B.S. Dandapat

Heat transfer in a liquid film on an unsteady stretching surface

Abstract The momentum and heat transfer in a laminar liquid film on a horizontal stretching sheet is analysed. The governing time-dependent boundary layer equations are reduced to a set of ordinary differential equations by means of an exact similarity transformation. The resulting two-parameter problem is solved numerically for some representative values of the unsteadiness parameter S for Prandtl numbers from 0.001 to 1000. The temperature is observed to increase monotonically from the elastic sheet towards the free surface except in the high diffusivity limit Pr→0 where the surface temperature approaches that of the sheet. A low stretching rate, i.e. high values of S, tends to reduce the surface temperature for all Prandtl numbers. The heat flux from the liquid to the elastic sheet decreases with S for Pr≲0.1 and increases with increased unsteadiness for Pr≳1.

Magnetohydrodynamic flow of a power-law fluid over a stretching sheet

Abstract Magnetohydrodynamic flow of an electrically conducting power-law fluid over a stretching sheet in the presence of a uniform transverse magnetic field is investigated by using an exact similarity transformation. The effect of magnetic field on the now characteristics is explored numerically, and it is concluded that the magnetic field tends to make the boundary layer thinner, thereby increasing the wall friction.

Flow and heat transfer in a viscoelastic fluid over a stretching sheet

Abstract The paper discusses the flow of an incompressible second-order fluid due to stretching of a plane elastic surface in the approximation of boundary layer theory. An exact analytical solution of the non-linear equation governing the self-similar flow is given. The skin friction decreases with increase in the elastic parameter k1. The analysis of heat transfer in this flow reveals that when the wall and the ambient temperature are held constant, temperature at a point increases with increase in k1, for fixed Prandtl number σ.

Thermocapillarity in a liquid film on an unsteady stretching surface

Abstract The influence of thermocapillarity on the flow and heat transfer in a thin liquid film on a horizontal stretching sheet is analysed. The time-dependent governing boundary layer equations for momentum and thermal energy are reduced to a set of coupled ordinary differential equations by means of an exact similarity transformation. The resulting three-parameter problem is solved numerically for some representative values of an unsteadiness parameter S and a thermocapillarity number M for Prandtl numbers from 0.001 to 100. The thermocapillary surface forces drag the liquid film in the same direction as the stretching sheet and a local velocity minimum occurs inside the film. The surface velocity, the film thickness, and the Nusselt number at the sheet increase with M for Pr ≲10. For higher Prandtl numbers, the thermal boundary layer is confined to the lower part of the liquid film and the temperature at the free surface remains equal to the slit temperature and the thermocapillary forces vanish.

Flow of a power-law fluid film on an unsteady stretching surface

Abstract Flow of a thin liquid film of a power-law fluid caused by the unsteady stretching of a surface is investigated by using a similarity transformation. This transformation reduces the unsteady boundary-layer equations to a non-linear ordinary differential equation governed by a nondimensional unsteadiness parameter S. The effect of S on the film thickness is explored numerically for different values of the power-law index n. A physical explanation for the findings is also provided.

Film cooling on a rotating disk

Abstract The unsteady flow of a liquid film on a cold rotating disk is analysed by means of matched asymptotic expansion under the assumptions of radially uniform film thickness that varies with time. The velocity, the temperature and the rate of heat transfer are determined. It is shown how the uniform film thins with time for fixed values of the cooling or heat dissipating parameter β, and the Prandtl number σ. When either β increases or σ decreases, the film thickness increases, which implies that a relative resistance to film thinning is developed inside the film- A zone S, bounded by a curve in the r-z plane, may be delineated such that the temperature is minimum on this curve. Thus, heat flows from the disk to the fluid, inside the zone S, and in the opposite direction outside S.

Flow of a thin liquid film on an unsteady stretching sheet

The stretching surface is assumed to be stretched impulsively from rest and the effect of inertia of the liquid is considered. Equations describing the laminar flow on the stretching surface are solved analytically by using the singular perturbation technique and the method of characteristics is used to obtain an analytic expression for film thickness. The results show that the final film thickness is independent of the amount of liquid distributed initially and on the initial film thickness be it uniform or nonuniform. It is also shown that the forceful stretching produces quicker thinning of the film on the stretching surface.

Waves on a film of power-law fluid flowing down an inclined plane at moderate Reynolds number

Waves that occur at the surface of a power-law fluid film flowing down an inclined plane are investigated. Using the method of integral relations, an evolution equation is derived for two types of wave equations which are possible under long wave approximation. This equation is valid for moderate Reynolds numbers and reveals the presence of both kinematic and dynamic wave processes which may either act together or singularly dominate the wave field depending on the order of different parameters. It is shown that, at a small flow rate, kinematic waves dominate the flow field and it acquires energy from the mean flow, while, for high flow rate, inertial waves dominate and the energy comes from the kinematic waves. This energy transfer from kinematic waves to inertial waves depends on the power-law index n. Linear stability analysis predicts the contribution of different terms in the wave mechanism. Further, it is found that surface tension plays a double role, for the kinematic wave process, it exerts dissipative effects so that a finite amplitude case may be established, but for the dynamic wave process it yields dispersion. The evolution equation is capable of predicting amplitudes, shapes, and interaction at the finite amplitude level. It is also shown that the results of the interaction may lead either to forward breaking waves or solitary waves with dark soliton depending on the flow rate, Weber number and the angle of inclination with the horizon. Power-law index n plays a vital role in the wave mechanism.

Waves on the surface of a falling power-law fluid film

Abstract Waves that occur at the surface of a falling film of thin power-law fluid on a vertical plane are investigated. Using the method of integral relations an evolution equation is derived for two types of waves equation which are possible under long wave approximation. This equation reveals the presence of both kinematic and dynamic wave processes which may either act together or singularly dominate the wave field depending on the order of different parameters. It is shown that, at a small flow rate, kinematic waves dominate the flow field and the energy is acquired from the mean flow during the interaction of the waves, while, for high flow rate, inertial waves dominate and the energy comes from the kinematic waves. It is also found that this exchange of energy between kinematic and inertial waves strongly depends on the power-law index n. Linear stability analysis predicts the contribution of different terms in the wave mechanism. Further, it is found that the surface tension plays a double role: for a kinematic wave process, it exerts dissipative effects so that a finite amplitude case may be established, but for a dynamic wave process it yields dispersion. Further, it is shown that the non-Newtonian character n plays a vital role in controlling the role of the term that contains surface tension in the above processes.

Solitary waves on the surface of a viscoelastic fluid running down an inclined plane

In this paper, we study the existence and the role of solitary waves in the finite amplitude instability of a layer of a second-order fluid flowing down an inclined plane. The layer becomes unstable for disturbances of large wavelength for a critical value of Reynolds number which decreases with increase in the viscoelastic parameter M. The long-term evolution of a disturbance with an initial cosinusoidal profile as a result of this instability reveals the existence of a train of solitary waves propagating on the free surface. A novel result of this study is that the number of solitary waves decreases with in crease in M. When surface tension is large, we use dynamical system theory to describe solitary waves in a moving frame by homoclinic trajectories of an associated ordinary differential equation.