Highly Viscous Flows…. P M V Subbarao Professor Mechanical Engineering Department I I T Delhi Creeping Flows

The Principle Characteristic Creeping Flows Re is indicative of the ratio of inertial to viscous forces. The assumption of small Re means that viscous forces dominate the dynamics. That suggests that to drop entirely the term Dv/Dt from the Navier-Stokes equations. This renders the linear system. The linearity of the problem will be a major simplification.

It is tempting to say that the smallness of Re means that the left-hand side of the first equation can be neglected. This leads to the reduced (linear) system

Modification of Non-Dimensional Terms Re define the dimensionless pressure as pL/(2μU) instead of p/(  U 2 ).

Stokes Flow The basic assumption of creeping flow was developed by Stokes (1851) in a seminal paper. This states that density (inertia) terms are negligible in the momentum equation. Under non-gravitational field. In dimensional form with

The Curl of Stokes Flow Equation Taking the curl of Stokes flow equation: Using the vector identity With Any viscous flow, whose curl of curl of vorticity is zero is called as stokes flow or creeping flow

Some Applications 1.How does sedimentation vary with the size of the sediment particles? 2.What electric field is required to move a charged particle in electrophoresis? 3.What g force is required to centrifuge cells in a given amount of time. 4.What is the effect of gravity on the movement of a monocyte in blood? 5.How rapidly do enzyme-coated beads move in a bioreactor? 6.The flow geometry of all above mentioned applications is flow past a sphere. 7.Define the term vorticity in spherical coordinate system.

Symmetry of the Geometry The flow will be symmetric with respect to .

Incident Velocity Component of incident velocity in the radial direction, Cartesian Uniform Velocity in Spherical System Component of incident velocity in the  - direction, Uniform velocity in spherical system:

Vorticity of 2-D Flow in Spherical System The only component of vorticity in this axisymmetric problem is ω ϕ, and is given by Strokes defined a stream function in spherical coordinate system (1851) as: The stokes solution to be evaluated is:

the momentum equation as a scalar equation for ψ. Stokes Flow in terms of Stream Function

Qualitative Solution to Stokes Problem

Imagination of Flow on the Surface of a Sphere Euler’s Imagination: Nonviscous Flow Larger velocity near the sphere is an inertial effect. Stokes Imagination : Creeping Flow Velocity

Boundary Conditions (B.C.s) for Flow around a Sphere Euler Flows Creeping Flows In both cases there is symmetry about the  -axis. Thus (a) nothing depends on , and (b) there is no  - velocity. Slip is allowed in ideal flow. Slip is not allowed in viscocus flow.

General Flow around a Sphere A more general case Shear region near the sphere caused by viscosity and no-slip. Increased velocity as a result of inertia terms. Incident velocity is approached far from the sphere.

The boundary conditions Stokes Problem

Far-field Boundary Conditions in Terms of  From definition: which suggests the  -dependence of the solution.

Separability Again, at r=R: and at r  : For a separable solution, we look for a form: Because the  -dependence holds for all , but the r- dependence does not, we must write: