Drainage of a micro-polar liquid on a flat stationary plate is investigated in this paper. Due to the presence of couple stresses, the Navier–Stokes equation now contains an additional term proportional to the fourth derivative of the fluid velocity. This equation is solved by the method of finite sine transforms and its solution is utilized to study the time development of the drainage film profiles. These profiles exhibit distinctive features which can be traced to the couple stresses term in the momentum equation.
The aim of this note is twofold: first, we attempt a unified description of the drainage, withdrawal, and postwithdrawal drainage of a liquid over a flat plate; second, we examine the question of which of two conditions that have appeared in the literature is to be employed for correctly obtaining the film thickness profiles, A knowledge of these profiles is of importance in a variety of applications, such as dipcoating, enameling, electroplating, and capillary viscometry (Tallmadge and Gutfinger, 1967). PLATE LIFTED WITH A GENERAL VELOCITYWe choose the yz and xz planes to coincide with the initial position of the plate and the initial surface of the bath, At t = 0, the plate is lifted along the y axis with a moderate velocity f ( t ) , and the bath is allowed to drain freely under the action of gravity. The solution of the problem will be obtained for arbitrary f ( t ) . From this, the results for the drainage, withdrawal, and post withdrawal processes emerge as special cases for appropriate choices of the plate velocity f ( t ) .We confine ourselves to the parallel flow region and neglect the effect of surface tension on the flow. With the assumption of laminar one-dimensional flow, permissible in this region (Groenveld, 1970), the Navier-Stokes equation in the y direction reduces to (Gutfinger and Tallmadge, 1964) a 2 0 au at ax2 (1) g -=v--The velocity ~( x , t ) of the liquid is subject to the usual conditions au ax ~( x , 0) = 0, -(h, t ) = 0 and ~( 0 , t ) = f ( t ) (2) where u ( x , s) is required to satisfy the conditions c dvu ( 0 , s ) = f (~) and -= 0 at x -h (4) dx -Equation (3) has the solution -( 5 ) The inverse of Equation (5) gives the solution of Equation (1): x > O = f ( t ) , for x = O (6) where an = ( n + 1 / 2 )~, and w is the supplementary variable brought in by the convolution theorem. In terms of nondimensional quantities, the flux of the entrained liquid is obtained as The film thickness profile is connected to the flow rate through the equation of continuity, which yields Taking the Laplace transform of Equation (I), we obtain d2U sg -v = -(3) dx2 v sv Y =J (S) T dT + 4(H) (8)
The paper presents an exact analysis of the dispersion of a solute in an electrically conducting fluid flowing between two parallel plates in the presence of a uniform transverse magnetic field. Using a generalized dispersion model, which is valid for all time after the injection of the solute in the flow we evaluate the longitudinal dispersion coefficients as functions of time. For small values of the Hartmann number M , the values of the dispersion coefficients show rapid fluctuations which decay with increase in M , and for moderate and large values of M , these coefficients monotonically decrease with increase in M . In a non-conducting fluid, such rapid fluctuations are absent and the dispersion coefficients for a pipe flow are less than the corresponding values for a channel flow.
A theoretical investigation of the effects of a transverse magnetic field on the combined problem of viscous lifting and drainage of a conducting fluid on a plate is presented. The effecis of inertia and transverse magnetic field on the liquid film thickness is studied for two cases namely a plate withdrawn with a constant velocity and one withdrawn with a constant acceleration. The expressions for the flow rate and the free surface profiles are obtained for the above two cases. It is found that the free surface profiles are convex in nature as in the non-magnetic case thus showing that the inertia does not effect tile general pattern of flow, and the effect of the magnetic field is to retard both the lifting and drainage of the fluid.
The paper is concerned with the dispersion of a solute in a Bingham plastic fluid flowing in a pipe or a parallel plate channel. For pipe flow. the dispersion coefficient K, first increases with to (the dimensionless radius of the plug flow region), reaches a maximum and then decreases. But in a channel Row. K, decreases monotonically with increasing 6. Further K, for channel flow is found to be larger than that for pipe flow for all values of b except 0.8 s to s 1.
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