In this paper, we study the unsteady hydromagnetic flow of a Walter's fluid (Model B') down an open inclined channel of width 2a and depth d under gravity, the walls of the channel being normal to the surface of the bottom under the influence of a uniform transverse magnetic field. A uniform tangential stress is applied at the free surface in the direction of flow. We have evaluated the velocity distribution by using Laplace transform and finite Fourier Sine transform technique. The velocity distribution has been obtained taking different form of time dependent pressure gradient g(t), viz., i) constant ii) exponential decreasing function of time and iii) Cosine function of time. The effects of magnetic parameter M, Reynolds number R and the viscoelastic parameter K are discussed on the velocity distribution in three different cases.
A flowing liquid is said to have a free surface when the upper part of the bounding surface of the liquid is in contact with the overlying atmosphere, rather than with a solid, as would be the case of the flow were in a pipe completely full of the liquid. A flow with a free surface proceeding in a natural or artificial channel or conduit is called an openchannels flow. Open-channels may be divided into two types namely (i) Natural channels and (ii) Artificial channels. Natural channels range inform from the bounder-strewn bed of a mountain torrent to the relatively uniform channel of a large river. Artificial channels are man-made and are constructed in many forms. A pipe in which water flows with a free surface and flume of rectangular cross section constructed of sheet iron are two kinds of artificial channels. Other types are canals excavated in earth or blasted in rock, which either are left unlined or lined with smooth concrete or another suitable material. Unlined or lined tunnels bored through rock may also contain free surface flows.
The flow of a liquid in an open inclined channel with a free surface has a wide application in the designs of drainage, irrigation canals, flood discharge channels and coating to paper rolls etc. Hence the flow of a liquid in an open inclined channel with a free surface under gravity has long been studied experimentally and several interesting empirical results have been reported by many investigators [3,6,7,10,11,14]. The steady laminar flow of a viscous fluid flowing down an open inclined channel has been discussed by Satyaprakash [13], Gupta et al [4] have studied the flow of a viscous fluid through a porous medium down an open inclined channel. Venkataramana and Bathaiah [18] have studied the flow of a hydromagnetic viscous fluid down an open inclined channel with naturally permeable bed under the influence of a uniform transverse magnetic field. Unsteady laminar flow of an incompressible viscous fluid between porous, parallel flat plates has been investigated by Singh [12], taking (i) both plates are at rest and (ii) Generalized plane Coutte flow. The free surface was exposed to atmospheric pressure and bottom was taken as impermeable. Bakhmeteff [1], Henderson [5] and Chow [2] have discussed many types of open channel flows. Recently, many authors [8, 9 and 14] have studied the flow of Walter’s B’ fluid.
The subject of Rheology is of great technological importance in many branches of industry. The problem arises of designing apparatus to transport or to process substances which cannot be governed by the classical stress-strain velocity relations. Example of such substances and process are many, the extrusion of plastics, in the manufacture of rayon, nylon or other textile fibres, viscoelastic effects transported or forced through spinnerts and in the manufacture of lubricating grease and rubber.
Non-Newtonian fluids have wide importance in the present day technology and industries; the Walter’s fluid is one of such fluid. The model of Walter’s B fluid is chosen for our study as it involves non-Newtonian parameter. The Cauchy stress tensor T in such a fluid is related to the motion in the following manner
In this equation P is the pressure, I is the Identity tensor and the rate of strain tensor e is defined by ( )
where v is the velocity vector, ∇ is the gradient operator and
and ( )
N τ being the relaxation spectrum as introduced by Walter’s [19,20]. This idealized model is a valid approximation of Walter’s fluid (model B ’ ) taking very short memory into account, so that terms involving
have been neglected.
In addition to equation (1.1), the equation of motion and continuity are
In this paper, we study the unsteady hydromagnetic flow of a Walter’s fluid (model B ’ ) down an open inclined channel under gravity of width 2a and depth d, the walls of the channel being normal to the surface of the bottom, under the influence of uniform transverse magnetic field. A uniform tangential stress is applied at the free surface in the direction of flow. We have evaluated the velocity distribution by using Laplace Transform and Finite Fourier Sine Transform techniques. Here it is assumed that (i) the fluid flows in the steady state for 0 t ≤ , (ii) Unsteady state occurs at The effects of magnetic parameter M, Reynolds number R and viscoelastic parameter K are investigated on the velocity distribution in three different cases.
We consider the unsteady Hydromagnetic flow of a Walter’s fluid (model B ’ ) down an open inclined channel of width 2a and depth d under gravity, the walls of the channel being normal to the surface of the bottom under the influence of uniform transverse magnetic field. A uniform tangential stress S is applied at the free surface. The bottom of the channel is taken at angle ( ) slightly conducting, the magnetic Reynolds number is much less than unity, so that the induced magnetic field can be neglected in comparison with the applied magnetic fi
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