Abstract
The stability of the conduction regime of natural convection in a porous vertical slab saturated with an OldroydB fluid has been studied. A modified Darcy’s law is utilized to describe the flow in a porous medium. The eigenvalue problem is solved using Chebyshev collocation method and the critical Darcy–Rayleigh number with respect to the wave number is extracted for different values of physical parameters. Despite the basic state being the same for Newtonian and OldroydB fluids, it is observed that the basic flow is unstable for viscoelastic fluids—a result of contrast compared to Newtonian as well as for powerlaw fluids. It is found that the viscoelasticity parameters exhibit both stabilizing and destabilizing influence on the system. Increase in the value of strain retardation parameter \(\Lambda _2 \) portrays stabilizing influence on the system while increasing stress relaxation parameter \(\Lambda _1\) displays an opposite trend. Also, the effect of increasing ratio of heat capacities is to delay the onset of instability. The results for Maxwell fluid obtained as a particular case from the present study indicate that the system is more unstable compared to OldroydB fluid.
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Abbreviations
 a :

Vertical wave number
 c :

Wave speed
 \(c_r\) :

Phase velocity
 \(c_i\) :

Growth rate
 2d :

Thickness of the porous layer
 \(\vec {g}\) :

Acceleration due to gravity
 \(\hat{{k}}\) :

Unit vector in zdirection
 K :

Permeability
 p :

Pressure
 P :

Modified pressure
 \(\vec {q}=(u,v,w)\) :

Velocity vector
 \(R_\mathrm{D}\) :

Darcy–Rayleigh number
 t :

Time
 T :

Temperature
 \(T_1\) :

Temperature of the left vertical wall
 \(T_2\) :

Temperature of the right vertical wall
 \(\left( {x,y,z} \right) \) :

Cartesian coordinates
 \(\alpha \) :

Ratio of heat capacities
 \(\beta \) :

Thermal expansion coefficient
 \(\theta \) :

Fluid temperature
 \(\Theta \) :

Disturbance fluid temperature
 \(\lambda _1 \) :

Stress relaxation time constant
 \(\lambda _2 \) :

Strain retardation time constant
 \(\Lambda _1 \) :

Relaxation parameter
 \(\Lambda _2 \) :

Retardation parameter
 \(\mu \) :

Fluid viscosity
 \(\kappa \) :

Effective thermal diffusivity
 \(\rho \) :

Fluid density
 \(\rho _0 \) :

Reference density at \(T_0 \)
 \(\psi \) :

Stream function
 \(\Psi \) :

Disturbance stream function
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Communicated by Patrick Jenny.
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Shankar, B.M., Shivakumara, I.S. On the stability of natural convection in a porous vertical slab saturated with an OldroydB fluid. Theor. Comput. Fluid Dyn. 31, 221–231 (2017). https://doi.org/10.1007/s0016201604158
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Keywords
 Natural convection
 Vertical porous layer
 OldroydB fluid
 Viscoelastic fluid