Shear rate

<h2 id="simple-shear">Simple shear</h2>
The shear rate for a fluid flowing between two parallel plates, one moving at a constant speed and the other one stationary (<a href="/facts/Couette_flow/pSBxDXGk">Couette flow</a>), is defined by

γ
              ˙
            
          
        
        =
        
          
            v
            h
          
        
        ,
      
    
    {\displaystyle {\dot {\gamma }}={\frac {v}{h}},}

where: 

<ul><li>
 
 
 
 
 
 
 γ
 ˙
 
 
 
 
 
 {\displaystyle {\dot {\gamma }}}
 
 is the shear rate, measured in <a href="/facts/Reciprocal_seconds/W0Iqf8EL">reciprocal seconds</a>;</li>
<li>v is the velocity of the moving plate, measured in meters per second;</li>
<li>h is the distance between the two parallel plates, measured in meters.</li></ul>
Or:

γ
                ˙
              
            
          
          
            i
            j
          
        
        =
        
          
            
              ∂
              
                v
                
                  i
                
              
            
            
              ∂
              
                x
                
                  j
                
              
            
          
        
        +
        
          
            
              ∂
              
                v
                
                  j
                
              
            
            
              ∂
              
                x
                
                  i
                
              
            
          
        
        .
      
    
    {\displaystyle {\dot {\gamma }}_{ij}={\frac {\partial v_{i}}{\partial x_{j}}}+{\frac {\partial v_{j}}{\partial x_{i}}}.}

For the <a href="/facts/Simple_shear/kSPvFWkE">simple shear</a> case, it is just a <a href="/facts/Gradient/TsR7uukj">gradient</a> of <a href="/facts/Velocity/Q3o1LVDe">velocity</a> in a flowing material. The <a href="/facts/SI_unit/S7YLglNz">SI unit</a> of measurement for shear rate is s−1, expressed as "reciprocal seconds" or "<a href="/facts/Inverse_seconds/W0Iqf8EL">inverse seconds</a>".<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a> However, when modelling fluids in 3D, it is common to consider a scalar value for the shear rate by calculating the <a href="/facts/Invariants_of_tensors/fwpws1vj">second invariant</a> of the <a href="/facts/Strain-rate_tensor/DAuV9bPz">strain-rate tensor</a>

γ
 ˙
 
 
 
 =
 
 
 2
 ε
 :
 ε
 
 
 
 
 {\displaystyle {\dot {\gamma }}={\sqrt {2\varepsilon :\varepsilon }}}
 
.
The shear rate at the inner wall of a <a href="/facts/Newtonian_fluid/hpzNyDH5">Newtonian fluid</a> flowing within a pipe<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a> is

γ
              ˙
            
          
        
        =
        
          
            
              8
              v
            
            d
          
        
        ,
      
    
    {\displaystyle {\dot {\gamma }}={\frac {8v}{d}},}

where:

<ul><li>
 
 
 
 
 
 
 γ
 ˙
 
 
 
 
 
 {\displaystyle {\dot {\gamma }}}
 
 is the shear rate, measured in reciprocal seconds;</li>
<li>v is the linear fluid velocity;</li>
<li>d is the inside diameter of the pipe.</li></ul>
The linear fluid velocity v is related to the volumetric flow rate Q by

v
        =
        
          
            Q
            A
          
        
        ,
      
    
    {\displaystyle v={\frac {Q}{A}},}

where A is the cross-sectional area of the pipe, which for an inside pipe radius of r is given by

A
        =
        π
        
          r
          
            2
          
        
        ,
      
    
    {\displaystyle A=\pi r^{2},}

thus producing

v
        =
        
          
            Q
            
              π
              
                r
                
                  2
                
              
            
          
        
        .
      
    
    {\displaystyle v={\frac {Q}{\pi r^{2}}}.}

Substituting the above into the earlier equation for the shear rate of a Newtonian fluid flowing within a pipe, and noting (in the denominator) that d = 2r:

γ
              ˙
            
          
        
        =
        
          
            
              8
              v
            
            d
          
        
        =
        
          
            
              8
              
                (
                
                  
                    Q
                    
                      π
                      
                        r
                        
                          2
                        
                      
                    
                  
                
                )
              
            
            
              2
              r
            
          
        
        ,
      
    
    {\displaystyle {\dot {\gamma }}={\frac {8v}{d}}={\frac {8\left({\frac {Q}{\pi r^{2}}}\right)}{2r}},}

which simplifies to the following equivalent form for wall shear rate in terms of volumetric flow rate Q and inner pipe radius r:

γ
              ˙
            
          
        
        =
        
          
            
              4
              Q
            
            
              π
              
                r
                
                  3
                
              
            
          
        
        .
      
    
    {\displaystyle {\dot {\gamma }}={\frac {4Q}{\pi r^{3}}}.}

For a <a href="/facts/Newtonian_fluid/hpzNyDH5">Newtonian fluid</a> wall, <a href="/facts/Shear_stress/aWWnzIhE">shear stress</a> (τw) can be related to shear rate by 
 
 
 
 
 τ
 
 w
 
 
 =
 
 
 
 
 γ
 ˙
 
 
 
 
 x
 
 
 μ
 
 
 {\displaystyle \tau _{w}={\dot {\gamma }}_{x}\mu }
 
 where μ is the dynamic <a href="/facts/Viscosity/QLVfrdCz">viscosity</a> of the fluid. For non-Newtonian fluids, there are different <a href="/facts/Constitutive_equation/IXTjEAbL">constitutive laws</a> depending on the fluid, which relates the <a href="/facts/Viscous_stress_tensor/ljeMKpDI">stress tensor</a> to the shear rate tensor.

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Shear_strain/xK7R59oI">Shear strain</a></li>
<li><a href="/facts/Strain_rate/QptTh1KW">Strain rate</a></li>
<li><a href="/facts/Non-Newtonian_fluid/jX8ldWJa">Non-Newtonian fluid</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">"Brookfield Engineering - Glossary section on Viscosity Terms". Archived from the original on 2007-06-09. Retrieved 2007-06-10. <a href="https://web.archive.org/web/20070609171914/http://www.brookfieldengineering.com/education/viscosity_glossary.asp" target="_blank">https://web.archive.org/web/20070609171914/http://www.brookfieldengineering.com/education/viscosity_glossary.asp</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Darby, Ron (2001). Chemical Engineering Fluid Mechanics (2nd ed.). CRC Press. p. 64. ISBN 9780824704445. <a href="9780824704445" target="_blank">9780824704445</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
</ol>

Shear rate open-in-new

Shear rate