Enhanced heat transfer in working fluids using nanoparticles with ramped wall temperature: Applications in engine oil

Abstract

The purpose of this article is to investigate the flow of Maxwell fluid with nanoparticles, that is, molybdenum disulfide and graphene with ramped temperature condition at the boundary, and engine oil is considered as base fluid. Furthermore, molybdenum disulfide and graphene nanoparticles are uniformly distributed in the base fluid. The problem is modeled in terms of partial differential equations with physical initial and boundary conditions. To make the system of governing equations dimensionless, we introduced some suitable non-dimensional variables. The obtained dimensionless system of equations is solved using the Laplace transform technique. From graphical analysis, it can be noticed that the velocity is high with isothermal wall temperature and lower for ramped wall temperature. These solutions are verified by comparing with the well-known published results. In addition, the physics of all parameters of interest is discussed through graphs. The mathematical expressions for skin friction and Nusselt number are mentioned and the obtained results are presented in tabular form. Finally, the effect of molybdenum disulfide and graphene nanoparticles is briefly discussed for the flow and heat profiles for Maxwell nanofluid.

Introduction

The topic related to nanofluids is getting more attention in the last three decades in the field of science as well as in engineering and biological sciences. The conventional base fluids, for example, kerosene oil, engine oil (EO), water, polyethylene glycol, and ethylene glycol, are used for thermal transport purpose. They have low thermal conductivities as a result of which the heat transfer rate is reduced. The first scientific attempts have been discussed by Maxwell ¹ who explained the process of suspending some micro-sized solid particles in the base fluids. However, there were some limitations in Maxwell’s process of suspended micro-sized solid particles. Afterward, Choi ² introduced the concept of nanofluid which was accepted, and now nanotechnology is used in biological and engineering sciences. Nanoparticles are dispersed uniformly in different base fluids to increase the heat transfer rate of the base fluids. The dispersion of nanometer-sized particles is better than micro-sized particles because of some valid reasons such as nanoparticles are more stable as compared to micro-sized particles. Nanofluids have many useful applications in physical phenomena due to high thermal conductivities. For example, nanofluids are used in different machinery and automobiles for coolant purpose, in biomedicines, food processing, and so on. Carbon nanotube is used for the purpose of heat transfer, biomedicines, and lubrication processes in the grinding parts of the machinery.^3,4 Loganathan et al. ⁵ were the first to investigate the closed-form solutions for the flow of nanofluids. Li et al. ⁶ discussed the applications of molybdenum disulfide (MoS₂) and graphene nanoparticles and used these nanoparticles for the hydrogen evolution reaction. Samnakay et al. ⁷ and Liang et al. ⁸ explained some applications of graphene and MoS₂ nanoparticles and proposed a mechanism for rechargeable batteries.

In the modern world, many simple and complex machines are used to perform different processes in industries and many other fields of sciences. Mostly, in all machinery, there are different lubricants used for lubrication purposes to reduce friction inside the machine parts. Furthermore, an enormous number of nanoparticles are used to enhance the thermal conductivities of lubricants due to which machines are able to work for a longer period of time.^9,10 Zhu et al. ¹¹ investigated strain tuning of optical emission energy using MoS₂ nanoparticles. Guo and Dong ¹² investigated molecular engineering synthesis using graphene nano-sheet and discussed some analytical and energy applications regarding the graphene nanoparticles. In addition, graphene nanoparticles are used for different purposes in biomedical sciences and material sciences.^13–15

Recently, in nanotechnology, different nanoparticles are used to improve the thermal performance of lubricants. In the present study, spherical shape MoS₂ and graphene nanoparticles have been dispersed in EO. The heat produced in the grinding parts of the machine is reduced due to these nanoparticles. MoS₂ and graphene nanoparticles are able to enhance the heat transfer rate and the lubrication of the EO.^16,17 Furthermore, some other applications and scientific works on MoS₂ are reported by Gul et al., ¹⁸ Kato et al., ¹⁹ and Mao et al. ²⁰ The capacity of heat in nanofluids, thermal performance of nanofluids, and the volume thermal expansion coefficients were investigated after some experiments in the laboratory by Liu et al. ²¹ and Ding and Xiao. ²² In addition, we also considered the effect of graphene nanoparticles in the base fluid EO. Some physical properties and applications related to graphene nanoparticles are discussed and explained by Katz et al. ²³ Yoon et al. ²⁴ discussed the stabilization of oil emulsions, and Wang et al. ²⁵ reported some applications of graphene in biological science. Kuila et al. ²⁶ investigated some applications and functions of graphene in chemical technology.

The problems resulting from the flows of incompressible non-Newtonian fluids have been of great and increasing interest. These non-Newtonian fluids have been discussed by researchers due to the increasing interest in such fluids in the last five decades. Maxwell fluid is one of the non-Newtonian fluids and it was originally modeled by Maxwell ²⁷ to obtain the visco-elastic behavior of air through the dynamical theory of gases. In physical situations, many fluids, such as glycerin, crude oils, EOs, and some other polymeric materials, behave like Maxwell fluids. The Maxwell fluids with some interesting applications and properties have been discussed by Jordan et al. ²⁸ Denn and Porteous ²⁹ studied the Couette flow of Maxwell fluid between two infinite parallel plates. Karra et al. ³⁰ investigated the flow of Maxwell fluid with relaxation time and viscosity. Vieru and Zafar ³¹ analyzed Couette flow of Maxwell fluid with slip boundary conditions. Some more physical uses and applications of Maxwell fluid are mentioned in the previous studies.^32–34

All the aforementioned works used uniform wall temperature conditions, but there are some real-world problems which are not completely explained by isothermal wall temperature. The non-uniform wall temperature (ramped temperature) is applied in industries and some other physical applications such as the phase transition process of different materials/production, air conditioning processes, photovoltaic instruments, the phenomenon of nuclear heat transfer control, heat transfer in building, heat transfer in the turbine blades, and some useful applications in electric circuits. Due to increasing interest in these issues, many researchers investigated different problems: such as Chandran et al. ³⁵ published their work on natural convection near a vertical plate with ramped wall temperature. Some other applications of non-uniform wall temperature were investigated by Ali et al., ³⁶ Kataria and Patel, ³⁷ and Narahari et al. ³⁸

From the given literature survey, it can be noted that no work is reported to have obtained the exact solutions of Maxwell nanofluid with the effect of non-uniform wall temperature at the plate. Therefore, this study focused on obtaining the closed-form solutions for Maxwell nanofluids. Furthermore, MoS₂ and graphene nanoparticles are uniformly distributed in EO, and the fluid flow is along an infinite vertical plate with ramped wall temperature conditions at the boundary. In this study, solutions of Maxwell nanofluids have been compared for ramped and isothermal temperature on the boundary. The effect of MoS₂ and graphene nanoparticles is shown through graphs which show that MoS₂ is more effective as compared to graphene in the EO. Various parameters of interest are discussed through graphs. The numerical values are obtained for skin friction and Nusselt number and mentioned in tabular form. Finally, a brief discussion describes the behavior of the heat transfer rate and fluid flow with the increment in the volume fraction parameter $\varphi$ .

Mathematical modeling

The unsteady flow of Maxwell nanofluid over an infinite vertical flat plate with ramped wall temperature at the boundary is considered in this study. The fluid occupies the space $(y\ge 0)$ or the fluid is in the $\mathrm{xz}$ plane. The plate is along x-axis and the fluid is moving along the direction of the plate. EO is selected as the base fluid, and MoS₂ and graphene nanoparticles are suspended in the base fluid. Initially, the fluid and plate are at rest at a constant temperature $T_{\infty }$ .

After some time when $t=0^{+}$ , the temperature of the plate rises gradually to $T_w$ and then approaches the value $T_{\infty }$ . In other words, when time $t=0^{+}$ , the temperature of the plate changes—it is lowered or raised with time $t$ . When time $t\le t_0$ , then the temperature of the plate is $T_{\infty }+(T_w-T_{\infty })t/t_0$ , and when time $t>t_0$ , then the temperature of the plate is maintained as constant temperature $T_w$ . The velocity of the fluid is zero and temperature of the fluid is ambient temperature $T_{\infty }$ when $y\to \infty$ (see Figure 1).

Figure 1.

Physical sketch of the model and coordinate system.

Under the assumed conditions in this study, the governing equations are expressed in the following form ³⁹

$$\begin{array}{c}{\rho }_{\mathrm{nf}}(1+{\lambda }_1\frac{\partial }{\partial t^{*}})\frac{\partial u({\xi }^{*},t^{*})}{\partial t^{*}}\hfill \\ ={\mu }_{\mathrm{nf}}\frac{{\partial }^2{u}^{*}({\xi }^{*},t^{*})}{\partial {{\xi }^{*}}^2}+(1+{\lambda }_1\frac{\partial }{\partial t^{*}}){\left(\rho {\beta }_T\right)}_{\mathrm{nf}}g(T^{*}-T_{\infty }^{*})\hfill \end{array}$$

(1)

$${\left(\rho C_p\right)}_{\mathrm{nf}}\frac{\partial T({\xi }^{*},t^{*})}{\partial t^{*}}=k_{\mathrm{nf}}\frac{{\partial }^{2}T({\xi }^{*},t^{*})}{\partial {{\xi }^{*}}^2}$$

(2)

the corresponding initial and boundary conditions are

$$\begin{array}{c}{u}^{*}({\xi }^{*},0)=0,T^{*}({\xi }^{*},0)=T_{\infty }^{*}\hfill \\ u^{*}(0,t^{*})=0,T^{*}(0,t^{*})=\{\begin{array}{c}{T_{\infty }}^{*}+(T_w^{*}-T_{\infty }^{*})t^{*}/t_0,0<t^{*}\le t_0\hfill \\ \multicolumn{1}{c}{T_w^{*},t^{*}\ge t_0}\end{array}\hfill \\ u(\infty ,t^{*})=0,T^{*}(\infty ,t^{*})=T_{\infty }^{*}\hfill \end{array}$$

(3)

where $u^{*}$ represents the velocity component of nanofluid along x-axis, $T^{*}$ represents the temperature profile of nanofluid, $T_w^{*}$ is the constant temperature of the wall where constant wall temperature is greater than temperature away from the bounding plate, $g$ is gravitational acceleration, and $k_{\mathrm{nf}}$ , ${\mu }_{\mathrm{nf}}$ , $(\rho C_p{)}_{\mathrm{nf}}$ , and ${\beta }_{\mathrm{nf}}$ represent the thermal conductivity of nanofluid, dynamic viscosity of nanofluid, the heat capacitance of nanofluid, and the coefficient of thermal expansion of nanofluids, respectively. The thermophysical properties of the base fluid and nanoparticles are given in Table 1.

Table 1.

Thermo-physical properties of engine oil, graphene, and MoS₂. ³⁹

Model	$\rho (\mathrm{kg}{\mathrm{m}}^{-3})$	$c_p(\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{K}}^{-1})$	$k(\mathrm{W}{\mathrm{m}}^{-1}{\mathrm{K}}^{-1})$	$\beta \times {10}^{-5}({\mathrm{K}}^{-1})$
Engine oil	884	1910	0.144	70
Graphene	2250	2100	2500	21
MoS2	5.06 × 103	397.21	904.4	2.8424

MoS₂: molybdenum disulfide.

The effective thermal conductivity of nanofluids is given by Kakac and Pramuanjaroenkij ⁴⁰ and Oztop and Abu-Nada ⁴¹ as

$$\begin{array}{c}{\rho }_{\mathrm{nf}}=(1-\varphi ){\rho }_f+\varphi {\rho }_s,{\mu }_{\mathrm{nf}}=\frac{{\mu }_f}{{(1-\varphi )}^{2.5}},\hfill \\ {\left(\rho {\beta }_T\right)}_{\mathrm{nf}}=(1-\varphi ){\left(\rho {\beta }_T\right)}_f+\varphi {\left(\rho {\beta }_T\right)}_s\hfill \\ {\left(\rho c_p\right)}_{\mathrm{nf}}=(1-\varphi )(\rho c_p{)}_f+\varphi (\rho c_p{)}_s,\hfill \\ \frac{k_{\mathrm{nf}}}{k_f}=\frac{(k_s+2k_f)-2\varphi (k_f-k_s)}{(k_s+2k_f)+\varphi (k_f-k_s)}\hfill \end{array}$$

(4)

where $\varphi$ is the volume fraction of nanoparticles, ${\rho }_f$ is the base fluid density, ${\rho }_s$ is the density of the solid particle, and $C_p$ is the specific heat at constant pressure.

To make the system of equations dimensionless, the following non-dimensional variables have been used ⁴²

$$\begin{array}{c}u=\frac{u^{*}}{U_0},\xi =\frac{{\xi }^{*}}{\sqrt{\upsilon t_0}},t=\frac{t^{*}}{t_0},\theta =\frac{T^{*}-T_{\infty }^{*}}{T_w^{*}-T_{\infty }^{*}},\hfill \\ t_0={\left[\frac{{\upsilon }^{1/2}}{g\beta (T_w^{*}-T_{\infty }^{*})}\right]}^{\frac{2}{3}}\hfill \end{array}$$

(5)

After dimensional analysis, the governing equations of the problem are given as

$${\varphi }_1(1+\lambda \frac{\partial }{\partial t})\frac{\partial u(\xi ,t)}{\partial t}={\varphi }_2\frac{{\partial }^{2}u(\xi ,t)}{\partial {\xi }^2}+{\varphi }_3(1+\lambda \frac{\partial }{\partial t})\mathrm{Gr}\theta$$

(6)

$$\mathrm{Pr}{\varphi }_4\frac{\partial \theta (\xi ,t)}{\partial t}={\varphi }_5\frac{{\partial }^2\theta (\xi ,t)}{\partial {\xi }^2}$$

(7)

From equations (6) and (7), we get the following form

$$\frac{{\partial }^{2}u(\xi ,t)}{\partial {\xi }^2}-a_0(1+\lambda \frac{\partial }{\partial t})\frac{\partial u(\xi ,t)}{\partial t}=-a_2(1+\lambda \frac{\partial }{\partial t})\theta$$

(8)

$$\frac{{\partial }^2\theta (\xi ,t)}{\partial {\xi }^2}-b\frac{\partial \theta (\xi ,t)}{\partial t}=0$$

(9)

Corresponding dimensionless initial and boundary conditions are given as follows

$$\begin{array}{c}u(\xi ,0)=0,\theta (\xi ,0)=0\hfill \\ u(0,t)=0,\theta (0,t)=\{\begin{array}{c}t,0<t\le 1\hfill \\ \multicolumn{1}{c}{1,t>1}\end{array}=\mathrm{tH}(t)-(t-1)H(t-1)\hfill \\ u(\infty ,t)=0,\theta (\infty ,t)=0\hfill \end{array}$$

(10)

where

$$\begin{array}{c}\lambda =\frac{{\lambda }_1}{t_0},\mathrm{Gr}=\frac{g{\beta }_T(T_w-T_{\infty })t_0}{U_0},\hfill \\ \mathrm{Pr}={\left(\frac{\mu C_p}{k}\right)}_f,a_0=\frac{{\varphi }_1}{{\varphi }_2},a_1=\frac{{\varphi }_3}{{\varphi }_2}\hfill \\ a_2=a_1\mathrm{Gr},a_3=\frac{{\varphi }_4}{{\varphi }_5},b=a_3\mathrm{Pr}\hfill \end{array}$$

$$\begin{array}{c}{\varphi }_1=(1-\varphi )+\varphi \frac{{\rho }_s}{{\rho }_f},{\varphi }_2=\frac{1}{{(1-\varphi )}^{2.5}},\hfill \\ {\varphi }_3=(1-\varphi )+\varphi \frac{{\left(\rho {\beta }_T\right)}_s}{{\left(\rho {\beta }_T\right)}_f}\hfill \end{array}$$

$${\varphi }_4=(1-\varphi )+\varphi \frac{{(\rho c_p)}_s}{{(\rho c_p)}_f},{\varphi }_5=\frac{(k_s+2k_f)-2\varphi (k_f-k_s)}{(k_s-2k_f)+\varphi (k_f-k_s)}$$

Exact solutions

Applying the Laplace transform to equations (8) and (9), using the given initial and boundary conditions from equation (10), we get the transform solutions as

$$\overline{\theta }(\xi ,q)=\frac{1-e^{-q}}{q^2}\exp (-\xi \sqrt{\mathrm{bq}})$$

(11)

$$\begin{array}{c}\overline{u}(\xi ,q)=-\left[\frac{a_2(\lambda q+1)(1-e^{-q})}{q^3[a_0\lambda q+(a_0-b)]}\right]\exp (-\xi \sqrt{a_{0}q(\lambda q+1)})\hfill \\ +\left[\frac{a_2(\lambda q+1)(1-e^{-q})}{q^3[a_0\lambda q+(a_0-b)]}\right]\exp (-\xi \sqrt{\mathrm{bq}})\hfill \end{array}$$

(12)

By taking the Laplace inverse transform technique to equations (11) and (12), we get the solutions for uniform and non-uniform (ramped wall) temperature at boundary which are mentioned in the following sections: “Solution for energy equation with Ramped wall temperature,”“Solution for energy equation with isothermal wall temperature,”“Solution of velocity profile for ramped wall temperature,” and “Solution of velocity profile for isothermal wall temperature.”

Solution for energy equation with ramped wall temperature

$$\theta {(\xi ,t)}_{\mathrm{ramp}}={\theta }_1(\xi ,t)-{\theta }_1(\xi ,t-1)H(t-1)$$

(13)

Solution for energy equation with isothermal wall temperature

$$\theta {(\xi ,t)}_{\mathrm{iso}}=\mathrm{erfc}\left(\frac{y}{2}\sqrt{\frac{b}{t}}\right)$$

(14)

where

$${\theta }_1(\xi ,t)=[(t+\frac{b{\xi }^2}{2})\mathrm{erfc}\left(\frac{\xi }{2}\sqrt{\frac{b}{t}}\right)-\xi \sqrt{b}\sqrt{\frac{t}{\pi }}{e}^{-\frac{b{\xi }^2}{4t}}]$$

(15)

Solution of velocity profile for ramped wall temperature

$$\overline{u}{(\xi ,q)}_{\left(\mathrm{ramp}\right)}=\frac{1}{q}\overline{F_1}(\xi ,q)(1-e^{-q})+\frac{1}{q}\overline{F_2}(\xi ,q)(1-e^{-q})$$

(16)

$$\begin{array}{c}\multicolumn{1}{c}{u{(\xi ,t)}_{\left(\mathrm{ramp}\right)}=\underset{0}{\overset{t}{\int }}[F_1(\xi ,\tau )-F_1(\xi ,\tau -1)H(\tau -1)]d\tau }\\ \multicolumn{1}{c}{+\underset{0}{\overset{t}{\int }}[F_2(\xi ,\tau )-F_2(\xi ,\tau -1)H(\tau -1)]d\tau }\end{array}$$

(17)

Solution of velocity profile for isothermal wall temperature

$$\overline{u}{(\xi ,q)}_{\left(\mathrm{iso}\right)}=\frac{1}{q}\overline{F_1}(\xi ,q)+\frac{1}{q}\overline{F_2}(\xi ,q)$$

(18)

$$u{(\xi ,t)}_{\left(\mathrm{iso}\right)}=\underset{0}{\overset{t}{\int }}{F}_1(\xi ,\tau )d\tau +\underset{0}{\overset{t}{\int }}{F}_2(\xi ,\tau )d\tau$$

(19)

where $u(\xi ,t{)}_{(\mathrm{ramp})}$ is the solution for velocity profile with ramped wall temperature and $u(\xi ,t{)}_{(\mathrm{iso})}$ is the solution for velocity profile with isothermal wall temperature, respectively.

$$\overline{F_1}(\xi ,q)=-\left[\frac{a_2(\lambda q+1)(1-e^{-q})}{q^3[a_0\lambda q+(a_0-b)]}\right]·\exp (-\xi \sqrt{a_{0}q(\lambda q+1)})$$

(20)

$$\overline{F_2}(\xi ,q)=\left[\frac{a_2(\lambda q+1)(1-e^{-q})}{q^3[a_0\lambda q+(a_0-b)]}\right]·\exp (-\xi \sqrt{\mathrm{bq}})$$

(21)

The inverse Laplace transform of equations (20) and (21) can be written as

$$F_1(\xi ,t)=-\frac{a_2}{a_0\lambda }\underset{0}{\overset{t}{\int }}[\frac{a\lambda -1}{a^2}H(t-\tau )+\frac{1}{a}(t-\tau )+\frac{1-a\lambda }{a^2}\exp (-a(t-\tau ))]f_1(\xi ,t-\tau )d\tau$$

(22)

$$F_2(\xi ,t)=\frac{a_2}{a_0\lambda }\underset{0}{\overset{t}{\int }}[\frac{a\lambda -1}{a^2}H(t-\tau )+\frac{1}{a}(t-\tau )+\frac{1-a\lambda }{a^2}\exp (-a(t-\tau ))]\frac{\xi \sqrt{b}\exp (-\frac{{\xi }^{2}b}{4t})}{2t\sqrt{\pi t}}d\tau$$

(23)

Nusselt number and skin friction

The mathematical expressions for Nusselt number and skin friction of the present study are evaluated for both the cases ramped and isothermal wall temperature which are given in the following sections: “Nusselt number” and “Skin Friction.”

Nusselt number

The expressions of Nusselt number for ramped and isothermal wall temperature conditions are determined from equations (13) and (14) and are given as follows

$$\mathrm{Nu}={-\frac{k_{\mathrm{nf}}}{k_f}\frac{\partial \theta }{\partial \xi }|}_{\xi =0}$$

(24)

$$N{u}_{\mathrm{iso}}=-\frac{k_{\mathrm{nf}}}{k_f}\mathrm{Re}\sqrt{\frac{b}{\pi t}}$$

(25)

Equivalently, we can write as

$$N{u}_{\mathrm{iso}}=-\frac{(k_s+2k_f)-2\varphi (k_f-k_s)}{(k_s+2k_f)+\varphi (k_f-k_s)}\mathrm{Re}\sqrt{\frac{b}{\pi t}}$$

(26)

$$N{u}_{\mathrm{ramp}}=-\frac{k_{\mathrm{nf}}}{k_f}\mathrm{Re}\eta [N{\theta }_1(\xi ,t)-N{\theta }_1(\xi ,t-1)H(t-1)]$$

(27)

where ${\mathrm{Re}}_x$ is Reynold’s number, ${\mathrm{Re}}_x=x{U}_0/v_f$

and

$$N_1\left(t\right)=\frac{1}{2}[\mathrm{erfc}\sqrt{\mathrm{bt}}-\mathrm{erf}\sqrt{b(t-1)}H(t-1)]$$

(28)

Skin friction

The corresponding expressions of skin friction for ramped and isothermal wall temperature conditions are determined from equations (17) and (19) and are given as follows

$$C_f={-(1+\lambda \frac{\partial }{\partial t})\frac{\partial u}{\partial \xi }|}_{\xi =0}$$

(29)

$$\begin{array}{c}{C}_{f_{\mathrm{ramp}}}=-(1+\lambda \frac{\partial }{\partial t})\hfill \\ \left[\begin{array}{c}\underset{0}{\overset{t}{\int }}[{F\prime }_1(0,\tau )-{F\prime }_1(0,\tau -1)H(\tau -1)]d\tau \hfill \\ \multicolumn{1}{c}{+\underset{0}{\overset{t}{\int }}[{F\prime }_1(0,\tau )-{F\prime }_2(0,\tau -1)H(\tau -1)]d\tau }\end{array}\right]\hfill \end{array}$$

(30)

$$C_{f_{\mathrm{iso}}}=-(1+\lambda \frac{\partial }{\partial t})[\underset{0}{\overset{t}{\int }}{F\prime }_1(0,\tau )d\tau +\underset{0}{\overset{t}{\int }}{F\prime }_2(0,\tau )d\tau ]$$

(31)

Limiting cases

The solutions obtained in the given problem can be verified by reducing these solutions to the already published papers. The limiting cases of this study are explained in the following sections.

In the absence of ramped temperature

By considering the isothermal boundary conditions at the plate, our obtained solutions were reduced to the following form

$$\overline{\theta }(\xi ,q)=\frac{1}{q}·\exp (-\xi \sqrt{\mathrm{bq}})$$

(32)

$$\begin{array}{cc}\multicolumn{1}{c}{\overline{u}(\xi ,q)=}& -\left[\frac{a_2(\lambda q+1)}{q^2[a_0\lambda q+(a_0-b)]}\right]·\exp (-\xi \sqrt{a_{0}q(\lambda q+1)})\hfill \\ \multicolumn{1}{c}{}& +\left[\frac{a_2(\lambda q+1)}{q^2[a_0\lambda q+(a_0-b)]}\right]·\exp (-\xi \sqrt{\mathrm{bq}})\hfill \end{array}$$

(33)

By taking the inverse Laplace transform using convolution theorem and some calculi, we get the following solutions which are quite identical to solutions obtained by Aman et al ³⁹

$$\theta (\xi ,t)=\mathrm{erfc}\left(\frac{y}{2}\sqrt{\frac{b}{t}}\right)$$

(34)

$$\begin{array}{c}\multicolumn{1}{c}{u(\xi ,t)=-\frac{a_2}{a_0\lambda }\underset{0}{\overset{t}{\int }}[\frac{a\lambda -1}{a^2}H(t-\tau )+\frac{1}{a}(t-\tau )+\frac{1-a\lambda }{a^2}\exp (-a(t-\tau ))]f_1(\xi ,t-\tau )d\tau }\\ +\frac{a_2}{a_0\lambda }\underset{0}{\overset{t}{\int }}[\frac{a\lambda -1}{a^2}H(t-\tau )+\frac{1}{a}(t-\tau )+\frac{1-a\lambda }{a^2}\exp (-a(t-\tau ))]\frac{\xi \sqrt{b}\exp (-\frac{{\xi }^{2}b}{4t})}{2t\sqrt{\pi t}}d\tau \hfill \end{array}$$

(35)

Newtonian viscous fluid

In the absence of Maxwell fluid parameter $\lambda =0$ , our solutions were reduced to the Newtonian viscous fluid which is quite identical to the solutions obtained by Khalid et al ⁴²

$$\overline{\theta }(\xi ,q)=\frac{1-e^{-q}}{q^2}·\exp (-\xi \sqrt{\mathrm{bq}})$$

(36)

$$\begin{array}{cc}\multicolumn{1}{c}{\overline{u}(\xi ,q)=}& -\left[\frac{a_2(1-e^{-q})}{q^3(a_0-b)}\right]·\exp (-\xi \sqrt{a_{0}q})\hfill \\ \multicolumn{1}{c}{}& +\left[\frac{a_2(1-e^{-q})}{q^3(a_0-b)}\right]·\exp (-\xi \sqrt{\mathrm{bq}})\hfill \end{array}$$

(37)

Velocity and temperature solutions for isothermal wall temperature

$$\overline{\theta }(\xi ,q)=\frac{1}{q}·\exp (-\xi \sqrt{\mathrm{bq}})$$

(38)

$$\begin{array}{cc}\multicolumn{1}{c}{\overline{u}(\xi ,q)=}& -\left[\frac{a_2}{q^2(a_0-b)}\right]·\exp (-\xi \sqrt{a_{0}q})\hfill \\ \multicolumn{1}{c}{}& +\left[\frac{a_2}{q^2(a_0-b)}\right]·\exp (-\xi \sqrt{\mathrm{bq}})\hfill \end{array}$$

(39)

By applying the inverse Laplace transform of equations (38) and (39), we get

$$\theta {(\xi ,t)}_{\mathrm{iso}}=\mathrm{erfc}\left(\frac{\xi }{2}\sqrt{\frac{b}{t}}\right)$$

(40)

$$\begin{array}{c}u{(\xi ,t)}_{\mathrm{iso}}=-\left(\frac{a_2}{a_0-b}\right).\hfill \\ [(t+\frac{{\xi }^2{a}_0}{2})\mathrm{erfc}\left(\frac{\xi }{2}\sqrt{\frac{a_0}{t}}\right)-\xi \sqrt{\frac{a_{0}t}{\pi }}{e}^{\frac{-{\xi }^2{a}_0}{4t}}]\hfill \\ +\left(\frac{a_2}{a_0-b}\right).[(t+\frac{{\xi }^{2}b}{2})\mathrm{erfc}\left(\frac{\xi }{2}\sqrt{\frac{b}{t}}\right)-\xi \sqrt{\frac{\mathrm{bt}}{\pi }}{e}^{\frac{-{\xi }^{2}b}{4t}}]\hfill \end{array}$$

(41)

Velocity and temperature solutions for ramped wall temperature

By taking the inverse Laplace transform of equations (36) and (39), we get

$$\theta {(\xi ,t)}_{\mathrm{ramp}}={\theta }_1(\xi ,t)-{\theta }_1(\xi ,t-1)H(t-1)$$

(42)

where

$${\theta }_1(\xi ,t)=[(t+\frac{b{\xi }^2}{2})\mathrm{erfc}\left(\frac{\xi }{2}\sqrt{\frac{b}{t}}\right)-\xi \sqrt{b}\sqrt{\frac{t}{\pi }}{e}^{-\frac{b{\xi }^2}{4t}}]$$

(43)

$$u{(\xi ,t)}_{\mathrm{ramp}}=u_1(\xi ,t)-u_1(\xi ,t-1)H(t-1)$$

(44)

where

$$\begin{array}{c}{u}_1(\xi ,t)=-\left(\frac{a_2}{a_0-b}\right)·\hfill \\ \underset{0}{\overset{t}{\int }}[(\tau +\frac{{\xi }^2{a}_0}{2})\mathrm{erfc}\left(\frac{\xi }{2}\sqrt{\frac{a_0}{\tau }}\right)-\xi \sqrt{\frac{a_0\tau }{\pi }}{e}^{\frac{-{\xi }^2{a}_0}{4\tau }}]d\tau \hfill \\ +\left(\frac{a_2}{a_0-b}\right)·\underset{0}{\overset{t}{\int }}[(\tau +\frac{{\xi }^{2}b}{2})\mathrm{erfc}\left(\frac{\xi }{2}\sqrt{\frac{b}{\tau }}\right)-\xi \sqrt{\frac{b\tau }{\pi }}{e}^{\frac{-{\xi }^{2}b}{4\tau }}]d\tau \hfill \end{array}$$

(45)

Special case

Solution for Newtonian viscous fluid

In this case, if we want to investigate pure viscous fluid without dispersing nanoparticles, we remove the thermal and mechanical parameters of nanofluids with those of viscous fluid and then the solutions obtained in equations (13)–(23) reduce to the solution independent of nanoparticles. These solutions will be reduced to Newtonian viscous fluid by putting $\lambda =0$ with the bounding plate being considered to be at rest all the time. Then the solution obtained in this case is quite similar to the results obtained by Chandran et al. ³⁵

Results and discussion

In this study, the thermal performance of MoS₂ and graphene nanoparticles is investigated in EO which is taken as the base fluid. The comparison of velocity and temperature profiles for MoS₂ and graphene nanoparticles is shown through graphs. The fluid flow is considered along an infinite vertical plate with non-uniform wall temperature. From graphical results, one can identify easily that the magnitude of Maxwell nanofluid velocity is greater for isothermal temperature and magnitude of fluid velocity is smaller for non-uniform or ramped wall temperature as shown in graphs. The Laplace transform technique is used to find the exact solutions for the governing equations with the corresponding boundary conditions. From the graphical analysis, various embedded parameters are discussed. Furthermore, the present study is focused at analyzing the effect of MoS₂ and graphene nanoparticles which are dispersed uniformly in EO. The behaviors of different parameters on velocity profiles are investigated. Figures 2–5 are the graphs that are plotted for volume friction parameter $\varphi$ , Maxwell parameter $\lambda$ , time $t$ , and comparison of MoS₂ and graphene nanoparticles for velocity profile, respectively. Figures 6–8 represent the graphs for energy profiles which depict the effect of volume friction parameter $\varphi$ , time $t$ , and comparison of MoS₂ and graphene nanoparticles based on the energy profile, respectively. Finally, Figure 9 depicts the comparison of the present study with the result obtained by Aman et al., ³⁹ and Figure 10 shows the comparison of Maxwell nanofluid with Newtonian viscous fluid.

Figure 2.

Variation of volume fraction parameter $\varphi$ on velocity profile when $\mathrm{Pr}=6400$ , $\mathrm{Gr}=62$ , $t_0=0.6$ , $t=1.6$ , and $\lambda =1$ .

Figure 3.

Effect of Maxwell parameter $\lambda$ on ramped and isothermal velocities when $\mathrm{Pr}=6400$ , $\mathrm{Gr}=62$ , $t_0=0.6$ , $t=1.6$ , and $\varphi =0.02$ .

Figure 4.

Effect of time $t_0$ and $t$ on velocity profile for ramped and isothermal velocities when $\mathrm{Pr}=6400$ , $\mathrm{Gr}=62$ , $\lambda =1$ , and $\varphi =0.02$ .

Figure 5.

Comparison of MoS₂ and graphene nanoparticles on velocity profile when $\mathrm{Pr}=6400$ , $\mathrm{Gr}=62$ , $\lambda =1$ , $t_0=0.6$ , $t=1.6$ , and $\varphi =0.02$ .

Figure 6.

Effect of different values of $\varphi$ on energy profile, when $\mathrm{Pr}=6400$ , $\mathrm{Gr}=62$ , $\lambda =1$ , $t_0=0.6$ , and $t=1.6$ .

Figure 7.

Effect of time $t_0$ and $t$ on temperature when $\mathrm{Pr}=6400$ , $\mathrm{Gr}=62$ , $\lambda =1$ , and $\varphi =0.02$ .

Figure 8.

Comparison of MoS₂ and graphene nanoparticles on temperature profile when $\mathrm{Pr}=6400$ , $\mathrm{Gr}=62$ , $\lambda =1$ , $t_0=0.6$ , $t=1.6$ , and $\varphi =0.02$ .

Figure 9.

Comparison of this study’s results with those obtained by Aman et al. ³⁹

Figure 10.

Comparison of Maxwell fluid and Newtonian viscous fluid.

Figure 2 depicts the variation in velocities with ramped temperature and isothermal temperature for different values of $\varphi$ . From the figure, it is clear that the variation is negligible in velocity profile for non-uniform wall temperature, and for constant wall temperature, the velocity profile shows decreasing behavior with increasing $\varphi$ . Physical sketch of $\varphi$ shows that by increasing the volume of nano-sized solid particles, viscous forces are produced in the nanofluid. This leads to creation of resistance between the fluid particles, as a result of which velocity decreases.

Figure 3 displays the effect of Maxwell fluid parameter $\lambda$ on velocity distribution for non-uniform and isothermal wall temperature velocities. The effect of Maxwell parameter $\lambda$ is that with the increase of the values of $\lambda$ , the velocity of the nanofluid is more near the plate. But, from the figure, it can be observed that when the fluid moved away from the bounding plate, then this pattern of $\lambda$ reverses.

Figure 4 discusses the effect of time $t_0$ for non-uniform wall temperature and $t$ for isothermal wall temperature on velocity distribution. From the figure, it is clear that for a time period that is longer, both the velocities for ramped and isothermal wall temperature increase.

Figure 5 displays the comparison of velocity profile for MoS₂ and graphene nanoparticles for ramped and isothermal wall temperatures. It is observed that the magnitude of velocity profile is more for graphene-based nanofluid near the plate. As this distance increases from the plate, the MoS₂-based nanofluid gained maximum velocity but graphene-based nanofluid velocity reduced. Furthermore, it can be noticed that the effect of MoS₂ and graphene-based nanofluid on velocity profile is quite similar for ramped and isothermal wall temperatures.

Figure 6 discusses the behavior of nanofluid temperature by increasing volume friction parameter $\varphi$ while keeping the other values constant. From this graph, it can be observed that by increasing the values of $\varphi$ , the collision between the molecules of fluid increases as a result of which the fluid temperature increases.

Figure 7 shows the behavior of time $t$ on temperature distribution. By increasing the time $t$ , ramped temperature increases, while isothermal temperature remains unchanged because isothermal temperature means constant temperature.

Figure 8 shows the comparison of MoS₂ and graphene nanoparticles on temperature distribution. From the figure, it can be seen that EO has higher temperature with MoS₂ as compared to graphene nanoparticles dispersed in EO.

Figure 9 shows the comparison of the present study and the solutions obtained by Aman et al. ³⁹ From the figure, it is clear that our solutions are very similar to the work of Aman et al, which verifies our solutions.

Figure 10 displays the comparison of Maxwell nanofluid with Newtonian viscous fluid. Graphically, we have observed that the velocity obtained by Newtonian fluid is higher as compared to Maxwell nanofluid.

Skin friction for Maxwell nanofluid is evaluated and mentioned in Table 2, which shows the effect of the parameters involved, namely, $\mathrm{Pr}$ , $\lambda$ , $\varphi$ , $t_0$ , and $t$ , on skin friction for non-uniform wall temperature $C_{f_{\mathrm{ramp}}}$ and isothermal wall temperature $C_{f_{\mathrm{iso}}}$ . From this table, we observed that with increase of $\lambda$ , $\varphi$ , $t_0$ , and $t$ , the skin fraction increases, whereas increase of $\mathrm{Pr}$ leads to decrease in skin friction. The corresponding Nusselt number for Maxwell nanofluid is evaluated and presented in Table 3, which shows the effect of volume fraction parameter $\varphi$ on the rate of heat transfer (Nusselt number $\mathrm{Nu}$ ). From Table 3, we see that the rate of heat transfer is higher for graphene nanoparticles dispersed in EO and is lower for MoS₂ nanoparticles in EO. Furthermore, for time $t_0=0.6$ , $N{u}_{\mathrm{ramp}}$ is the Nusselt number for ramped wall temperature, and for time $t=1.6$ , $N{u}_{\mathrm{iso}}$ is the corresponding Nusselt number for isothermal wall temperature. The Nusselt number increases with the increasing values of $\varphi$ . Consequently, the EO will carry more heat from the engine parts when graphene nanoparticles are dispersed in it. As a result, it will improve the anti-wear forces and will enhance the engine working ability.

Table 2.

Skin friction for ramped and isothermal wall temperature.

$\mathrm{Gr}$	$\mathrm{Pr}$	$\lambda$	$\varphi$	$t_0$	$t$	$C_{f_{\mathrm{ramp}}}$	$C_{f_{\mathrm{iso}}}$
62	6400	1	0.02	0.6	1.6	2695	23.457
62	6500	1	0.02	0.6	1.6	2932	25.212
62	6400	2	0.02	0.6	1.6	8111	72.547
62	6400	1	0.04	0.6	1.6	2406	21.309
62	6400	1	0.02	0.8	1.6	601.36	23.457
62	6400	1	0.02	0.6	1.8	2695	13.461

Note: The bold values in the table shows that in this row the values of the skin friction is changed because of the variation in parameter in this column.

Table 3.

Variation in the rate of heat transfer (Nusselt number for ramped and isothermal wall temperatures) for graphene and MoS₂ nanoparticles.

$\varphi$	$t_0$	$t$	Engine oil graphene nanofluid		Engine oil MoS2 nanofluid
	$t_0$ ramp	$t$ iso	$N{u}_{\mathrm{iso}}$	$N{u}_{\mathrm{ramp}}$	$N{u}_{\mathrm{iso}}$	$N{u}_{\mathrm{ramp}}$
0	0.6	1.6	1.163	1.749	1.163	1.749
0.01	0.6	1.6	1.191	1.791	1.182	1.777
0.02	0.6	1.6	1.219	1.834	1.200	1.805
0.03	0.6	1.6	1.248	1.877	1.219	1.834
0.04	0.6	1.6	1.277	1.921	1.238	1.862

MoS₂: molybdenum disulfide.

Concluding remarks

This study focused on calculating the closed-form solutions for heat transfer flow of Maxwell nanofluids with ramped wall temperature conditions at the boundary. The Laplace transform technique is applied to obtain the solutions for velocity and temperature distributions. The heat transfer rate of MoS₂ and graphene nanoparticles is analyzed in EO. Furthermore, the solutions obtained for non-uniform wall and constant wall temperatures are discussed through graphs. From the figures, it can be observed that the magnitude of velocity is smaller for non-uniform wall temperature and greater for isothermal wall temperature. The skin friction and Nusselt number are evaluated and presented in tabular form.

The following are the main results obtained from this study:

• Increasing the volume of nano-sized solid particles results in decrease in the Maxwell nanofluid velocity.

• The magnitude of velocity and energy profile are smaller for non-uniform or ramped wall temperature and greater for constant wall temperature.

• Heat transfer rate of MoS₂ is greater than graphene nanoparticles.

• By increasing $\mathrm{Gr}$ , $\lambda$ , and time $t$ , the nanofluid velocity increases.

• By increasing $\varphi$ , the fluid velocity decreases.

• Newtonian viscous fluid velocity is higher than Maxwell nanofluid velocity.