Impacts of Thermal Radiation and Heat Consumption/Generation on Unsteady MHD Convection Flow of an Oldroyd-B Fluid with Ramped Velocity and Temperature in a Generalized Darcy Medium

Abstract

This article analyzes the time-dependent magnetohydrodynamic flow of Oldroyed-B fluid in the presence of heat consumption/generation and thermal radiation. The flow is restricted to a vertical infinite plate saturated in porous material along with ramp wall velocity and ramp wall temperature conditions. This flow also incorporates the generalized Darcy’s law. In this paper, accurate equation of velocity field is presented first and then solutions of mass and energy equation are derived in Laplace domain. Real-time domain solutions are obtained by tackling the complexity of Laplace domain expressions through numerical Laplace inversion. Skin friction coefficient and Nusselt number are also calculated. A comparison for ramp wall temperature condition and isothermal temperature condition is also drawn to investigate the difference. A graphical study is conducted to analyze the influence of parameters on fluid flow and heat transfer. It is found that radiation parameter and heat generation elevate the energy profile, while flow is accelerated by increasing the retardation time and porosity parameter and an opposite behavior is noted for increasing relaxation time and magnetic parameter. Furthermore, heat transfer rate is higher for increasing Prandtl number and velocity on plate decreases with increase in relaxation time ${\lambda }_1$.

1. Introduction

In emerging and modern technologies, non-Newtonian fluids are gathering attraction due to their high practical significance. Non-Newtonian fluids involve honey, paints, toothpaste, polymer solutions, and greases. In modern days, flow of such fluids together with magnetohydrodynamic (MHD) having free and forced convection are of great use in energy generator purification of mineral oil and power generators. Moving forward, the addition of thermal radiation to such flows has important affects in the mechanisms of aerosol technology, solar collectors, and high temperature polymeric mixtures, which act at high and medium temperatures [1]. Hydrologists and engineers investigated the flows of fluid in porous materials ranging from fused Pyrex glass to sand packs to anticipate their behaviors in different kinds of reservoirs [2]. Incorporation of heat source or sink to magnetohydrodynamics (MHD) convective fluid has strong usage in welding mechanics and thermal engineering [3,4]. Oldroyd-B fluid lies in the category of non-Newtonian fluids and is completely capable of presenting viscoelastic fluids [5]. We considered this model in this work due to its simplicity, vastness, and complete resemblances with the fluids showing viscous and elastic profiles simultaneously. This model is an extension of the Maxwell model and the viscous model. Moreover, in a special case when viscosity of the solvent vanishes, it reduces to Upper Convected Maxwell fluid. Further details of these models can be approached in References [6,7,8].

In the literature, there is a dearth of articles when it comes to dealing with flows subjected to ramp wall temperature and ramp wall velocity conditions despite valuable physical significance. One of the main possibilities is the handling of complex resulted expressions. These combined wall conditions have significant applications like diagnoses of cardiovascular diseases using treadmil testing or ergometers. Analysis of blood vessel functioning and diagnoses of heart diseases also depend upon ramp wall velocity conditions. To cure cancer cells, a therapy based on ramp wall conditions is utilized as it has negligible side effects on the human body [9]. To further improve the cure of cancer, different types of boundary conditions were presented by Kundu [9]. More applications of ramped conditions can be studied in the contribution of Schetz [10], Hayday [11], and Malhotra [12].

The idea of simultaneous ramped conditions was proposed by Ahmed and Dutta [13] to study the flow passing an instinctively infinite vertical plate. Mass and heat transfer phenomenons were studied by Seth et al. for ramp temperature conditions in case of moving vertical plates [14,15,16]. In recent times, Chandran et al. studied the effect of ramp temperature condition on mixed convection fluid flow [17]. A study is conducted by Narahari et al. to analyze the fluid motion for infinite vertical plate with wall heating [18]. An extension of Khan’s [19] work on MHD flow of Jeffery fluid was provided by Zin et al. [20] for ramp wall temperature condition. Further, this work was given an extension by Maqbool et al. for simultaneous ramp wall conditions [21]. More applications of ramped conditions can be found in the work of Myers et al. [22] and Bruce [23].

On the basis of such strong motivation, we have considered the free convection flow inculcating thermal radiative effects along MHD. Moreover, heat suction/injection is also introduced to the flow with the existence of a porous medium. The ramp velocity and ramp temperature conditions are considered at the wall. Laplace transformation is implemented to reach the solutions.

2. Mathematical Modeling

The subsequent equations [24,25] describing unsteady, incompressible, and MHD motion of Oldroyd-B fluid over an infinite vertical plate are provided under Boussinesq’s approximations [26] as

$$\begin{array}{c}\hfill \nabla ·\mathbf{V}=0,\end{array}$$

(1)

$$\begin{array}{c}\hfill \rho \left[\frac{\partial \mathbf{V}}{\partial t}+(\mathbf{V}.\nabla )\mathbf{V}\right]=\mathrm{div}\mathbf{T}+\mathbf{J}\times \mathbf{B}+\mathbf{g}\rho \beta (T-T_{\infty })+\mathbf{r}.\end{array}$$

(2)

where $\rho$, $\mathbf{r}$, $\mathbf{J}$, $\mathbf{B}$, g, $\beta$, t, and T represent fluid density, Darcy’s resistence, current density, total magnetic field, gravitational acceleration, thermal volume expansion, time, and temperature of fluid, respectively. Moreover, velocity $\mathbf{V}$, accounting for one-dimensional and unidirectional flow, and the Cauchy stress tensor $\mathbf{T}$ are defined as

$$\mathbf{V}=[u(y,t),0,0],$$

(3)

$$\begin{array}{cc}\hfill \mathbf{T}& =-\mathcal{P}\mathbf{I}+\mathbf{S}.\hfill \end{array}$$

(4)

where $\mathbf{S}$ and $-\mathcal{P}\mathbf{I}$ denote the extra stress tensor and indeterminate stress tensor, respectively. Moreover, $\mathbf{S}$ holds the following relation:

$$\begin{array}{c}\hfill \mu \left(1+{\lambda }_r\frac{D}{Dt}\right){\mathbf{A}}_1=\mathbf{S}\left(1+\lambda \frac{D}{Dt}\right),\end{array}$$

(5)

where $\mu$ refers to dynamic viscosity of fluid. ${\lambda }_r$ and $\lambda$ refer to retardation and relaxation time, respectively. Additionally, material time derivative $\frac{D}{Dt}$ and Rivlin–Ericksen tensor ${\mathbf{A}}_1$ are defined as

$$\begin{array}{cc}\hfill \frac{D\mathbf{S}}{Dt}& =\frac{\partial \mathbf{S}}{\partial t}+u\frac{\partial \mathbf{S}}{\partial x}+v\frac{\partial \mathbf{S}}{\partial y}+w\frac{\partial \mathbf{S}}{\partial z},\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill {\mathbf{A}}_1& =\nabla \mathbf{V}+{(\nabla \mathbf{V})}^T=\left(\begin{array}{cc}0& u_y\\ u_y& 0\end{array}\right).\hfill \end{array}$$

(7)

where $x,y,z$ and $u,v,w$ are space variables and velocity components. For Oldroyd-B fluid, modified Darcy’s law is defined as

$$\begin{array}{c}\hfill -\frac{\mu \varphi }{k}\left(1+{\lambda }_r\frac{\partial }{\partial t}\right)\mathbf{V}=\left(1+\lambda \frac{\partial }{\partial t}\right)\mathbf{r},\end{array}$$

(8)

where $\varphi$ and k are porosity and permeability of the porous medium. The equations of Maxwell are given as

$$\begin{array}{c}\hfill \mathrm{div}\mathbf{B}=0,\mathrm{curl}\mathbf{B}={\mathbf{\mu }}_m\mathbf{J},\mathrm{curl}\mathbf{E}=-\frac{\partial \mathbf{B}}{\partial \mathrm{t}},\end{array}$$

(9)

$$\begin{array}{c}\hfill \mathbf{J}\times \mathbf{B}=-\left(\mathbf{\sigma }{\mathbf{B}}_{\mathbf{0}}^{\mathbf{2}}\mathbf{u},\mathbf{0},\mathbf{0}\right).\end{array}$$

(10)

where ${\mathbf{\mu }}_m$, $\mathbf{\sigma }$, and $\mathbf{E}$ refer to magnetic permeability, electrical conductivity, and electric field, respectively. The total magnetic field is given as $\mathbf{B}=B_0+b_0$. Here, $B_0$ denotes the magnetic field applied and $b_0$ denotes the magnetic field induced.

In the presence of Equations (3)–(8), a deduced form of Equation (2) can be presented as

$$\begin{array}{c}\hfill \rho \frac{\partial u}{\partial t}=\rho g\beta (T-T_{\infty })+{\mathbf{r}}_x+{(\mathbf{J}\times \mathbf{B})}_x+\frac{\partial S_{xy}}{\partial y},\end{array}$$

(11)

Using Maxwell’s equations and modified Darcy’s law in the above equation and multiplying it by $(1+\lambda {\partial }_t)$, we obtain the following form:

$$\begin{array}{cc}(1+\lambda {\partial }_t)\rho \frac{\partial u}{\partial t}& =(1+\lambda {\partial }_t)\rho g\beta (T-T_{\infty })-(1+{\lambda }_r{\partial }_t)\frac{\mu \varphi }{k}u\hfill \\ & -(1+\lambda {\partial }_t)\sigma B_0^{2}u+(1+\lambda {\partial }_t)\frac{\partial S_{xy}}{\partial y},\end{array}$$

(12)

Using relation $(1+\lambda {\partial }_t)S_{xy}=\mu (1+{\lambda }_r{\partial }_t)u_y$ in the above equation and rearranging the resulting equation in some fascinating manner, the correct presentation of velocity equation is obtained as

$$\begin{array}{cc}\rho (1+\lambda \frac{\partial }{\partial t})\frac{\partial u}{\partial t}& =\mu (1+{\lambda }_r\frac{\partial }{\partial t})\frac{{\partial }^{2}u}{\partial y^2}+\rho g\beta (1+\lambda \frac{\partial }{\partial t})(T-T_{\infty })\hfill \\ & -\sigma B_0^2(1+\lambda \frac{\partial }{\partial t})u-\frac{\mu \varphi }{k}(1+{\lambda }_r\frac{\partial }{\partial t})u.\end{array}$$

(13)

However, in the work of Mazhar et al. [27], the term $1+\lambda \frac{\partial }{\partial t}$ is missing with the coefficient of thermal expansion pointing out the deficiency of their model. The geometrical presentation of considered model is provided in Figure 1.

The governing equations for flow and energy under the assumptions of Boussinesq’s approximation and small Reynolds number are provided as

$$\begin{array}{cc}(1+\lambda \frac{\partial }{\partial t})\frac{\partial u}{\partial t}& =\nu (1+{\lambda }_r\frac{\partial }{\partial t})\frac{{\partial }^{2}u}{\partial y^2}+g\beta (1+\lambda \frac{\partial }{\partial t})(T-T_{\infty })\hfill \\ & -\frac{\sigma B_0^2}{\rho }(1+\lambda \frac{\partial }{\partial t})u-\frac{\nu \varphi }{k}(1+{\lambda }_r\frac{\partial }{\partial t})u,\end{array}$$

(14)

$$\begin{array}{cc}\hfill \rho c_p\frac{\partial T}{\partial t}& =k\frac{{\partial }^{2}T}{\partial y^2}-\frac{\partial q_r}{\partial y}+Q_0(T-T_{\infty }),\hfill \end{array}$$

(15)

where $\rho c_p,k,q_r$, and $Q_0$ refer to heat capacitance, thermal conductivity, radiative heat flux, and heat generation/absorption constant.

The associated initial and boundary conditions are defined as

$$\begin{array}{cc}\hfill u(y,0)& =0,T(y,0)=T_{\infty },\hfill \end{array}$$

$$\begin{array}{cc}\hfill y\ge 0:u_t(y,0)& =0,u_y(y,0)=0,\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill t>0:u(y,t)& \to 0,T(y,t)\to T_{\infty },\mathrm{for}y\to \infty ,\hfill \\ \hfill u(0,t)& =\left\{\begin{array}{c}{u}_0\frac{t}{t_0}0<t\le t_0\hfill \\ u_{0}t>t_0\hfill \end{array}\right.\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill T(0,t)& =\left\{\begin{array}{c}{T}_{\infty }+(T_w-T_{\infty })\frac{t}{t_0}0<t\le t_0\hfill \\ T_{w}t>t_0\hfill \end{array}\right.\hfill \end{array}$$

(18)

The radiation heat flux, after using Rosseland approximation comes out to be [28]

$$\begin{array}{c}\hfill q_r=-\frac{4{\sigma }_1\partial T^4}{3k_1\partial y}.\end{array}$$

(19)

where the Stafan–Boltzman constant and adsorption coefficient are represented by ${\sigma }_1$ and $k_1$, respectively. The term $q_r$ can be linearized by expansion of $T^4$ using Taylor series about $T_{\infty }$, keeping the supposition in mind that temperature differences are small enough to neglect the higher-order terms. After normalizing, $T^4$ comes out to be $T^4\approx 4T_{\infty }^{3}T-3T_{\infty }^4$.

Using this linearization in Equation (15) results in

$$\begin{array}{c}\hfill \rho c_p\frac{\partial T}{\partial t}=k\left(1+\frac{16{\sigma }_1{T}_{\infty }^3}{3k_{1}k}\right)\frac{{\partial }^{2}T}{\partial y^2}+Q_0(T-T_{\infty }).\end{array}$$

(20)

For the sake of braveity, dimensionless terms are defined as

$$\begin{array}{c}\hfill u^{*}=\frac{u}{u_0},\eta =\frac{y{u}_0}{\nu },\tau =\frac{t{u}_0^2}{\nu },{\tau }_0=\frac{\nu }{u_0^2},\theta =\frac{T-T_{\infty }}{T_w-T_{\infty }}.\end{array}$$

(21)

Introducing the above terms in the equations of flow and energy and dropping the * for convenience, they take the following form:

$$\begin{array}{cc}\hfill \left(a_1+{\lambda }_1\frac{\partial }{\partial \tau }\right)\frac{\partial u}{\partial \tau }& =\left(1+{\lambda }_2\frac{\partial }{\partial \tau }\right)\frac{{\partial }^{2}u}{\partial {\eta }^2}-b_{1}u+Gr\left(1+{\lambda }_1\frac{\partial }{\partial \tau }\right)\theta ,\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill \frac{\partial \theta }{\partial \tau }& =\left(\frac{1+Nr}{Pr}\right)\frac{{\partial }^2\theta }{\partial {\eta }^2}+Q\theta .\hfill \end{array}$$

(23)

where dimensionless parameters are defined as

$$\begin{array}{c}\hfill Gr=\frac{g\beta \nu (T_w-T_{\infty })}{u_0^3},Nr=\frac{16{\sigma }_1{T}_{\infty }^3}{3k{k}_1},M=\frac{\sigma B_0^2\nu }{\rho u_0^2},\\ \hfill Pr=\frac{\mu c_p}{k},b_1=M+\frac{1}{K},{\lambda }_1=\frac{\lambda u_0^2}{\nu },{\lambda }_2=\frac{{\lambda }_r{u}_0^2}{\nu },\\ \hfill \frac{1}{K}=\frac{\varphi {\nu }^2}{k{u}_0^2},Q=\frac{\nu Q_0}{\rho c_p{u}_0^2},a_1=1+{\lambda }_{1}M+\frac{{\lambda }_2}{K},\end{array}$$

(24)

The initial and boundary conditions in dimensionless form can be presented as

$$\begin{array}{cc}\hfill u(\eta ,0)& =0,\theta (\eta ,0)=0,\hfill \\ \hfill \eta \ge 0:u_{\tau }(\eta ,0)& =0,u_{\eta }(\eta ,0)=0,\hfill \end{array}$$

(25)

$$\begin{array}{cc}\hfill \tau >0:u(\eta ,\tau )& \to 0,\theta (\eta ,\tau )\to 0\mathrm{when}\eta \to \infty ,\hfill \end{array}$$

(26)

$$\begin{array}{cc}\hfill u(0,\tau )& =\theta (0,\tau )=\left\{\begin{array}{c}\tau 0<\tau \le 1\hfill \\ 1\tau >1,\hfill \end{array}\right.\hfill \end{array}$$

(27)

3. Analytical Solutions

To generate the solution of this problem, Laplace transform [29] is a convenient tool due to its effective applicability for nonuniform boundary conditions. Other convenient methods like Adomian decomposition, Homotopy analysis method, perturbation method, and separation of variables do not serve the purpose here due to complicated boundary conditions. We formulate the Laplace transform pair for the sake of the results of current problem as an integral of the following form:

$$\begin{array}{c}\hfill \overline{R}(\eta ,q)=\underset{0}{\overset{\infty }{\int }}{e}^{-q\tau }R(\eta ,\tau )d\tau =\mathcal{L}[R]\left(\tau \right),\tau \ge 0,\end{array}$$

(28)

where $R\in \{u,\theta \}$. The above integral is convergent for $Re\left(q\right)>{\gamma }_0$ where $q=\Psi +j\Omega$, ${\gamma }_o$ is a real number, and $j=\sqrt{-1}$. Transformation of the Laplace domain solutions back to the original time domain can be performed as

$$\begin{array}{c}\hfill R(\eta ,\tau )=\frac{1}{2\pi j}\underset{BR}{\int }{e}^{q\tau }\overline{R}(\eta ,q)dq={\mathcal{L}}^{-1}[\overline{R}]\left(q\right),\end{array}$$

(29)

Implementation of Laplace transform on mass and energy equations and putting $a_1=\frac{1+Nr}{Pr}$ in the energy equation yields

$$\begin{array}{c}\hfill q\overline{\theta }=a_1\frac{{\partial }^2\overline{\theta }}{\partial {\eta }^2}+Q\overline{\theta },\end{array}$$

(30)

$$\begin{array}{c}\hfill \frac{{\partial }^2\overline{u}}{\partial {\eta }^2}-\left(\frac{a_{1}q+q^2{\lambda }_1+b_1}{1+{\lambda }_{2}q}\right)\overline{u}=-Gr\left(\frac{1+{\lambda }_{1}q}{1+{\lambda }_{2}q}\right)\overline{\theta },\end{array}$$

(31)

Application of Laplace transform on initial and boundary condition responds as

$$\begin{array}{cc}\hfill \overline{u}(\eta ,0)& =0,\overline{\theta }(\eta ,0)=0,\hfill \end{array}$$

(32)

$$\begin{array}{cc}\hfill \overline{u}(\eta ,q)& \to 0,\overline{\theta }(\eta ,q)\to 0\mathrm{for}\eta \to \infty ,\hfill \end{array}$$

(33)

$$\begin{array}{cc}\hfill \overline{u}(0,q)& =\overline{\theta }(0,q)=\frac{1-e^{-q}}{q^2}.\hfill \end{array}$$

(34)

Solution of the energy equation in Equation (30) in Laplace domain subjected to the boundary conditions in Equations (33) and (34) is given as

$$\begin{array}{c}\hfill \overline{\theta }(\eta ,q)=\left(\frac{1-e^{-q}}{q^2}\right)e^{-\sqrt{\alpha (q-Q)}\eta }.\end{array}$$

(35)

Plugging in the value of $\overline{\theta }$ in mass in Equation (31) implies

$$\begin{array}{cc}\hfill \frac{{\partial }^2\overline{u}}{\partial {\eta }^2}-\left(\frac{a_{1}q+q^2{\lambda }_1+b_1}{1+{\lambda }_{2}q}\right)\overline{u}& =-Gr\left(\frac{1+{\lambda }_{1}q}{1+{\lambda }_{2}q}\right)\left(\frac{1-e^{-q}}{q^2}\right)e^{-\sqrt{\alpha (q-Q)}\eta }.\hfill \end{array}$$

(36)

The solution of the above equation subjected to boundary conditions is given as

$$\begin{array}{c}\hfill \overline{u}(\eta ,q)=\left(\frac{1-e^{-q}}{q^2}\right)\overline{H}(\eta ,q),\\ \hfill \overline{H}(\eta ,q)=e^{-\sqrt{\frac{a_{1}q+q^2{\lambda }_1+b_1}{1+{\lambda }_{2}q}}\eta }+\frac{Gr(1+{\lambda }_{1}q)e^{-\sqrt{\frac{a_{1}q+q^2{\lambda }_1+b_1}{1+{\lambda }_{2}q}}\eta }}{({\lambda }_2\alpha -{\lambda }_1)[{(q-m_1)}^2-m_2^2]}-\frac{Gr(1+{\lambda }_{1}q)e^{-\sqrt{\alpha (q-Q)}\eta }}{({\lambda }_2\alpha -{\lambda }_1)[{(q-m_1)}^2-m_2^2]},\end{array}$$

(37)

where

$$\begin{array}{c}\hfill m_1=\frac{a_1+{\lambda }_2\alpha Q-\alpha }{2({\lambda }_2\alpha -{\lambda }_1)},m_2=\sqrt{{\left(\frac{a_1+{\lambda }_2\alpha Q-\alpha }{2({\lambda }_2\alpha -{\lambda }_1)}\right)}^2+\frac{b_1+\alpha Q}{{\lambda }_2\alpha -{\lambda }_1}}.\end{array}$$

(38)

Since the mass and energy fields contain the complex combinations of Laplace parameter q, numerical inversion named as Durbin method [30] is applied to deduce the solutions in a real-time domain $\tau$.

The expressions for Nusselt number Nu and skin friction ${\tau }_w$ are given as

$$\begin{array}{cc}\hfill Nu& =-\frac{\partial \theta }{\partial \eta }(0,\tau ),\hfill \end{array}$$

(39)

$$\begin{array}{cc}\hfill Nu& ={\mathcal{L}}^{-1}\left[\sqrt{\alpha (q-Q)}\left(\frac{1-e^{-q}}{q^2}\right)\right],\hfill \end{array}$$

(40)

$$\begin{array}{cc}\hfill {\tau }_w& =\frac{\mu }{1+{\lambda }_1{\partial }_{\tau }}\left(1+{\lambda }_2{\partial }_{\tau }\right)\frac{\partial u}{\partial \eta }(0,\tau ),\hfill \end{array}$$

(41)

where

$$\begin{array}{c}\hfill \frac{\partial u}{\partial \eta }(0,\tau )=\frac{\partial H}{\partial \eta }(0,\tau )+\frac{\partial H(0,\tau -1)F(\tau -1)}{\partial \eta }.\end{array}$$

(42)

4. Special Cases

This section deals with some special cases of the current work.

4.1. Case 1

The solution of viscous fluid with ramp wall temperature can be fetched when ${\lambda }_1=0$ and ${\lambda }_2=0$ [31].

$$\begin{array}{cc}\hfill \theta (\eta ,q)& ={\mathcal{L}}^{-1}\left[\left(\frac{1-e^{-q}}{q^2}\right)e^{-\sqrt{\alpha (q-Q)}\eta }\right],\hfill \end{array}$$

(43)

$$\begin{array}{cc}\hfill u(\eta ,\tau )& ={\mathcal{L}}^{-1}\left[\frac{1}{q}{e}^{-\sqrt{q+b_1}\eta }+\left(\frac{Gr}{\alpha (q-Q)-(q+b_1)}\frac{1-e^{-q}}{q^2}\right)(e^{-\sqrt{q+b_1}\eta }-e^{-\sqrt{\alpha (q-Q)}\eta })\right].\hfill \end{array}$$

(44)

4.2. Case 2

The Maxwell fluid flow profiles can be derived when ${\lambda }_2=0$ [32].

4.3. Case 3

The simultaneous ramp wall conditional solutions of Oldroyd-B fluid can be deduced when $Nr$, Q → 0.

$$\begin{array}{c}\hfill \theta (\eta ,\tau )={\mathcal{L}}^{-1}\left[\frac{1-e^{-q}}{q^2}{e}^{-\sqrt{Prq}\eta }\right],\end{array}$$

(45)

$$\begin{array}{c}\hfill u(\eta ,\tau )={\mathcal{L}}^{-1}\left[\left(\frac{1-e^{-q}}{q^2}\right)\overline{H}(\eta ,q)\right].\end{array}$$

(46)

where

$$\begin{array}{cc}\hfill \overline{H}(\eta ,q)& =e^{-\sqrt{\frac{a_{1}q+q^2{\lambda }_1+b_1}{1+{\lambda }_{2}q}}\eta }+\frac{Gr(1+{\lambda }_{1}q)e^{-\sqrt{\frac{a_{1}q+q^2{\lambda }_1+b_1}{1+{\lambda }_{2}q}}\eta }}{({\lambda }_{2}Pr-{\lambda }_1)[{(q-m_1)}^2-m_2^2]}-\frac{Gr(1+{\lambda }_{1}q)e^{-\sqrt{Prq}\eta }}{({\lambda }_{2}Pr-{\lambda }_1)[{(q-m_1)}^2-m_2^2]},\hfill \\ \hfill m_1& =\frac{a_1-Pr}{2({\lambda }_{2}Pr-{\lambda }_1)},m_2=\sqrt{{\left(\frac{a_1-Pr}{2({\lambda }_{2}Pr-{\lambda }_1)}\right)}^2+\frac{b_1}{{\lambda }_{2}Pr-{\lambda }_1}}.\hfill \end{array}$$

(47)

4.4. Case 4

The mass and energy solutions of Oldroyd-B fluid for constant boundary conditions can be deduced as

$$\begin{array}{c}\hfill \theta (\eta ,\tau )=e^{-\eta \dot{ı}\sqrt{\alpha Q}}erfc\left(\frac{\eta \sqrt{Q}}{2\sqrt{\tau }}-\dot{ı}\sqrt{Q\tau }\right)+e^{\eta \dot{ı}\sqrt{\alpha Q}}erfc\left(\frac{\eta \sqrt{Q}}{2\sqrt{\tau }}+\dot{ı}\sqrt{Q\tau }\right),\end{array}$$

(48)

$$\begin{array}{c}\hfill u(\eta ,\tau )={\mathcal{L}}^{-1}\left[\left(\frac{1-e^{-q}}{q}\right)\overline{H}(\eta ,q)\right],\end{array}$$

(49)

where

$$\begin{array}{cc}\hfill \overline{H}(\eta ,q)& =e^{-\sqrt{\frac{a_{1}q+q^2{\lambda }_1+b_1}{1+{\lambda }_{2}q}}\eta }+\frac{Gr(1+{\lambda }_{1}q)e^{-\sqrt{\frac{a_{1}q+q^2{\lambda }_1+b_1}{1+{\lambda }_{2}q}}\eta }}{({\lambda }_2\alpha -{\lambda }_1)[{(q-m_1)}^2-m_2^2]}-\frac{Gr(1+{\lambda }_{1}q)e^{-\sqrt{\alpha (q-Q)}\eta }}{({\lambda }_2\alpha -{\lambda }_1)[{(q-m_1)}^2-m_2^2]}.\hfill \end{array}$$

(50)

5. Parametric Study

The meaningful role of several associated parameters on the mass and energy boundary layer is interpreted with the help of graphs in this section. These graphs of mass and energy profiles have two types of solution: (1) the solution with simultaneous ramp wall conditions and denoted by solid lines and (2) the solution with ramp wall velocity and isothermal temperature conditions and denoted by dashed lines.

First of all, a velocity profile comparison of the present work and the solution of the corrected model of Reference [27] is graphed in Figure 2, and a good agreement is observed in both solutions in the absence of heat consumption/generation and radiation effects. Figure 3 indicates that an increase in Gr values provides enhancement in the momentum boundary layer. It is supported by the physical fact that Gr is the fraction of buoyancy and viscous forces. An increase in Gr means that the buoyancy force gets stronger near the plate, that it overcomes the viscous force, and that the fluid gets accelerated. It is observed that the momentum boundary layer has more thickness in the case of isothermal temperature as compared to ramp temperature. Figure 4 presents the mass distribution under the influence of a magnetic parameter. It is witnessed that an increase in the magnetic parameter decreases the momentum boundary layer. The reason is the strong Lorentz force induced due to magnetic field. This strong Lorentz force acts as a resistance and decreases the velocity of fluid. Similar profiles are observed for ramped temperature and isothermal temperature. In Figure 5, the effect of changes in porosity parameter K is illustrated for isothermal and ramped temperature. It is observed that velocity for isothermal temperature is greater as compared to ramped temperature. Furthermore, It is also noticed that an increase in porosity reduces the friction of porous material which in turn increases the momentum development of the regime and, as a result, the velocity profile is enhanced. Figure 6 covers the influence of relaxation time ${\lambda }_1$ on velocity behavior. It is witnessed that the increasing variation of ${\lambda }_1$ reduces the thickness of momentum boundary layer which results in deceleration of the fluid. As a relaxation time increment implies that the fluid will take extra time to calm, it readily justifies the decrease in velocity. Afterwards, a comparison analysis describes that the ramped condition velocity is higher than the isothermal velocity. The role of Pr on the velocity curve is graphed in Figure 7. It is found that fluids having larger Pr values have lower velocity. The physical phenomenon behind this behavior is the increment in viscous force. It means the dragging force gets stronger with increase in Pr and results in retardation of fluid velocity. The contribution of Nr is shown in Figure 8 for both isothermal and ramped plates. It is found that the enhancement in Nr increases the thickness of the momentum boundary layer. This enhancement is physically justified by the high rate of transfer of energy when Nr is increased. This higher energy transfer rate weakens the bonds between fluid particles which results in the form of weaker resistance, and eventually, the fluid gets accelerated. It is also revealed that the velocity profile of the isothermal plate is higher than that of the ramped plate. Figure 9 reveals the impact of retardation time ${\lambda }_2$ on velocity solution. It is observed that both ramped and isothermal solutions get higher profiles in the case of increasing retardation time. This is justified by the fact that an increase in ${\lambda }_2$ decreases the effect of friction. Therefore, for large values of ${\lambda }_2$, the thickness of the momentum boundary layer increases. Figure 10 depicts that enlargement in time $\tau$ raises the velocity profiles of both ramped and isothermal wall.

Figure 11 shows a good agreement between the present work and the result of Mazhar et al. [27] when radiation and heat consumption/generation are removed from the thermal system. The function of Pr on temperature profiles is given in Figure 12. It is witnessed that temperature decreases for larger Pr values. The physical reason is the reduction of thermal conductivity. It means the fluid receives small amounts of heat when Pr increases. For lower Pr values, the fluid has higher thermal conductivity, and for large Pr values, the fluid faces more resistance. Moreover, temperature boundary layer thickness is observed to be greater in the case of the isothermal plate. Figure 13 sketches the significance of variating Nr on temperature curves. An elevation in temperature profile is witnessed for increasing Nr values. Physically, $k^{*}$ reduces due to the enhancement in divergence of radiative heat flux $\frac{\partial q_r}{\partial y}$. This leads to increases in the amount of radiative heat transfer to the fluid, and ultimately, the temperature of the fluid gets elevated. The influence of installed heat consumption and generation is illustrated in Figure 14. In the graph, negative values of Q are associated with heat consumption and positive values of Q refer to heat generation. Physically, an increase in negative value of Q refers to more consumption of heat and eventually a drop of temperature, as graphed in Figure 14. Similarly, an increase in positive values of Q refers to more heat generation and eventually a raise in temperature. Additionally, a temperature profile for ramped wall is observed lower as compared to the isothermal wall. Figure 15 describes that temperature gets elevated with the extension of time $\tau$.

Figure 16 and Figure 17 showcase the exact inverse behavior of rate of heat transfer for increasing Pr and Nr values. An effective behavior is witnessed; for increasing Pr, Nusselt number gets elevated very rapidly for $\tau <1$, but after this, it starts decreasing. On the other hand, increasing Nr leads to a decrease in the rate of heat transfer. As noticed for Pr, the Nusselt number for Nr increases for $\tau <1$ and, then, it starts decreasing. Additionally, for the case of the isothermal plate, only one kind of behavior is observed. In Figure 18, the role of heat consumption and generation in the transfer of heat is presented. It is noticed that, when large amounts of heat are consumed $(Q<0)$, the heat transfer rate from plate to fluid has a higher magnitude. Oppositely, when more heat is generated $(Q>0)$, the magnitude of heat transfer rate decreases.

Significance of relaxation and retardation time $({\lambda }_1,{\lambda }_2)$ in the behavior of shear stress is graphed in Figure 19. It is spotted that the magnitude of skin friction decreases for $\tau <1$ as a result of increasing ${\lambda }_1$, but after that, it starts increasing. This behavior is justified by the fact that small viscosity brings a significant decrease in skin friction. Contrarily, skin friction is elevated by the higher values of retardation time ${\lambda }_2$.

The numerical variation in heat transfer rate and skin friction is tabulated for influencing parameters in

Table 1. Varaition of Nusselt number for different values of parameters.

$\mathit{\tau }$	Pr	Nr	Q	Nu
0.3	7.0	0.5	0.5	1.2672
0.4	7.0	0.5	0.5	1.4366
0.5	7.0	0.5	0.5	1.5761
0.5	0.71	0.5	0.5	0.4984
0.5	2.0	0.5	0.5	0.8425
0.5	7.0	0.5	0.5	1.5761
0.5	7.0	0	0.5	1.9304
0.5	7.0	2	0.5	1.1145
0.5	7.0	3	0.5	0.9652
0.5	7.0	0.5	0	1.7235
0.5	7.0	0.5	1	1.4208
0.5	7.0	0.5	2	1.0814

and

Table 2. Varaition of skin friction for different values of parameters.

$\mathit{\tau }$	M	K	${\mathit{\lambda }}_{\mathbf{1}}$	${\mathit{\lambda }}_{\mathbf{2}}$	Gr	${\mathit{\tau }}_{\mathit{w}}$
0.3	2.0	0.5	1.0	1.0	1.0	−0.4013
0.4	2.0	0.5	1.0	1.0	1.0	−0.4941
0.5	2.0	0.5	1.0	1.0	1.0	−0.5845
0.5	3.0	0.5	1.0	1.0	1.0	−0.6320
0.5	4.0	0.5	1.0	1.0	1.0	−0.6768
0.5	5.0	0.5	1.0	1.0	1.0	−0.7192
0.5	2.0	0.2	1.0	1.0	1.0	−0.7192
0.5	2.0	0.5	1.0	1.0	1.0	−0.5845
0.5	2.0	0.9	1.0	1.0	1.0	−0.5396
0.5	2.0	0.5	0.5	1.0	1.0	−0.8244
0.5	2.0	0.5	0.8	1.0	1.0	−0.6567
0.5	2.0	0.5	1.0	1.0	1.0	−0.5845
0.5	2.0	0.5	1.0	0.5	1.0	−0.4252
0.5	2.0	0.5	1.0	0.8	1.0	−0.5219
0.5	2.0	0.5	1.0	1.3	1.0	−0.6763
0.5	2.0	0.5	1.0	1.0	0.5	−0.5845
0.5	2.0	0.5	1.0	1.0	0.8	−0.5455
0.5	2.0	0.5	1.0	1.0	1.3	−0.5065

. Table 1 shows that that Nusselt number gets elevated with an increase in $\tau$ and Pr. An interesting observation is made here; for ramp wall conditions, heat transfer rate initially increases for $0<\tau <1$ and then faces a decay after $\tau >1$. Moreover, heat transfer rate decreases with enhancement in Nr and Q. Table 2 presents that shear stress enhances with an increase in $\tau$, Gr, and K and that an inverse behavior is observed for increasing M and ${\lambda }_2$. In the tables, bold values are presented for the particular parameters to show how the results are changing with those parameters.

6. Conclusions

The main focus of this work is to examine the impact of ramp wall velocity and temperature together on time-dependent MHD convective flow of an Oldroyd-B fluid. The model also involves the porous medium, heat consumption/generation, and thermal radiation. During the modeling of flow, deficiency of Reference [27] modeling is removed and the correct equation of velocity is obtained. Laplace transformation is used as a solution finding tool. The simultaneous ramp wall conditions lead to complex combinations of Laplace paramter in solutions. To overcome this situation, numerical Laplace inversion named as Durbin Method is used and solutions are obtained in the original time domain. The behavioral study of mass and energy profiles under associated parameters is conducted and provided through graphs. Moreover, a comparison of ramp wall temperature and isothermal temperature is also obtained for different parameters. Lastly, expressions for shear stress and heat transfer rate are also calculated and the influence of parameters on them is given in the form of tables and graphs.

The important results of this investigation are as follows:

• An elevation in mass profile is seen with the enhancement in Gr, K, ${\lambda }_2$, and t. An increase in magnetic parameter (M) and relaxation time ${\lambda }_1$ leads to lowering of velocity curves.
• Energy boundary layer decreases with elevation in Pr and addition of heat sink to the system. Increases in Nr and t and addition of a heat source increase the energy boundary layer.
• Nusselt number indicates that high values of Pr provide resistance to heat transfer while small values of Pr have greater thermal conductivity. Moreover, the rate of heat transfer from plate to fluid decays with increase in Nr and Q.
• Velocity on the plate decreases with increase in relaxation time ${\lambda }_1$ and behaves oppositely for retardation time ${\lambda }_2$ (skin friction).