Heat transfer analysis of Walters’-B fluid with Newtonian heating through an oscillating vertical plate by using fractional Caputo–Fabrizio derivatives

  • Muhammad AbdullahEmail author
  • Asma Rashid Butt
  • Nauman Raza


This paper presents a study of Walters’-B fluid with Caputo–Fabrizio fractional derivatives through an infinitely long oscillating vertical plate by Newtonian heating under the action of transverse magnetic field. The fractional calculus approach is employed to obtain a system of fractional partial differential equations. The governing equations of momentum and energy are converted first into dimensionless form and then solved by employing Laplace transformation. The Laplace inverse transform has been evaluated both analytically and numerically. The graphical illustrations represent the behavior of material parameters on the solutions. A comparison between exact and numerical solutions is presented in tabular and graphical form. The variation in Nusselt number with the change in fractional and physical parameters is also presented. The velocity and temperature of the fluid decreases with the enhancement in the fractional parameter for small values of time, and it has the opposite behavior for greater values of time.


Walter’s-B fluid Newtonian heating Magnetic field Stehfest’s algorithm Caputo–Fabrizio derivatives 


\(u \)

Fluid velocity, [\(\mbox{m}\,\mbox{s}^{ - 1}\)]

\(T \)

Fluid temperature, [K]

\(g \)

Gravitational acceleration, [\(\mbox{m}\,\mbox{s}^{ - 2}\)]

\(c_{p} \)

Specific heat at a constant pressure, [\(\mbox{J}\,\mbox{kg}^{ - 1}\,\mbox{K}^{ - 1}\)]

\(T_{\infty } \)

Temperature of the fluid away from the plate, [K]

\(\mathit{Gr} \)

Thermal Grashof number, [\(\beta T_{w}\)]

\(k \)

Fluid thermal conductivity, [\(\mbox{W}\,\mbox{m}^{ - 2}\,\mbox{K}^{ - 1}\)]

\(\mathit{Pr} \)

Prandtl number, [\(= \mu c_{p}/k\)]

\(q \)

Laplace transforms parameter, [–]

\(h \)

Heat transfer coefficient, [\(\mbox{W}\,\mbox{m}^{ - 2}\,\mbox{K}^{ - 1}\)]

\(\mu \)

Dynamic viscosity, [\(\mbox{kg}\,\mbox{m}^{ - 1}\,\mbox{s}^{ - 1}\)]

\(\gamma _{T} \)

Coefficient of volumetric thermal expansion, [\(\mbox{K}^{ - 1}\)]

\(\rho \)

Fluid density, [\(\mbox{kg}\,\mbox{m}\,\mbox{s}^{ - 3}\)]

\(\nu \)

Kinematics viscosity of fluid, [\(\mbox{m}^{2}\,\mbox{s}^{ - 1}\)]

\(\beta \)

Walter’s-B fluid parameter


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Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  • Muhammad Abdullah
    • 1
    Email author
  • Asma Rashid Butt
    • 1
  • Nauman Raza
    • 2
  1. 1.Department of MathematicsUniversity of Engineering & TechnologyLahorePakistan
  2. 2.Department of MathematicsUniversity of the PunjabLahorePakistan

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