With the Navier-Stokes Equation in Cartesian form (in absence of body forces), Laplace Transforms provides a simple approach towards solving the unsteady flow of a viscous incompressible fluid over a suddenly accelerated flat plate. On comparing the results between Laplace Transforms and similarity methods, it reveals that Laplace Transforms is simple and effective.

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International Journal of Engineering & Technology, 7 (3.6) (2018) 267-269

International Journal of Engineering & Technology

Website: www.sciencepubco.com/index.php/IJET

Research paper

Exact Solution for Unsteady Flow of Viscous Incompressible

Fluid Over a Suddenly Accelerated Flat Plate (Stokes' First

Problem) Using Laplace Transforms

1Department Of Mathematics, Shillong College, Shillong , India.

Abstract

With the Navier-Stokes Equation in Cartesian form (in absence of body forces), Laplace Transforms provides a simple approach towards

solving the unsteady flow of a viscous incompressible fluid over a suddenly accelerated flat plate. On comparing the results between

Laplace Transforms and similarity methods, it reveals that Laplace Transforms is simple and effective.

1. Introduction

The problem of unsteady flow of viscous incompressible fluid

over a Suddenly accelerated flat plate was first solved by

G.G.Stokes in the year 1856 in his famous treatment of the

pendulum. Later in the year 1911, Lord Rayleigh also treated this

flow and it often called the Rayleigh problem in the Literature. In

many cases, this Stokes' First Problem is also known as 'the start-

up' flow problem i.e problem related with motion from rest.

The Laplace Transform or Laplace Transformation is a method for

solving many differential operations arising in Physics and

Engineering. By the way, the Laplace Transformation is just one

of many Integral transform in general use. Conceptually and

computationally, it is probably the simplest. Indeed, if one can

understand the Laplace Transform, then he or she will find it

much easier to pick up other transformations as needed.

The Laplace Transform is defined by

( ) = { } = 

   (1)

2. Some Basic Terminologies

1. Unsteady flow: The flow is said to be unsteady when

the conditions and fluid property (,say) at any point of

the flow field change with regard to the time. For

example, Water being pumped through a fixed system at

an increasing rate represents Unsteady flow.

Mathematically, 



2. 2. Viscosity: Each element of a fluid experiences stress

exerted on it by other elements of the fluid which

surround it. The stress at each part of the surface of the

element is resolved into two components: Normal and

Tangential to the surface( see figure 1.1), known as

pressure and shearing stress respectively. Pressure are

exerted whether the fluid is moving or at rest, but shear

stress occurs only in moving fluids. This feature is the

one which enables fluid to distinguished from solids.

The property which give rise to shear stress is called

Viscosity. Viscosity arise when there is a relative motion

between different fluid layers. It possessed by all real

fluids.

Viscous fluid is defined as the fluid where normal as

well as shearing stresses exist .For example, Syrup and

Heavy oil are treated as viscous fluids.

Figure 1.1

3. Incompressible fluid : The compressibility of a fluid is

defined as the variation of its density, with the variation

of pressure. For example, LPG, Nitrogen gas, etc.

Therefore, a fluid is said to be incompressible if it

requires a large variation in pressure to produce some

variation in density. For example, water, air(to some

extent),etc.

4. Newtonian fluids: The fluids in which the stress

components (Shearing stress,) are linear functions of

rate of strain components (rate of deformation, 

)

are termed as Newtonian fluids. For example, Real

fluids and Ideal fluids.

Mathematically, 



[ the constant of proportionality (or, Coefficient of

viscosity or, Dynamic Viscosity or, Viscosity]

3. Stokes' First Problem(Start-Up Flow

Problem)

Stokes' first problem is a fundamental solution in fluid dynamics

which represents one of the few exact solution to the Navier-

International Journal of Engineering & Technology

Stokes equations. Consider a Cartesian coordinate system of an

infinite long flat plate extending to a large distances in the and

-directions. Let there be an incompressible viscous fluid

occupying the half plane (i.   ). At , the

plate is suddenly set in motion at a constant velocity in the

direction.

This generates a one-dimensional parallel flow near the plate as

shown in figure 1.2

Figure 1.2

i.e  , ,

  (2)

(i.e the only non-zero velocity component will be

which will be

a function of )

The Continuity Equation reduces to



=0, so that = ( , ) (3)

Since the plate is situated in an infinite fluid, the Pressure must be

constant everywhere (i.e. Pressure (  is independent of



 

Hence, the Navier-Stokes equations in absence of body forces

reduces to



=

 (4 )

Which is to be solved under the following Initial condition and

Boundary Conditions of the problem:

when for all (5a)

  

 when (5b)

As we have the governing equation, Initial condition and

Boundary conditions ; therefore, the problem is well-posed. We

utilize the Laplace Transform method to reduce the two variables

into a single variable i.e transferring partial differential equation

into ordinary differential equation.

On using Laplace Transform technique, equation (4) and the

Boundary Conditions takes the following form:



 







 (6)

Where 

 

 

 



and



 =  = 

  



 

(on Integrating by parts)

or,

 = 0 + 



=  

=  0 = 

The general equation of (6) is



 

 (7 )

Since  must be bounded as , we must have

 also bounded as . It follows that we must choose

provided .

Therefore (7) reduces to



 (8)

Subjecting to boundary conditions , we have



So , equation (8) takes the form



 (9)

On taking inverse Laplace Transform in (9), the velocity profile

 is given by

     



 



Therefore,

0 erfc (

 

or,   (10)

(where

 )

Equation (10) can be written as :

 

 

(11)

On using General Leibnitz 's Rule ,

We have from (11),





 [ 



 ]

or, 

 

 

 

 







 

(on replacing 

 )

or, 

 0 

  









or, 

 



 



Therefore,



  





 

 

 (12)

Finally, it is of interest to determine the drag at the plate.

Therefore the Skin friction (the Shearing Stress at the plate) is

given by  where



  

 

 (using (12))

so, 

 (13)

From (13), We see that at initial point (i. e at ), the wall

Shear Stress is infinite and it decreases to Zero in proportion to

It is also proportional to . It means here that a large force

is needed to set the Fluid in motion.

and Coefficient of Skin friction











 .

4. Conclusion

In this connection, we successfully applied Laplace Transform to

solve the Stokes' first problem (A Newtonian Fluid Problem). The

result is identical to the one given in Literature. It gives a simple

International Journal of Engineering & Technology

and a powerful mathematical tool. This Result reveals that the

method is simple and effective.

References

[1] Schlichting H & Gersten K, Boundary Layer Theory (Springer), 8th

Edition, (2016) .

[2] Kleinstreuer C, Engineering Fluid Dynamics (An Interdisciplinary

Systems Approach, Cambridge University Press, (1997).

[3] Achenson DJ, Elementary Fluid Dynamics, Oxford University

Press, (2009), pp.26-40.

[4] Raisinghania MD, Fluid Dynamics, S.Chand and Company Ltd.,

(2005), pp.717 -720.

[5] Swarup S, Fluid Dynamics, Krishna Prakashan Media(P) Ltd,

(2009), pp.570 -572

[6] Raisinghania MD, Saxena HC & Dass HK, Integral

Transforms(Laplace & Fourier Transforms), S.Chand and

Company Ltd., (2003).

[7] Vasishtha AR & Gupta RK, Integral Transforms, Krishna

Prakashan Media(P) Ltd. ,Meerut, (2008).

  • Clement Kleinstreuer Clement Kleinstreuer

This text provides a thorough treatment of the fundamental principles of fluid mechanics and convection heat transfer and shows how to apply the principles to a wide variety of fluid flow problems. The focus is on incompressible viscous flows with special applications to non-Newtonian fluid flows, turbulent flows, and free-forced convection flows. A special feature of the text is its coverage of generalized mass, momentum, and heat transfer equations, Cartesian tensor manipulations, scale analyses, mathematical modeling techniques, and practical solution methods. The final chapter is unique in its case-study approach, applying general modeling principles to analyze nonisothermal flow systems found in a wide range of engineering disciplines. The author provides numerous end-of-chapter problems, solutions, and mathematical aids to enhance the reader's understanding and problem-solving skills.

  • S Swarup
  • Fluid Dynamics

Swarup S, Fluid Dynamics, Krishna Prakashan Media(P) Ltd, (2009), pp.570-572

  • M D Raisinghania
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Raisinghania MD, Saxena HC & Dass HK, Integral Transforms(Laplace & Fourier Transforms), S.Chand and Company Ltd., (2003).

  • A R Vasishtha
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  • Integral Transforms

Vasishtha AR & Gupta RK, Integral Transforms, Krishna Prakashan Media(P) Ltd.,Meerut, (2008).