Four types of unsteady flows of a viscous fluid over a plane wall bounded by two side walls are considered. They are flow caused by impulsive motion of a plate, flow due to oscillation of a plate, flow induced by constantly accelerating plate, and flow imposed by a plate that applies a constant tangential stress to the fluid. In order to solve these problems, the sine and cosine transformations are used, and exact solutions for the velocity distribution are found in terms of definite integrals. The cases for which the time goes to infinity and the distance between two side walls goes to infinity are compared with the cases for flows over a plane wall in the absence of the side walls. These provide to know the required time to attain the steady-state and what is the distance between the side walls for which the measured value of the velocity or the stress would be unaffected by the presence of the side walls.

The flow of a viscous fluid over a plane wall with different boundary and initial conditions has been investigated by many authors. The fluid over a plane wall is initially at rest and is set in motion under the application of a body force considered by Erdoğan [

The aim of this paper is to investigate the effects of the side walls on the unsteady flow over a plane wall. Four illustrative examples are considered. They are flow due to the impulsive motion of a plate, flow caused by oscillation of a plate, flow due to the constant acceleration of the plate and the flow generated by tangential stress on the fluid. It is well known that the solution of the governing equation for the flow of second-grade fluid over a plane wall obtained by the Laplace transform method is failed [

The first problem considered is the flow due to impulsive motion of a plate. The velocity and the flux across a plane normal to the flow are given in terms of definite integrals. In the limiting cases when time goes to infinity, the solution reduces to the steady-state, and when the distance between two side walls goes to infinity, the solution reduces to the flow over a plate. The second problem considered is the flow due to the oscillation of a plane wall. Two solutions are given in terms of definite integrals. It is important fact that the starting solution can be represented as the sum of the steady-state and transient solutions. For large values of time, the transient solutions disappear. The limiting cases when time goes to infinity, the distance between two side walls goes to infinity and the frequency of the oscillation goes to zero are discussed. The third problem is the flow induced by a constantly accelerating plate. In the limiting case when the distance between the side walls goes to infinity, the solution reduces to the flow over a plate, which can be expressed in terms of a tabulated function. The fourth problem is the flow induced by a plate that applies a constant stress to the fluid. The limiting cases when the time goes to infinity and when the distance between the side walls goes to infinity are discussed. It is a very important fact that these four examples show that the required time to attain the steady-state is affected by the side walls.

The fluid is over a plane wall and between two side walls perpendicular to the plate, The

Flow geometry and coordinate system.

The first boundary condition suggests that

When

The variation of

The volume flux across a plane normal to the flow is given by

The shear stress at the bottom wall can be calculated by (

The flow over a plane wall which is initially at rest and the plate begins to oscillate in its own plane is termed Stokes’ second problem [

The variation of

The plate is initially at rest and has, after time zero, a constant acceleration

The variation of

Suppose that the plane wall suddenly applies a constant tangential stress

The variation of

The variation of

Four types of unsteady flows of incompressible viscous fluids over a plane wall bounded by two side walls are considered. In order to understand the effects of the side walls, the cases for which time goes to infinity and the distance between two side walls goes to infinity are compared with the cases for flows over a plane wall in the absence of the side walls. These provide to know the required time to attain the steady-state and what is the distance between the side walls for which the measured value of the velocity or the stress or the temperature would be unaffected by the presence of the side walls. The flow problems which are the flow generated by impulsive motion of a plate, flow due to oscillation of the plate, flow induced by constantly accelerating plate, and flow due to imposed by a plate that applies a constant tangential stress to the fluid are solved by the application of the Fourier transform method to the governing equation of motion. The solutions can be expressed in terms of the definite integrals. It is shown that the starting solution for the flow of the oscillating plate can be presented as the sum of the steady-state and the transient solutions. The limiting cases when the time goes to infinity and the distance between two side walls goes to infinity are obtained and discussed. For flow induced by a plate that applies a constant stress, for large times the flow becomes steady in the case of the side walls, but the flow remains time-dependent in the absence of the side walls.

The authors are grateful to Professor K. R. Rajagopal for many stimulating correspondences and suggestions. The authors would like to thank the referees for their valuable comments.