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Exact solutions for the unsteady flow of a Burger’s fluid in a duct induced by time-dependent prescribed volume flow rate

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Abstract

In this investigation, some unsteady flows in a circular duct have been studied. The fluid obeys viscoelastic non-Newtonian model with the Burgers’ constitutive equation and all fluid properties are constant. The flows in a duct are due to the prescribed arbitrary time dependent inlet volume flow rates. Four types of flow situations are considered. The governing equations are first developed and then solved using Laplace transform technique. Results indicate the strong effect of Burgers’ fluid parameter on the velocity fields and pressure gradients.

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Abbreviations

\(\vec{A}\) :

Rivlin-Ericksen tensor

a p :

constant acceleration

H(t):

Heaviside unit step function

\(\vec{I}\) :

identity tensor

J0, J1:

Bessel functions of zeroth and first order

\(\vec{k}\) :

unit vector in the z-coordinate

p :

static fluid pressure

Q(t):

inlet volume flow rate varying with time

r :

coordinates along the radius direction

R :

radius of circular duct

s :

new coordinate after taking Laplace Transform

t :

time

t0, t1, t2:

time period of acceleration, constant velocity and deceleration, respectively

\(\vec{T}\) :

stress tensor

u :

velocity in z direction

u p (t):

average inlet velocity

U p :

constant inlet piston velocity

\(\vec{V}\) :

velocity vector

z :

coordinate along the centerline of the circular duct

λ, μ, θ, β:

material properties

ϕ:

coordinate along the circumferential direction

ν:

kinematic viscosity coefficient

ϑ:

frequency

ρ:

fluid density

τ rr :

shear stress acting on the r plane toward r direction

τ rz :

shear stress acting on the r plane toward z direction

τ zz :

shear stress acting on the z plane toward z direction

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Authors and Affiliations

  1. Department of Industrial Engineering and Management, I-Shou University, No. 1, Sec. 1, Syuecheng Rd., Dashu Township, Kaohsiung County, Taiwan, ROC, 84041

    Chun-I Chen

  2. Department of Mathematics, Quaid-i-Azam University, 45320, Islamabad, Pakistan

    Tasawar Hayat

  3. Department of Mechanical and Automation Engineering, I-Shou University, No. 1, Sec. 1, Syuecheng Rd., Dashu Township, Kaohsiung County, 84041, Taiwan, ROC

    Jiann-Lin Chen

Authors
  1. Chun-I Chen
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  2. Tasawar Hayat
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  3. Jiann-Lin Chen
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Correspondence to Chun-I Chen.

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Chen, CI., Hayat, T. & Chen, JL. Exact solutions for the unsteady flow of a Burger’s fluid in a duct induced by time-dependent prescribed volume flow rate. Heat Mass Transfer 43, 85–90 (2006). https://doi.org/10.1007/s00231-006-0092-z

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