Unsteady natural convection boundary layer heat transfer of fractional Maxwell viscoelastic fluid over a vertical plate

In classical study on viscoelastic fluids, researchers commonly ignored the effects of nonlinear convection and dealt only with the cases when the governing equations are linear. This paper presents a research on unsteady boundary layer natural convection heat transfer of Maxwell viscoelastic fluid over a vertical plate. The fractional boundary layer governing equations are firstly formulated and derived. From such derivation, the model constitutes nonlinear coupled equations with mixed time–space derivatives in the convection terms, which are solved by a newly developed finite difference method combined with an L1-algorithm. Results show that both the velocity and the temperature boundary layer manifest a short-term memory and basic relaxation time characteristics. For example, for specified velocity and temperature fractional derivative αα and ββ, the velocity profile has a maximum, which increases and goes from a far-field region to the vertical plate as a decreasing function of αα, implying a loss in velocity boundary layer thickness. But the values of such maxima behave conversely for ββ. More effects of the fractional order parameters αα and ββ on velocity and temperature fields are shown graphically and analyzed in detail.

[1]  H. A. Luther,et al.  Applied numerical methods , 1969 .

[2]  Corina Fetecau,et al.  Unsteady flow of a Maxwell fluid with fractional derivatives in a circular cylinder moving with a nonlinear velocity , 2014 .

[3]  D. Y. Song,et al.  Study on the constitutive equation with fractional derivative for the viscoelastic fluids – Modified Jeffreys model and its application , 1998 .

[4]  Constantin Fetecau,et al.  Flow of a viscoelastic fluid with the fractional Maxwell model between two side walls perpendicular to a plate , 2008, Appl. Math. Comput..

[5]  W. Glöckle,et al.  Fractional relaxation and the time-temperature superposition principle , 1994 .

[6]  Haitao Qi,et al.  Unsteady Rotating Flows of a Viscoelastic Fluid with the Fractional Maxwell Model Between Coaxial Cylinders , 2006 .

[7]  I. Podlubny.  Fractional differential equations , 1998 .

[8]  S. Ostrach.  An analysis of laminar free-convection flow and heat transfer about a flat plate parallel to the direction of the generating body force , 1953 .

[9]  P. Ganesan,et al.  Finite difference analysis of unsteady natural convection MHD flow past an inclined plate with variable surface heat and mass flux , 2004 .

[10]  H. R. Hicks,et al.  Numerical methods for the solution of partial difierential equations of fractional order , 2003 .

[11]  Constantin Fetecau,et al.  Unsteady flow of a generalized Maxwell fluid with fractional derivative due to a constantly accelerating plate , 2009, Comput. Math. Appl..

[12]  Youbing Yin,et al.  Oscillating flow of a viscoelastic fluid in a pipe with the fractional Maxwell model , 2006, Appl. Math. Comput..

[13]  Haitao Qi,et al.  Unsteady flow of viscoelastic fluid with fractional Maxwell model in a channel , 2007 .

[14]  Wen Chen,et al.  A fractional-order model on new experiments of linear viscoelastic creep of Hami Melon , 2013, Comput. Math. Appl..

[15]  Constantin Fetecau,et al.  Decay of potential vortex for a viscoelastic fluid with fractional Maxwell model , 2009 .

[16]  Exact Solutions for an Unsteady Flow of Viscoelastic Fluid in Cylindrical Domains Using the Fractional Maxwell Model , 2015 .

[17]  Xu Mingyu,et al.  Plane surface suddenly set in motion in a viscoelastic fluid with fractional Maxwell model , 2002 .

[18]  Shaowei Wang,et al.  Analytical solution of the transient electro-osmotic flow of a generalized fractional Maxwell fluid in a straight pipe with a circular cross-section , 2015 .

[19]  D. Pal,et al.  Hydromagnetic convective–radiative boundary layer flow of nanofluids induced by a non-linear vertical stretching/shrinking sheet with viscous–Ohmic dissipation , 2015 .

[20]  N. Heymans.  Hierarchical models for viscoelasticity: dynamic behaviour in the linear range , 1996 .

[21]  Dharmendra Tripathi,et al.  Peristaltic transport of fractional Maxwell fluids in uniform tubes: Applications in endoscopy , 2011, Comput. Math. Appl..

[22]  Tan Wen-chang,et al.  A note on unsteady flows of a viscoelastic fluid with the fractional Maxwell model between two parallel plates , 2003 .

[23]  C. Friedrich,et al.  Generalized Cole-Cole behavior and its rheological relevance , 1992 .

[24]  S. Das,et al.  Peristaltic flow of viscoelastic fluid with fractional Maxwell model through a channel , 2010, Appl. Math. Comput..

[25]  Tasawar Hayat,et al.  Periodic unidirectional flows of a viscoelastic fluid with the fractional Maxwell model , 2004, Appl. Math. Comput..

[26]  Fawang Liu,et al.  Stability and convergence of the difference methods for the space-time fractional advection-diffusion equation , 2007, Appl. Math. Comput..

[27]  C. Friedrich.  Relaxation and retardation functions of the Maxwell model with fractional derivatives , 1991 .