Consider an unsteady axisymmetric boundary layer flow of an incompressible viscous fluid along a horizontal cylinder which is considered to be continuously stretching. The cylinder is contracting or expanding according to the relation \( a(t)={a}_0\sqrt{1-\beta t} \), where *a*(*t*) is the radius of the cylinder at any time *t*, *a*_{0} is the initial value of the cylinder radius, and *β* is a constant which indicates to contraction (*β* > 0) or expansion (*β* < 0).

The stretching time-dependent velocity of the surface of the cylinder is assumed to be \( {U}_w\left(x,t\right)=\frac{4\ \nu\ {U}_0x}{a^2(t)} \), and the fluid is assumed to move along the axial direction *x* while the radial coordinate *r* is perpendicular to the axis of the cylinder. The temperature of the cylinder surface is assumed to be time dependent in the form \( {T}_w\left(x,t\right)={T}_{\infty }+\frac{a_0\ {T}_0x}{a(t)} \). Figure 1 shows the considered model.

The governing equations [13, 14] are:

$$ \frac{\partial }{\ \partial\ x}\left(r\ u\right)+\frac{\partial }{\partial\ r}\left(r\ v\right)=0 $$

(1)

$$ \frac{\partial\ u}{\partial\ t}+u\ \frac{\partial\ u}{\partial\ x}+v\ \frac{\partial\ u}{\partial\ r}=\frac{\nu }{r}\ \frac{\partial }{\partial\ r}\left(r\ \frac{\partial\ u}{\partial\ r}\right) $$

(2)

$$ \frac{\partial\ T}{\partial\ t}+u\ \frac{\partial\ T}{\partial\ x}+v\ \frac{\partial\ T}{\partial\ r}=\frac{\alpha }{r}\ \frac{\partial }{\partial\ r}\left(r\ \frac{\partial\ u}{\partial\ r}\right)-\frac{\alpha }{\kappa }\ \frac{1}{r}\frac{\partial }{\partial\ r}\left(r\ {q}_r\right) $$

(3)

the boundary conditions are:

$$ u={U}_w\left(x,t\right),v=\frac{a_{0\ V}}{a(t)},T={T}_w\left(x,t\right),\kern0.5em \mathrm{at}\ r=a(t)\kern0.5em $$

(4)

$$ u=0,T={T}_{\infty },\kern0.5em \mathrm{as}\ r\to \infty $$

(5)

where *u* and *v* are the components of the fluid velocity along *x* axis and *r* axis respectively. *ν* is the fluid kinematic viscosity, *α* is the fluid thermal diffusivity, *κ* is the thermal conductivity, *V* is the constant of suction (*V* < 0) or injection (*V* > 0), and \( {q}_r=-\frac{4\ \sigma }{3\ {\alpha}^{\ast }}\ \frac{\partial\ {T}^4}{\partial\ r} \) is the radiation heat flux such that *σ* and *α*^{∗} are the Stefan-Boltzman constant and the mean absorption coefficient, respectively.

The temperature differences within the flow are assumed to be sufficiently small such that *T*^{4} is expressed as a linear function of *T*; hence, the Taylor expansion of *T*^{4} about *T*_{∞}, neglecting higher order terms, is given by

$$ {T}^4=4\ {T}_{\infty}^3T-3{T}_{\infty}^4 $$

(6)

Considering the similarity transformations

$$ \eta =\frac{r^2}{a^2(t)}-1,u=\frac{U_w}{U_0}\ {f}^{\prime}\left(\eta \right),v=-\frac{2\ \nu }{r}\ f\left(\eta \right),\theta =\frac{T-{T}_{\infty }}{T_w-{T}_{\infty }} $$

(7)

along with Eq. (6), the system of partial differential Eqs. (1)–(3) with the boundary conditions (4)–(5) is transformed into the following system of ordinary differential equation

$$ \left(1+\eta \right){f}^{\prime \prime \prime }+{f}^{\prime \prime }+f{f}^{\prime \prime }-f{\prime}^2-A\left[\left(1+\eta \right){f}^{\prime \prime }+{f}^{\prime}\right]=0 $$

(8)

$$ \frac{1}{\mathit{\Pr}}\left[\left(1+{N}_R\right)\left(\left(1+\eta \right){\theta}^{\prime \prime }+{\theta}^{\prime}\right)\right]+f{\theta}^{\prime }-{f}^{\prime}\theta -A\left[\left(1+\eta \right){\theta}^{\prime }+\frac{\theta }{2}\right]=0 $$

(9)

subject to the boundary conditions:

$$ f(0)=-{f}_0,{f}^{\prime }(0)={U}_0,\theta (0)=1 $$

(10)

$$ {f}^{\prime}\left(\infty \right)\to 0,\theta \left(\infty \right)\to 0 $$

(11)

while primes denote differentiation with respect to *η*, \( A=\frac{\beta\ {a}_0^2}{4\ \nu } \) is the unsteadiness parameter. Where the negative values of *A* correspond to contraction and the positive values of *A* correspond to expansion, \( \mathit{\Pr}=\frac{\nu }{\alpha } \) is the Prandtl number, \( {f}_0=\frac{a_0V}{2\ \nu } \) is the suction (*f*_{0} < 0) or injection (*f*_{0} > 0) parameter, and \( {N}_R=\frac{16\ \sigma {T}_{\infty}^3}{3\ \kappa\ {\alpha}^{\ast }} \) is the thermal radiation parameter.

Two important physical quantities of interest are the skin friction, *C*_{f}, and the local Nusselt number, *Nu*_{x}, which are defined as:

$$ {C}_f=\frac{2\ {\tau}_w}{\rho\ {U}_w^2},N{u}_x=\frac{x\ \left({q}_w+{q}_r\right)}{\kappa \left({T}_w-{T}_{\infty}\right)} $$

(12)

where \( {\tau}_w=\mu {\left(\frac{\partial u}{\partial r}\right)}_{r=a(t)} \) is the cylinder surface sheer stress and \( {q}_w=-\kappa {\left(\frac{\partial T}{\partial r}\right)}_{r=a(t)} \) is the cylinder surface heat flux. Using the dimensionless similarity transformations (7), we get:

$$ \frac{U_0}{2}{C}_f\sqrt{U_0\ R{e}_x}={f}^{\prime \prime }(0),N{u}_x\ \sqrt{U_0/R{e}_x}=-\left(1+{N}_R\right){\theta}^{\prime }(0) $$

(13)

where \( R{e}_x=\frac{x\ {U}_w}{\nu } \) is the Reynolds number.