Coefficients of viscosity

Set up fixed coordinate axes in a liquid that is flowing in a laminar manner. Despite the dissipation of energy against the viscous forces, assume that the temperature of the liquid remains constant. Assume further that conditions are steady and let a and h, respectively, be two streamlines close together in a small region of the liquid. Now let two light particles A and B be placed, one on each streamline, at a time /, in such positions that the line joining 

The speed of liquid flow will be slightly different along the two streamlines so that, after a further time St, the line joining A and B will make an angle Sy with its direction at time t. The liquid between the streamlines has suffered a shear of 67 and 5y/St measures the rate of shear or shear strain rate. 

Now assume further that the liquid is flowing in the x direction in layers normal to the y axis. The streamlines are then straight and parallel to the x axis and the flow velocity of the liquid is constant along each streamline, the only variation in flow velocity being in the y direction. Assume that streamhnes a and b lie in the same xy plane. If the speed of liquid flow along streamline a is u, that along streamline b is  

Therefore, when a liquid is flowing in planar layers and conditions are steady, the rate of shear is equal to the velocity gradient normal to the direction of flow. 

For many liquids undergoing steady flow in the manner described, the shear stress r acting in the direction of flow needed to maintain a given shear rate is proportional to that shear rate. Then  

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For example, the definition of a system as 10.0 g FI2O at 10.0°C at an applied pressure p= 1.00 atm is sufficient to specify that the water is liquid and that its other properties (energy, density, refractive index, even non-thennodynamic properties like the coefficients of viscosity and themial condnctivify) are uniquely fixed. 

The dynamic viscosity, or coefficient of viscosity, 77 of a Newtonian fluid is defined as the force per unit area necessary to maintain a unit velocity gradient at right angles to the direction of flow between two parallel planes a unit distance apart. The SI unit is pascal-second or newton-second per meter squared [N s m ]. The c.g.s. unit of viscosity is the poise [P] 1 cP = 1 mN s m . The dynamic viscosity decreases with the temperature approximately according to the equation log rj = A + BIT. Values of A and B for a large number of liquids are given by Barrer, Trans. Faraday Soc. 39 48 (1943). 

In Sec. 2.2 we saw that the coefficient of viscosity is defined as the factor of proportionality between the shearing force per unit area = F /A and the velocity gradient dv/dy within a liquid [Eq. (2.2)] ... 

Solutes in Aqueous Solution. As mentioned in See. 88, when we say that we expect to find a correlation between the /1-coefficients of viscosity of various species of ions, and their entropy of solution, this refers only to the unitary part of the entropy, the part associated with the ionic co-sphere. We are inclined to adopt the view that a negative //-coefficient for a pair of ions should be accompanied by a positive increment in entropy, while a positive //-coefficient should be accompanied by a decrease in entropy. The values of AS0, the conventional entropy of solution, to be found in the literature, do not, give a direct answer to this question, since they contain the cratic term, which in water at room temperature amounts to 16 e.u. This must be subtracted. 

This procedure is especially useful when the temperature coefficient of viscosity is small and the temperature coefficient of the association constant is large. 

The Maxwell model is also called Maxwell fluid model. Briefly it is a mechanical model for simple linear viscoelastic behavior that consists of a spring of Young s modulus (E) in series with a dashpot of coefficient of viscosity (ji). It is an isostress model (with stress 5), the strain (f) being the sum of the individual strains in the spring and dashpot. This leads to a differential representation of linear viscoelasticity as d /dt = (l/E)d5/dt + (5/Jl)-This model is useful for the representation of stress relaxation and creep with Newtonian flow analysis. 

R is the shear stress in the fluid and divelocity gradient or the rate of shear. It may be noted that R corresponds to r used by many authors to denote shear stress similarly, shear rate may be denoted by either dw,/dy or y. The proportionality sign may be replaced by the introduction of the proportionality factor n, which is the coefficient of viscosity, to give ... 

This is a statement of Newton s law of viscosity and the constant of proportionality fi is known as the coefficient of dynamic viscosity or, simply, the viscosity, of the fluid. The rate of change of the shear strain is known as the rate of (shear) strain or the shear rate. The coefficient of viscosity is a function of temperature and pressure but is independent of the shear rate y. 

For a Newtonian fluid, the shear stress is proportional to the shear rate, the constant of proportionality being the coefficient of viscosity. The viscosity is a property of the material and, at a given temperature and pressure, is constant. Non-Newtonian fluids exhibit departures from this type of behaviour. The relationship between the shear stress and the shear rate can be determined using a viscometer as described in Chapter 3. There are three main categories of departure from Newtonian behaviour behaviour that is independent of time but the fluid exhibits an apparent viscosity that varies as the shear rate is changed behaviour in which the apparent viscosity changes with time even if the shear rate is kept constant and a type of behaviour that is intermediate between purely liquid-like and purely solid-like. These are known as time-independent, time-dependent, and viscoelastic behaviour respectively. Many materials display a combination of these types of behaviour. 

When the plot of shear stress versus shear rate is linear, the liquid behaviour is simple and the liquid is Newtonian2 with the coefficient of viscosity, rj, being the proportionality constant. 

The viscosity of petroleum fractions increases on the application of pressure, and this increase may be very large. The pressure coefficient of viscosity correlates with the temperature coefficient even when oils of widely different types are compared. At higher pressures the viscosity decreases with increasing temperature, as at atmospheric pressure in fact, viscosity changes of small magnitude are usually proportional to density changes, whether these are caused by pressure or by temperature. 

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