1 Introduction

In the study of thermal convection in a porous medium, the elemental volume-averaged temperatures of the fluid phase and the solid phase are identical or different and heat transfer between them is a matter of concern and it depends on the situation in hand. The former contemplation is called the local thermal equilibrium (LTE) model, and the latter is called the local thermal nonequilibrium (LTNE) model. It is experimentally observed that the LTE model requires numerous constraints and this assumption is no longer valid when the particles or pores are not small enough, when the thermal properties differ widely, or when the convective transport is important [1,2,3]. These restrictions enforced the inevitability of switching over to LTNE model in the study of thermal convection in porous media.

Thermogravitational convection in fluid-saturated porous media using a LTNE model has been studied extensively due to its applications in various fields of science and engineering, for example, building thermal insulation, nuclear reactor maintenance, nuclear waste disposal, oil reservoir, geothermal energy utilization and porous insert for thermal enhancement, to mention a few (Virto et al. [4]). In a detailed manner, Banu and Rees [5] studied buoyancy-driven convection in a layer of Darcy porous medium. Subsequently, many researchers extended this study under various additional effects [6,7,8,9,10,11,12,13]. An exhaustive bibliography on this topic can be found in the review article by Rees and Pop [14] and in the book by Neild and Bejan [15].

The study of non-Newtonian fluids in a porous medium is of prime significance in numerous fields such as petroleum, nuclear and chemical industries, reservoir engineering and bioengineering. The heavy crude is non-Newtonian and the rheology of such fluids depends on a generalized Darcy equation, which considers non-Newtonian behavior of fluids. Specifically, some oil sand contains waxy crude at shallow depth in the reservoir which is viewed as a viscoelastic fluid. Generalized Darcy equation is beneficial to the investigation of portability control in the displacement of oil mechanism, which increases the productivity of the oil recuperation. The onset of convection in a viscoelastic fluid-saturated porous layer has been studied extensively using a LTE model, and little consideration has been given toward the study using a LTNE model. Thermal convective instability of an Oldroyd-B fluid-saturated porous layer using a LTNE model was investigated by Malashetty et al. [16] and Shivakumara et al. [17]. The convective instability of Maxwell fluid-saturated porous layer using a LTNE model was studied by Malashetty and Kulkarni [18]. Recently, Shankar and Shivakumara [19] investigated the effect of LTNE on the stability of natural convection in an Oldroyd-B fluid-saturated vertical porous layer. All these studies confirmed that the oscillatory convection is the preferred mode of instability.

Copious literature is available on mixed (forced and free) convective flow in a porous medium. However, the studies on the impact of applied pressure gradient on buoyancy-driven convection in a fluid-saturated porous layer are still in infancy. These studies are found to be more relevant to model realistic effects occurring in several applications such as heat exchanger, sink simulation and cooling/heating design of the system, to mention a few. The influence of horizontal fluid motion on thermally induced convection currents in a layer of porous medium was investigated by Prats [20] using a LTE model. He showed that the basic flow and convection pattern move with the same speed in the same direction. The effect of inertia on the onset of mixed convection in a porous medium using a LTNE model was considered by Postelnicu [21]. Later, Postelnicu [22] studied the effect of a constant horizontal pressure gradient on the onset of Darcy–Bénard convection in a Newtonian fluid-saturated porous layer with LTNE model.

The existing studies on the onset of mixed convection are limited to Newtonian fluids-saturated porous media. Nonetheless, the consideration of non-Newtonian fluids is warranted in many applications such as ceramic processing, enhanced oil recovery, filtration and liquid composite molding. The intent of the current study is to examine the effect of a constant horizontal pressure gradient and viscoelasticity of the fluid on the onset of thermal convection in a fluid-saturated porous medium using a LTNE model. A modified Darcy–Oldroyd-B formulation is adopted to analyze the flow in the porous medium. The presence of constant pressure gradient affects the basic velocity, and also the stability equations involve complex coefficients. The stability eigenvalue problem is numerically solved using the Galerkin method, and the results are tabulated and exhibited graphically for different values of governing parameters. Attempts are also made to obtain the solution analytically for the onset of convection using a single-term Galerkin expansion technique. The results obtained from the single-term Galerkin method are found to be in close agreement with those computed from the higher order Galerkin method.

2 Mathematical formulation

The schematic representation consists of an incompressible Oldroyd-B fluid-saturated porous layer bounded by the horizontal surfaces \(z = 0\) and \(z = d\)(see Fig. 1), and these surfaces are held at fixed temperatures \(T_{l}\) and \(T_{u} ( < T_{l} )\), respectively. The x-axis is taken along the horizontal direction, and the gravity acts vertically downward. In the horizontal direction, a constant pressure gradient is applied. The fluid and solid phases of the porous medium possess different temperatures as the LTNE model is invoked.

Fig. 1
figure 1

Schematic representation of physical configuration

Full size image

The governing equations under the Oberbeck–Boussinesq approximation are [16, 17, 19]:

$$\nabla \cdot {\mathbf{q}} = 0,$$
(1)
$$\left( {1 + \lambda_{1} \frac{\partial }{\partial t}} \right)\left( { - \rho_{f} {\mathbf{g}} + \nabla p} \right) = - \frac{{\mu_{\text{f}} }}{K}\left( {1 + \lambda_{2} \frac{\partial }{\partial t}} \right){\mathbf{q}},$$
(2)
$$(\rho c)_{\text{f}} \left( {\varepsilon \frac{{\partial T_{\text{f}} }}{\partial t} + ({\mathbf{q}}.\nabla )T_{\text{f}} } \right) = \varepsilon k_{\text{f}} \nabla^{2} T_{\text{f}} - (T_{\text{f}} - T_{\text{s}} )h,$$
(3)
$$(\rho c)_{\text{s}} (1 - \varepsilon )\frac{\partial Ts}{\partial t} = k_{\text{s}} (1 - \varepsilon )\nabla^{2} T_{\text{s}} + (T_{\text{f}} - T_{\text{s}} )h,$$
(4)
$$\rho_{f} = \rho_{0} [1 - \beta (T_{\text{f}} - T_{\text{u}} )],$$
(5)

where \({\mathbf{q}} = (u,0,w)\) is the velocity vector, \(p\) is the pressure, \(\rho_{\text{f}}\) is the density of the fluid, \({\mathbf{g}}\) is the gravitational acceleration, \(\mu\) is the fluid viscosity, \(\lambda_{1}\) and \(\lambda_{2}\) are the stress relaxation and strain retardation time constants, K is the permeability, \(\varepsilon\) is the porosity, \(T_{\text{f}}\) and \(T_{\text{s}}\) are, respectively, the temperature of fluid and solid phases,\(h\) is the interphase heat transfer coefficient, \(\beta\) is the thermal expansion coefficient, \(c\) is the specific heat, \(\tilde{\mu }_{\text{f}}\) is the effective viscosity, \(k_{\text{f}}\) and \(k_{\text{s}}\) are the thermal conductivity of fluid and solid phases, respectively, and \(\rho_{0}\) is the reference density.

The bounding horizontal surfaces of the porous layer are isothermal and impermeable. The appropriate boundary conditions on velocity and temperature are:

$$\begin{aligned} {\mathbf{q}} = 0,\,\,T_{\text{f}} = T_{\text{s}} = T_{\text{l}} \,\,{\text{at}}\,\,z = 0, \hfill \\ {\mathbf{q}} = 0,\,\,T_{\text{f}} = T_{\text{s}} = T_{\text{u}} \,\,{\text{at}}\,\,z = d. \hfill \\ \end{aligned}$$

The quantities are rendered to dimensionless form using the following transformations:

$$\begin{aligned} (x,y,z) = d(x^{*} ,y^{*} ,\,z^{*} ),\,\,\,\,{\mathbf{q}} = \frac{{\varepsilon k_{\text{f}} }}{{d(\rho c)_{\text{f}} }}{\mathbf{q}}^{*} ,\,\,\,t = \frac{{(\rho c)_{\text{f}} d^{2} }}{{k_{\text{f}} }}t^{*} ,\,\,p = \frac{{\varepsilon \mu k_{\text{f}} }}{{(\rho c)_{\text{f}} }}p^{*} , \hfill \\ \kappa_{\text{f}} = \frac{{k_{\text{f}} }}{{(\rho c)_{\text{f}} }},\,\,\,\,T_{\text{f}} = (T_{\text{l}} - T_{\text{u}} )\theta + T_{\text{u}} ,\,\,\,\,T_{\text{s}} = (T_{\text{l}} - T_{\text{u}} )\phi + T_{\text{u}} ,\,\,\nabla = \frac{{\nabla^{*} }}{d}. \hfill \\ \end{aligned}$$
(6)

Equations (1)–(5), using Eq. (6), become (after neglecting asterisks)

$$\nabla \cdot {\mathbf{q}} = 0,$$
(7)
$$\left( {1 + \varLambda_{1} \frac{\partial }{\partial t}} \right)\left( { - R_{D} \theta \,{\hat{\mathbf{k}}} + \nabla p} \right) = - \left( {1 + \varLambda_{2} \frac{\partial }{\partial t}} \right){\mathbf{q}},$$
(8)
$$\frac{\partial \theta }{\partial t} + (\mathbf{q}.\nabla )\theta = \nabla^{2} \theta + H(\phi - \theta ),$$
(9)
$$\alpha \frac{\partial \phi }{\partial t} = \nabla^{2} \phi + \gamma H(\theta - \phi ),$$
(10)

where

$$R_{\text{D}} = \frac{{\rho_{0} g\beta \Delta TKd(\rho c)_{\text{f}} }}{{\varepsilon \mu_{\text{f}} k_{\text{f}} }},H = \frac{{hd^{2} }}{{\varepsilon k_{\text{f}} }},\alpha = \frac{{k_{\text{f}} (\rho c)_{\text{s}} }}{{k_{\text{s}} (\rho c)_{\text{f}} }} = \frac{{\kappa_{\text{f}} }}{{\kappa_{\text{s}} }},\gamma = \frac{{\varepsilon k_{\text{f}} }}{{(1 - \varepsilon )k_{\text{s}} }},\varLambda_{1} = \frac{{\lambda_{1} k_{\text{f}} }}{{\left( {\rho c} \right)_{\text{f}} d^{2} }},\varLambda_{2} = \frac{{\lambda_{2} k_{\text{f}} }}{{\left( {\rho c} \right)_{\text{f}} d^{2} }}.$$

are the Darcy–Rayleigh Number, scaled interphase heat transfer coefficient, ratio of diffusivities, ratio of porosity-modified conductivities, relaxation and retardation viscoelastic parameters, respectively.

The boundary conditions become

$${\mathbf{q}} = 0\,\,\,{\text{at}}\,\,z = 0,\,\,1,\,\,\,\theta = \phi = 1\,\,\,\,{\text{at}}\,\,z = 0,\,\,\theta = \phi = 0\,\,\,{\text{at}}\,\,z = 1.$$
(11)

3 Linear instability analysis

The basic state is characterized by

$${\mathbf{q}}_{\text{b}} = u_{\text{b}} \,{\hat{\mathbf{i}}} = \varPi \,{\hat{\mathbf{i}}},\theta_{\text{b}} = \phi_{\text{b}} = 1 - z,$$
(12)

where \(\varPi\) is the dimensionless constant horizontal pressure gradient and \({\hat{\mathbf{i}}}\) is the unit vector in the horizontal \(x\)-direction. The basic state is perturbed in the form

$${\mathbf{q}} = \varPi \,{\hat{\mathbf{i}}} + {\mathbf{q}}',\,\,\,\theta = 1 - z + \theta ',\,\,\,\phi = 1 - z + \phi ',$$
(13)

where \({\mathbf{q}}'\), \(\theta '\) and \(\phi '\) are the perturbed quantities and assumed to be small. Equation (13) is substituted back in Eqs. (7)–(10), linearized, curl is operated on the momentum equation to discard the pressure term and the stream functions \(\psi^{\prime}(x,z,t)\) are introduced through

$$\left( {u^{\prime},\,\,0,\,\,w^{\prime}} \right) = \left( { - \frac{{\partial \psi^{\prime}}}{\partial z},\,\,0,\,\,\frac{{\partial \psi^{\prime}}}{\partial x}} \right),$$
(14)

to obtain finally the stability equations in the form (after ignoring the primes)

$$\left( {1 + \varLambda_{2} \frac{\partial }{\partial t}} \right)\left( {\nabla^{2} \psi } \right) = R_{\text{D}} \left( {1 + \varLambda_{1} \frac{\partial }{\partial t}} \right)\frac{\partial \theta }{\partial x},$$
(15)
$$\frac{\partial \theta }{\partial t} + \varPi \frac{\partial \theta }{\partial x} - \frac{\partial \psi }{\partial x} = \nabla^{2} \theta + H(\phi - \theta ),$$
(16)
$$\alpha \frac{\partial \phi }{\partial t} = \nabla^{2} \theta + \gamma H(\theta - \phi ).$$
(17)

The boundary conditions become

$$\psi = \,\,\theta = \phi = 0\,\,\,\,{\text{at}}\,\,\,z = 0\,,1.$$
(18)

The normal mode analysis is employed in the form

$$(\psi ,\theta ,\phi ) = \,[\varPsi (z),\varTheta (z),\varPhi (z)]\exp \{ ia(x - \omega t\} .$$
(19)

where \(\omega\)\(( = \omega_{r} + i\omega_{i} )\) is the growth term and \(a\) is the horizontal wave number. Equation (19) is substituted back in Eqs. (15)–(18) to obtain, respectively,

$$\left( {1 - ia\omega \varLambda_{2} } \right)\left( {D^{2} - a^{2} } \right)\varPsi = iaR_{\text{D}} \left( {1 - ia\omega \varLambda_{1} } \right)\varTheta ,$$
(20)
$$ia\varPsi + [(D^{2} - a^{2} ) - H + ia\omega - ia\varPi ]\varTheta + H\varPhi = 0,$$
(21)
$$\gamma H\,\varTheta + [(D^{2} - a^{2} ) - \gamma H + ia\omega \alpha ]\varPhi = 0.$$
(22)

where \(D = {\text{d}}/{\text{d}}z.\)

The associated boundary conditions are

$$\varPsi = \,\,\varTheta = \varPhi = 0\,\,\,{\text{at}}\,\,\,z = 0\,,1.$$
(23)

4 Numerical solution of the eigenvalue problem

Equations (20)–(23) form a complex stability eigenvalue problem and solved numerically to extract the critical eigenvalue \(R_{\text{Dc}}\) with respect to the wave number \(a\) as a function of \(\varPi ,\,\,\alpha ,\,\,H,\,\,\varLambda_{1} ,\,\,\varLambda_{2} \,\,{\text{and}}\,\,\gamma\). The Galerkin technique is utilized to solve the ensued stability eigenvalue problem, and accordingly \(\varPsi (z),\,\,\varTheta (z)\) and \(\varPhi (z)\) are expanded as follows [23]:

$$\varPsi = \,\sum\limits_{i\, = 1}^{N} {A_{i} \,\varPsi_{i} (z)} \,,\,\,\varTheta = \sum\limits_{i\, = 1}^{N} {B_{i} \,\varTheta_{i} (z)\,,\,\,\varPhi = \sum\limits_{i\, = 1}^{N} {C_{i} \,\varPhi_{i} (z)\,} } .$$
(24)

Substituting Eq. (24) in Eqs. (20), (21) and (22) and multiplying the resulting equations by \(\varPsi_{j} (z)\),\(\,\varTheta_{j} (z)\) and \(\varPhi_{j} (z)\), respectively, and integrating with respect to \(z\) between \(z = 0\) and \(1\), we get the following system of algebraic equations

$$L_{ji} A_{i} + M_{ji} B_{i} = \omega \{ N_{ji} A_{i} + O_{ji} B_{i} \} ,$$
(25)
$$P_{ji} A_{i} + Q_{ji} B_{i} + R_{ji} C_{i} = \omega S_{ji} B_{i} ,$$
(26)
$$T_{ji} B_{i} + U_{ji} C_{i} = \omega V_{ji} C_{i} .$$
(27)

The coefficients \(L_{ji} - V_{ji}\) involve inner products of the base functions and are given by

$$\begin{aligned} & L_{ji} = a^{2} \left\langle {\varPsi_{j} \varPsi_{i} } \right\rangle + \left\langle {D\varPsi_{j} D\varPsi_{i} } \right\rangle ,\,\,\,M_{ji} = iaR_{D} \left\langle {\varPsi_{j} \varTheta_{i} } \right\rangle ,\,\\ & N_{ji} = ia\varLambda_{2} \{ a^{2} \left\langle {\varPsi_{j} \varPsi_{i} } \right\rangle + \left\langle {D\varPsi_{j} D\varPsi_{i} } \right\rangle \}, \\ & O_{ji} = ia\varLambda_{1} R_{D} \left\langle {\varPsi_{j} \varTheta_{i} } \right\rangle ,\,\,\,P_{ji} = - ia\left\langle {\varTheta_{j} \varPsi_{i} } \right\rangle ,\,\,\\ & Q_{ji} = (a^{2} + H + ia\varPi )\left\langle {\varTheta_{j} \varTheta_{i} } \right\rangle + \left\langle {D\varTheta_{j} D\varTheta_{i} } \right\rangle ,\,\,\, \\ & R_{ji} = - H\left\langle {\varTheta_{j} \varPhi_{i} } \right\rangle ,\,\,\,S_{ji} = ia\left\langle {\varTheta_{j} \varTheta_{i} } \right\rangle ,\,\,\,T_{ji} = - H\gamma \left\langle {\varPhi_{j} \varTheta_{i} } \right\rangle ,\, \\ & U_{ji} = (a^{2} + H\gamma )\left\langle {\varPhi_{j} \varPhi_{i} } \right\rangle + \left\langle {D\varPhi_{j} D\varPhi_{i} } \right\rangle ,\,\,\,V_{ji} = ia\alpha \left\langle {\varPhi_{j} \varPhi_{i} } \right\rangle , \\ \end{aligned}$$
(28)

where \(\left\langle \cdots \right\rangle = \int\nolimits_{0}^{1} {( \cdots )\,{\text{d}}z} .\) Equations (25)–(27) can be written in the matrix form

$$M\,X = \omega \,NX$$
(29)

where

$$M = \left[ {\begin{array}{*{20}c} {L_{ji} } & {M_{ji} } & 0 \\ {P_{ji} } & {Q_{ji} } & {R_{ji} } \\ 0 & {T_{ji} } & {U_{ji} } \\ \end{array} } \right],\,\,N = \left[ {\begin{array}{*{20}c} {N_{ji} } & {O_{ji} } & 0 \\ 0 & {S_{ji} } & 0 \\ 0 & 0 & {V_{ji} } \\ \end{array} } \right]\,\,{\text{and}}\,\,X = \left[ {\begin{array}{*{20}c} {A_{i} } \\ {B_{i} } \\ {C_{i} } \\ \end{array} } \right].$$

We observe that \(M\) and \(N\) are complex matrices of order \(3N \times 3N\), \(X\) is the characteristic vector and \(\omega\) is the eigenvalue. By using the subroutine GVLRG of the ISML library, the complex eigenvalue problem \(\omega\) is determined when the other parameters are specified. Then, one of the parameters, say \(R_{\text{D}}\), is varied until the real part of \(\omega \,( = \omega_{r} )\) vanishes. The zero crossing of \(\omega_{r}\) is achieved by Newton’s method for a fixed-point determination, and the imaginary part of \(\omega \,\,( = \omega_{i} )\) indicates whether the instability onsets into steady convection \((\omega_{i} = 0)\) or into growing oscillations \((\omega_{i} \ne 0)\). The corresponding values of \(R_{\text{D}}\) and \(a\) are the critical conditions for neutral stability. Then, the critical Darcy–Rayleigh number with respect to the wave number is calculated using golden section search method. The base functions \(\varPsi_{i} (z)\), \(\varTheta_{i} (z)\) and \(\varPhi_{i} (z)\) are chosen such that they satisfy the respective boundary conditions

$$\varPsi_{i} = z^{i} - z^{i + 1} = \varTheta_{i} = \varPhi_{i} .$$
(30)

Equation (29) is a generalized eigenvalue problem and numerically solved following the procedure explained in the work of Shivakumara et al. [24].

5 Results and discussion

The impact of constant horizontal pressure gradient on the criterion for the onset of thermal convection in an Oldroyd-B fluid-saturated porous layer with LTNE temperatures is investigated. The critical Darcy–Rayleigh number \(R_{\text{Dc}}\), the corresponding critical wave number \(a_{\text{c}}\) and the critical frequency of oscillations \(\omega_{ic}\) are computed numerically, and the convergence is achieved by considering six terms in the Galerkin expansion. The progress of convergence is shown in Table 1 for various values of governing parameters. From the table, it is evident that there is not much deviation in the critical values between the first and higher order Galerkin methods. Hence, it is intuitive to look at the analytical solution for a single-term Galerkin method as it gives satisfactory results with minimum mathematical computations. Taking N = 1 with \({ \sin }\;\pi z\) as the trial function, an analytical expression for \(R_{\text{D}}\) is obtained from Eq. (30) in the form

$$R_{\text{D}} = \frac{{\delta^{2} }}{{a^{2} }}\left[ {\frac{{(1 - ia\omega_{i} \varLambda_{2} )[( \delta^{2}+H+ia(\varPi-\omega))(\delta^{2}+H\gamma -\alpha ia \omega_{i})-H^2\gamma] }}{{(H\gamma + \delta^{2} - \alpha ia\omega_{i} )( 1 -ia \omega_{i} \varLambda_{1} )}}} \right]$$
(31)

where \(\delta^{2} = a^{2} + \pi^{2} .\) After removing the complex quantities from the denominator of Eq. (31), we obtain

$$R_{\text{D}} = \frac{1}{{a^{2} \left\{ {k_{8}^{2} + a^2\alpha^{2} \omega_{i}^{2} } \right\}(1 +a^2 \varLambda_{1}^{2} \omega_{i}^{2} )}}\left[ {\Delta_{1} + i\omega_{i} \Delta_{2} } \right]$$
(32)

where

$$\begin{aligned} & \Delta_{1} = \delta^{2} [(\delta^{2}+a^2\alpha^{2}\omega_{i}^{2})(-a^2(\varLambda_{1} - \varLambda_{2} )(\varPi-\omega_{i})\omega_{i}+\delta^{2}(1+a^2\varLambda_{1}\varLambda_{2}\omega_{i}^{2}))+H^2\gamma(a^2(\varLambda_{1} - \varLambda_{2})\omega_{i}(-\gamma \varPi+(\alpha+\gamma)\omega_{i})+(1+\gamma)\delta^{2}(1+a^2\varLambda_{1}\varLambda_{2}\omega_{i}^{2}))+H(-2a^2\delta^{2}\gamma(\varLambda_{1}-\varLambda_{2})(\varPi-\omega_{i})\omega_{i}+(1+2\gamma)\delta^{4}(1+a^2\varLambda_{1}\varLambda_{2}\omega_{i}^{2})+a^2\alpha_{2}\omega_{i}^{2}(1+a^2\varLambda_{1}\varLambda_{2}\omega_{i}^{2}))] \\ \end{aligned}$$
(33)
$$\begin{aligned} & \Delta_{2} = a \delta^{2} [ H^2\gamma(\gamma \varPi-(k_{10}+\gamma k_{11})\omega_{i}+a^2\gamma \varLambda_{1}\varLambda_{2}\varPi\omega_{i}^{2}-a^2(\alpha+\gamma)\varLambda_{1}\varLambda_{2}\omega_{i}^{3})+(\delta^{4}+a^2\alpha^{2}\omega_{i}^{2})(\varPi(1+a^2\varLambda_{1}\varLambda_{2}\omega_{i}^{2})-\omega_{i}(k_{11}+a^2\varLambda_{1}\varLambda_{2}\omega_{i}^{2}))+H((\varLambda_{1}-\varLambda_{2})\omega_{i}(\delta^{2}+a^2\alpha^{2}\omega_{i}^{2})+2\gamma \delta^{2}(\varPi(1+a^2\varLambda_{1}\varLambda_{2})\omega_{i}^{2}-\omega_{i}(k_{11}+a^2\varLambda_{1}\varLambda_{2}\omega_{i}^{2})))] . \\ \end{aligned}$$
(34)
Table 1 Comparison of critical stability parameters calculated from single-term and different higher order Galerkin methods
Full size table

The condition \(\Delta_{2} = 0\) gives a dispersion relation of the form

$$k_{1} \omega_{i}^{5} + k_{2} \omega_{i}^{4} + k_{3} \omega_{i}^{3} + k_{4} \omega_{i}^{2} + k_{5} \omega_{i} + k_{6} = 0$$
(35)

where

$$\begin{aligned} & k_{1} = - \alpha^{2} a^{2} \varLambda_{1} \varLambda_{2} ,\,\,k_{2} = -\varPi k_{1} ,\\ & k_{3} = \delta^{2} [\alpha^{2} k_{7} - (H^{2} \alpha \gamma \varLambda_{1} \varLambda_{2} + k_{8}^{2} \varLambda_{1} \varLambda_{2} )], \\ & k_{4} = a^{2} (\alpha^{2} + k_{8}^{2} \varLambda_{1} \varLambda_{2} )\varPi ,\,\, \\ & k_{5} = a^{2} [k_{9} + H(2\gamma\delta^{2}k_{9}++\delta^{4}(\varLambda_{1}-\varLambda_{2}))-H^2\gamma(k_{10}+\gamma k_{11})], \\ & k_{6} = k_{8}^{2} \varPi ,\,\,k_{7} = [ - 1 + (H + \delta^{2} )(\varLambda_{1} - \varLambda_{2} )], \\ & k_{8} = (H\gamma + \delta^{2} ),\,k_{9} = [-1+\delta^{2}(\varLambda_{1}-\varLambda_{2})], \\ & k_{10} = [\alpha+\delta^{2}(-\varLambda_{1}+\varLambda_{2})], \\ & k_{11} = [1 +\delta^{2}(-\varLambda_{1}+\varLambda_{2})] . \\ \end{aligned}$$
(36)

The critical values of \(R_{\text{D}}\) and \(\omega_{i}\) over \(a\) are computed numerically, and the values are given in the end of Table 1, and it is seen that there is an excellent agreement between the results obtained from single-term and higher order Galerkin methods. The results obtained under the limiting case of \(\varLambda_{1} = \varLambda_{2} = 0\)(Newtonian fluid) by taking six terms in the Galerkin expansion are compared with those of Postelnicu [22] obtained using the numerical solver dsolve from Maple in Table 2. We note that the results obtained from two different methods are in close agreement indicating the validity of numerical method employed.

Table 2 Comparison of critical stability parameters for different values of governing parameters
Full size table

The viscoelastic parameters influence only the oscillatory onset and not the stationary convection. This is because viscoelastic fluid of simple type becomes Newtonian when the flow is steady and weak and hence the viscoelasticity produces nothing new on the onset of stationary convection. Thus, there is no distinction between viscous fluid and viscoelastic fluid as far as the stationary convection is concerned. However, in the case of oscillatory convection (time-dependent motion), the viscoelastic relaxation and retardation time parameters influence the oscillatory onset. The neutral stability curves in the (\(R_{\text{D}} ,\,\,a\))-plane for various values of \(\varLambda_{1}\), \(\varLambda_{2}\), \(\gamma\), \(\varPi\), \(H\) and \(\alpha\) are displayed in Fig. 2a–f. It is noted that the neutral curves are unimodal and akin to those seen in the classical Darcy–Bénard problem. In addition, \(\varLambda_{1}\) and \(\varLambda_{2}\) have contradictory influences on the instability characteristics of the system. In particular, increase in \(\varLambda_{1}\) (Fig. 2a) and \(\varLambda_{2}\) (Fig. 2b) is to increase and decrease the instability region, respectively. The neutral curve shown for \(\varLambda_{2} = 0\) in Fig. 2b corresponds to that for a Maxwell fluid. Furthermore, increase in \(\gamma\) (Fig. 2c) and \(\varPi\) (Fig. 2d) is to increase the instability region, whereas an opposite behavior is noticed for decreasing \(H\) (Fig. 2e) and \(\alpha\) (Fig. 2f). Besides, the oscillatory neutral stability curves tilt toward lower values of the wave number when the values of \(\varLambda_{1}\), \(\gamma\) and \(\varPi\) increase. This amounts to a reduction in the critical wave number indicating an increase in the cell width.

Fig. 2
figure 2

Neutral stability curves for different values of a \(\varLambda_{1}\) with \(\varLambda_{2} = 0.2\), b \(\varLambda_{2}\) with \(\varLambda_{1} = 1\) when \(H = 10\), \(\alpha = 0.5\), \(\,\gamma = 1\,\) and \(\varPi = 10\), c \(\gamma\) with \(\varPi = 10\), d \(\varPi\) with \(\gamma = 1\) when \(H = 10\), \(\alpha = 0.5\), \(\varLambda_{1} = 0.5\) and \(\varLambda_{2} = 0.2\), e \(H\) with \(\alpha = 0.5\) and f \(\alpha\) with \(H = 10\) when \(\varPi = 10\), \(\gamma = 1\), \(\varLambda_{1} = 0.5\,\) and \(\varLambda_{2} = 0.2\)

Full size image

Figure 3a–c demonstrates the way in which \(R_{\text{Dc}}\), \(a_{\text{c}}\) and \(\omega_{ic}\) vary with \(\varPi\) for different values of \(H\) when \(\alpha = 0.5\), \(\,\varLambda_{1} = 0.5\) and \(\varLambda_{2} = 0.2\). Figure 3a suggests that \(R_{\text{Dc}}\) is inversely proportional to \(\varPi\) and it remains invariant with increasing \(\varPi\). It is noted that an increase in \(H\) is to increase \(R_{\text{Dc}}\) due to an increase in rapid heat exchange between the fluid and solid phases of the porous medium and also increase in residence of time of heat in solid phase. Hence, its effect is to delay the onset of convection. Figure 3b illustrates that \(a_{\text{c}}\) diminishes initially up to certain values of \(\varPi\), but remains unchanged with increasing values of \(\varPi\). Also note that increasing \(H\) is to increase \(a_{\text{c}}\) and as a result the size of the convection cell decreases. Figure 3c displays the variation in \(\omega_{ic}\) as a function of \(\varPi\), and note that all the curves increase gradually irrespective of values of \(H\). The variation in \(R_{\text{Dc}}\), \(a_{\text{c}}\) and \(\omega_{ic}\) as a function of \(\varPi\) for different values of \(\gamma\) is shown in Fig. 4a–c. The influence of \(\gamma\) on the critical stability parameters with \(\varPi\) turns out to be opposite to that of \(H\) as observed in Fig. 3a–c.

Fig. 3
figure 3

Variation in a critical Darcy–Rayleigh number vs the pressure gradient \(\varPi\), b critical wave number vs the pressure gradient \(\varPi\) and c critical frequency vs the pressure gradient \(\varPi\), when \(\alpha = 0.5\), \(\gamma = 1\), \(\varLambda_{1} = 0.5\) and \(\varLambda_{2} = 0.2\)

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Fig. 4
figure 4

Variation in a critical Darcy–Rayleigh number vs the pressure gradient \(\varPi\), b critical wave number vs the pressure gradient \(\varPi\) and c critical frequency vs the pressure gradient \(\varPi\), when \(\alpha = 0.5\), \(H = 10\), \(\varLambda_{1} = 0.5\) and \(\varLambda_{2} = 0.2\)

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The impact of viscoelastic parameters \(\varLambda_{1}\) and \(\varLambda_{2}\) on the critical values of \(R_{\text{Dc}}\), \(a_{c}\) and \(\omega_{ic}\) is illustrated in Fig. 5a–c as a function of \(\varPi\) when \(\alpha = 0.5\), \(H = 10\) and \(\gamma = 1\). From Fig. 5a, it is clear that \(R_{\text{Dc}}\) passes through a minimum with increasing \(\varPi\) before attaining a constant value at higher values of \(\varPi\). This trend is found to be the same for all the values of \(\varLambda_{1}\) and \(\varLambda_{2}\) considered. It is also noticed that increase in \(\varLambda_{1}\) is to decrease \(R_{\text{Dc}}\) because of allowing the applied stress to act for a longer time on the fluid. In fact, the increase in relaxation ceases the stickiness of the fluid and hence the effect of friction will be lesser so that convection sets in at lower values of \(R_{\text{Dc}}\). An opposite phenomenon is observed with increase in \(\varLambda_{2}\). That is, increase in the value of \(\varLambda_{2}\) is to delay the onset of convection because of increase in the retardation effect. Moreover, the curves of \(\varLambda_{2} \ne 0\) lie above the curve of \(\varLambda_{2} = 0\), which indicates that the stickiness of Maxwell fluid is less compared to an Oldroyd-B fluid. Figure 5b demonstrates that \(a_{\text{c}}\) decreases initially up to certain values of \(\varPi\), but gradually increases for increasing values of \(\varPi\). Eventually, all the curves of \(a_{\text{c}}\) for different \(\varLambda_{1}\) and \(\varLambda_{2}\) coalesce at higher values of \(\varPi\). From Fig. 5c, we observe that an increase in \(\varPi\) is to increase \(\omega_{ic}\) rapidly for various values of \(\varLambda_{1}\) and \(\varLambda_{2}\).

Fig. 5
figure 5

a Critical Darcy–Rayleigh number vs the pressure gradient \(\varPi\), b critical wave number vs the pressure gradient \(\varPi\) and c critical frequency vs the pressure gradient \(\varPi\), when \(\alpha = 0.5\), \(H = 10\) and \(\gamma = 1\)

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Figure 6a–c demonstrates the variation in \(R_{\text{Dc}}\), \(a_{\text{c}}\) and \(\omega_{ic}\) with \(\varLambda_{2}\) for various values of \(\varPi\) and \(\varLambda_{1}\) when \(\alpha = 0.5\), \(H = 10\) and \(\gamma = 1\). Figure 6a shows that increase in \(\varPi\) and \(\varLambda_{1}\) is to decrease the critical Darcy–Rayleigh number for a fixed value of \(\varLambda_{2}\), and therefore its effect is to hasten the onset of oscillatory convection. On the contrary, increasing \(\varLambda_{2}\) delays the onset of oscillatory convection. This is because increasing \(\varLambda_{2}\) amounts to increase in time taken by the fluid element to respond to the applied stress. For a fixed values of \(\varLambda_{1}\), there exists a threshold value of \(\varLambda_{2} = \varLambda_{2}^{*}\) beyond which the stationary convection prevails, and note that increase in \(\varPi\) is to increase \(\varLambda_{2}^{*}\). From Fig. 6b, it is observed that the curves of \(a_{\text{c}}\) drop suddenly and remain invariant for different values of \(\varPi\) and \(\varLambda_{1}\) at those values of \(\varLambda_{2}\) at which the preferred mode of instability switches over from oscillatory to stationary convection. Also, increase in \(\varPi\) decreases \(a_{\text{c}}\) and hence increases the convection cells size, whereas an opposite trend could be seen with increasing \(\varLambda_{1}\). The value of \(\omega_{ic}\) decreases with increasing \(\varLambda_{2}\) and increases with increasing \(\varPi\) and \(\varLambda_{1}\) (Fig. 6c) due to an increase in the elasticity of the fluid.

Fig. 6
figure 6

Variation in a \(R_{\text{Dc}}\), b \(a_{\text{c}}\) and c \(\omega_{i\,c}\) with \(\varLambda_{2}\) for different \(\varPi\) when \(H = 10\), \(\gamma = 1\) and \(\alpha = 0.5\)

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The plots of \(R_{{{\text{Dc}}\,}}\), \(a_{\text{c}}\) and \(\omega_{ic}\) with \({ \log }_{10} H\) are depicted in Fig. 7a–c for different values of \(\varPi\) when \(\alpha = 0.5\), \(\gamma = 1\), \(\varLambda_{1} = 0.5\) and \(\varLambda_{2} = 0.2\). From Fig. 7a, it is observed that \(R_{{{\text{Dc}}\,}}\) increases gradually with \(H\), reaches a maximum and remains constant subsequently with further increase in \(H\). Also, \(R_{{{\text{Dc}}\,}}\) decreases with increasing \(\varPi\) because increase in \(\varPi\) leads to prominent heat transfer through both the phases which in turn eases the stabilizing effect of \(H\) and speeds up the onset of oscillatory convection. Figure 7b indicates that \(a_{\text{c}}\) remains unaffected in the small-\(H\) and large-\(H\) limits, whereas at intermediate values of \(H\) it reaches maximum values for various values of \(\varPi\). We note that \(a_{\text{c}}\) decreases with increasing \(\varPi\) at intermediate values of \(H\) and as a result the convection cells size increases. From Fig. 7c, increase in \(\omega_{ic}\) is noted as \(\varPi\) increases.

Fig. 7
figure 7

Variation in a \(R_{\text{Dc}}\), b \(a_{\text{c}}\) and c \(\omega_{i\,c}\) with \({ \log }_{10} H\) for different \(\varPi\) when \(\alpha = 0.5\), \(\varLambda_{1} = 0.5\), \(\gamma = 1\) and \(\varLambda_{2} = 0.2\)

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To know distinctly the effect of individual and simultaneous presence of horizontal pressure gradient and the viscoelasticity of the fluid on the convective instability, the values of \(R_{{{\text{Dc}}\,}}\), \(a_{\text{c}}\) and \(\omega_{ic}\) computed for these cases are given in Table 3. It is observed that oscillatory convection is possible even in the isolation presence of pressure gradient and viscoelasticity: a result of contrast in which stationary convection is only possible in their absence. In the case of Newtonian fluids (\(\varLambda_{1} = 0 = \varLambda_{2}\)), the effect of increasing \(\varPi\) is found to increase \(R_{{{\text{Dc}}\,}}\) marginally and hence to delay the onset of oscillatory convection, while \(R_{{{\text{Dc}}\,}}\) decreases slightly with increasing \(\varPi\) if the fluid is viscoelastic and hence to hasten the onset of oscillatory convection. Nevertheless, the frequency of oscillations increases with increasing \(\varPi\) irrespective of whether the fluid is Newtonian or viscoelastic.

Table 3 Critical values for different values of pressure gradient and viscoelastic parameters
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6 Conclusions

The implications of a constant horizontal pressure gradient and LTNE temperatures on the onset of thermal convective instability of an Oldroyd-B fluid-saturated porous medium are explored. The stability eigenvalue problem with complex coefficients is solved numerically.

The important findings of the present study may be summarized as follows:

  1. 1.

    Contrary to the observed phenomenon in Newtonian fluids, the effect of constant horizontal pressure gradient \(\varPi\) is to hasten the onset of oscillatory convection in an Oldroyd-B fluid-saturated porous layer.

  2. 2.

    The extent to which the relaxation (\(\varLambda_{1}\)) and the retardation (\(\varLambda_{2}\)) viscoelastic parameters encompass opposite contributions on the onset of oscillatory convection is to diminish in the presence of pressure gradient. The impact of increasing \(\varLambda_{1}\) is to advance marginally while increasing \(\varLambda_{2}\) is to suppress noticeably the onset of oscillatory convection.

  3. 3.

    The range of values of retardation parameter \(\varLambda_{2}\) up to which the instability sets in as oscillatory convection increases with increasing \(\varPi\).

  4. 4.

    Increasing \(\varPi\) is to increase the convection cells size of oscillatory onset and also the critical frequency of oscillations.

  5. 5.

    Both large and small values of interphase heat transfer coefficient \(H\) have no noticeable impact on the oscillatory onset but the intermediate values of \(H\) exhibit strong influence on the onset of oscillatory convection.

  6. 6.

    The system is more unstable for Maxwell fluid than that of Oldroyd-B type of viscoelastic fluid.