In recent decades, technology has been growing exponentially in numerous fields of science and engineering such as power generation and electronics where heat transport is an essential phenomenon. The extraordinary growth in technologies needs revolutionary and innovative heat transport management. In this case, advanced and innovative cooling procedures are required to be invented. As the heat transfer management in industries and electronic devices gained prominence, the research community tried to develop an efficient heat transfer medium. Numerous conventional techniques such as mini channel and increasing the surface area had been applied but failed due to scientific limitations. Despite that, enrichment in material science provides specialness to heat transfer fluid by dissolving nano-sized particles in the base fluid. This phenomenon of dispersion of nano-sized particles in host fluid had been revealed by Choi et al. [1]. Thereby, numerous investigators characterized, prepared, and examined different nanofluids in heat transport applications [2,3]. Many experimental and numerical studies indicated a different combination of nanoparticle and base fluids. Amongst the many types of nanoparticles, carbide ceramic, metal oxide, metals, nitride ceramic, semiconductor, and carbon in the various forms are commonly used in the traditional fluids such as water, oil, alcohol, and polymer solutions [4]. However, some recent studies indicated that carbon nanotubes (CNTs) are more efficient in convection heat transfer due to their high thermal conductivity.

CNTs nanoparticles are in a cylindrical shape with superior thermal characteristics (higher electrical and thermal conductivities and low density) compared to spherical shape nanoparticles. CNTs have incredible applications in various areas of science, which include electronics, health care, energy, cooling, heat exchanger, bio-medicine, solar collector, diesel engine, and environment [5]. Khalid et al. [6,7] investigated the magneto-hydro-dynamic (MHD) flow of Blood-CNTs nanofluid in a porous medium. In these studies, SWCNTs and MWCNTs had been considered. They reported that the temperature field increases with increasing volume concentration whereas the velocity field indicated an opposite trend. Aman et al. [8] analyzed the flow of CNTs nanofluid in a microchannel by using the perturbation method. Their results revealed that the temperature field of SWCNTs is higher than MWCNTs in different based fluids. Some more recent studies on nanofluids can be found in [9–14] and the references therein.

In the open literature, many studies had focused on convection heat transfer with constant wall temperature. But several physical situations follow arbitrary thermal conditions at the boundary. The convection heat transfer problems are effective to study with step-change thermal boundary conditions (Ramped wall thermal boundary conditions). The problems with ramped wall thermal boundary conditions can be implemented in thin-film photovoltaic devices to accomplish a certain finish of the system [15]. Ramped wall temperature is also essential for heat management in buildings such as air conditioning where the constant wall thermal conditions lead to noticeable error. Keeping in mind the importance of step-change thermal boundary conditions, this study investigates convection heat transfer with ramped boundary condition.

In the open literature, nanofluids are extensively studied without ramped wall conditions. The studies on nanofluid flow with ramped wall temperature are very rare and the flow of CNTs nanofluid with Caputo–Fabrizio time-fractional Brinkman type fluid model are not as yet reported in literature. Thereby, to fill this research gap, a Caputo time-fractional Brinkman typed fluid model is developed for the flow of CNTs (SWCNTs and MWCNTs) under the influence of the external magnetic field. In the discussed literature, the governing equations that describe the convection heat transfer in nanofluid had been modeled in terms of integer order constitutive relationship. However, with the growth of technologies, fractional calculus has reached terrific advancement due to wide range of applications. As accepted in the open literature, integer-order derivatives are local in nature, but time-fractional derivatives are non-local, which retain memory property [16]. Having such motivation, this study deals with convection heat transfer in MHD CNTs nanofluid based on the time-fractional Brinkman type fluid model. The classical Brinkman model together with energy equation have been transformed into fractional forms using Caputo time-fractional derivative. The generalized energy equation is partially coupled with the momentum equations to study the memory effect. The sinusoidal oscillating and ramped wall conditions have been imposed. The fractional model has been solved for exact solutions using the Laplace transform. The physical impact of pertinent flow parameters have been highlighted in various graphs with physical interpretation.

Consider the convection heat transfer in water-CNTs nanofluid near an infinite plate with ramped wall temperature in x,y plane where the plate is selected along x-axis and y-axis is chosen normal to it. It is assumed that the fluid is electrically conducting. The magnetic Reynold is assumed to be very small, thereby, the induced magnetic field is neglected [17]. Initially, at t≤0, both the fluid and plate are at rest with ambient temperature T∞. After a short interval of time t = 0⁺, the temperature of the plate rises from T∞ to T0+(TW-T∞)t/t0 for 0 < t < t₀ or T_W for t > t₀. In addition, the plate starts oscillations with U0H(t)cos(ωt) where U₀ is the amplitude of oscillation, H(t) is the Heaviside unit step function, ω is the frequency of oscillation and t is the time period. Due to the temperature gradient and plate oscillations, the mixed convection is taken place and the nanofluid begins to move in x-direction. The governing equations of the problem are given by

(1) ρnf(∂u(y,t)∂t+β*u(y,t))=μnf∂2u(y,t)∂y2-σnfB02u(y,t)+g(ρβT)nf(T(y,t)-T∞),

(2) (ρCp)nf∂T(y,t)∂t=knf∂2T(y,t)∂y2,

where ρnf is the density, u(y,t) is the velocity, β* is the Brinkman type fluid parameter, μnf is the dynamic viscosity, σnf is the electrical conductivity, B₀ is a uniform magnetic field, g gravitational acceleration, (βT)nf is the thermal expansion, T(y,t) is the temperature, (Cp)nf is the heat capacitance, and k_nf is the thermal conductivity of nanofluid. The appropriate physical initial and boundary conditions are given as

(3) u(y,0)=0,T(y,0)=T∞,∀y≥0,

(4) t_{0}\end{array}\right. \\[20pt] u \left(y,t\right)\rightarrow 0~\mathrm{and}~T \left(y,t\right)\rightarrow T_{\infty };\quad {\rm if}\,y\rightarrow \infty \end{array}\right\}, \label{eqn-4} \end{equation}$$]]> u(0,t)=U0H(t)cos(ωt);∀t≥0+T(0,t)={T0+(TW-T∞)tt0;if0<t<t0TW;ift>t0 u(y,t)→0andT(y,t)→T∞;ify→∞},

where mathematical expressions for ρnf, μnf, σnf, (βT)nf, (Cp)nf and k_nf are given by [18].

(5) ρnf=(1-ϕ)ρf+ϕρs,μnf=μf(1-ϕ)2.5,σnfσf=1+3(σsσf-1)ϕ(σsσf+2)-(σsσf-1)ϕ,(ρβT)nf=(1-ϕ)(ρβT)f+ϕ(ρβT)s,(ρCp)nf=(1-ϕ)(ρCp)f+ϕ(ρCp)s,knfkf=(1-ϕ)+2ϕksks-kf lnks-kf2kf(1-ϕ)+2ϕksks-kf lnks+kf2kf },

The subscripts nf, f, and s in Eqs. (5)–(10) are referred to as nanofluid, base fluid, and solid CNTs nanoparticles, respectively. The numerical values of thermophysical properties are given in Tab. 1 [6].

The dimensionless variables are introduced into governing Eqs. (1) and (2) and initial and boundary conditions (4) and (5) to minimize the number of variables and to simplify the mathematical model. Incorporating the following dimensionless variable u*=uU0,y*=U0υfy,t*=tt0,t0=υfU02,θ=T-T∞TW-T∞ into Eqs. (1) and (4) and dropping*sign for simplicity yield to

(6) ϕ0{∂u(y,t)∂t+βu(y,t)}=ϕ1∂2u(y,t)∂y2-ϕ2Mu(y,t)+ϕ3Grθ(y,t),

(7) ϕ4 Pr∂θ(y,t)∂t=ϕ1∂2θ(y,t)∂y2,

and

(8) u(y,0)=0,θ(y,0)=0,∀y≥0,

(9) 1 \end{array}\right. \\[16pt] u \left(y,t\right)\rightarrow 0~\mathrm{and}~T \left(y,t\right)\rightarrow 0;\quad \mathrm{if}~y\rightarrow \infty \end{array}\right\}, \label{eqn-9} \end{equation}$$]]> u(0,t)=H(t)cos(ωt);∀t≥0+T(0,t)={t;if0<t<11;if t>1 u(y,t)→0andT(y,t)→0;ify→∞},

where β=υfβ*U02,M=υfσfB02U02ρf,Gr=g(βTυ)f(TW-T∞)U03,Pr=(μCpk)f,ϕ0=(1-ϕ)+ϕρsρf,ϕ1=1(1-ϕ)2.5,ϕ2=σnfσf,ϕ3=(1-ϕ)+ϕ(ρβT)s(ρβT)f,ϕ4=(1-ϕ)+ϕ(ρCp)s(ρCp)f,ϕ5=knfkf, is the Brinkman type fluid parameter, magnetic number, thermal Grashof number, and Prandtl number respectively. In addition, ϕ0, ϕ1, ϕ2, ϕ3, ϕ4 and ϕ5 are constant terms which are produced during the calculi. The dimensionless governing Eqs. (6) and (7) are transformed to Caputo–Fabrizio times-Fractional form as

(10) ϕ0{Dtαu(y,t)+βu(y,t)}=ϕ1∂2u(y,t)∂y2-ϕ2Mu(y,t)+ϕ3Grθ(y,t),

(11) ϕ4 PrDtαθ(y,t)=ϕ1∂2θ(y,t)∂y2,

where Dtα(,.,) is the Caputo–Fabrizio time-fractional operator (CFTO) defined by [19]

(12) Dtαf(y,t)=N(α)1-α∫ 0t exp(-α(t-τ)1-α)∂f(y,τ)∂τdτ;0<α<1,

where

(13) N(1)=N(0)=1.

The Laplace transform of Eq. (12) using Eq. (13) yield to

(14) L{Dtαf(y,t)}(q)=qf¯(y,q)-f(y,0)(1-α)q+α,0<α≤1,

such that

(15) limα→1[L{Dtαf(y,t)}(q)]= limα→1{qf¯(y,q)-f(y,0)(1-α)q+α}=qf¯(y,q)-f(y,0)=L{∂f(y,t)∂t}.

Eq. (15) clearly indicates that the fractional order CFTO can be reduced to integer order partial differential derivatives.

In this section, the fractional Laplace transform method is adopted to solve Eqs. (10) and (11) using the initial and boundary conditions (8) and (9).

The solutions for velocity and temperature fields are computed for various flow parameters such as fractional parameter α, volume fractional of CNTs ϕ, Brinkman type fluid parameter β, magnetic parameter M, and thermal Grashof number Gr. To explore the physical aspects of the problem, the temperature and velocity fields are graphically demonstrated in Figs. 1–9. It is important to mention here that t = 0.5 and t = 1.5 are chosen for ramped and isothermal temperature and velocity, respectively.

Fig. 1 depicts the effect of α on temperature field for both ramped and isothermal wall temperature. It is noticed that the temperature field decreased with increasing values of α in case of ramped temperature. However, in the isothermal case, the temperature field near the plate increased with increasing α. This trend reverses away from the plate. The trend of temperature field can be physically justified, as an increase in α causes a decrease in the thermal boundary layer, as a result, the temperature field increased. Fig. 2 presents the influence of α on velocity profile. This figure indicates that the velocity retarded when α is increased. This is because the velocity boundary layer decreases with an increase in α. The trend of α on velocity and temperature field is similar to that of Ahmed et al. [20] in the case of Caputo fractional derivative.

The effect of ϕ on the temperature field is illustrated in Fig. 3. It is noticed that the temperature field enhanced with enhancement in ϕ for ϕ=0.0,0.02,0.03,0.04. This is because of the effective thermal conductivity of CNTs. When ϕ increased, it caused an increment in thermal conductivity, as a result, the temperature filed increased. The impact of ϕ on the velocity field is highlighted in Fig. 4. The velocity field decreased with an increase in ϕ due to the effective density of CNTs. Increment in ϕ results in an increase in the density of nanofluid, which retarded the velocity field. The effects of ϕ on temperature and velocity field can be validated with [21] (Figs. 2 and 6).

In Fig. 5 the temperature field is compared for SWCNTs and MWCNTs. It is found that the temperature field of SWCNTs is higher than MWCNTs. This is due to the difference in the thermal conductivities of SWCNTs and MWCNTs. The thermal conductivity of SWCNTs (6600 k (W/mK)) is higher than the thermal conductivity of MWCNTs (3000 k (W/mK)) which correspond to the higher temperature field of SWCNTs than MWSNTs. It is noticed in Fig. 6 that the velocity of MWCNTs is higher than the velocity of SWCNTs. This is because the density and thermal conductivity of SWCNTs is greater than MWCNTs, which absorb more heat energy and are at the same time denser. Thereby, the velocity field of MWCNTs is higher than SWCNTs. The same behavior of velocity and temperature field were presented by Aman et al. [8] (Figs. 2 and 5).

The consequence of β on the velocity field is portrayed in Fig. 7. The β is directly related to the drag force of the porous medium. Higher the β corresponds to strengthen the drag forces which reduced the velocity field. The trend of β is similar to that of [22].

The influence of M on the velocity field is studied in Fig. 8. Increasing M strengthens the Lorentz forces which resist the velocity. Therefore, the nanofluid velocity decreases with increasing values of M. Finally, the result of Gr is illustrated in Fig. 9. The Gr represent the ratio of buoyancy forces to the viscous forces. The greater the values of Gr, the stronger will be the buoyancy forces that cause an increase in the velocity field.

In this article, the Caputo time-fractional Brinkman type nanofluid model is constructed to investigate the convection heat transfer in a water-CNTs nanofluid. The time-fractional model have been solved for exact solutions using the Laplace transform method. The exact expressions for velocity and temperature field have been computed and displayed in numerous graphs with the physical explanation. The velocity and temperature fields are decreasing with increasing fractional parameters due to the reduction in the thickness of velocity and temperature boundary layers. The temperature field increased with an increase in volume concentration of CNTs whereas the velocity field performed oppositely due to thermal conductivity and density factors of CNTs. It is noticed that the temperature field of SWCNTs is higher than MWCNTs because of the thermal conductivity factor. However, the velocity field of MWCNTs is higher than SWCNTs. The velocity field decreased with an increase in Brinkman type fluid parameter due to drag forces. The magnetic parameter decreased the velocity because of Lorentz forces. Thermal Grashof number is the ratio of buoyancy force to the viscous force which increases the velocity field.