The National Portrait Gallery caption, that comes across Wikimedia, is "by William Henry Mote, after William Salter Herrick, stipple engraving, published 1885. Her oldest son married Queen Victoria's daughter Princess Louise.
Her Wikipedia article is here.
Duchess Elizabeth's dress has flared, pagoda-like sleeves, and a neckline reminiscent of the Louis XV period.
Keywords: 1851, Herrick, Mote, Elizabeth Georgina Campbell, Duchess of Argyll, Leveson Gower family, Campbell family, Duchess, British, straight coiffure, trapezoidal neckline, bertha, lace, modesty piece, bodice, long bottom flared slose sleeves, engageantes, buttons, darts, full skirt, bracelets