The time dependent, fully developed, and incompressible circumferential flow between two infinite co-axial horizontal porous “cylinders containing fluid and porous layer separated by a permeable thin interface with fluid occupying the interval \(d^{^{\prime}} \le r^{\prime } \le r_{0}\) while the interval \(r_{{{\text{in}}}} \le r^{\prime } \le d^{^{\prime}}\) is occupied by a fluid-saturated porous material of uniform permeability. *r*_{in} and *r*_{0} are the radii of the inner and the outer cylinder, respectively, as shown in Fig. 1. The fluid is set in motion due to sudden application of constant circumferential pressure gradient and the angular, rotating with an angular velocity \(\omega_{{{\text{in}}}}\) and \(\omega_{0}\).” The velocity of the fluid is showed to be a function of the radial coordinate \(r^{\prime }\) and time *t* only. The flow in the porous region is governed by the Darcy law while the flow in the clear region is governed by the usual Navier–Stokes equations.
Continuity of velocity and shear stress has been used at the fluid–porous interface. In cylindrical coordinates, these equations are given as

$$\frac{{\partial u_{{\text{p}}}^{\prime } }}{{\partial t^{\prime } }} + u_{1} \left[ {\frac{1}{{r^{\prime } }}\frac{{\partial u_{{\text{p}}}^{\prime } }}{{\partial r^{\prime } }} - \frac{{u_{{\text{p}}}^{\prime } }}{{r^{\prime 2} }}} \right] = \nu_{{{\text{eff}}}} \left[ {\frac{{\partial^{2} u_{{\text{p}}}^{\prime } }}{{\partial r^{\prime 2} }} + \frac{1}{{r^{\prime } }}\frac{{\partial u_{{\text{p}}}^{\prime } }}{{\partial r^{\prime } }} - \frac{{u_{{\text{p}}}^{\prime } }}{{r^{\prime 2} }}} \right] - \frac{\nu }{{k^{\prime } }}u_{{\text{p}}}^{\prime } - \frac{1}{\rho }\frac{\partial p}{{\partial \theta }}\frac{1}{{r^{\prime } }}\quad {\text{for}}\quad r_{{{\text{in}}}} \le r^{\prime } \le d^{\prime }$$

(1)

$$\frac{{\partial u_{f}^{^{\prime}} }}{{\partial t^{\prime } }} + u_{1} \left[ {\frac{1}{{r^{\prime } }}\frac{{\partial u_{f}^{^{\prime}} }}{{\partial r^{\prime } }} - \frac{{u_{f}^{^{\prime}} }}{{r^{^{\prime}2} }}} \right] = \nu \left[ {\frac{{\partial^{2} u_{f}^{^{\prime}} }}{{\partial r^{^{\prime}2} }} + \frac{1}{{r^{\prime } }}\frac{{\partial u_{f}^{^{\prime}} }}{{\partial r^{\prime } }} - \frac{{u_{f}^{^{\prime}} }}{{r^{^{\prime}2} }}} \right] - \frac{1}{\rho }\frac{\partial p}{{\partial \theta }}\frac{1}{{r^{\prime } }}\quad {\text{for}}\quad d^{\prime } \le r^{\prime } \le r_{0}$$

(2)

The initial and the condition at the surfaces for the problem under consideration are

$$\begin{aligned} & t^{\prime } \le 0:u_{{\text{p}}}^{^{\prime}} = u_{{\text{f}}}^{^{\prime}} = 0\quad {\text{for}}\quad r_{{{\text{in}}}} \le r^{\prime } \le r_{0} \\ & t^{\prime } > 0:\quad \left[ {\begin{array}{*{20}c} {u_{{\text{p}}}^{^{\prime}} = r_{{{\text{in}}}} \omega_{{{\text{in}}}} \;\;{\text{at}}\;\;r^{\prime } = r_{{{\text{in}}}} } \\ {u_{{\text{f}}}^{^{\prime}} = r_{0} \omega_{0} \;\;{\text{at}}\;\;r^{\prime } = r_{0} } \\ \end{array} } \right] \\ \end{aligned}$$

(3)

With the dimensional matching condition at the interface given as

$$t^{\prime } > 0:\left[ {\begin{array}{*{20}l} {u_{{\text{p}}}^{^{\prime}} = u_{{\text{f}}}^{^{\prime}} = u_{{\text{i}}}^{^{\prime}} } \\ {\nu_{{{\text{eff}}}} \left[ {\frac{{\partial u_{{\text{p}}}^{^{\prime}} }}{{\partial r^{\prime } }} - \frac{{u_{{\text{p}}}^{^{\prime}} }}{{r^{\prime } }}} \right] - \nu \left[ {\frac{{\partial u_{{\text{f}}}^{^{\prime}} }}{{\partial r^{\prime } }} - \frac{{u_{{\text{f}}}^{^{\prime}} }}{{r^{\prime } }}} \right] = \frac{\beta \nu }{{\sqrt {k^{\prime } } }}u_{{\text{p}}}^{^{\prime}} } \\ \end{array} } \right]\;\;{\text{at}}\;\;r^{\prime } = d^{\prime }$$

(4)

Using the following non-dimensional quantities:

$$\begin{aligned} R & = \frac{{r^{\prime } }}{{r_{{{\text{in}}}} }} ;t = \frac{{\nu t^{^{\prime}} }}{{r_{{{\text{in}}}}^{2} }} ; \gamma = \frac{{\nu_{{{\text{eff}}}} }}{\nu } ;{\text{Da}} = \frac{{r_{{{\text{in}}}}^{2} }}{{k^{\prime } }} ;U_{{\text{p}}} = U_{{\text{f}}} = \frac{{\left( {u_{{\text{p}}}^{^{\prime}} ; u_{{\text{f}}}^{^{\prime}} } \right)}}{{u_{0} }} ; \\ u_{0} & = - r_{{{\text{in}}}} \frac{\partial p}{{\partial \theta }}\frac{1}{\rho \nu }; d = \frac{{d^{\prime } }}{{r_{{{\text{in}}}} }} ; \lambda = \frac{{r_{0} }}{{r_{{{\text{in}}}} }} ; U_{{\text{i}}} = \frac{{u_{{{\text{in}}}}^{^{\prime}} }}{{U_{0} }}, S = \frac{{u_{{\text{i}}} r_{{{\text{in}}}} }}{\nu } \\ \end{aligned}$$

(5)

Using the following dimensionless parameters defined in Eq. (5) on Eqs. (1) to (4)

$$\frac{{\partial U_{{\text{p}}} }}{\partial t} = \gamma \frac{{\partial^{2} U_{{\text{p}}} }}{{\partial R^{2} }} + \left( {\gamma - S} \right)\frac{1}{R}\frac{{\partial U_{{\text{p}}} }}{\partial R} - \frac{{\left( {\gamma + S} \right)}}{{R^{2} }}U_{{\text{p}}} - \frac{{U_{{\text{p}}} }}{{{\text{Da}}}} + \frac{P}{R}\quad {\text{for}}\quad 1 \le R \le d$$

(6)

$$\frac{{\partial U_{{\text{f}}} }}{\partial t} = \frac{{\partial^{2} U_{{\text{f}}} }}{{\partial R^{2} }} + \left( {1 - S} \right)\frac{1}{R}\frac{{\partial U_{{\text{f}}} }}{\partial R} - \frac{{\left( {1 + S} \right)}}{{R^{2} }}U_{{\text{f}}} + \frac{P}{R}\quad {\text{for}}\quad d \le R \le \lambda$$

(7)

Subject to the following dimensionless initial and boundary conditions

$$\begin{aligned} t \le 0:\;\;U_{{\text{p}}} = U_{{\text{f}}} = 0\quad {\text{for}}\quad 1 \le R \le \lambda \hfill \\ t > 0:\;\;\left[ {\begin{array}{*{20}c} {U_{{\text{p}}} = A^{*} \;\;{\text{at}}\;\;R = 1} \\ {U_{{\text{f}}} = \omega \lambda \;\;{\text{at}}\;\;R = \lambda } \\ \end{array} } \right] \hfill \\ \end{aligned}$$

(8)

With the matching condition at the interface given as

$$t > 0:\;\;\left[ {\begin{array}{*{20}l} {U_{{\text{p}}} = U_{{\text{f}}} = U_{{\text{i}}} } \\ {\gamma \left[ {R\frac{\partial }{\partial R}\left( {\frac{{U_{{\text{p}}} }}{R}} \right)} \right] - \left[ {R\frac{\partial }{\partial R}\left( {\frac{{U_{{\text{f}}} }}{R}} \right)} \right] = \frac{\beta }{{\sqrt {{\text{Da}}} }}U_{{\text{p}}} } \\ \end{array} } \right] \;\;{\text{at}}\;\;R = d$$

(9)

The Laplace domain of Eqs. (6) to (9) can be obtained using the Laplace transform technique \(\overline{U}_{{\text{p}}} \left( {R,\vartheta } \right) = \int_{0}^{\infty } {U_{{\text{p}}} \left( {R,t} \right)e^{ - \vartheta t} {\text{d}}t} \;\;{\text{and}}\;\; \overline{U}_{{\text{f}}} \left( {R,\vartheta } \right) = \int_{0}^{\infty } {U_{{\text{f}}} \left( {R,t} \right)e^{ - \vartheta t} {\text{d}}t}\) (where \(\vartheta > 0\) is the Laplace parameter) subject to initial condition gives:

$$\frac{{{\text{d}}^{2} \overline{U}_{{\text{p}}} }}{{{\text{d}}R^{2} }} + \frac{1}{R}\left( {\gamma - S} \right)\frac{{{\text{d}}\overline{U}_{{\text{p}}} }}{{{\text{d}}R}} - \left( {\gamma + S} \right)\frac{{\overline{U}_{{\text{p}}} }}{{R^{2} }} - \frac{1}{\gamma }\left[ {\frac{1}{{{\text{Da}}}} + \vartheta } \right]\overline{U}_{{\text{p}}} = - \frac{P}{\gamma \vartheta R}$$

(10)

$$\frac{{{\text{d}}^{2} \overline{U}_{{\text{f}}} }}{{{\text{d}}R^{2} }} + \frac{1}{R}\left( {1 - S} \right)\frac{{{\text{d}}\overline{U}_{{\text{f}}} }}{{{\text{d}}R}} - \left( {1 + S} \right)\frac{{\overline{U}_{{\text{f}}} }}{{R^{2} }} - \vartheta \overline{U}_{{\text{f}}} = - \frac{P}{\vartheta R}$$

(11)

Subject to the following boundary condition

$$t > 0:\;\;\left[ {\begin{array}{*{20}c} {\bar{U}_{{\text{p}}} = \frac{{A^{*} }}{\vartheta }\;\;{\text{at}}\;\;R = 1} \\ {\bar{U}_{{\text{f}}} = \frac{{\omega \lambda }}{\vartheta }\;\;{\text{at}}\;\;R = \lambda } \\ \end{array} } \right]$$

(12)

With the matching condition at the interface given as

$$t > 0:\;\;\left[ {\begin{array}{*{20}l} {\overline{U}_{{\text{p}}} = \overline{U}_{{\text{f}}} = U_{{\text{i}}} } \\ {\gamma \left[ {\frac{{{\text{d}}\overline{U}_{{\text{p}}} }}{{{\text{d}}R}} - \frac{{\overline{U}_{{\text{p}}} }}{R}} \right] - \left[ {\frac{{{\text{d}}\overline{U}_{{\text{f}}} }}{{{\text{d}}R}} - \frac{{\overline{U}_{{\text{f}}} }}{R}} \right] = \frac{\beta }{{\sqrt {{\text{Da}}} }}U_{{\text{i}}} } \\ \end{array} } \right] \;\;{\text{at}}\;\;R = d$$

(13)

Equations (10) and (11) can be transform to Bessel function using the following equations;

$$\overline{U}_{{\text{p}}} = \overline{U}_{{{\text{ph}}}} R^{S/2} + \frac{P}{{\vartheta \left( {\frac{1}{{{\text{Da}}}} + \vartheta } \right)R}}\;\;{\text{and}}\;\;\overline{U}_{{\text{f}}} = \overline{U}_{{{\text{fh}}}} R^{S/2} + \frac{P}{{\vartheta^{2} R}}$$

(14)

Letting \(\gamma = 1\) and then applying Eq. (14) on Eqs. (10) and (11), the solutions in terms of the modified Bessel function obtained are given as:

$$\overline{U}_{{\text{p}}} = R^{S/2} \left[ {C_{5} I_{n} \left( {\eta R} \right) + C_{6} K_{n} \left( {\eta R} \right)} \right] + \frac{P}{{\vartheta \left( {\frac{1}{{{\text{Da}}}} + \vartheta } \right)R}}$$

(15)

$$\overline{U}_{{\text{f}}} = R^{S/2} \left[ {C_{7} I_{n} \left( {\sqrt \vartheta R} \right) + C_{8} K_{n} \left( {\sqrt \vartheta R} \right)} \right] + \frac{P}{{\vartheta^{2} R}}$$

(16)

where \(\eta = \sqrt {\left( {\frac{1}{Da} + \vartheta } \right)} , n = \left( {\frac{S}{2} + 1} \right)\) and *I*_{n}, *K*_{n} are the modified Bessel function of a first and second kind, respectively, of order *n*.

using Eq. (12) and on Eqs. (15) and (16) the constants \(C_{5} ,{ }C_{6} ,{ }C_{7} ,{ }C_{8} { }\) and *U*_{i} are obtained as

$$\begin{aligned} C_{5} & = a_{2} + U_{{\text{i}}} a_{3} ,\quad C_{6} = a_{4} - U_{{\text{i}}} a_{5} ,\quad C_{7} = a_{7} + U_{{\text{i}}} a_{8} ,\quad C_{8} = a_{9} - U_{{\text{i}}} a_{10} \\ U_{{\text{i}}} & = \frac{{ - a_{16} }}{{a_{17} }} \\ \end{aligned}$$

$$\begin{aligned} a_{1} & = d^{{\frac{S}{2} + 1}} \vartheta \left( {{\text{Da}}\vartheta + 1} \right) \left( {I_{n} \left( \eta \right)K_{n} \left( {d\eta } \right) - I_{n} \left( {d\eta } \right)K_{n} \left( {d\eta } \right)} \right), \\ a_{2} & = \frac{1}{{a_{1} }}\left[ {{\text{Da}}PK_{n} \left( \eta \right) + A^{*} d^{{\frac{S}{2} + 1}} K_{n} \left( {d\eta } \right) - {\text{Da}}Pd^{{\frac{S}{2} + 1}} K_{n} \left( {d\eta } \right) + A^{*} Da\vartheta d^{{\frac{S}{2} + 1}} K_{n} \left( {d\eta } \right)} \right], \\ a_{3} & = - \frac{{\left( {d\vartheta + {\text{Da}} d \vartheta^{2} } \right)K_{n} \left( \eta \right)}}{{a_{1} }}, \\ a_{4} & = - \frac{1}{{a_{1} }}\left[ {{\text{Da}}PI_{n} \left( \eta \right) + A^{*} d^{{\frac{S}{2} + 1}} I_{n} \left( {d\eta } \right) - {\text{Da}}Pd^{{\frac{S}{2} + 1}} I_{n} \left( {d\eta } \right) + A^{*} {\text{Da}}\vartheta d^{{\frac{S}{2} + 1}} I_{n} \left( {d\eta } \right)} \right], \\ a_{5} & = - \frac{{\left( {d\vartheta + {\text{Da }}d \vartheta^{2} } \right)I_{n} \left( \eta \right)}}{{a_{1} }} , \\ a_{6} & = \lambda^{{\frac{S}{2} + 1}} d^{{\frac{S}{2} + 1}} \vartheta \left( { I_{n} \left( {\lambda \sqrt \vartheta } \right)K_{n} \left( {d\sqrt \vartheta } \right) - I_{n} \left( {d\sqrt \vartheta } \right)K_{n} \left( {\lambda \sqrt \vartheta } \right)} \right), \\ a_{7} & = \frac{1}{{a_{6} }}\left[ {\lambda^{2} d^{{\frac{S}{2} + 1}} \vartheta wK_{n} \left( {d\sqrt \vartheta } \right) + \lambda^{{\frac{S}{2} + 1}} PK_{n} \left( {\lambda \sqrt \vartheta } \right) - Pd^{{\frac{S}{2} + 1}} K_{n} \left( {d\sqrt \vartheta } \right) } \right], \\ a_{8} & = - \frac{{\lambda^{{\frac{S}{2} + 1}} d\vartheta^{2} K_{n} \left( {\lambda \sqrt \vartheta } \right)}}{{a_{6} }}, \\ a_{9} & = \frac{1}{{a_{6} }}\left[ {Pd^{{\frac{S}{2} + 1}} I_{n} \left( {d\sqrt \vartheta } \right) - \lambda^{2} d^{{\frac{S}{2} + 1}} \vartheta wI_{n} \left( {d\sqrt \vartheta } \right) - \lambda^{{\frac{S}{2} + 1}} PI_{n} \left( {\lambda \sqrt \vartheta } \right) } \right] , \\ a_{10} & = - \frac{{\lambda^{{\frac{S}{2} + 1}} d\vartheta^{2} I_{n} \left( {\lambda \sqrt \vartheta } \right)}}{{a_{6} }}, \\ a_{11} & = \eta d^{n - 1} I_{n - 1} \left( {d\eta } \right) - 2d^{n - 2} I_{n} \left( {d\eta } \right) - \frac{\beta }{{\sqrt {{\text{Da}}} }}d^{n - 1} I_{n} \left( {d\eta } \right), \\ a_{12} & = \eta d^{n - 1} K_{n - 1} \left( {d\eta } \right) + 2d^{n - 2} K_{n} \left( {d\eta } \right) - \frac{\beta }{{\sqrt {{\text{Da}}} }}d^{n - 1} K_{n} \left( {d\eta } \right), \\ a_{13} & = 2d^{n - 2} I_{n} \left( {d\sqrt \vartheta } \right) - \vartheta d^{n - 1} I_{n - 1} \left( {d\sqrt \vartheta } \right), \\ a_{14} & = 2d^{n - 2} K_{n} \left( {d\sqrt \vartheta } \right) + \vartheta d^{n - 1} K_{n - 1} \left( {d\sqrt \vartheta } \right), \\ a_{15} & = \frac{2P}{{\vartheta^{2} d^{2} }} - \frac{2P}{{\vartheta \eta^{2} d^{2} }} - \frac{P\beta }{{\eta^{2} d\vartheta \sqrt {{\text{Da}}} }}. \\ \end{aligned}$$