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Unsteady natural convection flow of a dusty non-Newtonian Casson fluid along a vertical wavy plate: numerical approach

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Abstract

The present contribution provides a numerical treatment for the flow of unsteady natural convection of non-Newtonian Casson fluid model loaded with dusty particles over a vertical wavy plate. It is assumed that the fluid is incompressible and the dusty particles are considered to be spheres with the same size. Via the help of primitive variable formulation, the dimensionless boundary layer governing equations are reduced into a convenient coordinate system. The transformed resulting system of governing equations is tackled numerically employing the fully implicit finite difference method. The impact of various controlling physical parameters such as the amplitude of the wavy plate, the dimensionless time and fluid–particle interaction parameters on the velocity and temperature of both fluid and particle phase is analyzed. It is found that the magnitude of the velocity of both fluid and particle phase diminishes with increasing amplitude of the wavy plate parameter, while an opposite behavior is observed on temperature distributions as the amplitude of the wavy plate parameter rises. A significant impact is noticed for the heat transfer rate with enhancement values of fluid–particle interaction parameter. The numerical solutions are compared with the available data in the literature. Quantitative comparison illustrates good compatibility between the current and previous results.

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Abbreviations

\({\hat{a}}\) :

Amplitude of the wavy surface

a :

Dimensionless amplitude of the wavy surface

C :

Concentration of fluid

\(C_{\mathrm{{fx}}}\) :

Local skin-friction coefficient

\(c_{\mathrm{p}}\) :

Specific heat at constant pressure

\(c_s\) :

Specific heat of particle phase

D :

Mass diffusivity of fluid

\(D_\rho \) :

Mass concentration of the dust particle

Ec :

Eckert number

g :

Acceleration of gravity vector

Gr :

Grashof number

k :

Thermal conductivity of fluid

l :

Characteristics of wavy length

n :

Power index

N :

Buoyancy ratio parameter

Nu :

Local Nusselt number

\({\hat{P}}\) :

Pressure

P :

Dimensionless pressure

Pr :

Prandtl number

\(P_{\mathrm{{y}}}\) :

Yield stress of fluid

Sc :

Schmidt number

Sh :

Local Sherwood number

\({\hat{t}}\) :

Time

t :

Dimensionless time

T :

Temperature

(\({\hat{u}}, {\hat{v}}\)):

Velocity components

(u, v):

Dimensionless velocity components

(\({\hat{x}},{\hat{y}}\)):

Cartesian coordinates

(x, y):

Dimensionless Cartesian coordinates

\(\mu \) :

Dynamic viscosity

\(\mu _{\mathrm{{B}}}\) :

Plastic dynamic viscosity

\(\beta \) :

Thermal expansion coefficient

\(\beta ^{*}\) :

Concentration expansion coefficient

\(\tau _{\mathrm{{m}}}\) :

Velocity relaxation time

\(\tau _{\mathrm{{T}}}\) :

Thermal relaxation time

\(\rho \) :

Fluid density

\(\theta \) :

Dimensionless temperature

\(\nu \) :

Kinematic viscosity

\(\gamma \) :

Non-Newtonian Casson parameter

\(\Gamma \) :

Specific heat ratio of the mixture

\(\epsilon \) :

Successive under-relaxation parameter

\(\phi \) :

Dimensionless concentration of fluid

\({\hat{\delta }}\) :

Wavy surface coordinate

\(\delta \) :

Dimensionless wavy surface coordinate

w :

Conditions at the surface

\(\infty \) :

Conditions far away from the surface

p :

Particle phase

f :

Fluid

l, k :

Grid points

m :

Time step

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Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Assiut University, Assiut, 71515, Egypt

    F. M. Hady

  2. Mathematics Department, Faculty of Science, South Valley University, Qena, 83523, Egypt

    A. Mahdy, R. A. Mohamed, S. E. Ahmed & Omima A. Abo-zaid

  3. Department of Mathematics, Faculty of Science, King Khalid University, Abha, Saudi Arabia

    S. E. Ahmed

Authors
  1. F. M. Hady
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  2. A. Mahdy
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  3. R. A. Mohamed
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  4. S. E. Ahmed
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  5. Omima A. Abo-zaid
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Correspondence to Omima A. Abo-zaid.

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Technical Editor: Cezar Negrao, Ph.D.

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Hady, F.M., Mahdy, A., Mohamed, R.A. et al. Unsteady natural convection flow of a dusty non-Newtonian Casson fluid along a vertical wavy plate: numerical approach. J Braz. Soc. Mech. Sci. Eng. 41, 472 (2019). https://doi.org/10.1007/s40430-019-1966-6

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  • DOI: https://doi.org/10.1007/s40430-019-1966-6

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