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Research article

The effects of different modes of foreign bank entry in the Turkish banking sector during the 2007–2009 Global financial crisis

  • This paper provides insights on how foreign bank entry modes (acquisition vs. greenfield investment) in an emerging market (Turkey) influenced bank strategies during the 2007–2009 global financial crisis. Using a comprehensive dataset comprising twenty-nine accounting variables from Turkish banks' financial statements during 2005–2010, we find important differences between foreign-acquired banks and foreign bank branches in lending and sourcing funds. We find that foreign bank branches continued to support international trade by issuing import loans during 2007–2009 global financial crisis, whereas foreign-acquired banks focused on issuing consumer and credit card loans. In terms of bank sourcing funds, we find that foreign-acquired banks were able to continue to use foreign currency deposits of Turkish residents and local interbank funding including participation (Islamic) banks. Foreign bank branches, on the other hand, relied on sourcing funds from international interbank funding and foreign currency deposits of residents abroad, which led to the necessity for them to change their strategies because of funding shortage in international markets. Our results show that the presence of foreign banks in Turkish banking sector enabled the continuity of bank lending activities in host market during the turmoil of 2007–2009 global financial crisis. Our findings on foreign bank entry mode provide new evidence and have important implications for both policy makers and practitioners in emerging markets.

    Citation: Nereida Polovina, Ken Peasnell. The effects of different modes of foreign bank entry in the Turkish banking sector during the 2007–2009 Global financial crisis[J]. Quantitative Finance and Economics, 2023, 7(1): 19-49. doi: 10.3934/QFE.2023002

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  • This paper provides insights on how foreign bank entry modes (acquisition vs. greenfield investment) in an emerging market (Turkey) influenced bank strategies during the 2007–2009 global financial crisis. Using a comprehensive dataset comprising twenty-nine accounting variables from Turkish banks' financial statements during 2005–2010, we find important differences between foreign-acquired banks and foreign bank branches in lending and sourcing funds. We find that foreign bank branches continued to support international trade by issuing import loans during 2007–2009 global financial crisis, whereas foreign-acquired banks focused on issuing consumer and credit card loans. In terms of bank sourcing funds, we find that foreign-acquired banks were able to continue to use foreign currency deposits of Turkish residents and local interbank funding including participation (Islamic) banks. Foreign bank branches, on the other hand, relied on sourcing funds from international interbank funding and foreign currency deposits of residents abroad, which led to the necessity for them to change their strategies because of funding shortage in international markets. Our results show that the presence of foreign banks in Turkish banking sector enabled the continuity of bank lending activities in host market during the turmoil of 2007–2009 global financial crisis. Our findings on foreign bank entry mode provide new evidence and have important implications for both policy makers and practitioners in emerging markets.



    Transport phenomena in porous media have widespread applications throughout manifold aspects of nature and industrial engineering, such as groundwater utilization, oil and gas extraction, chemical industry, building materials, biomedical development, etc. [1,2]. The relevant studies on heat transfer theory also permeate many scientific and technical fields [3]. On account of the structural complexity inside porous media and the diversity of fluids, convective flow and heat transfer in porous media have strongly nonlinear characteristics, and the classical linear Darcy's law is not applicable. Moreover, the coupled effects of velocity and temperature fields have important influences on the heat transfer process. Because of this, a novel constitutive equation and heat transfer model should be derived to characterize and simulate the internal physical mechanism inside the porous medium.

    For the last few years, constitutive relationship equations with fractional derivatives have been frequently adopted in the description of complex dynamics [4,5,6,7], and simultaneously relevant research on fractional porous media has also been carried out widely. Wei et al. [8] proposed two fractional derivative models to describe non-Darcy flow in porous media. Yang et al. [9] discussed anomalous imbibition of non-Newtonian fluid in a porous medium by a spatiotemporal fractional model. Ahmed [10] used Caputo fractional derivatives to govern the mixed convective flow of hybrid nanofluids together with the Darcy model. Ai et al. [11] presented a viscoelastic model with fractional derivatives to simulate multilayered cross-anisotropic porous media. Jiang et al. [12] solved the governing equations of fractional second-grade fluid through a porous medium. Jiang et al. [13] developed a fractional Burgers model to simulate non-Newtonian fluid in porous media with complex physical-chemical conditions.

    Nevertheless, the restrictions of the mentioned research are fixed fractional derivative parameters that manifest a rather limited class of memory characteristics and nonlocal properties [14]. Recent developments in various fields, such as thermal management, manufacturing, blood flow, etc., emphasize the need to resolve the limitations of conventional models. The convective flow and heat transfer problems of viscoelastic fluids involve complex molecule internal structures and multi-scale effects. Therefore, the distributed order fractional derivative that was proposed by Caputo to develop the fractional derivative models [15] has been frequently applied to boundary layer flow and heat transfer in view of the advantage of integrating over a given range within a continuum. Liu et al. [16] utilized distributed order fractional derivatives to investigate boundary layer flow and heat transfer through a moving plate. Qiao et al. [17] solved the pipe flow governing equations of viscoelastic Maxwell fluids with distributed order time fractional derivatives. Long et al. [18] developed the generalized Maxwell and Cattaneo model with distributed order time fractional derivative to study unsteady Marangoni convective boundary layer flow and heat transfer. Yang et al. [19] discussed boundary layer flow and heat transfer by way of the space distributed-order constitution relationship. Feng et al. [20] investigated anomalous transport in a binary medium on finite domains and analyzed the properties of the distributed-order space fractional derivatives with nonlocal operators. Chen et al. [21] presented a study on start-up pipe flow with double time-distributed order Maxwell model. Niu et al. [22] proposed a fast numerical algorithm for the fractional Sobolev model, with non-smooth solutions for the nonlinear distributed-order equations in porous media. Liu et al. [23] constructed a novel distributed order time fractional model to solve heat conduction, anomalous diffusion, and viscoelastic flow problems. Hu et al. [24] applied a distributed order time fractional Maxwell constitutive fluid model to simulate viscoelastic blood flowing in a narrowing blood vessel with a uniform magnetic field. Zhang et al. [25] introduced the distributed-order time fractional derivative to establish the generalized fractional Bloch-Torrey equation in heterogeneous biological tissues.

    On the other hand, there are very few studies on natural convective flow and heat transfer in porous media with distributed order time fractional derivatives. This is because the momentum and energy governing equations are strongly coupled, and the analytical solutions are difficult to obtain for the nonlinear fractional convection terms. Given this, the paper aims to further discuss the internal transport mechanism in natural convective flow and heat transfer of porous media within distributed order time fractional derivative models. The following Section 2 presents unsteady natural convective flow and heat transfer governing equations using the generalized Maxwell model and Darcy's law with initial and boundary conditions. Section 3 presents the fractional finite volume method together with the fractional L1 scheme to solve the nonlinear governing equations. A mid-point quadrature rule is employed to discretize the distributed order time fractional derivatives. Section 4 validates the acceptable accuracy of the numerical solutions after the grid independence test and comparisons with exact solutions. Section 5 analyzes and discusses the effects of active physical parameters on velocity and temperature distributions in detail. Section 6 summarizes the main conclusions of this paper.

    Consider unsteady convective flow and heat transfer of viscoelastic fluid over a vertical plate embedded in a porous medium. The wall temperature is denoted as Tw and is always maintained as a constant. The ambient temperature far away from the plate is T. Particularly, the direction of the vertical plate is selected as the x-axis, while the y-axis is horizontal accordingly. It is assumed that the fluid is incompressible, and the Boussinesq approximation can apply to the external body force. The porous medium is also assumed to be isotropic and homogeneous. Under these assumptions, the momentum conservation equation in the porous medium follows

    ρ(ut+uux+vuy)=σxyy+gβT(TT)+R, (2.1)

    where ρ is the fluid constant density, u and v are the corresponding velocity components, σxy is the shear stress tensor component, g is the acceleration due to gravity, βT is the thermal expansion coefficient, T is the temperature, and R is the Darcy resistance.

    The microscopic structure of the porous medium and associated environments introduce complexity via porosity and permeability. The distributed order fractional derivative provides more flexibility for simulating anomalous diffusion behaviors with multi-scale characteristics [26]. Thus, the generalized distributed order Maxwell model and Darcy's law are developed by integrating the order of the time-fractional derivative, respectively:

    σxy+10ω1(α)λα1ασxytαdα=μuy, (2.2)
    R+10ω1(α)λα1αRtαdα=μεKu, (2.3)

    where α is the velocity fractional derivative parameter that characterizes the frequency-dependent complex modulus, λ1 is the velocity relaxation time, ω1(α) is the weight coefficient that follows ω1(α) ≥ 0 and 10ω1(α)dα=1 [27], μ is the dynamic viscosity of the fluid, Κ and ε are the permeability and porosity of the porous medium, respectively, and αtα denotes the Caputo fractional derivative operator defined as in (2.4) [28].

    αtαf(x,y,t)=1Γ(mα)t0(tη)mα1fm(x,y,η)ηmdη,m1<α<m, (2.4)

    where m∈+, Γ(∙) denotes the gamma function, and f(x, y, t) is simplified as f(t). Combining Eqs (2.1)-(2.3) to eliminate the terms σxy and R, the nonlinear coupled momentum equation with distributed order fractional derivative is derived as follows:

    ut+x(uu)+y(vu)+10ω1(α)λα1α+1utα+1dα+10ω1(α)λα1αtα(x(uu))dα+10ω1(α)λα1αtα(y(vu))dα=νfεKu+νf2uy2+gβT(TT)+10ω1(α)λα1αtα[gβT(TT)]dα. (2.5)

    Similarly, the homologous energy equation with distributed order fractional derivative is obtained in the following:

    Tt+x(uT)+y(vT)+10ω2(β)λβ2β+1Ttβ+1dβ+10ω2(β)λβ2βtβ(x(uT))dβ+10ω2(β)λβ2βtβ(y(vT))dβ=αf2Ty2, (2.6)

    where ω2(β) denotes the weight coefficient that is suitable for certain conditions as ω2(β) ≥ 0 and 10ω2(β)dβ=1, β is the temperature fractional derivative parameter that improves the efficiency of the thermoelectric material figure-of-merit, λ2 is the temperature relaxation time, and αf is the thermal diffusion coefficient.

    The initial conditions and boundary conditions of the natural convective flow in the porous medium are assumed as follows:

    u(x,y,0)=0,u(x,y,0)t=0,v(x,y,0)=0,T(x,y,0)={Tw,y=0T,y0,T(x,y,0)t=0,u(x,0,t)=0,v(x,0,t)=0,T(x,0,t)=Tw,u(0,y,t)=0,T(0,y,t)=T,u(x,,t)=0,T(x,,t)=T.

    We employ non-dimensional quantities as follows:

    x=xL,y=yLGr1/4,u=uLνfGr1/2,v=vLνfGr1/4,t=νftL2Gr1/2,λ1=λ1νfL2Gr1/2,λ2=λ2νfL2Gr1/2,Da=KL2,Gr=gβT(TwT)L3ν2f,θ=TTTwT,Pr=νfαf,

    where L is the total length of the plate, Da is the Darcy number, Gr is the Grashof number, and Pr is the Prandtl number.

    Non-dimensional distributed order governing equations for the porous medium are derived with time-fractional derivatives after omitting the dimensionless sign "*":

    ux+vy=0, (2.7)
    ut+x(uu)+y(vu)+10ω1(α)λα1α+1utα+1dα+10ω1(α)λα1αtα(x(uu))dα+10ω1(α)λα1αtα(y(vu))dα=εuDaGr1/2+2uy2+θ+10ω1(α)λα1αθtαdα, (2.8)
    θt+x(uθ)+y(vθ)+10ω2(β)λβ2β+1θtβ+1dβ+10ω2(β)λβ2βtβ(x(uθ))dβ+10ω2(β)λβ2βtβ(y(vθ))dβ=1Pr2θy2. (2.9)

    The non-dimensional initial conditions and boundary conditions are subject to

    u(x,y,0)=0,u(x,y,0)t=0,v(x,y,0)=0,θ(x,y,0)={1,y=00,y0,θ(x,y,0)t=0,u(x,0,t)=0,v(x,0,t)=0,θ(x,0,t)=1,u(0,y,t)=0,θ(0,y,t)=0,u(x,,t)=0,θ(x,,t)=0.

    Figure 1 shows the collocated grid system of the physical problem, where the computational domain is divided into discrete control volumes, and the general central nodal point is denoted as ΡP. The grid distances between the inner nodes in the x and y directions are denoted as Δx and Δy, respectively. The time step is denoted as Δt.

    Figure 1.  The grid system of the control volume.

    First, the following integral equation is derived by the integration of Eq (2.9):

    t+ΔttΔVθtdVdt+t+ΔttΔVx(uθ)dVdt+t+ΔttΔVy(vθ)dVdt+t+ΔttΔV10ω2(β)λβ2β+1θtβ+1dβdVdt+t+ΔttΔV10ω2(β)λβ2βtβ(x(uθ))dβdVdt+t+ΔttΔV10ω2(β)λβ2βtβ(y(vθ))dβdVdt=t+ΔttΔV1Pr2θy2dVdt, (3.1)

    where ΔV is the control volume that equals Δx·Δy.

    The backward difference scheme is implemented to discretize the first item of the time term in Eq (3.1) as follows:

    t+ΔttΔVθtdVdt=(θkPθk1P)ΔV+O(Δt). (3.2)

    The volume integrals inside the convective terms and diffusive term are substituted by surface integrals, which are carried out by the first-order upwind difference scheme. In addition, the gradient at the interface is approximated by the central difference format. After these methods of processing, the following discretized equations of integer derivatives are formulated:

    t+ΔttΔVx(uθ)dVdt=[(uAθ)kn(uAθ)ks]Δt=(uk1PAnθkPuk1SAsθkS)Δt+O(Δx), (3.3)
    t+ΔttΔVy(vθ)dVdt=[(vAθ)ke(vAθ)kw]Δt=(vk1EAeθkEvk1PAwθkP)Δt+O(Δy), (3.4)
    t+ΔttΔV2θy2dVdt=[(Aθy)ke(Aθy)kw]Δt=(AeθkEθkPΔyAwθkPθkWΔy)Δt+O(Δy2), (3.5)

    where A represents the face area of the control volume, and Aw = Ae = Δx, An = As = Δy.

    Second, the mid-point quadrature rule is adopted to approximate the distributed order time fractional derivatives. Distributed-order fractional partial differential equations (PDEs) can be regarded as the limiting case of multi-term fractional PDEs [29]. Diethelm and Ford [30] have observed that small changes in the order of a fractional PDE lead to only slight changes in the final solution, which gives initial support to the employed numerical integration method. Thus, the summations of multi-fractional terms are acquired [31]:

    10ω2(β)λβ2β+1θtβ+1dβhβKs=1ω2(βs)λβs2βs+1θtβs+1, (3.6)
    10ω2(β)λβ2βtβ(x(uθ))dβhβKs=1ω2(βs)λβs2βstβs(x(uθ)), (3.7)
    10ω2(β)λβ2βtβ(y(vθ))dβhβKs=1ω2(βs)λβs2βstβs(y(vθ)), (3.8)

    where hβ=1K denotes the fractional parameter step, and βs=(s1)hβ+shβ2.

    The fractional L1 scheme is adopted to estimate the time-fractional derivative of order βs, as 0 < βs < 1 [32,33] under the assumption that f (tk) ∈ C2[0, t]:

    βstβsf(tk)=ΔtβsΓ(2βs)[f(tk)βsk1f(t0)k1j=1(βsj1βsj)f(tkj)]+O(Δt2βs), (3.9)

    where βsj=(j+1)1βsj1βs. For the order of βs+1, the Caputo time fractional derivative has the following property [28]:

    βs+1tβs+1f(tk)=βstβsf(tk), (3.10)

    where f(tk)=[f(tk)f(tk1)]/Δt.

    Utilizing Eqs (3.9) and (3.10), Eqs (3.6)-(3.8) can be transformed into the discretized forms

    10ω2(β)λβ2β+1θtβ+1dβ=hβΔVKs=1ω2(βs)λβs2ΔtβsΓ(2βs)[θkPθk1Pk1j=1(βsj1βsj)(θkjPθkj1P)]+R1, (3.11)
    10ω2(β)λβ2βtβ(x(uθ))dβ=hβKs=1ω2(βs)λβs2Δtβs+1Γ(2βs)[uk1PAnθkPuk1SAsθkSk1j=1(βsj1βsj)(ukj1PAnθkjPukj1SAsθkjS)]+R2, (3.12)
    10ω2(β)λβ2βtβ(y(vθ))dβ=hβKs=1ω2(βs)λβs2Δtβs+1Γ(2βs)[vk1EAeθkEvk1PAwθkPk1j=1(βsj1βsj)(vkj1EAeθkjEvkj1PAwθkjP)]+R3, (3.14)

    where the truncation errors are R1C(Δt), R2C(Δt+Δx), R3C(Δt+Δy).

    Finally, the iterative difference equation of Eq (2.9) is established:

    aPθkP=aSθkS+aWθkW+aEθkE+SP, (3.15)
    aP=(1+Ks=1rs)(ΔVΔt+Δyuk1PΔxvk1P)+2ΔxPrΔy,
    aS=(1+Ks=1rs)Δyuk1Saw=ΔxPrΔyaE=ΔxPrΔy(1+Ks=1rs)Δxvk1E
    SP=(1+Ks=1rs)ΔVΔtθk1P+ΔVΔtKs=1rsk1j=1(βsj1βsj)(θkjPθkj1P)+ΔyKs=1rsk1j=1(βsj1βsj)(ukj1PθkjPukj1SθkjS)+ΔxKs=1rsk1j=1(βsj1βsj)(vkj1EθkjEvkj1PθkjP),

    where rs=hβω2(βs)λβs2ΔtβsΓ(2βs).

    The velocity terms of Eq (2.8) are carried out to deduce the discretized forms similarly. The semi-implicit scheme is utilized for the integration of the single velocity and temperature terms:

    t+ΔttΔVudVdt=ΔtΔV2(ukP+uk1P)+O(Δt), (3.16)
    t+ΔttΔVθdVdt=ΔtΔV2(θkP+θk1P)+O(Δt), (3.17)
    t+ΔttΔV10ω1(α)λα1αθtαdαdVdthαΔV2Ks=1ω1(αs)λαs1Δtαs+1Γ(2αs)[θkP+θk1Pk1j=1(αsj1αsj)(θkjP+θkj1P)]+O(Δt), (3.18)

    where hα=1K, αs=(s1)hα+shα2, αsj=(j+1)1αsj1αs.

    Meanwhile, Eq (2.7) is integrated to obtain the velocity component in the y direction as follows:

    0=t+ΔttΔVux dVdt+t+ΔttΔVvy dVdt=Δt2[(uA)kn(uA)ks+(uA)k1n(uA)k1s]+Δt2[(vA)ke(vA)kw+(vA)k1e(vA)k1w], (3.19)
    vkP=vk1P+vkW+vk1WΔyΔx(ukPukS+uk1Puk1S)+O(Δx+Δy). (3.20)

    Theorem 1. The iterative difference equation (3.14) is uniquely solvable.

    Proof. The values of θ at the kth time level are acquired from the iterative difference equation (3.14). We summarize the difference scheme on a particular i-level in matrix form:

    Giθki=Hiθk1i+Ski, (4.1)
    θki=[θki,2,θki,3,,θki,j,,θki,N1]T,θk1i=[θk1i,2,θk1i,3,,θk1i,j,,θk1i,N1]T,
    Gi=(aP(i,2)00 aE(i,3) aw(i,j1) aP(i,j) aE(i,j+1) aW(i,N2) 00aP(i,N1)),
    Hi=[as(i,2),as(i,3),,as(i,j),,as(i,N1)]T,
    Ski=[SP(i,2)+aW(i,2)θki,1,SP(i,3),,SP(i,j),,SP(i,N1)+aE(i,N)θki,N]T,

    where Gi is a tri-diagonal and full-rank matrix.

    Therefore, the difference scheme (3.20) is uniquely solvable. The numerical solutions of uki and vki are similarly proved.

    The computational domain is regarded as a rectangle with sides Xmax = 1 and Ymax = 12, respectively, where Ymax represents y→∞ approximately. The coordinate mesh sizes are fixed as Δx = 0.05 and Δy = 0.1, respectively, and the time step is selected as Δt = 0.1. To compromise between numerical precision and computation time, the absolute differences between two consecutive time steps are set as smaller than 10−3 within the whole domain. To measure the grid independence, Figure 2 presents the numerical solutions through different groups of grid systems. The velocity curves are in good coincidence, which demonstrates that the selected mesh sizes are appropriate for the calculations.

    Figure 2.  Grid independence test.

    To verify the numerical error accuracy of the presented fractional finite volume method, comparisons are implemented between numerical solutions and exact solutions with two source terms:

    ut+x(uu)+y(vu)+10ω1(α)λα1α+1utα+1dα+10ω1(α)λα1αtα(x(uu))dα+10ω1(α)λα1αtα(y(vu))dα=εuDaGr1/2+2uy2+θ+10ω1(α)λα1αθtαdα+f1(x,y,t), (4.12)
    θt+x(uθ)+y(vθ)+10ω2(β)λβ2β+1θtβ+1dβ+10ω2(β)λβ2βtβ(x(uθ))dβ+10ω2(β)λβ2βtβ(y(vθ))dβ=1Pr2θy2+f2(x,y,t). (4.13)

    The homogeneous initial conditions and boundary conditions are given as follows:

    u(x,y,0)=0,u(x,y,0)t=0,v(x,y,0)=0,θ(x,y,0)=0,θ(x,y,0)t=0,u(x,0,t)=0,v(x,0,t)=0,θ(x,0,t)=0,u(0,y,t)=0,θ(0,y,t)=0,u(x,1,t)=0,θ(x,1,t)=0.

    The eligible exact solutions are chosen specially as

    u(x,y,t)=x2(1x)2y2(1y)2t2,
    v(x,y,t)=2x(1x)(12x)(13y312y4+15y5)t2,
    θ(x,y,t)=x2(1x)2y2(1y)2t2.

    The two source terms are constructed by reverse derivation:

    f1(x,y,t)=[10ω1(α)2λα1t1αΓ(2α)dα+2t10ω1(α)2λα1t2αΓ(3α)dαt2]x2(1x)2y2(1y)2+(10ω1(α)24λα1t4αΓ(5α)dα+t4)[(1x)3(2x34x4)(y4y5)(1323y+35y215y3)]2x2(1x)2(16y+6y2)t2+εx2(1x)2y2(1y)2t2DaGr1/2,
    f2(x,y,t)=[10ω2(β)λβ2t1βΓ(2β)dβ+t]2x2(1x)2y2(1y)22Prx2(1x)2(16y+6y2)t2+(10ω2(β)24λβ2t4βΓ(5β)dβ+t4)[(1x)3(2x34x4)(y4y5)(1323y+35y215y3)].
    Figure 3.  Comparison between numerical solutions and exact solutions.

    Figure 3 illustrates the comparisons between the exact solutions and numerical solutions of u and θ. The velocity and temperature distributions are both in fine consistency, which verifies that the presented fractional finite volume method has reliable convergence accuracy.

    In this section, the effects of the active physical parameters on velocity and temperature distributions are illustrated graphically: namely, the porosity, Darcy number, and the relaxation time parameters. The distributed order fractional derivatives are integrated over a given range [0, 1], and the summations of multi-fractional terms are acquired by the mid-point quadrature rule in Eqs (3.6)-(3.8). For this reason, the effects of the fractional derivative parameters are not discussed in this paper. The Prandtl number and Grashof number are both fixed as constants for brevity.

    Figure 4.  Temperature profiles with different ε.
    Figure 5.  Velocity profiles with different ε.

    Figure 4 shows temperature profiles with different values of porosity. The temperature distribution is monotone decreasing for a single value of porosity. With the increase of ε, the temperature distribution increases, and the thickness of the thermal boundary layer gets bigger. The velocity profiles with different ε are depicted in Figure 5. Each velocity profile is not monotone and possesses a maximum value. As the porosity increases, the velocity distribution declines, and the velocity boundary layer becomes thinner. In addition, the point location of the maximum value gets closer to the vertical plate. These results demonstrate that larger porosity imposes negative restrictions on momentum transfer.

    The effects of Darcy number on temperature and velocity profiles are illustrated in Figures 6 and 7. It is found in Figure 6 that the temperature distribution shifts down with the increase of Da, but the thickness of the temperature boundary layer declines slightly. In Figure 7, the velocity distribution rises with the increase of Da, but the increasing magnitude lessens distinctly. The result proves that the permeability accelerates the development of the velocity. It can be concluded that the porosity and permeability have opposite influences on temperature and velocity fields.

    Figure 6.  Temperature profiles with different Da.
    Figure 7.  Velocity profiles with different Da.
    Figure 8.  Temperature profiles with different λ2.
    Figure 9.  Velocity profiles with different λ1.

    Figures 8-10 present the effects of the relaxation time λ1 and λ2 on velocity and temperature. It can be seen from Figure 8 that the temperature distribution decreases apparently with the increase of λ2. In Figure 9, the velocity distribution rises as the relaxation time λ1 increases, where the Newtonian fluid that responds to λ1 = 0 has the smallest velocity values. The result verifies that the velocity relaxation time promotes the momentum transfer. In Figure 10, the velocity distribution descends with the augment of λ2, which has inverse effects compared with the relaxation time λ1. The velocity profiles stay close to each other near the plate and separate evidently at a certain distance from the plate. Meanwhile, the velocity boundary layer thickness reduces slightly with the increase of λ2.

    Figure 10.  Velocity profiles with different λ2.

    This paper discussed the unsteady natural convective flow and heat transfer of generalized Maxwell viscoelastic fluids with distributed order time fractional derivatives embedded in a porous medium. The finite volume method combined with the fractional L1 scheme rule was applied to solve the nonlinear governing equations. Numerical solutions of velocity and temperature were discussed after the grid independence test and comparisons with exact solutions. Results are concluded as follows: (i). The temperature distribution rises with the increase of ε but declines as Da and λ2 become larger. (ii). The velocity distribution ascends with the augment of Da and λ1 but reduces with the increase of ε and λ2.

    The work is supported by the Outstanding Youth Research Project of Anhui Universities (2023AH030075), the National Natural Science Foundations of China (No. 12102093), the Natural Science Research Project of Anhui Universities (2022AH051336), the Talent Project of Fuyang Normal University (rcxm202105).

    The authors declare that they have not used artificial intelligence (AI) tools in the creation of this article.

    The author declares that he has no conflict of interest.



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