 − A2)y(A12 − A22 − A42)(e2yA4C7 + C8)))−A12 + A22 + A42, (26)

(27)φ(y, t) = eA1yC1 + e−A2yC2 + ε e−nt( e(A1 − A2)yC3 + e−(A1 + A2)yC4). (27)

Here,

A1 = k s  ,  A2 = n s , A3 = −F + αT  , A4 = −F + αT − nPrA5 = β + kM2 βk (1 + β),  A6 = k(−Gr αc + Gc (F + ks − αT)),A = β (β + k(F − αT + (F + M2 − αT))β),  A8 = β(1 + kM2 − kn)k (1 + β),A9 = (−Gr αc + Gc (F + n (Pr − s) + ks − αT)),A10 =  β (β + k(F + n Pr − αT + (F + M2 + n(−1 + Pr) − αT))β),

C1 = eA1e2 A1 − e2 h  A1,C2 = eA1 + 2 h  A1e2 A1 − e2 h  A1,C3 = 1eA1−A2 + eh(A1 − A2),C4 = e(A1−A2) + h(A1−A2)eA1−A2 + eh(A1−A2),

C5 = 1(e2A3 − e2hA3)(B2 − A32)(−A3(C2(ehA1 + A3 − eA1 + hA3) + C1(−e(2 + h)A1 + A3 + e(1 + 2 h)A1 + hA3))2 + A12)e−(1+h)A1A1αc+B(C2αc(ehA1 + A3−eA1 + hA3) + C1αc(e(2 + h)A1 + A3−e(1 +2  h)A1 + hA3) + C1e(1 + h)A1 + A3(A12 − A32))(B + A3),

C6=(e−(1 + h)A1 + A3(−(C2ehA1 + 2 h A3 − eA1 + (1 + h)A3) + C1(−e(2 + h)A1 + 2 h A3 + e(1 + 2 h)A1 + (1 + h)A3)A1αc−B(C2(ehA1 + 2 h A3 − eA1 + (1 + h)A3) + C1αc(e(2 + h)A1 + 2 h A3−e(1 + 2 h)A1 + (1 + h)A3) + e(1 + h)A1 + 2 h A3(A12 − A32)))/((e2 A3 − e2 h A3)(A12 − A32)( B + A3)),

C7 = e−(1 + h)(A1 − A2)(−(C4(eh(A1 − A2 + A4)−e(A1 − A2 + hA4)) + C3(−e(2 + h)(A1 − A2 + A4)−e(1 + 2 h)(A1 − A2 + hA4)))((A1 − A2)αc−B(A12 − A22 − A42)(C4eh(A1 − A2 + A4)αc(1−C3e2) + αc(C4e(A1 − A2 + hA4)(+C3e(1 + 2 h)(A1 − A2 + h A4)) + − e(1 + h)(A1 − A2 +  A4) + e(1 + h)(A1 − A2 + h A4))/((e2A4 − e2hA4)(A12 − A22 − A42)(−B + A4)),

C8 =  e−(1 + h)(A1 − A2 + A4)(−(C4(eh(A1 − A2 + 2 h A4) − e(A1 − A2 + (1 + h)A4)) + C3(−e(2 + h)(A1 − A2 + 2 h A4) + e(1 + 2 h)(A1 − A2 + (1 + h)A4)))((A1 − A2)αc + B( eh(A1 − A2 + 2 h A4)αc(−C4 − C3e2) + αc(C4e(A1 − A2 + (1 + h)A4)+C3e(1 + 2 h)(A1 − A2 + (1 + h)A4))−e(1 + h)(A1 − A2 +  A4))−e(1 + h)(A1 − A2 + 2 h  A4)(A12 − A22 − A42))/((e2A4 − e2 h A4)(A12 − A22 − A42)(B + A4))),

C9 = 1k(1 + β)(A52 − A12)ehA5(−k(1 + β)(A52 − A12)(A6β(c2e−A1 + c1eA1)(A12 − A32)k(1 + β)(A52 − A12)+ Grkβ2(C5 eA3 − C6 e−A3)A7)+A6β(c1ehA1 + C2e−hA1−A5)(A12 − A32) + (Grkβ2(c6 e−hA3 + c5 ehA3)(A12 − A32))/(e−(1 − h)A5−e−(1 + h)A5),

C10 = 1−k(1 + β)(A52 − A12)eA5(k(1 + β)(A52 − A12)(A6β(c1eA1 + c2e−A1)(A12 − A32)k(1 + β)(A52 − A12)+ Grkβ2(C5 eA3 + C6 e−A3) A7)−(C2e((2−h)A5−A1)A6β)(A12 − A32)+c1eA1A6β(A12 − A32)k(1 + β)(A52 − A12) + (e−A1Grkβ2(c6 +  c5)(A12 − A32)) + e−A5(( A6β(c2e−hA1 + c1ehA1)(A12 − A32)k(1 + β)(A52 − A12))+(Grkβ2(C6e−hA3 + C5ehA3)A7))/2e−A5(1 + h),

C11 = −(e−hA8)(Grkβ2(C6e−A4 + C5eA4)A10 + kβA9(C4e−(A1 − A2) + C3e(A1 − A2))(A12 − A22 − A42)(k(1 + β)(A52 − A12) + A22(1 + β)−nβ))−e−A8(Grkβ2(C6e−hA4 + C5ehA4)A7 + kA9 β(C4e−h(A1 − A2) + C3eh(A1 − A2))(A12 − A22 − A42)(k(1 + β)(A52 − A12) + A22(1 + β)−nβ))/(eA8(1−h)−e−A8(1−h)),

C12 = −e−hA8((Grkβ2(C6e−hA4 + C5ehA4)A10) + kβA9(C4e−h(A1 − A2) + C3eh(A1 − A2))(A12 − A22 − A42)(k(1 + β)(A52 − A12) + A22(1 + β)−nβ)+(e2hA8(e−hA8(Grkβ2(C6e−A4 + C5eA4)A10 + kβA9(C4e−(A1 − A2) + C3e(A1−A2))(A12 − A22 − A42)(k(1 + β)(A52 − A12) + A22(1 + β) − nβ))−e−A8((Grkβ2(C6e−hA4 + C5ehA4)A10) + kβA9(C4e−h(A1 − A2) + C3eh(A1 − A2))(A12 − A22 − A42)(k(1 + β)(A52 − A12) + A22(1 + β)−nβ)))) /(eA8(1−h)−e−A8(1−h)),

4 Results and Discussion

This section provides graphs, the tabulated values for the skin friction, Nusselt number, Sherwood number, and physical behaviour of different involved parameters on velocity, temperature, and mass concentration. The values of the parameters K_r=2, α_c=2, α_T=2, G_r=2, G_C=1, P_r=0.71, B_i=0.5, F=1, S_c=0.96, t=2, and ε=2 are kept fixed except the varying one. Figure 2 reveals that the velocity decreases with an increase in the numerical value of the Biot number for both Newtonian and Casson fluids in cooling case as we are taking G_r>0. The effects of variation in radiation absorption parameter α_c on velocity profile are illustrated in Figure 3. It can be seen that by increasing the value of the radiation absorption parameter, the velocity also increases. Figure 4 depicts that increasing the value of the thermal radiation parameter has a decreasing impact on velocity. This was expected that as the momentum boundary layer thickness decreases for increasing the thermal radiation parameter, as a result, velocity decreases. Figure 5 shows the influence of solutal Grashof number on velocity profile, as solutal Grashof number is the ratio of species buoyant force to the viscous hydrodynamic force. An increase in the solutal Grashof number dominates species buoyancy force on viscous hydrodynamic force and as a result velocity increases significantly both for Casson and Newtonian fluids. The positive value of Grashof number physically represents the cooling of the plates. Actually, buoyant force suppresses viscous force, and the presence of this buoyancy force complicates the flow problem, due to coupling with thermal and mass problems. Therefore, an increase in the thermal Grashof number G_r results in an increase in the velocity due to enhancement of buoyancy force as shown in Figure 6. Overall, we notice that the velocity of Newtonian fluid is higher than the velocity of Casson fluid due to the reason that Casson fluid has a large viscosity as compared to Newtonian fluid.

Figure 2:

Velocity profile for different values of the Biot number.

Figure 3:

Velocity profile for different values of the radiation absorption parameter.

Figure 4:

Velocity profile for different values of thermal radiation parameter.