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On unsteady flow of an elastico-viscous fluid past an infinite plate with variable suction

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Abstract

Approximate solutions of the Navier-Stokes equations are derived through the Laplace transform for two dimensional, incompressible, elastico-viscous flow past a flat porous plate. The flow is assumed to be independent of the distance parallel to the plate. General formulae for the velocity distribution, skin friction and displacement thickness as functions of the given free stream velocity and suction velocity are obtained.

The response of skin friction to the impulsive perturbations in the stream and suction velocities is studied. It is found that the order of singularity in the skin friction at t=0 increases due to the elastic property of the fluid in the impulsive case. When the stream is accelerated the skin friction still anticipates the velocity but the time of anticipation is reduced from 1/4 to (1/4) (1—k), where k is the elastic parameter of the fluid. It is found that in general the resistance of the elastico-viscous fluids to an impulsive increase in the stream velocity is greater than the viscous fluids, the elasticoviscous fluids also reach the steady state earlier than the viscous fluids.

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Abbreviations

\(C\surd \overline {\omega t}\) :

Fresnal's cosine integral of arg. \(\surd \overline {\omega t}\)

e (1) ik , e (1)ik :

covariant and contravariant components of the rate of strain tensor

f(t):

an arbitrary function of t

\(\bar f\)(p):

Laplace transform of f(t), in this context p is the parameter of the transform

F(t):

an arbitrary function of time representing the variable part of the superimposed velocity in the free stream

G(t):

an arbitrary function representing the variable part of the superimposed suction velocity

\(\bar F(p), \bar G(p)\)(p):

Laplace transforms of F(t) and G(t)

g ik :

metric tensor

H(t):

Heaviside step function

i, j, k, m, r :

indices

k 0 :

coefficient of the elasticity of the fluid

k :

a nondimensional parameter of the elasticity of the fluid

N(τ):

distribution function of relaxation time τ

p, p′ :

isotropic pressure

p ik :

stress tensor

p′ ik :

pressure tensor

p 1 :

undisturbed skin friction

p 0 :

skin friction at any time after disturbation

p s :

steady state skin friction

p xy :

shearing stress

\(S\surd \overline {\omega t}\) :

Fresnel's sine integral of arg. \(\surd \overline {\omega t}\)

t :

non-dimensional time

t′ :

time

t c :

characteristic time

u :

nondimensional velocity of the fluid parallal to the plate

u′ :

velocity of the fluid parallal to the plate

U(t′):

free stream velocity

U 0 :

constant free stream velocity

V′ :

velocity of the fluid perpendicular to the plate

V 0 :

suction velocity

w :

frequency

x :

non-dimensional co-ordinate in the direction of the free stream flow parallal to the plate

x′ :

co-ordinate in the direction of the flow

y :

non-dimensional co-ordinate perpendicular to the plate

y′ :

co-ordinate perpendicular to the plate

δ(t):

Dirac delta function

η 0 :

limiting viscosity at small rates of shear

λ :

a constant

ν :

kinematic viscosity

ρ′ :

density of the fluid

τ :

relaxation time

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Authors and Affiliations

  1. Louisiana State University in New Orleans, 70122, New Orleans, Louisiana, USA

    P. Puri

Authors
  1. P. Puri
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Puri, P. On unsteady flow of an elastico-viscous fluid past an infinite plate with variable suction. Appl. Sci. Res. 28, 111–133 (1973). https://doi.org/10.1007/BF00413061

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  • DOI: https://doi.org/10.1007/BF00413061

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