Newtonian die-swell phenomenon revisited: Theory and simulations

The laminar jet of a Newtonian liquid emanating from a long circular nozzle into a gaseous environment is studied. Provided that gravity effects can be neglected, the flow depends solely on the Reynolds number ( <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mtext>Re</a:mtext> </a:math> ) and the Capillary number ( <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"> <b:mtext>Ca</b:mtext> </b:math> ) based on the nozzle bulk velocity ( <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"> <c:msub> <c:mi>U</c:mi> <c:mi>b</c:mi> </c:msub> </c:math> ). At high <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"> <d:mtext>Re</d:mtext> </d:math> , the jet contracts downstream of the nozzle, while at low <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"> <e:mtext>Re</e:mtext> </e:math> , the jet expands, which is known as the Newtonian die-swell phenomenon. To study the influence of <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"> <f:mtext>Re</f:mtext> </f:math> on the jet behavior, an integral momentum balance is theoretically derived for the entire flow, both in the nozzle and in the jet. A useful decomposition is made of the viscous force in the integral momentum balance into a contribution from the excess wall pressure drop ( <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"> <g:msub> <g:mi>P</g:mi> <g:mtext>exc</g:mtext> </g:msub> </g:math> ) and a contribution from the excess integral wall shear stress ( <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"> <h:msub> <h:mi>T</h:mi> <h:mtext>exc</h:mtext> </h:msub> </h:math> ) relative to, respectively, the pressure drop and integral wall shear stress for Poiseuille flow. The results of the numerical simulations for <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"> <i:mrow> <i:mtext>Re</i:mtext> <i:mo>∈</i:mo> <i:mo>[</i:mo> <i:mn>4.2</i:mn> <i:mo>:</i:mo> <i:mn>47.4</i:mn> <i:mo>]</i:mo> </i:mrow> </i:math> show that <j:math xmlns:j="http://www.w3.org/1998/Math/MathML"> <j:msub> <j:mi>P</j:mi> <j:mtext>exc</j:mtext> </j:msub> </j:math> is larger than zero for <k:math xmlns:k="http://www.w3.org/1998/Math/MathML"> <k:mrow> <k:mtext>Re</k:mtext> <k:mo>≲</k:mo> <k:mn>6</k:mn> </k:mrow> </k:math> , while negative for <l:math xmlns:l="http://www.w3.org/1998/Math/MathML"> <l:mrow> <l:mtext>Re</l:mtext> <l:mo>≳</l:mo> <l:mn>6</l:mn> </l:mrow> </l:math> . Furthermore, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>T</m:mi> <m:mtext>exc</m:mtext> </m:msub> </m:math> is always negative and is the dominant contribution to the overall viscous force for <n:math xmlns:n="http://www.w3.org/1998/Math/MathML"> <n:mrow> <n:mtext>Re</n:mtext> <n:mo>≲</n:mo> <n:mn>12</n:mn> </n:mrow> </n:math> . Normalized by the integral wall shear stress for Poiseuille flow, <o:math xmlns:o="http://www.w3.org/1998/Math/MathML"> <o:msub> <o:mi>T</o:mi> <o:mtext>exc</o:mtext> </o:msub> </o:math> appears to scale with <p:math xmlns:p="http://www.w3.org/1998/Math/MathML"> <p:msup> <p:mrow> <p:mtext>Re</p:mtext> </p:mrow> <p:mrow> <p:mo>−</p:mo> <p:mn>1</p:mn> <p:mo>/</p:mo> <p:mn>3</p:mn> </p:mrow> </p:msup> </p:math> , implying that axial velocity perturbations near the nozzle wall scale with <q:math xmlns:q="http://www.w3.org/1998/Math/MathML"> <q:msup> <q:mrow> <q:mtext>Re</q:mtext> </q:mrow> <q:mrow> <q:mo>−</q:mo> <q:mn>1</q:mn> <q:mo>/</q:mo> <q:mn>3</q:mn> </q:mrow> </q:msup> </q:math> too when normalized by <r:math xmlns:r="http://www.w3.org/1998/Math/MathML"> <r:msub> <r:mi>U</r:mi> <r:mi>b</r:mi> </r:msub> </r:math> . Self-similar behavior is also found for the axial velocity perturbation at the nozzle centerline, which scales with <s:math xmlns:s="http://www.w3.org/1998/Math/MathML"> <s:msup> <s:mrow> <s:mtext>Re</s:mtext> </s:mrow> <s:mrow> <s:mo>−</s:mo> <s:mn>2</s:mn> <s:mo>/</s:mo> <s:mn>3</s:mn> </s:mrow> </s:msup> </s:math> when normalized by <t:math xmlns:t="http://www.w3.org/1998/Math/MathML"> <t:msub> <t:mi>U</t:mi> <t:mi>b</t:mi> </t:msub> </t:math> . The observed power-law scalings in <u:math xmlns:u="http://www.w3.org/1998/Math/MathML"> <u:mtext>Re</u:mtext> </u:math> are explained theoretically from the development of a boundary layer along the jet interface, originating from the stick-slip transition at the nozzle lip, and the constant integral mass flux along the jet. The boundary-layer flow is responsible for axial viscous stresses at the nozzle exit plane that result in a perturbation flow within the nozzle over an axial distance on the order of the nozzle radius. The die-swell phenomenon at low <v:math xmlns:v="http://www.w3.org/1998/Math/MathML"> <v:mtext>Re</v:mtext> </v:math> is caused by the radially outward flow at the nozzle exit and the accompanying compressive normal viscous stress that pushes the interface outwards till equilibrium is achieved. Self-similar behavior is also found for the perturbation flow far downstream of the nozzle with respect to the plug flow at equilibrium. With increasing <w:math xmlns:w="http://www.w3.org/1998/Math/MathML"> <w:mtext>Re</w:mtext> </w:math> , the exponential decay rate of the perturbation in the centerline velocity converges toward a prediction first made by Bohr for a related problem [N. Bohr, ], while the exponential decay rate of the perturbation in the jet radius is approximately twice as high. Finally, a discussion is given of the implications of our findings for applications in practice.