Unsteady simple shear flow

Flow is generally classified as shear flow and extensional flow [2]. Simple shear flow is further divided into two categories Steady and unsteady shear flow. Extensional flow also could be steady and unsteady however, it is very difficult to measure steady extensional flow. Unsteady flow conditions are quite often measured. Extensional flow differs from both steady and unsteady simple shear flows in that it is a shear free flow. In extensional flow, the volume of a fluid element must remain constant. Extensional flow can be visualized as occurring when a material is longitudinally stretched as, for example, in fibre spinning. When extension occurs in a single direction, the related flow is termed uniaxial extensional flow. Extension of polymers or fibers can occur in two directions simultaneously, and hence the flow is referred as biaxial extensional or planar extensional flow. 

Flow is broadly classified as shear flow and extensional flow. A catalog of various t5qjes of shear flow has been given by Bird et al. [4]. In the present book, the discussion is restricted to only simple shear flow that occurs when a fluid is held between two parallel plates. Simple shear flow could be of the steady or unsteady type. Similarly extensional flow could be steady or unsteady. In the case of extensional flow, it is often difflcult to keep the measuring apparatus running for a long enough time to achieve steady state conditions and therefore unsteady conditions are quite often encountered. 

Unsteady simple shear flow would occur when the stresses involved are time-dependent. Small-amplitude oscillatory flow, stress growth, stress relaxation, creep and constrained recoil are some examples of such types of flows [4]. In the following, small-amplitude oscillatory flow is treated in sufficient detail while others are briefly described... 

Solution. For an unsteady simple shear flow the terms in Eq. 3.42 are given below ... 

A few additional comments about when and under what conditions one must use a nonlinear viscoelastic constitutive equation are discussed here. At this time it seems that whenever the flow is unsteady in either a Lagrangian DvIDt 0) or a Eulerian (9v/9r 0) sense, then viscoelastic effects become important. In the former case one finds flows of this nature whenever inhomogeneous shear-free flows arise (e.g., flow through a contraction) and in the latter case in the startup of flows. However, even in simple flows, such as in capillaries or slit dies, viscoelastic effects can be important, especially if the residence time of the fluid in the die is less than the longest relaxation time of the fluid. Then factors such as stress overshoot could lead to an apparent viscosity that is higher than the steady-state viscosity. In line with these ideas one defines a dimensionless group referred to as the Deborah number ... 

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