Steady mixed convection flow of Maxwell fluid over an exponentially stretching vertical surface with magnetic field and viscous dissipation

Abstract

The steady mixed convection flow and heat transfer from an exponentially stretching vertical surface in a quiescent Maxwell fluid in the presence of magnetic field, viscous dissipation and Joule heating have been studied. The stretching velocity, surface temperature and magnetic field are assumed to have specific exponential function forms for the existence of the local similarity solution. The coupled nonlinear ordinary differential equations governing the local similarity flow and heat transfer have been solved numerically by Chebyshev finite difference method. The influence of the buoyancy parameter, viscous dissipation, relaxation parameter of Maxwell fluid, magnetic field and Prandtl number on the flow and heat transfer has been considered in detail. The Nusselt number increases significantly with the Prandtl number, but the skin friction coefficient decreases. The Nusselt number slightly decreases with increasing viscous dissipation parameter, but the skin friction coefficient slightly increases. Maxwell fluid reduces both skin friction coefficient and Nusselt number, whereas buoyancy force enhances them.

Appendices

Appendix 1

Chebyshev polynomials have been used widely in CFD [39, 41, 42]. Eldabe et al. [42] have used Chebyshev finite-difference method (ChFDM) for the solution of the MHD flow and heat transfer over a porous stretching surface in an otherwise quiescent micropolar fluid. This method requires the definition of grid points and it is applied to satisfy the differential equations and the boundary conditions at these grid points. This method can be regarded as a non-uniform finite-difference scheme. The derivatives of the function F(X) at a point X _j is a linear combination from the values of the function F(X) at the Gauss–Lobatto points X _k  = cos(kπ/L), where $k = 0,\; 1, \ldots , \;L$, and j is an integer 0 ≤ j ≤ L.

The derivatives of the function F(X) at the points X _k are given by [39]

$$ F^{(n)} \left( {X_{k} } \right) = \mathop{\sum}\limits_{\begin{subarray}{c} j = 0\\ \end{subarray}}^{L}d_{k, j}^{\left( n \right)} F\left( {X_{j} } \right) ,\quad n = 1, \;\;2, \;\;3 $$

(13)

where

$$ d_{k, j}^{(1)} = \mathop{\frac{{4\Upgamma_{j} }}{L} \mathop{\sum} \limits_{n = 0}^{L}}\limits_{(n+1) odd} \mathop \sum \limits_{l = 0}^{n - 1} \frac{{nF_{n} }}{{c_{l} }} \bar{T}_{n} \left( {X_{j} } \right) \bar{T}_{l} \left( {X_{k} } \right) ,\quad k, \;\;j = 0,\;\;1, \ldots ,\;\;L, $$

$$ d_{k, j}^{(2)} = \mathop{\frac{{2\Upgamma_{j} }}{L}{\sum\limits_{n = 0}^{L}}}\limits_{(n+1)\,even} {\sum\limits_{l = 0}^{n - 2} } \frac{1}{{c_{l} }} \Upgamma_{n} n\left( {n^{2} - l^{2} } \right) \bar{T}_{n} \left( {X_{j} } \right) \bar{T}_{l} \left( {X_{k} } \right) , \quad k, \;\;j = 0,\;\;1, \ldots ,\;\;L, $$

$$ d_{k, j}^{(3)} = \mathop{\frac{{4\Upgamma_{j} }}{L} {\sum\limits_{n = 0}^{L}}}\limits_{{( {{n} + {\text{l}}}){\text{even }}}} \mathop{{\sum\limits_{l = 0}^{n - 2}}}\limits_{{\left( {{\text{i}} + {\text{l}}} \right){\text{ odd}}}} \mathop \sum \limits_{i = 0}^{l - 1} \frac{1}{{c_{i} c_{l} }} \Upgamma_{n} nl\left( {n^{2} - l^{2} } \right) \bar{T}_{n} \left( {X_{j} } \right) \bar{T}_{i} \left( {X_{k} } \right) $$

$$ k, \;j = 0, 1, \ldots , L ,\quad \Upgamma_{0} = \Upgamma_{1} = 1/2,\quad \Upgamma_{j} = 1 ,\quad {\text{for }}\,\, j = 1, \;2, \ldots , L - 1 . $$

(14)

For the present problem the domain is 0 ≤ η ≤ η _∞, where η _∞ is the edge of the boundary layer. The algebraic mapping

$$ \xi = 2\left( {\eta /\eta_{\infty } } \right) - 1 , $$

(15)

has been used to convert $\left[ {0, \eta_{\infty } } \right]$ to the computational domain $\left[ { - 1, 1} \right]$. The governing Eqs. (9) and (10) can be expressed as

$$ f^{\prime\prime\prime}\left( \xi \right) + \left( {\eta_{\infty } /2} \right)f\left( \xi \right)f^{\prime\prime}\left( \xi \right) - \eta_{\infty } \left( {f^{\prime}\left( \xi \right)} \right)^{2} + \left( {\lambda_{1} /8} \right)\eta_{\infty }^{3} \theta \left( \xi \right) - \left( {\eta_{\infty } /4} \right)Mf^{\prime}\left( \xi \right)\, - \lambda \left[ {\left( {f^{\prime}\left( \xi \right)} \right)^{2} \left\{ {4f^{\prime}\left( \xi \right) + 5\left( {1 + \xi } \right)f^{\prime\prime}\left( \xi \right) + \;\left( {1 + \xi } \right)^{2} f^{\prime\prime\prime}\left( \xi \right)} \right\}} + \left\{ {f\left( \xi \right) + \left( {1 + \xi } \right)f^{\prime}\left( \xi \right)} \right\}^{2} f^{\prime\prime\prime}\left( \xi \right) -\; 2f^{\prime}\left( \xi \right) {\left\{ {\left( {f\left( \xi \right) + \left( {1 + \xi } \right)f^{\prime}\left( \xi \right)} \right)\left( {3f^{\prime\prime}\left( \xi \right) +\; \left( {1 + \xi } \right)f^{\prime\prime\prime}\left( \xi \right)} \right)} \right\}} \right] = 0 , $$

(16)

$$ \theta^{''} \left( \xi \right) + \left( {\Pr \eta_{\infty } /2} \right)\left[ {f\left( \xi \right)\theta^{'} \left( \xi \right) - 4f^{'} \left( \xi \right)\theta \left( \xi \right)} \right] \hfill + 2 \Pr \,\,Gb \left[ {2\eta_{\infty }^{ - 2} \left( {f^{''} \left( \xi \right)} \right)^{2} +\; 2^{ - 1} M\left( {f^{'} \left( \xi \right)} \right)^{2} } \right] = 0 . \hfill \\ $$

(17)

The boundary conditions are given by

$$ f\left( { - 1} \right) = 0 , \, \, f^{'} \left( { - 1} \right) = 2^{ - 1} \eta_{\infty } , \theta \left( { - 1} \right) = 1 ,\;\; \hfill f^{'} \left( 1 \right) = \theta \left( 1 \right) = 0 . \hfill \\ $$

(18)

Using (13) and (14) in (16) and (17), we get the following Chebyshev finite difference equations

$$\mathop \sum \limits_{j = 0}^{L} d_{k, j}^{(3)} f\left( {\xi_{j} } \right) + 2^{ - 1} \eta_{\infty } f\left( {\xi_{k} } \right)\mathop \sum \limits_{j = 0}^{L} d_{k, j}^{(2)} f\left( {\xi_{j} } \right) + \lambda_{1} \left( {\eta_{\infty } /2} \right)^{3} \theta \left( {\xi_{k} } \right) - \left( {\eta_{\infty } /4} \right)M\mathop \sum \limits_{j = 0}^{L} d_{k, j}^{(1)} f\left( {\xi_{j} } \right) - \eta_{\infty } \left( {\mathop \sum \limits_{j = 0}^{L} d_{k, j}^{(1)} f\left( {\xi_{j} } \right)} \right)^{2} - \lambda \left[ {\left( {\mathop \sum \limits_{j = 0}^{L} d_{k, j}^{(1)} f\left( {\xi_{j} } \right)} \right)^{2} \times } \right. \hfill \left\{ {4\mathop \sum \limits_{j = 0}^{L} d_{k, j}^{(1)} f\left( {\xi_{j} } \right) + 5\left( {1 + \xi_{k} } \right)\mathop \sum \limits_{j = 0}^{L} d_{k, j}^{(2)} f\left( {\xi_{j} } \right) + \left( {1 + \xi_{k} } \right)^{2} \mathop \sum \limits_{j = 0}^{L} d_{k, j}^{(3)} f\left( {\xi_{j} } \right)} \right\} + \left\{ {f\left( {\xi_{k} } \right) + \left( {1 + \xi_{k} } \right)\mathop \sum \limits_{j = 0}^{L} d_{k, j}^{(1)} f\left( {\xi_{j} } \right)} \right\}^{2} \mathop \sum \limits_{j = 0}^{L} d_{k, j}^{(3)} f\left( {\xi_{j} } \right) - 2\mathop \sum \limits_{j = 0}^{L} d_{k, j}^{(1)} f\left( {\xi_{j} } \right)\left\{ {\left( {f\left( {\xi_{k} } \right) + \left( {1 + \xi_{k} } \right)\mathop \sum \limits_{j = 0}^{L} d_{k, j}^{(1)} f\left( {\xi_{j} } \right)} \right)} \right. \times \left. {\left. {\left( {3\mathop \sum \limits_{j = 0}^{L} d_{k, j}^{(2)} f\left( {\xi_{j} } \right) + \left( {1 + \xi_{k} } \right)\mathop \sum \limits_{j = 0}^{L} d_{k, j}^{(3)} f\left( {\xi_{j} } \right)} \right)} \right\}} \right] = 0 ,\quad k = 2, 3, \ldots, L - 1, $$

(19)

$$ \mathop \sum \limits_{j = 0}^{L} d_{k, j}^{(2)} \theta \left( {\xi_{j} } \right) + 2^{ - 1} \Pr \eta_{\infty } \left[ {f\left( {\xi_{k} } \right)\mathop \sum \limits_{j = 0}^{L} d_{k, j}^{(1)} \theta \left( {\xi_{j} } \right) - \;4\theta \left( {\xi_{k} } \right)\mathop \sum \limits_{j = 0}^{L} d_{k, j}^{(1)} f\left( {\xi_{j} } \right)} \right]\, + 2 \Pr \,\,\, Gb \left[ {2\eta_{\infty }^{ - 2} \left( {\mathop \sum \limits_{j = 0}^{L} d_{k, j}^{(2)} f\left( {\xi_{j} } \right)} \right)^{2} + \;2^{ - 1} M\left( {\mathop \sum \limits_{j = 0}^{L} d_{k, j}^{(1)} f\left( {\xi_{j} } \right)} \right)^{2} } \right] = 0 ,\quad k = 1, \;2, \ldots , \;L - 1 . $$

(20)

The boundary conditions given by (18) can be expressed as

$$ f = 0 , \;\; \mathop \sum \limits_{j = 0}^{L} d_{0, j}^{(1)} f\left( {\xi_{j} } \right) = 2^{ - 1} \eta_{\infty } ,\quad \theta = 1\quad \text{at}\;\xi = - 1,\;\;\mathop \sum \limits_{j = 0}^{L} d_{N, j}^{(1)} f\left( {\xi_{j} } \right) = 0 , \theta = 0\quad \text{at} \; \xi = 1 . $$

(21)

The system of nonlinear algebraic equations which contains 2L − 1 equations for the unknown f(ξ _i ), $i = 1, \;\;2, \ldots , \;\;L$ and θ(ξ _i ), $i = 1,\;\; 2, \ldots , \;\;L - 1$ has been linearised by Newton’s method and then solved by Varga algorithm [45]. Here L is taken as 50 and η _∞ = 15. Further increase in L and η _∞ changes the results by less than 0.5 %.

Appendix 2

The boundary layer equations can be derived for any viscoelastic fluid from Cauchy equations of motion. For steady two-dimensional flows, these equations can be expressed as [17]

$$ \rho \left( {u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y}} \right) = - \frac{\partial p}{\partial x} + \frac{{\partial \tau_{xx} }}{\partial x} + \frac{{\partial \tau_{xy} }}{\partial y}, $$

(22)

$$ \rho \left( {u\frac{\partial v}{\partial x} + v\frac{\partial v}{\partial y}} \right) = - \frac{\partial p}{\partial y} + \frac{{\partial \tau_{yx} }}{\partial x} + \frac{{\partial \tau_{yy} }}{\partial y}, $$

(23)

where ρ is the density of the fluid. The terms ∂τ _xx /∂x and ∂τ _yy /∂y are elastic terms and the terms ∂τ _xy /∂y and ∂τ _yx /∂x are viscous terms. Using the boundary layer theory [1], one can write

$$ u = o\left( 1 \right) ,\;\; v = o\left( \delta \right) ,\;\; x = o\left( 1 \right) ,\;\; y = o\left( \delta \right) . $$

(24)

For these orders of magnitudes, it can be shown that in Eqs. (22) and (23), the two elastic terms are of the same order as the two viscous terms, if we have

$$ \frac{{\tau_{xx} }}{\rho } = o\left( 1 \right) ,\;\;\frac{{\tau_{xy} }}{\rho } = o\left( \delta \right) ,\;\; \frac{{\tau_{yy} }}{\rho } = o\left( {\delta^{2} } \right) . $$

(25)

This implies that elastic effects should be considered in the boundary layer only for those viscoelastic fluids for which τ _xx is of an order larger than τ _xy and τ _yy . Not all viscoelastic fluid models can satisfy this condition. Assuming that the stress components of a UCM fluid satisfy the order estimates as given by Eq. (25), the equations of motion can be simplified to

$$ \rho \left( {u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y}} \right) = - \frac{\partial p}{\partial x} + \frac{{\partial \tau_{xx} }}{\partial x} + \frac{{\partial \tau_{xy} }}{\partial y} , $$

(26)

$$ \frac{\partial p}{\partial y} = 0 . $$

(27)

Equation (27) shows that, like Newtonian fluids, pressure variation across the boundary layer can be neglected for viscoelastic fluids [46]. Hence, for an incompressible fluid, the equations to be solved are

$$ \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0 , $$

(28)

$$ \rho \left( {u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y}} \right) = - \frac{\partial p}{\partial x} + \frac{{\partial \tau_{xx} }}{\partial x} + \frac{{\partial \tau_{xy} }}{\partial y} . $$

(29)

Equations (28) and (29) contain four unknowns: u, v, τ _xx and τ _xy , and pressure can be obtained from the potential flow. Therefore, a constitutive equation relating stress components to the deformation field is required to make the number of unknowns equal to the number of equations. For a Maxwell fluid, the stress tensor, d _ij , can be related to the deformation-rate tensor, d _ij , as [17, 18]

$$ \tau_{ij} + \lambda^{*} \frac{\delta }{\delta t}\tau_{ij} = 2 \mu d_{ij} , $$

(30)

where μ is the coefficient of viscosity and λ ^* is the relaxation time. The time derivative δ/δt appearing in the above equation is the upper-convected time derivative which is required to satisfy the requirements of continuous mechanics (i.e., material objectivity and frame indifference). This time derivative, when applied to the stress tensor, reads as follows [17, 18, 47]

$$ \frac{\delta }{\delta t}\tau_{ij} = \frac{D}{Dt}\tau_{ij} - L_{jk} \tau_{ik} - \tau_{kj} L_{ik} , $$

(31)

where L _ij is the velocity gradient tensor. After inserting the stress components for a Maxwell fluid [17, 18, 47] into Eq. (29), one obtains

$$ u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y} + \lambda^{*} \left( {u^{2} \frac{{\partial^{2} u}}{{\partial x^{2} }} + 2uv\frac{{\partial^{2} u}}{\partial x\partial y} + v^{2} \frac{{\partial^{2} u}}{{\partial y^{2} }}} \right) = - \frac{\partial p}{\partial x} + \nu \frac{{\partial^{2} u}}{{\partial y^{2} }} . $$

(32)

Equation (32) along with Eq. (28) is the boundary layer equation for a UCM fluid for a two-dimensional flow. However, to solve any problem, relevant boundary conditions are required. For our problem, boundary conditions are given in Eq. (7). The source terms due to the magnetic field, buoyancy force and pressure can easily be added to Eq. (32). In a similar manner, Eq. (6) can be obtained.