{"product_id":"weissenberga-rod-climbing","title":"Weissenberga – The Weissenberg Effect \/ Rod Climbing Physics Sticker","description":"\u003cp\u003eWeissenberga, a polymer gel-elemental, spins a rod in a vat of goo, and the goo climbs up it.\u003c\/p\u003e\u003ch3\u003eThe Physics\u003c\/h3\u003e\u003cp\u003eStir water with a spinning rod and the surface dips into a vortex, because inertia flings the liquid outward. Spin a rod in a polymer solution, such as dough or a dilute polyacrylamide solution, and the liquid climbs up the rod instead. Stretched along the circular streamlines, the long polymer chains act like stretched rubber bands. The resulting tension around the streamlines squeezes the fluid inward, towards the rod, and the only way it can go is up. Karl Weissenberg described this behaviour in “A continuum theory of rheological phenomena” (Nature 159, 310, 1947).\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eN\u003csub\u003e1\u003c\/sub\u003e = τ\u003csub\u003e11\u003c\/sub\u003e − τ\u003csub\u003e22\u003c\/sub\u003e \u0026gt; 0\u003c\/strong\u003e\u003c\/p\u003e\u003cul\u003e\n\u003cli\u003eN\u003csub\u003e1\u003c\/sub\u003e: first normal stress difference (Pa)\u003c\/li\u003e\n\u003cli\u003eτ\u003csub\u003e11\u003c\/sub\u003e: normal stress along the flow direction (Pa)\u003c\/li\u003e\n\u003cli\u003eτ\u003csub\u003e22\u003c\/sub\u003e: normal stress across the flow, in the direction of the velocity gradient (Pa)\u003c\/li\u003e\n\u003c\/ul\u003e\u003cp\u003eSimple liquids such as water have no normal stress differences in steady shear, so they cannot climb. A positive N\u003csub\u003e1\u003c\/sub\u003e is the signature of an elastic, viscoelastic fluid.\u003c\/p\u003e\u003cp\u003e\u003cem\u003eHonest caveat:\u003c\/em\u003e N\u003csub\u003e1\u003c\/sub\u003e \u0026gt; 0 is the main driver, but whether a given fluid climbs a given rod also depends on the second normal stress difference and on inertia, which pushes fluid outward and wins at high spin rates.\u003c\/p\u003e\u003ch3\u003eOn the sticker\u003c\/h3\u003e\u003cp\u003eWeissenberga grips a spinning rod as a spiral of glowing amber fluid climbs up it. Printed text: \u003cstrong\u003eWEISSENBERGA\u003c\/strong\u003e with the line \u003cstrong\u003eN1 \u0026gt; 0\u003c\/strong\u003e. Pixel-art kiss-cut vinyl sticker, white, in 2″, 3″, 4″ and 6″ sizes.\u003c\/p\u003e","brand":"Printify","offers":[{"title":"2\" × 2\" \/ White","offer_id":50963803668583,"sku":"86793099988844148771","price":4.99,"currency_code":"USD","in_stock":true},{"title":"3\" × 3\" \/ White","offer_id":50963803701351,"sku":"89202637691963610625","price":6.99,"currency_code":"USD","in_stock":true},{"title":"4\" × 4\" \/ White","offer_id":50963803734119,"sku":"15395025792047523543","price":8.99,"currency_code":"USD","in_stock":true},{"title":"6\" × 6\" \/ White","offer_id":50963803766887,"sku":"23901837859288641138","price":11.99,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0596\/8746\/9159\/files\/14326432099850986514_1200.jpg?v=1793655308","url":"https:\/\/pixelatedphysics.com\/it\/products\/weissenberga-rod-climbing","provider":"Pixelated Physics","version":"1.0","type":"link"}